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UBC Theses and Dissertations

Localized pattern formation in continuum models of urban crime Tse, Wang Hung 2015

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Localized Pattern Formation inContinuum Models of Urban CrimebyWang Hung Tsea thesis submitted in partial fulfillmentof the requirements for the degree ofDoctor of Philosophyinthe faculty of graduate and postdoctoral studies(Mathematics)The University of British Columbia(Vancouver)April 2016c© Wang Hung Tse, 2016AbstractIn this thesis, the phenomenon of localized crime hotspots in models capturingthe features of repeat and near-repeat victimization of urban crime was studied.Stability, insertion, slow movement of crime hotspots and the effect of police pa-trol modelled by an extra equation derived from biased random walk were studiedby means of matched asymptotic expansions, nonlocal eigenvalue problem (NLEP)stability analysis, and numerical computations.In the absence of police, we confirmed the linear stability of the far-from equi-librium steady-states with crime hotspots in the original parameters regimes asobserved in [47]. The results hold for both the supercritical and subcritical regimesdistinguished by a Turing bifurcation (cf. [48, 49]). Moreover, the phenomenonof peak insertion was characterized by a simple nonlinear equation computable byquadratures and a normal form equation identical to that of the self-replication ofMesa patterns [28] was derived. Slow dynamics of unevenly-spaced configurationsof hotspots were described by a system of differential-algebraic systems (DAEs),which was derived from resolving an intricate triple-deck structure of boundarylayers formed between the hotspots and their neighbouring regions.In the presence of police, which was modelled by a simple interaction with crim-inals, single and multiple hotspots patterns were constructed in a near-shadow limitof criminal diffusivity. While a single hotspot was found to be unconditionally stable,the linear stability behaviour of multiple-hotspot patterns was found to depend ontwo thresholds, between which we also observe a novel Hopf bifurcation phenomenonleading to asynchronous oscillations. For one particular, but representative, param-eter value in the model, the determination of the spectrum of the NLEP was foundto reduce to the study of a quadratic equation for the eigenvalue. For more generalparameter values, where this reduction does not apply, a winding number analysison the NLEP was used to determine detailed stability properties associated withmultiple hotspot steady-state solutions.iiPrefaceThe research presented in this thesis was the result of independent and collaborative workby the author Wang Hung Tse, supervised by Prof. Michael J. Ward.Chapter 1 was written independently by the author for the purpose of this dissertation. Thederivation of the mathematical model presented in Sections 1.4 was an unpublished work bythe author, based upon previous work referenced in the chapter. The specific case of the modelthat was studied Chapter 3 was first suggested by the thesis supervisor, who also attributedthe current form to influence from the literature and his collaborators, and suggested a greatportion of the literature reviewed in this chapter.Chapter 2 was largely an expanded version of the paper by the author and the thesissupervisor: W.-H. Tse, M. J. Ward, Hotspot Formation and Dynamics for a Continuum Modelof Urban Crime, European Journal of Applied Mathematics, accepted for publication. Thischapter includes all results and figures of the paper with further editing, and also a significantamount of new and original mathematical details and numerical studies from the author’s ownwork.Chapter 3 was also the joint work of the author and the thesis supervisor Prof. MichaelJ. Ward. Most results are unpublished with an exception of the content of Section 3.6, whichappears in a section of the paper: I. Moyles, W.-H. Tse, M. J. Ward (2015), Explicitly SolvableNonlocal Eigenvalue Problems and the Stability of Localized Stripes in Reaction-Diffusion Sys-tems, Studies in Appl. Math., to appear. Only the portion of the paper which were the jointwork of the author and the thesis supervisor was included in Section 3.6 with further editingby the author.All figures in this dissertation were independently produced by the author using publiclyavailable data or numerical computations, and with references where necessary. The strategiesof the numerical computations leading to the figures contain ideas from the thesis supervisor,who also assisted in the preparation of the manuscript and furnished suggestions in literaturereview and points for discussion.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Introduction to the Urban Crime Model . . . . . . . . . . . . . . . . . . . . . . 11.1 A Brief History of the Urban Crime Model . . . . . . . . . . . . . . . . . . . . . 11.2 Comparisons to Other Systems of Reaction-Diffusion Type . . . . . . . . . . . . . 61.3 The Structure of This Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 An Agent-Based Model of Urban Crime with Police Patrol . . . . . . . . . . . . . 101.4.1 A City with a Boundary Modelled by a Lattice . . . . . . . . . . . . . . . 121.4.2 Key Probabilities that Determines the Actions of the Criminals and thePolicemen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.4.3 Discrete Evolution of Agents’ State and the Localized Spread of the Riskof Crime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.5 Deriving the Continuum Limit as a System of PDEs . . . . . . . . . . . . . . . . 201.5.1 Derivation of the Individual PDEs . . . . . . . . . . . . . . . . . . . . . . 201.5.2 The Final Form of the PDE system and a Special Identity for the Non-linear Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 Crime Hotspot Formation and Long-time Behaviours . . . . . . . . . . . . . . 27iv2.1 Linear, Weakly Nonlinear and Far-from Equilibrium Regimes . . . . . . . . . . . 292.2 Super- and Sub-critical Crime Hotspots: Leading-Order Steady-State Theory . . 322.2.1 Monotonicity of the Outer Problem . . . . . . . . . . . . . . . . . . . . . . 362.2.2 Reduction to a Quadrature and Existence of a Maximum Threshold . . . 402.2.3 Crime Hotspot Insertion - the One-Sixth Rule and Fold Bifurcation . . . . 412.2.4 Determination of Attractiveness Amplitude at the Hotspot . . . . . . . . 442.3 NLEP Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.3.1 Derivation of Jump Conditions . . . . . . . . . . . . . . . . . . . . . . . . 472.3.2 The Outer Problem and Analyticity of Coefficients . . . . . . . . . . . . . 482.3.3 Conclusions on Stability for Various Patterns . . . . . . . . . . . . . . . . 502.4 Bifurcation Diagrams of Hotspot Equilibria: Numerical Continuation Computa-tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.4.1 Supercritical Regime - Fold Points for Spike Insertion . . . . . . . . . . . 512.4.2 Subcritical Regime - Fold Points for Spike Type Switching . . . . . . . . . 572.5 Refinements of the Steady State Solution: Higher-Order Theory . . . . . . . . . . 572.5.1 Improved Approximation of the Fold Point for the Supercritical Regime . 702.6 A Normal Form for Hotspot Insertion . . . . . . . . . . . . . . . . . . . . . . . . 712.7 Slow Dynamics of Crime Hotspots . . . . . . . . . . . . . . . . . . . . . . . . . . 802.7.1 The Slow Dynamics of One Hotspot . . . . . . . . . . . . . . . . . . . . . 802.7.2 A DAE System for Repulsive Hotspot Dynamics . . . . . . . . . . . . . . 862.7.3 Comparison of Asymptotic and Full Numerical Results for Slow HotspotDynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 882.8.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 882.8.2 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913 Police Intervention - a Simple Interaction Model . . . . . . . . . . . . . . . . . 953.1 Asymptotic Construction of a Multiple Hotspot Steady-State . . . . . . . . . . . 973.1.1 A Symmetric Pattern of Hotspots of Equal Amplitude . . . . . . . . . . . 983.2 NLEP Stability of Multiple Spike Steady-State for General Power 1 < q <∞ . . 102v3.2.1 Linearization with Floquet B.C. . . . . . . . . . . . . . . . . . . . . . . . 1033.2.2 The Jump Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043.2.3 The Competition Instability Threshold . . . . . . . . . . . . . . . . . . . . 1093.2.4 Interpretation of the Threshold . . . . . . . . . . . . . . . . . . . . . . . . 1113.2.5 Stability of a Single Spike . . . . . . . . . . . . . . . . . . . . . . . . . . . 1123.3 Analysis of the NLEP - Competition Instability and Hopf Bifurcation . . . . . . 1143.3.1 Combining the Nonlocal Terms . . . . . . . . . . . . . . . . . . . . . . . . 1143.3.2 The Zero Eigenvalue Crossing Revisited . . . . . . . . . . . . . . . . . . . 1173.3.3 Solution to NLEP as Zeros of a Meromorphic Function . . . . . . . . . . . 1193.4 Explicitly Solvable Case q = 3 and Asynchronous Oscillations . . . . . . . . . . . 1203.4.1 Explicit Determination of Hopf Bifurcation and Stability Region . . . . . 1243.5 General case q 6= 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1333.5.1 Determining the Number of Unstable Eigenvalues by the Argument Prin-ciple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1333.5.2 The Starting and Ending Point of the Path and the Two Main Cases . . . 1363.5.3 Key Global and Asymptotic Properties of C(λ) and F(λ) . . . . . . . . . 1393.5.4 Above Competition Instability ThresholdDj,2 > D∗j,2; A Unique UnstableReal eigenvalue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1433.5.5 Below Competition Instability Threshold Dj,2 < D∗j,2 . . . . . . . . . . . . 1443.5.6 A Gap Between the Lower and Upper Thresholds: Dj,2,min < Dj,2 < D∗j,2,Existence of Hopf Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . 1503.5.7 Conclusions on the Stability of a Symmetric K−Hotspot Steady-State . . 1533.5.8 Asymptotic Determination of Hopf Bifurcation Threshold . . . . . . . . . 1543.6 Stability of a Stripe Pattern, Explicitly Solvable Case. . . . . . . . . . . . . . . . 1573.6.1 Extension of a 1-D Spike Solution to a 2-D Stripe Solution . . . . . . . . 1583.6.2 The Stability of a Stripe . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1613.6.3 Analysis of the NLEP - Stripe Breakup Instability . . . . . . . . . . . . . 1663.7 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1683.7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1703.7.2 Open Problems and Future Directions . . . . . . . . . . . . . . . . . . . . 171viBibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .173A Lemmas and General Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . .179A.1 A Floquet Boundary Condition Approach to Neumann NLEP Problems on aBounded Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179A.1.1 Converting a Neumann problem to a Periodic Problem with Twice theDomain Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180A.1.2 Converting a Periodic Problem to a Floquet Problem . . . . . . . . . . . . 180A.1.3 The Floquet Eigenvalue Problem for the Stability of aK-spike SymmetricPattern with Neumann Boundary Conditions . . . . . . . . . . . . . . . . 181A.2 Properties of the Local Operator L0 in One Spatial Dimension . . . . . . . . . . 183A.2.1 Applications to Explicitly Solvable NLEP . . . . . . . . . . . . . . . . . . 187A.3 Miscellaneous Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187A.3.1 Formulas for the Lq-Norm of the Ground State w(y): p = 3 Case . . . . . 187A.3.2 Formulas for the Lq-Norm of the Ground State w(y): General Case . . . . 188viiList of Tables3.1 Values of the constant rj,K defined in (3.84). . . . . . . . . . . . . . . . . . . . . 128viiiList of Figures1.1 Burglary hotspots in West Vancouver, B.C., Canada (left) and Santa Clara-Sunnyville, California, US (right). Generated by raidsonline.com [44] using pub-licly available crime data for residential and commercial burglaries. Shown pic-tures are for a full year beginning August 2014. . . . . . . . . . . . . . . . . . . . 22.1 A Turing instability leading to a localized steady-state solution. Parameter valuesare  = 0.05, D = 2, ` = 1, γ = 2, α = 1, so that γ > 3α/2. The initial condition(left panel) for the numerical solution of (2.1) is a small random perturbation ofthe spatially uniform state given by A(x, 0) = Ae+rand∗0.1, ρ(x, 0) = ρe, whereAe = γ = 2 and ρe = 1− α/γ = 0.5. The right panel shows the hotspot solutionat the final time t = 105 with A (solid curve) and ρ (dotted curve). Notice thatthe range of A and ρ are on different scales. . . . . . . . . . . . . . . . . . . . . 322.2 Small initial bumps in A quickly evolve into hotspots, which then move slowly totheir steady-state locations. Parameter values are  = 0.05, D = 2, ` = 1, γ = 2,α = 1, so that γ > 3α/2. The initial condition (left panel) for the numericalsolution of (2.1) is A(x, 0) = Ae + ∑2i=1 sech(x−x0,i ) and ρ(x, 0) = ρe, wherex0,1 = −0.7, x0,2 = −.7, Ae = γ = 2, and ρe = 1− α/γ = 0.5. We only plot A. . . 332.3 Phase portraits of (2.19) with trajectories emanating from the line u = α. Thereare two types of trajectories: (i) those that hit the u−axis are admissible solutionsto (2.19) satisfying v(`) = 0 for some ` > 0, (ii) those that do not, but goes tothe line u = 1.5 develops singularity for finite values of x. The model parameterschosen were D = 1, α = 1 for both plots and γ = 2.0 and γ = 1.25 respectivelyfor the left and right plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38ix2.4 The function χ(µ), from (2.4a), that determines the outer solution for any µ ≡a0(`) on α < µ < min{γ, 3α/2}. Common parameter values are D = 1 and α = 1. 422.5 χmax against γ for five values of α (1.0, 1.25, 1.5, 1.75, 2.0), given by formulas(2.26) and (2.28). The curves were evaluated on the respective ranges of γsatisfying ((3α/2) ·1.01, (3α/2) ·2). It is evident from the plots that χmax → +∞as γ → (3α/2)+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.6 Comparison of numerical and asymptotic results for A(0) (left panel) and for−2V (0) (right panel) as  is decreased for a single hotspot centred at the origin.The parameter values are α = 1, γ = 2, ` = 1, and D = 2. The dashed horizontallines are the leading-order prediction of A(0) ≈ √2/√V0 and −2V (0) ≈ V0,where V0 is given in (2.35). The thin dotted curves are from the improvedasymptotic theory with V0 now given by (2.81). . . . . . . . . . . . . . . . . . . 462.7 Stability of O(−1)-amplitude and O(1)-amplitude hotspots. Parameter valuesare  = 0.01, D = 1, α = 1, γ = 1.25. Left plot: the O(−1)-amplitude hotspotpersists in this subcritical regime. Right plot: the O(1)-amplitude hotspot istransient and dissipates into the Turing-stable homogeneous state. . . . . . . . . 502.8 The plots (a), (b) and (c) show the continuation of steady states starting witheither one, two, or four, interior hotspots, respectively, for  = 0.01. The otherparameters are γ = 2, α = 1, and ` = 1. The solid and dashed curves inthe subplots show the profiles of A and V , respectively, at various values of Dspecified on top of the plots. These values of D correspond to the marked pointson the bifurcation diagram as shown on the left. Notice that the range of A andV are on different scales. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53x2.9 For the parameter set γ = 2, α = 1, and ` = 1, the solid curves on the left of eachof (a), (b), and (c), for K = 1, K = 2, and K = 4, interior hotspots, respectively,show the fold point values Dfold,K() associated with the small norm solutionbranch of A(`) versus D. The top tick-mark on the vertical axes in these plotsare the approximate values Dcrit,1 ≈ 1.793, Dcrit,2 ≈ 0.448, and Dcrit,4 ≈ 0.112from the leading-order theory of §2.2. The dashed curves in each of (a), (b), and(c), are the asymptotic results (2.82) for the fold point value for D, as predictedby the higher-order asymptotic theory of §2.5. For each of the three sets, thenumerically computed A versus x is plotted on |x| ≤ 1 at four values of . Atthe larger values of  the pattern is essentially sinusoidal. . . . . . . . . . . . . . 552.10 Plots of criminal density ρ near the onset of spot insertion as indicated in Fig.2.9. The parameter values are the same, i.e.  = 0.01, γ = 2, α = 1, ` = 1. Thepurple curves correspond to ρ at the numerically computed fold point, while thered and blue curves correspond to the upper and lower branch solutions at someidentical values of D close to, but respectively above and below the fold pointsshown in Fig. 2.9. Such chosen values of D are 2.0, 0.5 and 0.1 respectively forone, two and four interior spikes (before the insertion event), corresponding tothe sub-figures on the left, centre, and right, respectively. The fold point valuesof D are given in (2.49). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.11 Time evolution of criminal density ρ for spatial patterns found by continuationbeyond the fold point. Parameter values are the same as in Fig. (2.9), i.e. = 0.01, γ = 2, α = 1, ` = 1. Also, D = 2.0, 0.5, 0.1 for the left, centre and rightsub-figures, respectively. The dotted, dashed and solid blue curves shows theevolution of the upper solution at t = 0, 15, 100, respectively, while the heavysolid red curves shows the lower branch solution. We find that the plots att = 100 all overlap exactly with those of the lower branch counterparts, whichare the linearly stable patterns proved in Section 2.3. . . . . . . . . . . . . . . . . 56xi2.12 A closed homotopy of equilibria with a single hotspot (or two boundary hotspots)from the continuation in γ from the subcritical Turing bifurcation. Model pa-rameters are:  = 0.01, D = 1, α = 1. Observe that the amplitude of A at PointI and IV are O(1) and equal but different to that of Point II and III, which areof O(−1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582.13 Plot of the full numerical solution to the steady-state problem (2.11) showingthe knee behaviour of V for γ = 2, α = 1, D = 2,  = 0.01, and ` = 1. The figureon the right is a zoom of the one on the left. . . . . . . . . . . . . . . . . . . . . . 602.14 The knee-shaped function F (z) defined at (2.68). Parameter choices are the sameas in Fig. 2.13 which showed the numerical solution of V . . . . . . . . . . . . . . 642.15 Left: Plot of the bifurcation diagram of κ versus s = U(0) for solutions to thenormal form equation (2.96). Right: the solution U(y) (solid curves) and thederivative U ′(y) (dashed curves) versus y at four values of s on the bifurcationdiagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752.16 Numerically computed solution (solid curve) to (2.105) with asymptotic bound-ary condition (2.109) imposed at x = `− δ, with δ = 0.000001, and with param-eter values γ = 2, α = 1, ` = 1, and  = 0.01. We obtain H` = limx→`− H(x) ≈0.303. The dashed curve is the local Frobenius series approximation for H, validnear x = `, with leading terms given in (2.108). . . . . . . . . . . . . . . . . . . . 782.17 The asymptotic results (dashed curves) for A (left) and V (right) in the bound-ary layer region near x = ` at the fold point value for D are compared withcorresponding full numerical results (solid curves). The asymptotic fold pointvalue is given by (2.107). The top row is for  = 0.00273 and the bottom row isfor  = 0.005. The other parameter values are γ = 2, α = 1, and ` = 1. . . . . . . 792.18 The asymptotic result (dashed curve) for A(`) versus D in a narrow interval nearthe fold point, as obtained from (2.107) with H` ≈ 0.303, is compared with thecorresponding full numerical result (solid curve) computed using AUTO-07p. Theparameter values are γ = 2, α = 1, ` = 1, and  = 0.01. . . . . . . . . . . . . . . 80xii2.19 For parameter values γ = 2, α = 1, ` = 1, D = 4, and with initial statex0(0) = 0.3, the asymptotic result (2.120) for slow hotspot dynamics is plottedfor three values of . These results are compared with the corresponding resultwhen the switchback term − log  is neglected, so that x0± = x0 in (2.120b).The plot on the right is a zoom of that on the left. . . . . . . . . . . . . . . . . . 852.20 The slow dynamics of a single hotspot on the slow time-scale σ, as predicted bythe asymptotic theory (2.120) (dashed curves), are compared with correspondingfull numerical results of the PDE system (2.10) (solid curves) computed usingPDEPE of MATLAB R2013b. The domain is |x| ≤ 1 and the parameter values areγ = 2, α = 1, and D = 4. Left:  = 0.005. Middle:  = 0.01. Right:  = 0.02. . . 882.21 Comparison of slow dynamics predicted from the asymptotic theory (2.125)(dashed curves) and from full numerical simulations (solid curves) of the PDEsystem (2.10). The domain is |x| ≤ 1 and the parameter values are γ = 2, α = 1,and  = 0.01. Left: a two-hotspot evolution with D = 2, with initial locationsx0 ≈ −0.300, 0.299. Right: a four-hotspot evolution with D = 0.3, with initiallocations are x0 ≈ −0.794, −0.346, 0.151, 0.698. . . . . . . . . . . . . . . . . . . 892.22 The continuation of the function values at the core of the unstable crime hotspotshown in Fig. 2.12. Other parameter values are D = 1, α = 1, γ = 1.25. Theseshow that maxA = A(0) and minV = V (0) ∼ V0 are indeed O(1) as  → 0+.Note that minV < 427 = 0.1˙48˙. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 922.23 Continuation of O(1) amplitude spike in the A component in domain length `naturally connects to the O(−1)-amplitude spike at a fold bifurcation. We plotsthe range of possible A(`) and check possible overlapping in their range. This isto throw light on the possibility of asymmetric patterns with neighbouring O(1)and O(−1) spikes. Model parameters are  = 0.01, D = 1, γ = 1.25, α = 1. . . . 923.1 Competition instability threshold nonlinearity g(U0) against police deploymentU0 at various focus degrees q. Other model parameters are S = 2, γ = 2, α = 1,so U0,max = 2 as shown in the right-most tick of the figure. The competitioninstability threshold D∗0 is simply a positive scaling of g(U0) according to (3.47). 113xiii3.2 Regions of stability (shaded) and Hopf curves as function of D0 for K = 2, 3, 4according to (3.92). Model parameters are S = 4, γ = 2, α = 1, U0 = 1. Thevertical dotted lines denote D0,lower and D0,upper respectively. . . . . . . . . . . . 1293.3 Schematic plot of the Nyquist contour used for determining the number of un-stable eigenvalues of the NLEP in Re(λ) > 0. . . . . . . . . . . . . . . . . . . . . 1353.4 An image of the Nyquist contour transformed by ζ. Notice that ζ(CR) shrinks tothe complex infinity∞ as R→∞, and both ζ(Γ+) and ζ(Γ−) are asymptoticallyparallel to the imaginary axis due to (3.101). . . . . . . . . . . . . . . . . . . . . 1353.5 Close-up of two distinctive cases of the Nyquist plots for K = 2. For the modelparameters stated in (3.100), and for τu = 1, we show the Nyquist plot for twovalues of D0. The upper threshold, corresponding to the competition instability,is given by D0,upper = D+0,1 ≈ 3.103. The curve with ζ(0) < 0 and ζ(0) > 0correspond to the choice D0 = 3.0 and D0 = 3.2, respectively. The number ofunstable eigenvalues are respectively N = 0 and N = 1. . . . . . . . . . . . . . . 1383.6 Close-up of two distinctive cases of Nyquist plots for K = 2. Parameter valuescommon to both are α = 1, γ = 2, U0 = 1, D0 = 3 and ` = 1 (but S =2K` = 4). The curves that are clockwise and anticlockwise with respect to theorigin correspond to the choices τu = 1 and τu = 4, respectively. The numberof unstable eigenvalues are numerically determined to be N = 0 and N = 2 forthese particular cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1383.7 Plots of FR and FI for q = 2, 3, 4, 5. Note that FR(0) = 1/2 and FI(0) = 0, andthat the maximum of FI occurs near λI = 3. In fact, the maximum does occurexactly at λI = 3 when q = 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1413.8 Plots of CR and CI for τu = 1 and the model parameters given in (3.100). Thevalue for D0 was chosen as follows for each curve: Left figure: three cases ofCR(λI) are plotted: (i) b > 3 (D0 = 0.4), (ii) 3 − 3/(2a) < b < 3 (D0 = 0.6),(iii) b < 3− 3/(2a) (D0 = 1.0). Right figure: two cases of CI(λI) are plotted: (i)b < 3/ (1 + 1/(3τ˜j)) (D0 = 1.0), b > 3/ (1 + 1/(3τ˜j)) (D0 = 0.5), and the rightend point of the interval of negative values for CI is at λII ≈ 1.6204, indicatedby a heavy dot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143xiv3.9 Numerical verification of a Nyquist path that predicts stability. Left: a plot ofthe unique roots to ζR(λI) = 0 as τu varies. For very large τu, λI can be quiteclose to zero. Right: a plot of the possible range of values of ζI(λI) when theNyquist contour hits the imaginary axis, where λI was chosen to vary from zeroto the maximum value plotted on the left. The model parameters chosen wereD0 = 0.4 and the parameters specified in (3.100). . . . . . . . . . . . . . . . . . 1473.10 Locations of the three special points λIR , λII and λIM on the graphs of CR andCI versus λI , as discussed in the text. The model parameter values τu = 1and D0 = 0.4, together with those stated in (3.100), were used. The valueb ≈ 3.1279 > 3 was also numerically confirmed. . . . . . . . . . . . . . . . . . . . 1483.11 Numerical results for the location of λ∗I (indicated by a star) relative to λIR , λIM ,and λII (indicated by heavy dots), as stated in (3.115). The model parameters,τu = 1 and D0 = 0.4, together with the parameter values stated in (3.100), wereused. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1493.12 Locating the Hopf bifurcation point. The two curves shows the locus of theroots of ζR(λI) and ζI(λI) as functions of τu. Hopf bifurcation is determinednumerically to occur at τu ≈ 3.33 and λI ≈ 0.307, at the intersection of these twocurves. The model parameters are D0 = 0.6 and those stated in (3.100). For thisparameter set we confirm numerically that 3− 3/(2a) ≈ 2.397 < b ≈ 2.714 < 3,which satisfies (3.118). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1513.13 Plot of the steady-state spike from Proposition 3.22 in the inner region on the x1axis for  = 0.05, ` = 1.0, γ = 2, and α = 1. The x2 direction is omitted becauseit extends trivially. Left plot: The attractiveness A ∼ w (x1/) /(√v0) + α(heavy solid curve) and criminal density ρ ∼ w (x1/) (solid curve), these curvedo not depend on q. Right plot: The police density U from (3.136) for q = 2(heavy solid curve) and for q = 3 (solid curve). . . . . . . . . . . . . . . . . . . . 162xv3.14 Principal eigenvalue λ as a function of frequency m. Left: plot of λ versus m,as given in (3.153), for  = 0.05, D0 = 1, γ = 2, α = 1, l = 1, and U0 = 1.The asymptotic prediction as  → 0 for the instability band from (3.154) is0.158 < m < 34.64. The corresponding numerical result is 0.131 < m < 34.56.Right: plot of λ versus m near the lower threshold m− for U0 = 1 (solid curve)and U0 = 1.5 (heavy solid curve). The lower edge of the instability band increasesas U0 increases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1683.15 Spontaneous breakup of a stripe into one spot, obtained from a full numericalsimulations of (3.124) using VLUGR2. The model parameter values were  =0.05, D0 = 1, γ = 2, α = 1, U0 = 1, d = 2 and ` = 1. The time instants chosenwere t = 0.000, 0.7299, 8.220, 9.985 from top to bottom respectively. . . . . . . . . 169xviChapter 1Introduction to the Urban CrimeModel1.1 A Brief History of the Urban Crime ModelThere has been an increasing interest in recent years on developing mathematical tools tounderstand and predict spatial patterns of urban crime (cf. [47, 48, 49] and the survey [17]).One main impetus underlying this effort is the increased availability of residential burglarydata, partially due to improved mapping technology and digital record-keeping. Moreover, notonly are the authorities releasing more data to the public, but also some commercial and non-profit organizations are starting to utilize such data to create publicly available visualizationsof crime patterns. For example, Fig. 1.1 shows two crime density maps that illustrate thephenomenon of crime hotspots, where high levels of crime are often concentrated in certainspatial regions that may evolve slowly over time (cf. [4]). Many studies have hypothesized thatsuch hotspots are due to a repeat or near-repeat victimization effect, which postulates thatcrime in a certain region induces more crime in that and nearby regions (cf. [21, 55]). Suchtheoretical descriptions of nonlinear feedback mechanisms that diffuses risk by proximity haveresulted in various modelling efforts, that aim to quantify such mechanisms of criminology toprovide an explanation for the emergence and persistence of crime hotspots.In [47, 49], Short et al. introduced an agent-based model of urban crime that takes into1Figure 1.1: Burglary hotspots in West Vancouver, B.C., Canada (left) and Santa Clara-Sunnyville, California, US (right). Generated by raidsonline.com [44] using publicly availablecrime data for residential and commercial burglaries. Shown pictures are for a full year begin-ning August 2014.account the features of repeat or near-repeat victimization of urban crime, and derived a con-tinuum limit using methods similar to those used in analyzing biased random walk models inmathematical biology. In dimensionless form, the continuum limit of this agent-based model isthe two-component reaction-diffusion (RD) systemAt = 24A−A+ ρA+ α , ρt = D∇ ·(∇ρ− 2ρA∇A)− ρA+ γ − α , (1.1)with no-flux boundary conditions.Here, ρ represents the density of criminals, A measures the “attractiveness” of the environ-ment to burglary, and the term −D∇ · (2ρ∇A/A) models the tendency of criminals to movetowards sites with a higher attractiveness. In addition, α > 0 is the constant baseline attrac-tiveness, while γ − α, with γ > α, models a constant rate of background re-introduction ofcriminals. The constant 0 <   1 models the small diffusive spread of the attractiveness,which is relatively much slower compared to the movements of criminals, while the constant D,which represents the diffusivity of criminals, was not present in the form of the model originallyproposed in [47]. It is added here for uniformity in the subsequent presentations of results ofthis dissertation, and the choice of the form presented in (1.1) will be justified in due course ofour analysis. Further details of the model are given in [47] and in Section 1.4 as well, wherewe describe an agent-based model whose continuum limit is a three-component PDE extending(1.1).2We now turn to review briefly the history of the development of the model and the surround-ing literature. In [47], the authors performed numerical simulations of the discrete agent-basedmodel, and crime hotspots were clearly observed for some parameter choices. To study the rangeof parameters that would give rise to crime hotspots, they deduced the two-component PDEsystem (1.1) from the formal continuum limit of the agent-based model and performed Turingstability analysis from the constant equilibrium state. They were able to produce numericalsimulations of the full deterministic PDE system to exhibit crime hotspots when the parameterspredicts Turing instability. Furthermore, they observed that hotspots are formed only if areasof high criminal activity are separated far enough, and only when attractiveness diffuses overshorter distances. Otherwise, hotspots cannot be numerically observed. In other words, thediffusivity of the attractiveness, the parameter 2, is necessarily small for crime hotspot to existnumerically. This suggests that the system (1.1) possesses hotspots in the steady state onlywhen it is singularly perturbed.An admitted insufficiency in the initial work [47] was the lack of consideration of the impactof law enforcement on crime patterns. Thus, in a subsequent work [48], Short et al. stud-ied the effect of suppression and investigated the conditions for dissipation of crime hotspots.Expounding on the finding that their model predicts hotspots only when the diffusion rate ofrisk is small, they performed weakly nonlinear analysis to determine that Turing bifurcationwas subcritical and classified hotspots in the full numerics as being either subcritical or su-percritical. By manually introducing large perturbations at the hotspots in their numericalsimulations, they found that suppression of supercritical hotspots will cause risk to diffuse ra-dially, forming a transient “ring-like structure”, which subsequently disintegrated into multiplehotspots (see Fig. 10 of [48]). This phenomenon was referred to as “hotspot displacements”. Incontrast, when suppressing a subcritical hotspot, no such transient states were formed and thesuppression effect was observed to persist without causing displacement or production of newhotspots. The intention of this numerical experiment was to distinguish the qualitative featuresof supercritical and subcritical hotspots, which have apparently similar localized structures inthe solution profile, but differ markedly in their response to hotspot suppression. This is aninteresting observation especially for criminologists, among which contains disagreeing groups,some doubting and some affirming the virtue of hotspot policing.3Shortly afterwards, Jones, Brantingham and Chayes, the latter two being co-authors of theinitial work [47], published another work [22] which augments the initial work by consider-ing possible models of police deployment strategies by means of introducing an extra compo-nent to the two-component PDE. They modelled it in two ways: firstly, by modifying crime-attractiveness perception by criminals when police are present; secondly, by the deterrent effectacting on the criminals directly when the police are located at the same place (they assumedthe amount of immediate apprehension of lawbreakers was negligible). They focused more ondiscussing what deployment strategy could maximize the deterrent effect on the criminals, andthey proposed three variants of police patrolling strategies: “random walkers”, “cops on thedots” and “peripheral interdiction”. By considering the continuum limits, both the former twomodels resulted in an extra PDE modelling the police movements, while the same effort proveddifficult for the last model. This work has a strong influence on our work when we developedthe form of our model (1.20) in the presence of police.Meanwhile, the mathematical modelling of crime is gaining momentum in terms of publicinterest and recognition (cf. [11]). From within the mathematical community, an increasingamount of effort was invested to establish the theoretical validity of the models, and to extendthe original model, or even to consider alternative mathematical basis of crime hotspots (cf.[17]).For the continuous Short’s model (1.1), there have been a few other previous studies, inaddition to [47, 48, 49], on its pattern formation properties. For the parameter regime D  1,hotspot equilibria and their stability properties were analyzed in [29] by a combination offormal asymptotic methods and results from the spectral theory of nonlocal eigenvalue problems(NLEP). In [2] the existence of these hotspot equilibria for the regime D  1 was establishedrigorously using a Lyapunov-Schmidt reduction. In [46] the local existence of solutions to thecrime model in multidimensional domains was established. In [7], the branching behaviour nearthe Turing point associated with the spatially homogeneous steady-state was characterizedrigorously. Similar bifurcation theoretic results characterizing branching behaviour from thespatially homogeneous steady-state for some extensions of the basic crime model (1.1) aregiven in [18]. There are also new mathematical models that attempt to extend Short’s model(1.1) by incorporating the action of the police (see [22, 41, 45] for such PDE models). Other4types of non-PDE mathematical models that attempt to reformulate the mathematical basisof crime hotspots, or introducing police deterrence in a different way also exist (cf. [1, 17, 56]and the references therein). In this dissertation, we introduce a three-component PDE model(3.3) that extends (1.1). The model is heavily influenced by the work in [22] and [45], wherethe police interaction introduced is a variant of hotspot policing that was discussed in [56].As mentioned above, the key question that [49] attempts to address is whether crimehotspots can be eradicated by one-time suppression. They conclude that whether hotspot polic-ing will be a successful strategy or not depends on the system parameters, and they classifycrime hotspots as either supercritical or subcritical by the parameter regime that such solutionsreside in. This will become one of the key themes in our analysis presented in Chapter 2 in thatwe work towards a more complete description of crime hotspots in both the supercritical andsubcritical regimes, and at the same time shed light on the issue of crime hotspot displacementand annihilation. In Chapter 3, we use a continuous parameter that spans the spectrum be-tween purely random policing to focused hotspot policing to seek an optimal policing strategy.The optimal policing strategy was measured by its efficacy in destabilizing preexisting crimehotspots. However, compared to Chapter 2, the results in this chapter are, in comparison, morepreliminary in terms of application, and the central focus is to examine how much we couldextend the tool-set used in the study of other reaction-diffusion (RD) system by dint of theirstructural similarities. For example, while the hotspot solution profile construction by meansof matched asymptotic expansions are quite similar, the associated NLEP stability analysis isnow rather intricate except for a particular parameter choice where it simplifies considerably.For the methodology of this dissertation, the closest predecessor is the article by Kolokol-nikov et al. in [29], which contains the first attempt of studying the stability of crime hotspotsfor the model by Short et al. in [47] in the far-from equilibrium regime. Realizing that the PDEsystem is fundamentally singularly perturbed, they constructed steady-states with hotspots di-rectly using matched asymptotic expansions and then studied their stability with respect tokey model parameters. Their far-from equilibrium results also connect with those found by lin-earity stability and weakly nonlinear analysis in [47, 48], by means of a numerically computedbifurcation diagram (cf. Fig. 7 of [29]). Our new work follows the approach in [29] and extendsthe analysis by solving the conjectures concerning the dynamics of hotspots, and augmenting5the equations by deriving a police PDE equation from scratch, and then studying the equi-librium and stability problems. As a result of the research in the course of the developmentof this dissertation, a large portion of Chapter 2 and the Section 3.6 of Chapter 3 have beenpublished respectively in [51] and [37], and with further work being organized into new papersto be submitted in the future.1.2 Comparisons to Other Systems of Reaction-Diffusion TypeThe urban crime model with a new police component, given in (1.20), is also interesting math-ematically because it possesses various properties that are not observed in similar systems ofreaction-diffusion (RD) type. There are sufficient similarities between (1.20) and the RD modelsthat exhibits localized patterns, for e.g. the Gierer-Meinhardt, Gray-Scott, Brusselator and theSchnakenburg models, to name a few (cf. [36, 54] and the references therein), so that the toolsdeveloped for such models are also applicable to a large extent. For example, in this disserta-tion, the study of the existence and the linear stability (in O(1) time-scales) of multiple crimehotspots has drawn much influence from the corresponding study of multiple-spike solutions(cf. [13], [20], [24], [52], [53], [54] and also the references therein), and the methodology extendsquite naturally from previous work. However, there are various novelties both in terms of themathematical techniques developed and the results, which we describe briefly in this section.Firstly, in terms of the asymptotic construction of a steady state pattern with localizedstructures of high amplitude, which are usually located in an “inner region” in a matchedasymptotic expansion procedure, the interaction with the corresponding “outer region” is crucialin determining various properties of the localized structure, and notably the amplitude of thestructure. In the references cited above, the background states in the “outer region” are usuallyquite simple, e.g. constant or solutions of a linear differential equation. However, in Chapter2, we see that the background state is highly nonlinear and such a nonlinearity in the “outerregion” lends to the possibility of “peak insertion” governed by thresholds, which explainsthe emergence to further localized crime hotspots. Such a threshold phenomenon cannot bedescribed by elementary functions, which was the case when the shadow limit or a weakercoupling (cf. [29]) is considered.6Secondly, the interaction between the inner and outer regions is also considerably morecomplex in technical terms. In particular, to construct an approximation of a quasi-steadystate that evolves slowly in time, new nested boundary layers in between the inner and outerregions are required and switchback terms were necessary to complete the matching procedure.The use of switchback terms also arise in the singular perturbation analysis of some otherproblems, including model problems of low Reynolds number flows (cf. [31], [32], [42]), andthe analysis in [33] of singular solutions to a PDE model for the deflection of a micro-platecapacitor. The increasing technical effort that is required to produce a satisfactory correction tothe leading order theory also prompted us to consider alternatives. In particular, we devised analgebraically simpler procedure, which was somewhat inspired by the renormalization approachin [8]. This approach is presented in this dissertation in Section 2.5 and the result was that thekey effects from the switchback correction terms could be effectively incorporated in a relativelysimple way.The resolution of the nested boundary layers between the inner and outer regions also ledto a novel form of equations governing slow dynamics of crime hotspots. For the dynamicsof a single spike in standard singularly perturbed RD systems, such as the Gierer-Meinhardt,Schnakenberg, and Gray-Scott systems, studied in [9], [14], [15], [19], [43], [50] (see also thereferences therein), the governing equations were ODEs. In contrast, for the slow movement ofcrime hotspots studied in Section 2.7, the dynamics of a single hotspot is found to be governed bya system of differential algebraic equations (DAEs). For the case of multiple hotspots, a coupledsystem of DAEs was derived, and these asymptotic results were found to agree favourably withfull numerical simulations of the PDEs.With regards to the problem of studying the linear stability on O(1) time scales of a multi-hotspot steady-state, the results in both Chapter 2 and 3 are also quite novel. In Section 2.3for the D = O(1) regime for the basic crime model with no police, we found unconditionalstability for any number of crime hotpots (as long as they exist as steady states). This standsin sharp contrast to many other RD systems defined on a finite domain, and also with regardsto the stability analysis of [29] for the near-shadow (large diffusivity) regime D  1, whereexplicit stability thresholds in D were obtained. In terms of instability, the results in Chapter3 for the urban crime model with police intervention also point to new possibilities not seen, to7the best of our knowledge, in other RD systems. For instance, in Section 3.4, at a particularchoice of system parameter for the three-component system with the police component, a noveloscillatory instability that is asynchronous was found and characterized explicitly. We believeasynchronous oscillatory instability is a general trait exhibited by this system, though a fullcharacterization is currently lacking due to the complexity of the NLEP in the general case. Incontrast, for the case of Gray-Scott and Gierer-Meinhardt model studied in [9, 52, 50], wherea similar analysis was performed, the dominant oscillatory instabilities of the spike amplitudeswere all found to be synchronous.The difficulty of the stability problem for the urban crime model with police is also exem-plified by the fact that the NLEP studied in Section 3.2 has two nonlocal terms, while most ofthe RD systems studied in the past have only one nonlocal term. The novelty in the structureof the NLEP problem requests that new techniques be developed to analyze them, and alsopoints to potentially richer behaviours that were not observed in previous studies of other RDsystems.Another feature of the rich structure of the systems studied in this dissertation is theirrather intricate bifurcation diagrams of steady-state solutions. By using well established nu-merical continuation and evolution software (cf. [12, 3]), we were able to compute globalbifurcation diagrams for the “far-from equilibrium” steady states and determine their stabil-ity. The structure we observed is rather intricate, yet possessing symmetry. In particular, weperformed path-following numerical study of the steady states, similar to what is done in [35],but for a finite domain instead of a (truncated) infinite domain. Our conclusions were quitedifferent from that in [35], which we believe to be caused by the nature of the domain topology.In an infinite domain, the phenomenon of homoclinic snaking is well-studied in many other RDsystems, such as the Swift-Hohenberg model (see [5] and the references therein). In [35], theauthors demonstrated numerically the existence of wave-packet type localized solutions whichleads to a pattern of well-separated hotspots. However, in a finite domain with other parame-ters in the same regime as in [35], we observed a closed homotopy of localized equilibria witha single hotspot bifurcating out of a subcritical Turing point. We remark that it has beenshown recently in [6] that homoclinic snaking behavior is a rather generic feature of solutionsto certain types of RD systems near a subcritical Turing bifurcation point.81.3 The Structure of This DissertationTo provide a more thorough introduction to the model we study in this dissertation, in Section1.4, we relate principles in criminology to the definition of a new agent-based model whichextends that of [47] to include police interactions with the criminals. A key feature of thissection is the idea to use a new form of biased random walk to model police patrol with differentdegrees of focus, modelled by a parameter k, to sites attractive to criminals. Then, in Section1.5, we derive the continuum limit of this agent-based model to arrive at a three-componentsystem which will become the centre of the study of this dissertation.The remainder of the dissertation is divided into two chapters.In Chapter 2, we study the case when police are absent, i.e. the same PDE that is derivedin [47] but defined on a bounded 1-D domain. Unlike the analysis in [29], we study the regime  1, D = O(1) where there is stronger interaction between the two components in the RDsystem. The key results are that the far-from equilibrium existence and stability theory donot depend on whether the criminal reintroduction rate is in the Turing stable or unstableregime. However, a peak insertion phenomenon was found to occur precisely when the crimereintroduction rate is higher than half the baseline attractiveness (which is the Turing unstableregime). Moreover, applicable to both regimes, we derived an asymptotic description of theslow dynamics of crime hotspots where the hotspots undergo mutual repulsion.In Chapter 3, we study the case when policemen are present, who interact with the criminalsvia a simple interaction dynamics. We also study the problem on a bounded 1-D domain butin the weaker interaction asymptotic regime   1 and D = O(−2) as considered in [29] forthe basic urban crime model. The key results are the construction of multiple crime hotspotsusing matched asymptotic expansions, the unconditional linear stability of a single hotspot,lower and upper thresholds for stability and instability, respectively, of multiple hotspots, andfinally the existence of a Hopf bifurcation for multiple hotspots for parameters between thelower and upper thresholds. This instability is shown to lead to asynchronous oscillations inthe amplitudes of the hotspots.In both chapters, the key results from, for instance, formal asymptotic expansions, werealso stated as propositions. However, the results may be formal and we do not imply a level9of mathematical rigour usually expected for proofs from a careful consideration of functionspaces. Instead, we give evidence to the validity of our principal results by performing numericalsimulations at appropriately chosen model parameters.1.4 An Agent-Based Model of Urban Crime with Police PatrolThis section follows the approach in the original work in by Short et al. in [47], a result of theUC MaSC Project group located at University of California, Los Angeles, which includes bothmathematicians and criminologists. Following [47], we will also formally derive the continuumlimit of the discrete model, which leads to a coupled PDE system of reaction-diffusion type.In [47], the authors drew support from both evidence from field data and well-establishedcriminological theories of urban crimes to devise a model that aimed at reproducing the emer-gence, dynamics and steady-state properties of spatio-temporal clusters of crime, which is knownas crime hotspots.They chose household burglary as the prototypical crime to model due to the relative sim-plicity arising from the fact that sites of criminal activity are basically immobile. Their discretemodel was based on three key ideas:Firstly, two agents were assumed on a lattice which represents possible sites of burglary. Thefirst “agent” was an intangible one called the attractiveness, measuring the crime susceptibilityof each site, and it is perceptible by the second agent, the criminals, who were assumed to berandom walkers, but with bias in that they are more likely to burglarize and roam towards siteswith higher attractiveness. This results in a nonlinear drift term in the diffusion of criminals.Secondly, a burglary event triggers a positive feedback via the increase of attractiveness ofthe same site or nearby-sites which is to model the criminological theory of repeat or near-repeatvictimization. This results in a nonlinear coupled dynamics between the attractiveness andcriminals. Moreover, criminals were assumed to be removed from the system after a successfulburglary, but a constant background rate of introduction replenishes the number of criminalagents that roam on the lattice.Lastly, corresponding to the observation of a slow proliferation of illegal activities towardsneighbourhoods of crime hotspots, possibly due to various environmental cues and known pop-10ularly as the “broken windows” effect, the attractiveness agent is also assumed to diffuse, butin a much slower rate compared to the criminals. It is by this assumption that the eventualcontinuum model is naturally singularly perturbed by the diffusivity of attractiveness (2  1in our subsequent model parameters).Our work begins with a similar agent-based model which extends Short’s model in [47] de-scribed above by incorporating a police agent. The model was modified from previous attemptsto model police influence to criminal behaviours by deterrence but not direct apprehension. Theclosest model is particular case by Jones et al. in [22], where they model the criminal-policeinteraction by “behaviour modification”, while the police movement was modelled to be of the“cops on the dot” type. The “cops on the dot” terminology also appeared in [56] where insteadof deriving a PDE as a continuum limit, they assumed the police effect to be centred at the lo-cations of crime hotspots, and look for solutions to a constrained minimization problem definedon an appropriately chosen function space. See also [41] for another method of incorporatingpolice by means of an extra PDE.The model in [22] is a natural extension of the initial model by Short et al., and the keyidea that we are interested in this work is that we can consider patrolling policemen as randomwalkers biased towards sites with higher attractiveness, like criminals, for which Jones et al.termed “cops on the dot” in [22]. We attempt to generalize this idea to accommodate differentdegrees of focused patrolling strategy using a degree parameter k on top of the attractivenessfield, which we found flexible to accommodate the whole spectrum of police patrolling behaviourfrom pure random walk, to a more containment approach, and to very focused patrolling thatdeploys most of the policemen towards the centres of vulnerable sites.Similar to [22], we will derive in Section 1.5 a continuum limit of the agent-based modelwe define in this section. The result will be an extra PDE for the police term coupled to theoriginal continuous Short’s model (1.1) by the PDE for the criminals only. It is the subjectof the subsequent chapters by which we answer questions concerning the emergence, dynamicsand stability of crime hotspots and the effects of police patrol.Finally, we make a few remarks that compares our model to other related models in theliterature. Our model and the variants in [22], do not make the assumption that the presenceof police would alter the movements of criminals, in contrast to the models presented in [41].11Moreover, we also take a more conservative approach in the efficacy of police deployment. Unlikethe identically named, but mathematically different version of “cops on the dots” by Zipkin in[55], who emphasized the advantage of police over criminals due to a more centralized planningand higher degree of collaboration, we do not envision the action of the police to be necessarilyminimizing the the crime at each crime hotspots. Instead, we begin with the assumption thatpolice agents are essentially a type of biased random walkers not unlike the criminals in theirmovement, both of which may bias their movements due to the perception of attractiveness(or, “risk”, in police’s perspective). Moreover, we mention in passing that according to ourleading order steady-state theory in Section 3.1, such a conservative model can already predicta critical value of police deployment that would preclude the existence of steady-state crimehotspots.We now give the details of our agent-based model by both summarizing the work in [47, 22]and augmenting it with our new modelling efforts for the police term.1.4.1 A City with a Boundary Modelled by a LatticeWe describe the targets of burglary in an urban area using a square lattice x = (i, j), where0 ≤ i ≤ M and 0 ≤ j ≤ N , upon which we define moving “agents” with numerical valuesevolving with a discrete time variable t. The distance between two lattice point is fixed tobe l and the time-step of the simulation is fixed to be δt. Note that by nature of this spatialconfiguration, no agent can leave or enter the lattice. This will be connected to the type ofboundary condition we assume for the final PDE that we derive as a continuum limit of thefollowing model.At each point x and time instant t we associate four discretely evolving values: attractivenessA(x, t), number of crimes E(x, t), criminals N(x, t) and policemen R(x, t). Here, we begin bya overall sketch of the ideas that define our understanding of the relationships between thesefour numbers.The attractiveness at the site is defined as the sum of a static term A0 and a dynamic termB, i.e.A(x, t) = A0(x) +B(x, t).12The static term can represent the sum of a variety of factors including the security design ofurban homes etc., while the dynamic term refers to how the average burglar perceives of thedesirability of the site. This dynamic term can change over time and is significantly affectedby the number of crimes that occur close in time and space.The number of crimes E(x, t) is incremented when the criminals, with number recorded byN(x, t), decide to burglarize. This is assumed to happen with higher probability depending onthe attractiveness A(x, t) at the same site. The increase in the number of crimes E(x, t) will inturn contribute to the increase of the dynamic portion of attractiveness, i.e. B(x, t).After committing a burglary, the criminals involved will return home, leading to a decreaseof the number N(x, t). Unlike the discrete model in [47], this is not the only mechanism throughwhich the number of criminals is decreased in our model. Instead, we introduce the new agentR(x, t), which records the number of policemen located at a particular site. The number ofcriminal N(x, t) will be decreased at a rate that increase with the number of police R(x, t) thatis present at the same site.While the number of criminals N(x, t) will be replenished uniformly on the lattice at aconstant rate, we do not assume any mechanisms for either the increase or decrease of policemenR(x, t), i.e. the total number of policemen is conserved throughout.In the next subsections, we describe in more detail the working mechanisms of this agent-based model and relate them to the criminology being modelled.1.4.2 Key Probabilities that Determines the Actions of the Criminals andthe PolicemenThere are two key human agents: the criminal and the policeman. The probabilities that governtheir actions is central to the formulation of this agent-based model.We describe first the probabilities that determine the actions of the criminal which we call“windows-breaking” and “roaming”. More precisely, each criminal is allowed to actively performone of these two actions at every time step, i.e. burglarize the site or leave for a neighbouringsite.13Windows-breakingThe probability for which a certain criminal located at x will burglarize the site is defined bythe Poisson processpv(x, t) = 1− e−A(x,t)δt. (1.2)Note that as A and δt vanish, pv vanish as well. Also, as A→∞, we have pv → 1− as stipulatedby the meaning of attractiveness. Moreover, after each burglary, we assume that the burglardoes not commit another crime, but instead return home so we increase the count of crimesE(x, t) by 1 but reduce the count of criminal N(x, t) by 1. Finally, the number of burglars aremaintained by a uniform background re-introduction rate Γ.RoamingSince the lattice is square, there are four neighbouring sites for the criminal to move to. Wedefine the probability of moving to one of the neighbouring sites x′ from x at discrete timeinstant t bypm(x′, t; x) =A(x′, t)∑x′′∼xA(x′′, t), (1.3)where x′′ ∼ x denotes the four neighbouring sites to x and includes x′. The form above meansthat the random movement towards the four immediate neighbours is biased in favour of higherattractiveness A. More precisely, for a particular neighbour x′, pm(x′, t; x) is larger when A(x′, t)is larger.Next, we describe the probabilities that determine the actions of the policemen, which wecall “deterring” and “patrolling”. The reason for modeling deterrence as the only positive effectof police presence is that, according to [22] and the references there-in, it is documented that thevast majority of residential burglaries go unsolved. Therefore, we make the same assumptionas in [22] that the main effect of police modelled is not intercepting but deterring crime. Interms of our variables, that means we do not assume any direct effect on the number of crimes,E(x, t). It is reasonable to posit that the presence of policemen may decrease the attractivenessof the site, and in particular the dynamic component B(x, t). However, as in [22], we focusour modelling effort on the deterrence effect and the way policemen patrol on the grid, which14effectively means we disregard any coupling effect that could exist between the attractivenessA(x, t) and police presence R(x, t), but focus on the coupling between the criminals N(x, t) andthe policemen R(x, t).In conclusion, we will only consider the effect of the number of policemen present, R(x, t),on the number of criminals present, N(x, t). Together with a biased random walk of the policethat mimics the mentality of the criminals, the two major actions of the police term R(x, t) aresummarized as follows.DeterringThere are many possible ways to model the interaction of the policemen R(x, t) and criminalsN(x, t). We present two of the most intuitive and simple interaction mechanisms.The arguably simplest way is to assume that the deterrence effect is directly proportionalto the number of policemen present, with no stochastic elements. This implies the number ofcriminals N(x, t) is decremented by a term proportional to R(x, t) at every time instant, whichwe simply express as− νR(x, t), (1.4)where ν is the constant of proportionality describing the deterrent effect of the policemen.Another possibly more realistic way to model the deterrence effect is to imitate the classicalpredator-prey dynamics of Lotka–Volterra type and to regard the police as the “predator” andcriminals as the “prey”. Therefore, the deterrence effect is assumed to be directly proportionalto the product of the number of criminals and the number of policemen. However, we wouldlike to draw a key distinction here that we do not assume the totality of criminals either at aparticular site or on the whole lattice to be actually visible to policemen. Instead, potentialcriminals (who have yet to commit a burglary), at their random encounter with policemen at thesame site will return home by the deterrence effect of the police, which is directly proportionalto the number of policemen present. However, we assume that the policemen can perceive thesame environmental cues of crime attractiveness as the criminals do. In other words, whilst thecriminals (predator) are foraging for vulnerable sites and moving towards attractive sites (prey),the police (predator) also moves towards attractive sites (but not the criminals themselves) to15deter away criminals (prey).Therefore, we model the probability that criminals will decide to go home upon contactwith police also as a Poisson process with probabilitypu(x, t) = 1− e−µR(x,t), (1.5)where µ is measures the strength of the deterrence in such close contact events. We remark herethat in both events if a criminal burglarizes, i.e. “meets its prey”, or encounters a policeman,i.e. “meet its predator”, the criminal is removed from the system.PatrollingHere, we model another significant action of the police, which draws inspiration from formermodels, and in particular, we consider the following as a generalization of the “cops on thedots” policing strategy devised by Jones et al. in [22].Since policemen are assumed to detect the attractiveness of the sites in the same way ascriminals, we assume that their patrolling behaviour is also directed towards neighbouring sitesin favour of sites with higher risk of crime. However, the police may act in a more (or less)aggressive manner in focusing their patrolling efforts to more vulnerable sites. In the twoextreme ends, a policeman can wander aimlessly like a random walker, or move always to themost vulnerable site in the neighbourhood.To model this variation of patrolling behaviour, we introduce a simple degree parameter kon top of A to define the probability that the police will move from x to a neighbouring site x′bypk(x′, t; x) =Ak(x′, t)∑x′′∼xAk(x′′, t)(1.6)(cf. the paragraph immediately above formula (2.7) in [22]).When k = 1, this coincide exactly with the “cops on the dots” strategy, which we also callmimicry patrolling, and the police is roaming in a biased random walk in the exact same fashionas criminals. Next, we observe that, in particular, as k → 0+, the probabilities will result in apure random walk, which was mentioned as the less favourable strategy in [22]. On the other16hand, if k →∞, clearly we havepk(x′, t; x) =1 if x′ = max{A(x′′, t) : x′′ ∼ x}0 otherwise .This means that the policeman has both the perfect competence to understand the attractive-ness of the neighbourhood and the total willingness to patrol there.We will not study both of these extreme cases in isolation, but instead allow the parameterk to range from 0 to ∞, so that we may study how the degree of focus may affect both policemovement and the resulting deterring effect on the criminals, and ultimately the existence andstability of crime hotspots. We also remark that this is rather different from the assumptionsmade by Zipkin et al. concerning their version of “cops on the dot” strategy in [56]. The authors’main assumption about the policemen is that they are able to move in a highly coordinated wayto actively suppress crime hotspots. In contrast, we do not assume any coordination among themembers of the police force, but instead consider them as completely independent individuals.Finally, note that if neighbouring sites are burglarized, thus raising the dynamic attractive-ness, the possibility that nearby police agents also move towards those sites is increased in amanner similar to criminals. In other words, the “broken windows effect” also acts upon thepolice.1.4.3 Discrete Evolution of Agents’ State and the Localized Spread of theRisk of CrimeHaving defined the key probabilities that determine the active and passive actions of the crimi-nals and police, we turn to calculate the values of each of the agents at the next time step. Mostimportantly, the criminological theory of repeat or near-repeat victimization is introduced inthis subsection.Fading memory of criminal eventsAs in [47], we assume that the dynamic attractiveness decays with time exponentially as memoryof crime fades, which is a feature of many social and ecological phenomena. Thus, for a site17without crime for a long time the dynamic attractiveness B should decay to zero as well, andthe total attractiveness A should return to the baseline value A0. This is modelled byB(x, t+ δ) = B(x, t)(1− ωδt), (1.7)where ω > 0 controls the rate of memory fade.Repeat and near-repeat victimizationAnother crucial feature of the discrete model by Short et al. in [47] is that the burglary eventraises the perception of all criminals in terms of the dynamic attractiveness of the very samesite. Moreover, it is theorized that the increased risk also diffuses to the neighborhood as well.This is modelled by the updateB(x, t+ δt) =(1− η)B(x, t) + η4 ∑x′′∼xB(x′′, t) (1− ωδt) + θE(x, t). (1.8)The parameter θ > 0 measures the effect of each crime on the dynamic attractiveness of thevery site. In other words, the θE(x, t) term models the repeat victimization effect resulting fromeach burglary. On the other hand, the parameter 0 ≤ η ≤ 1 measures the amount of perceivedattractiveness that is transferred to the immediate neighbours. This models the near-repeatvictimization phenomenon, where a site with high attractiveness is thought to also cause morecrimes in the neighbourhood.We believe that very often a small η > 0 is reasonable, which will subsequently lead toour PDE in the continuum limit to be singularly perturbed. In particular, a value higher thanη = 12 is rather counter-intuitive because location drifts of crime hotspots were documented tobe rather slow in time, which effectively means the diffusivity-like constant η should also below. In [47, 48], the authors also used a relatively small value of η.Biased random walk and the dynamics of criminal movementSince at each site, we assumed that the criminals would either burglarize or roam to a nearbysite, the new criminal count N(x, t+δt) is the sum of criminals coming in from all the neighbour-18ing site x′ ∼ x. These criminals are those who did not commit a burglary (with the probability1 − pv) and have chosen to move to this site x from their original site x′ by a biased randomwalk (with probability pm(x; t, x′)). In this way, we getN(x, t+ δt) =∑x′∼xN(x′, t)(1− pv(x′, t))pm(x; t, x′) + Γδt .Deterrent effect and mimicry movement of police agentsIn Short et al’s discrete model in [47], they stopped at the formula above for N(x, t). Now, wemust include the presence of the police agents R(x, t). As mentioned in the previous subsection,we will discuss two cases of the effect of police.For the simple interaction case (1.4), we assume that the number of criminals that go homeis directly proportional to the number of police present at the same site, and thus the numberof criminals at the next time-step is given by:N(x, t+ δt) =∑x′∼xN(x′, t)(1− pv(x′, t))pm(x; t, x′)− νR(x′, t) + Γδt . (1.9)For the predator-prey type interaction (1.5), we assume that each criminal located at x willgo home with the probability pu(x, t) upon encountering each patrolling police officers, and sowe arrive atN(x, t+ δt) =∑x′∼x[N(x′, t)(1− pv(x′, t))pm(x; t, x′)](1− pu(x, t)) + Γδt, (1.10)where we note that the number of police R(x′, t) is implicit in the definition of pu(x, t).As for police officers, since they are patrolling, we also assume that each officer at site x willproceed to patrol the neighbouring site, while the police at any neighbouring site x′ will cometo the site x also by the focused patrolling mechanism described above. Therefore, we deducethatR(x, t+ τδt) =∑x′∼xR(x′, t)pk(x; t, x′), (1.11)where we also multiplied δt by τ to reflect the relative speed of police to the criminals. Recall19also that we do not assume any mechanisms for removing and introducing police so that thetotal number of police should be conserved.1.5 Deriving the Continuum Limit as a System of PDEsIn what follows, we formally derive a continuum limit of this agent-based model to motivatethe form of the specific PDEs considered in this work. We remark that the derivation isformal (referred to as “naïve continuum limit” in [22]) and we will assume certain correlationsbetween certain terms to be negligible, as mentioned in Remark (1.1) below.Before we proceed, let us define the “discrete Laplacian”, which we will need to apply severaltimes to simplify algebra and the presentation:4F (x, t) = 1l2∑x′∼x(F (x′, t)− F (x, t)) = 1l2∑x′∼xF (x′, t)− 4F (x, t) ,where F (x, t) is some agent that evolves on the grid. Therefore, we have∑x′∼xF (x′, t) = l24F (x, t) + 4F (x, t) . (1.12)This will allow us to write our formulas entirely in terms of the variables x and t.We will derive a formal continuum limit from the discrete model, with attractiveness givenby (1.8), the criminals given by (1.9) or (1.10) (simple interaction and predator-prey typeinteraction respectively), and the police given by (1.11).1.5.1 Derivation of the Individual PDEsFor (1.8), we apply (1.12) to B(x′, t) and replace E(x, t) = N(x, t)pv(x, t) to getB(x, t+ δt) =(B(x, t) + ηl24 4B(x, t))(1− ωδt) + θN(x, t)pv(x, t) .Remark 1.1. As pointed in [22], and also quoted in [56], the assumption that the randomvariables N and pv are independent may not hold. However, as in [22], we assume that thecorrelation does not contribute a leading-order effect on the total number of crimes, which is20eventually represented by ρA in the continuum limit. The same point is also relevant to oursubsequent derivation of the police deterrent effect in the predator-prey type interaction case.We also assume that the random variables N and pu are essentially independent, so that theeffect of criminal deterrence is represented eventually by ρU .Note that the argument of each function is (x, t), so we now subtract B(x, t) from both sidesand divide by δt to find, (dropping the arguments for notational simplicity)∂B∂t∼ η4l2δt4B − ωB + l2δt(Nl2)θAδt ,using pv = 1− e−Aδt ∼ Aδt.Since δt→ 0, we maintain the the ratios of parameters that depend on the length and timescales of the model to be constants and make the following replacement:l2δt→ D, θδt→ e, Nl2→ ρ .In this way, we arrive at the following PDE:Bt = ηD4 4B − ωB + eρA .In terms of the total attractiveness A = A0 + B, and assuming A0 to be uniform in spacealso, then we can writeAt = ηD4 4A− ωA+ eρA+ ωA0For the probability pm associated with the biased random walk of criminals, which appearsin both (1.9) and (1.10), we obtain thatpm(x, t; x′)= A(x, t)∑x′′∼x′ A(x′′, t)= A(x, t)l24A(x′, t) + 4A(x′, t) , (1.13)upon using the discrete Laplacian formula (1.12).For the simple interaction case (1.9), we then have21N(x′, t+ δt) = A(x, t)∑x′∼xN(x′, t) (1− pv(x′, t))l24A(x′, t) + 4A(x′, t) − νR(x′, t) + Γδt. (1.14)In contrast, for the predator-prey interaction case (1.10), we have instead thatN(x′, t+ δt) = A(x, t)∑x′∼xN(x′, t) (1− pv(x′, t)) (1− pu(x, t))l24A(x′, t) + 4A(x′, t) + Γδt ,, A(x, t)∑x′∼xS(x′, t) + Γδt ,where we defined S(x, t) in the second equality.Therefore, we haveA(x, t)(l24S(x, t) + 4S(x, t))+ Γδt . (1.15)We first focus on the more complicated case (1.15). We subtract N(x, t) from both sidesof (1.15), and divide by l2, applying Nl2 → ρ, and then by δt to find that (dropping commonarguments again):∂ρ∂t∼ 1δt(A4S + 4Al2S − ρ)+ γ . (1.16)Here we made the replacement Γ/l2 → γ for the parameter Γ which depends on the length-scale.Now recall that pv ∼ Aδt as well as for pu = 1 − e−µR, since R is small when l → 0. Wethen apply the replacement Rl2 → U to findpu ∼ µR = µUl2 .By using this relation, we can simply the key term S in (1.16) as follows:S = N (1− pv) (1− pu)l24A+ 4A ,∼ l24Aρ (1−Aδt) (1− µUl2)1 + l24A4A,∼ l24Aρ (1−Aδt)(1− µUl2)(1− l24A4A).This yields thatSδt∼ l24δtρA= D4ρA,22and consequentlyA4Sδt∼ D4 A4(ρA),which gives the first term in (1.16).Next, observe that4ASl2− ρ ∼ ρ (1−Aδt)(1− µUl2)(1− l24A4A)− ρ ,∼ −ρAδt− l2µρU − l24ρA4A,which implies to1δt(4ASl2− ρ)∼ −ρA−DµρU − D4ρA4A.Thus, by combining with the previous term, we arrive at the PDE∂ρ∂t= D4 A4(ρA)− ρA−DµρU − D4ρA4A= D4(A4(ρA)− ρA4A)− ρA−DµρU. (1.17)Similarly, for the easier case (1.9) when simple interaction is assumed, we have instead that∂ρ∂t= D4(A4(ρA)− ρA4A)− ρA−DνU. (1.18)We pause to remark that there is an applicable identity A4 ( ρA)− ρA4A = ∇·(∇ρ− 2ρA∇A),for which a more general case that also applies to an analogous term in the police equation willbe derived in the next subsection.Next, we turn to (1.11), where we rewrite the expression aspk(x; t, x′) =Ak(x′, t)∑x′′∼xAk(x′′, t)= Ak(x, t)l24Ak(x′, t) + 4Ak(x′, t) ,23upon using the discrete Laplacian (1.12). Then, we calculate thatR(x, t+ τδt) = Ak(x, t)∑x′∼xR(x′, t)l24Ak(x′, t) + 4Ak(x′, t) ,, Ak(x, t)∑x′∼xT (x′, t) ,= Ak(x, t)(l24T (x, t) + 4T (x, t)).Similar to the criminal equation, after subtracting both sides by R(x, t), dividing by l2δtand replacing Rl2 → U , we findτ∂U∂t∼ 1δt(Ak4T + 4Akl2T − U),where there is an extra τ on the time derivative due to chain rule.Next, we observe thatT = Rl24Ak + 4Ak =l24AkU1 + l24Ak4Ak∼ l24AkU(1− l24Ak4Ak),which gives Tδt ∼ D4 UAk . In this way, we getAk4Tδt∼ D4(Ak4(UAk)).Next, observe that1δt(4Akl2T − U)∼ 1δt[U(1− l24Ak4Ak)− U]∼ −D4UAk4Ak.This leads to the PDE for the policeτ∂U∂t= D4(Ak4(UAk)− UAk4Ak).241.5.2 The Final Form of the PDE system and a Special Identity for theNonlinear DiffusionHaving converted the discrete models to their corresponding continuum limits, we now establishthe following identity that converts the PDEs to our final form:Ak4(UAk)− UAk4Ak ≡ ∇ ·(∇U − 2kUA∇A). (1.19)To derive this identity, we proceed directly by using the Leibniz rule: ∇2 (fg) = f∇2g +2∇f · ∇g + g∇2f . We obtain thatAk4(UAk)= Ak(4UAk+ 2∇U · ∇( 1Ak)+ U4( 1Ak)),= Ak(4UAk− 2kAk+1∇U · ∇A+ k(k + 1)Ak+2U |∇A|2 − kAk+1U4A)= 4U − 2kA∇U · ∇A+ k(k + 1)A2|∇A|2 − kA4A ,UAk4Ak = UAk∇ ·(kAk−1∇A),= UAk(Ak−2|∇A|2 + kAk−14A),= k(k − 1)A2U |∇A|2 + kA4A .By combining these results together we obtain thatAk4(UAk)− UAk4Ak = 4U − 2kA∇U · ∇A+ 2kA2U |∇A|2 − 2kAU4A ,= 4U − 2k(∇(UA)· ∇A+ UA4A),= 4U − 2k∇ ·(UA∇A)= ∇ ·(∇U − 2kUA∇A),as desired.Finally, for notational convenience, we let q = 2k and after nondimensionalization and25renaming of constants, we arrive at the following three-component RD system:At = 2Axx −A+ ρA+ α, (1.20a)ρt = D (ρx − 2ρAx/A)x − ρA+ γ − α− I(ρ, U), (1.20b)τuUt = = D (Ux − qUAx/A)x , (1.20c)where two cases of I(ρ, U) were given by:(i) Simple interaction case (corresponding to (1.18))I(ρ, U) = U, and(ii) Predator-prey type interaction case (corresponding to (1.17))I(ρ, U) = ρU.There are two further remarks concerning the form of (1.20) before we present various resultsabout (1.20).Firstly, we remark that due to a slightly different way of nondimensionalization, we obtaineda model with an extra parameter D as compared to the model in [22, 47], where D = 1. Anon-unit value of D can also be derived from the case D = 1 using a simple domain scalingargument. We will retain the diffusivity parameter D for ease in our subsequent analysis.Concerning the criminal-police interaction term I(ρ, U), we also remark that the predator-preytype interaction is to be preferred over the simple interaction case for purposes of application.However, due to the difficulty in the NLEP stability analysis with multiple nonlocal terms (seethe introductory paragraphs to Chapter 3 on page 95). The simple interaction case is favouredfor mathematical feasibility to begin our research in our study of police-criminal interactionand its effect on crime hotspots.26Chapter 2Crime Hotspot Formation andLong-time BehavioursIn this chapter, we consider (1.20) when U ≡ 0, i.e. with no police involvement in the crimedynamics and consider the finite one-dimensional domain −` < x < `. This reduces to thetwo-component reaction-diffusion (RD) system originally introduced by Short et al. in [47],which we express in the following formAt = 2Axx −A+ ρA+ α , ρt = D(ρx − 2ρAAx)x− ρA+ γ − α , (2.1)with no-flux boundary conditions ρx = Ax = 0 at x = ±`. Note that 2  1 is a singularperturbation parameter, and we assume D = O(1).We remark that by the simple rescaling x˜ = x/√D, (2.1) can be recast to the interval(−`/√D, `/√D), with the new diffusivities ˜ = /√D and D˜ = 1. This recovers the exactsame form given in [47] and at (1.1) without the factor D. However, in our analysis, ratherthan using the domain length as a bifurcation parameter, we will consider (2.1) on a fixeddomain, but allow the criminal diffusivity D = O(1) to vary while treating the diffusivity 2of the attractiveness field as an asymptotically small quantity. This has an advantage both inthe notational simplicity in the asymptotic analysis of multiple crime hotspots, and also thenumerical computations, where our formulation avoids problems arising out of domain rescalingin scenarios where the numerically computable range in terms of  is too restrictive.27The study of (2.1) in this chapter is directly related to that of [29], where D = O(−2) wasassumed and a similar tool-set of matched asymptotic expansions, supplemented by numericalcomputations were used in [29] as well. The work in this chapter is a variation of what is studiedin [29] when the coupling of the two PDEs in (2.1) is stronger, which will lead to a couple ofnovel phenomena.In addition to [29], the previous work that is most relevant to our study is that of [35]. ForD = 1, in [35] the bifurcation software AUTO-07p (cf. [12]) was used to numerically show thatthere is an intricate homoclinic snaking bifurcation structure for the steady-states of (2.1) onthe infinite line when the parameter γ is below the Turing bifurcation threshold γc, for whichγc ∼ 3α/2 as → 0. At finite , the localized states for A that were computed in [35] are wave-packet type solutions consisting of closely spaced pulses. In the singular limit → 0, and withα < γ < γc ∼ 3α/2, these wave-packet localized solutions were found in [35] to lead to a patternof well-separated hotspots. An asymptotic analysis, based on geometric singular perturbationtheory, was given in [35] for the construction of a solitary hotspot solution on the infinite linein the subcritical case α < γ < 3α/2. However, as remarked in [35], there were some issues inthe asymptotic matching procedure in this construction that were left unresolved. Homoclinicsnaking behaviour has been well-studied in other systems, such as the Swift-Hohenberg model(see [5] and the references therein). Moreover, it has been shown recently in [6] that homoclinicsnaking behaviour is a rather generic feature of solutions to certain types of RD systems neara subcritical Turing bifurcation point.In contrast to [35], our study of hotspot equilibria and the stability in the singular limit → 0will focus on the finite domain problem, and our method of matched asymptotic expansion andNLEP stability analysis work equally well for both the supercritical regime γ > 3α/2 and thesubcritical regime, for which the spatially homogeneous steady-state is linearly unstable andstable respectively. Moreover, our numerical studies in Section (2.4.2), which is an extensionof Fig. 7 in [29], show that the crime hotspot solutions follow a closed-loop which both beginsand ends at the Turing bifurcation point from the subcritical side, instead of a snaking path inthe infinite domain case of [35].The outline of this chapter is as follows. In §2.2, we provide a leading-order construction ofa single hotspot steady-state solution in the limit → 0 for D = O(1). By reflecting and gluing28this single hotspot solution, equilibria with multiple hotspots are obtained. We emphasize adistinction between the supercritical and subcritical regime by a formula of the outer problem(2.23). In §2.3 we show that steady-state patterns with K interior hotspots are linearly stableon an O(1) time-scale regardless of D = O(1). In §2.4 we present numerical bifurcation resultsfor hotspot equilibria computed using the bifurcation software AUTO-07p (cf. [12]). For the su-percritical regime, we present results that exhibit a saddle-node bifurcation structure of hotspotequilibria and a peak insertion behaviour near the saddle-node bifurcation point. For the sub-critical regime, we present a closed homotopy of homoclinic in a three-dimensional diagramand show the existence of a new type of unstable hotspot that connects the weakly nonlinearregime to the far-from equilibrium regime through a saddle-node bifurcation. In §2.5 we presenta higher-order asymptotic theory to construct a steady-state hotspot solution, and in §2.6 westudy analytically the onset of peak insertion behaviour near the saddle-node bifurcation point.In §2.7 we derive a system of differential-algebraic equations (DAE) characterizing the slowdynamics of a collection of hotspots for (2.10). Finally, in §2.8, we briefly discuss a few openproblems that warrant further study.2.1 Linear, Weakly Nonlinear and Far-from Equilibrium RegimesIn this section, we review results from linear stability analysis of [49, 29] and recall conclusionsfrom the weakly nonlinear analysis in [48] that lead to the classification of the parameterregime into supercritical (γ > 3α/2) and subcritical (γ < 3α/2) regimes. Finally, we motivateour study in the far-from equilibrium regime by a few numerical computations to highlight theexistence of crime hotspots away from the Turing bifurcation point. This will be the subject ofthe remainder of this chapter.The system (2.1) has the unique spatially homogeneous steady-state solution given byAe = γ and ρe = 1− α/γ > 0. (2.2)The linear stability analysis of [49] showed that this solution is linearly unstable in the limit29→ 0 whenγ >32α , (2.3)and that a spatially heterogeneous solution bifurcates from the spatially homogeneous steady-state at the bifurcation point γ ∼ 32α when → 0.Here, reproduce some details about the Turing bifurcation that has been done in [49, 29].First, we linearizing (2.1) around the steady-state by introducing the perturbationA = Ae + a0eimx+λt, ρ = ρe + ρ0eimx+λt .This leads to the dispersion relation(λ+Ae +Dm2) (λ+ 2m2 + (1− ρe))+Ae(ρe − 2DAeρem2)= 0 . (2.4)To calculate the instability band, we set λ = 0 to (2.4) and let  1, with m = O(1), whichyields the lower thresholdDm2lower =Ae3ρe − 1 , mlower =γ√D√2γ − 3α . (2.5)To obtain the upper edge of the band we let m 1 with 2m2 = O(1) to find2m2upper + (1− ρe) = 2ρe, mupper =1√2γ − 3α√γ. (2.6)To find the most unstable mode, we differentiate (2.4) with respect to m and set dλ/dm = 0to findλ(1 + 2D)= 3ρe − AeD2 − 22m2 .In this way, we obtain that the dominant growth rateλdom ∼ 3ρe − 1 = 2− 3α/γ , (2.7)30and that the most unstable mode ismdom ∼ −1/2[(2ρeD)(32Ae + λdom)]1/4= −1/2[2(1− α/γD)(32γ + λdom)]1/4. (2.8)In other words, for an initial condition condition consisting of a random perturbation ofa spatially uniform steady-state, we can calculate the characteristic half length of the patternfrom the Turing instability to be`Turing ∼ pimdom= 1/2pi[2(1− α/γD)(32γ + λdom)]−1/4.For α = 1, γ = 2, D = 2,  = 0.05, we calculate `Turing ≈ 0.59. We observe from the fullnumerical results computed from the PDE shown in Fig. 2.1 and 2.2 that this Turing instabilitycan lead to the creation of either a single or a double hotspot pattern on the domain (−1, 1).Moreover, the weakly nonlinear analysis of [48] showed that this Turing bifurcation is sub-critical when  1 (see also Fig. 7 of [29]). The theory in [48], based on a normal form equationderived from a multiple-scales approximation, is able to characterize the development of spatialpatterns near the Turing bifurcation point. However, it is not capable of describing the highlylocalized spatial patterns observed in full numerical simulations of (2.1) when the parametervalues are not near the Turing point. More specifically, an initial random perturbation closeto an unstable spatially homogeneous steady-state typically leads to highly localized spatialpatterns, consisting of the concentration of criminal activity in localized spatial regions. Werefer to such patterns as hotspot patterns. A localized hotspot solution, not amenable to ananalytical description by a weakly nonlinear analysis, was observed in the numerical solutionsof [48].To illustrate these hotspot patterns, we perform full numerical simulations of (2.1) using thesoftware PDEPE in MATLAB R2013b. For  = 0.05, D = 2, γ = 2, and α = 1, in the right panelof Fig. 2.1 we plot A and ρ at some large time when the initial condition is a small randomperturbation of the unstable spatially uniform state as shown in the left panel of Fig. 2.1. In thesupercritical regime, γ > 3α/2, the spatially uniform state is unstable and we observe, for largetime, that the solution approaches a steady-state pattern with one interior hotspot. For the31same parameter set, but where the initial condition for A has two localized bumps, in Fig. 2.2we show the initial formation of a two-hotpot pattern on an O(1) time-scale, followed by a veryslow dynamics of the two-hotspot pattern towards its steady-state limit. One of the main goalsof this chapter is to give an explicit analytical characterization of the slow dynamics of suchquasi-steady state hotspot patterns.For the subcritical regime α < γ < 3α/2, we also observe from (2.7) that a crime hotspotborn in the supercritical regime persists into subcritical regime without dissipating, even thoughthe spatially homogeneous equilibrium (2.2) is stable in this regime. The leading order theory inthe next section will show that asymptotic structures of crime hotspot in both the supercriticaland subcritical regimes are largely rather similar, which also lead to identical linear stabilitypredictions.−1.0 −0.5 0.0 0.5 A0.470.480.490.500.510.520.53Crime Density ρ−1.0 −0.5 0.0 0.5 1.0024681012Attractiveness A0. Density ρFigure 2.1: A Turing instability leading to a localized steady-state solution. Parameter valuesare  = 0.05, D = 2, ` = 1, γ = 2, α = 1, so that γ > 3α/2. The initial condition (left panel)for the numerical solution of (2.1) is a small random perturbation of the spatially uniform stategiven by A(x, 0) = Ae + rand ∗ 0.1, ρ(x, 0) = ρe, where Ae = γ = 2 and ρe = 1 − α/γ = 0.5.The right panel shows the hotspot solution at the final time t = 105 with A (solid curve) andρ (dotted curve). Notice that the range of A and ρ are on different scales.2.2 Super- and Sub-critical Crime Hotspots: Leading-OrderSteady-State TheoryIn our analysis of (2.1) it is convenient to introduce the new variable V , as first introduced in[29], defined byV ≡ ρ/A2 . (2.9)32−1.0 −0.5 0.0 0.5 1.0Location (x) (A)−1.0 −0.5 0.0 0.5 1.0Location (x)1234567Attractiveness (A)t = 1.292e+01t = 2.783e+02t = 1.000e+04Figure 2.2: Small initial bumps in A quickly evolve into hotspots, which then move slowlyto their steady-state locations. Parameter values are  = 0.05, D = 2, ` = 1, γ = 2, α = 1,so that γ > 3α/2. The initial condition (left panel) for the numerical solution of (2.1) isA(x, 0) = Ae+ ∑2i=1 sech(x−x0,i ) and ρ(x, 0) = ρe, where x0,1 = −0.7, x0,2 = −.7, Ae = γ = 2,and ρe = 1− α/γ = 0.5. We only plot A.In terms of A and V , (2.1) transforms to the PDE systemAt = 2Axx −A+ V A3 + α ,(A2V)t= D(A2Vx)x− V A3 + γ − α , (2.10)on −` < x < `, with Ax = Vx = 0 at x = ±`. The corresponding steady-state problem for(2.10) is2Axx −A+ V A3 + α = 0 , D(A2Vx)x− V A3 + γ − α = 0 . (2.11)We use the method of matched asymptotic expansions in the limit  → 0 to constructa leading-order approximation of a steady-state solution with a single hotspot, or spike, for(2.11) on a domain of length 2`. For a fixed D, we will show that such a solution exists forany ` regardless of whether α < γ < 3α/2 (subcritical) or γ > 3α/2. We emphasize that theconstruction does not depend on whether γ is in the subcritical or supercritical case, but onlyon the positivity of γ − α, i.e. nonzero crime reintroduction rate.Moreover, we characterize a threshold phenomenon that occurs if and only if γ > 3α/2,so that a critical length `max exists so that ` < `max is required for a solution to exist. Thehotspot solution for (2.11) is an even solution with a spike in the profile of A centred at the33midpoint of the interval −` < x < `, and the solution satisfies the no-flux boundary condition:Ax = Vx = 0 at x = ±`.By reflecting and gluing copies of this single hotspot solution defined on (−˜`, ˜`) and ˜`= `/K,we can readily obtain a Khotspot solution on the original interval (−`, `). The key differencebetween our analysis and that in [29] is that here we consider the D = O(1) regime, whichleads below to a nonlinear ODE characterizing the outer solution. In [29], the limit D  O(1)was considered, which leads to a linear outer problem and, consequently, a more elementaryconstruction of the hotspot solution than for the case D = O(1).As in [29], a hotspot is characterized by a localized region of width O() near x = 0 whereA  1 and V  1. In the outer region, where |x|  O(), we have from (2.11) that both Aand V are O(1) when D = O(1).To determine the scaling for the inner region, we introduce the inner variable y = −1x sothat (2.11) becomesAyy −A+ V A3 − α = 0 , D−2(A2Vy)y− V A3 + γ − α = 0 . (2.12)In this inner region, we pose A = O(−p), with p > 0, and V = O(q). In order to obtain ahomoclinic solution solution characterizing the hotspot core, we must balance O(A) = O(V A3)in the first equation of (2.12), which yields q = 2p. To determine the second scaling relation, weintegrate the second equation in (2.11) over |x| ≤ ` to obtain that ´ `−` V A3 dx = O(1). In orderthat the inner region makes an O(1) contribution to this integral, we require that 1+q−3p = 0.With q = 2p, this yields that p = 1 and q = 2.One of the key differences between the analysis for our D = O(1) regime and that in [29]for D  O(1), is that here V is characterized by a rapid transition of scale from V = O(2) inthe inner region to V = O(1) in the outer region. In the analysis of [29] for the D  1 case, itwas found that V  1 uniformly across −` < x < `. A detailed analysis of this rapid transitionof scale for V , which requires introducing an intermediate layer between the inner and outerscales, is essential for characterizing the slow dynamics of a collection of hotspots. This morerefined analysis is given below in §2.5. In the remainder of this section we only construct aleading-order hotspot solution.34The simple scaling analysis above motivates the following inner expansion for (2.12):A ∼ −1A0 +A1 + . . . , V ∼ 2V0 + 3V1 + . . . .Upon substituting this expansion into (2.12) we obtain that V0 is a constant independent of yand that A0 satisfies A0yy −A0 + V0A30 = 0. This yields thatA0 =w(y)√V0, (2.13)where w(y) =√2 sech y is the unique homoclinic solution towyy − w + w3 = 0 , −∞ < y <∞ ; w(0) > 0 , wy(0) = 0 , w → 0 as y → ±∞ .(2.14)A remarkable fact is that the crime density ρ is a constant independent of all model parametersat the hotspot, and is given to leading order byρ = A2V ∼(A0)2 (2V0)= w2(x/) . (2.15)Moreover, the maximum crime density is at the centre of the hotspotmax ρ ∼ w2(0) = 2 . (2.16)To determine V0, we first need to construct an outer solution on the intervals 0+ < x < ` and−` < x < 0−. Since the hotspot solution is even, we need only consider the range 0+ < x < `.On this range, we expand A ∼ a0 + o(1) and V ∼ v0 + o(1). Upon substituting this expansioninto (2.11) we obtainD(a20v0x)x− v0a30 + γ − α = 0 , v0 = g(a0) ≡(a0 − α)a30. (2.17)In order that v0 can match to the inner solution, we require that v0(0+) = 0, so that a0(0+) = α.Then, upon combining the two equations in (2.17), it follows that in the outer region a0 satisfies35the nonlinear BVPD (f(a0)a0x)x = a0 − γ , 0+ < x < ` ; a0(0+) = α , a0x(`) = 0 , (2.18a)where we have defined f(a0) byf(a0) ≡ (3α− 2a0)a20= a20g′(a0) . (2.18b)2.2.1 Monotonicity of the Outer ProblemFor the well-posedness of this outer problem (2.18a) we require that f(a0) is non-vanishing on0+ < x < `. Since the only zero of f(a0) is at 3α/2 and f(α) = α−1 > 0, we shall considerthe range α < a0 < 3α/2. If one assumes γ > 3α/2, which implies that we are in the Turing-unstable regime, it follows that the right-hand side of the differential equation in (2.18a) isalways negative on the range α < a0 < 3α/2. This yields that a0x > 0 on 0+ < x < ` whenα < a0 < 3α/2. However this is not a necessary condition and we will show that γ > αis sufficient to guarantee that a0x > 0 whenever a solution exists to (2.18a). This is a crucialproperty of the solution a0(x) that will enable us to reduce the problem (2.18a) to a quadrature.Before solving the ODE, we first illustrate the structure of the leading-order outer solutionfrom a phase plane viewpoint, which will lead naturally to an argument showing that a0x > 0whenever γ > α. To do so, we introduce the temporary notation u ≡ a0 and v ≡ f(a0)a0x =f(u)u′ to rewrite (2.18a) in the following usual form of a dynamical system:u′ = vf(u) , v′ = (u− γ)D. (2.19)We seek to identify trajectories in the u, v phase-plane that satisfy u(0) = α and v(`) = 0,corresponding to the boundary conditions of (2.18a). From the two phase portraits shown inFig. 2.3, corresponding to the supercritical case where γ > 3α/2 and subcritical case whereα < γ < 3α/2 respectively, we clearly observe two types of trajectories and a critical thresholdvc for v(0). If v(0) < vc, then the trajectory converges to v(`) = 0 for some finite positive `.36However, if v(0) > vc, then the solution develops a singularity asu→(3α2)−where u′ → +∞ ,according to (2.19).For the supercritical case, near where this singularity occurs a new local boundary layer nearx = ` must be constructed. As motivated by the numerical bifurcation results in §2.4, in §2.6 weanalytically show that this singularity behaviour is associated with a fold-point bifurcation ofequilibrium hotspot solutions, and it characterizes the onset of a hotspot insertion phenomenaat x = ` when γ > 3α/2, i.e. when we are in the Turing-unstable regime.On the other hand, for the subcritical case, we observe that admissible trajectories thatsatisfies v(`) = 0 share a common characteristic that u(`) < γ, and so the values of u are awayfrom the line of singularity u = 3α/2. Therefore, for both cases, assuming only γ > α, wesee that the value of u never exceeds min{γ, 3α/2}. It turns out this can be proved rigorouslywhich we will present as a lemma. This important observation implies that in particular a0x(i.e. u′) is always positive.In other words, a nonzero crime reintroduction rate, i.e. γ > α, which is a rather minimalassumption for the urban crime model, is sufficient to guarantee a monotonically increasingbehaviour in the attractiveness field A away from the crime hotspot, and regardless of whetherwe are in the supercritical (γ > 3α/2) or subcritical regime (α < γ < 3α/2).We now state and prove the claim.Lemma 2.1. Any solution to (2.19) on the interval x ∈ [0, `] subject to the conditions u(0) = αand v(`) = 0 must satisfy that u(x) < γ for all x ∈ [0, `).We also state an obvious corollary about the monotonicity of the solution which is essentialfor our subsequent calculations to reduce (2.18a) to a quadrature, which is valid for any γ > α.Corollary 2.2. u(x) is monotonically increasing and α < u(x) < γ for all x ∈ (0, `).Proof: First, we claim that if x0 ∈ (0, `) is a point such that u(x0) < γ, then we must alsohave v(x0) > 0. We prove this by contradiction.37(a) Supercritical case: γ > 3α/2 (b) Subcritical case: α < γ < 3α/2Figure 2.3: Phase portraits of (2.19) with trajectories emanating from the line u = α. Thereare two types of trajectories: (i) those that hit the u−axis are admissible solutions to (2.19)satisfying v(`) = 0 for some ` > 0, (ii) those that do not, but goes to the line u = 1.5 developssingularity for finite values of x. The model parameters chosen were D = 1, α = 1 for bothplots and γ = 2.0 and γ = 1.25 respectively for the left and right plots.Recall that for well-posedness of (2.18a), we required f(u) > 0 which holds only if u(x) <3α/2 for all x ∈ (0, `) because f(3α/2) = 0. Now, if a solution exists to (2.19), and u(x0) < γ,then we must have u(x0) < min{γ, 3α/2} so that f(u(x0)) > 0.Then, if to the contrary v(x0) ≤ 0, then u′(x0) = v(x0)/f(u(x0)) ≤ 0 as well. However,since x0 was chosen to give u(x0) < γ, we have v′(x0) = (u(x0)−γ)/D < 0. Thus, by continuity,there is a number x1 such that v′(x) < 0 for all x in [x0, x1). Moreover, we can choose x1 ≤ `to be the maximal one, so that i.e. v′(x1) = 0.If x1 < `, then note that v(x0) ≤ 0 and v′(x) < 0 for x ∈ [x0, x1) implies v(x) < 0 and thusu′(x) < 0 throughout the interval [x0, x1). Therefore, u(x1) < u(x0) < γ. However, substitutingu(x1) < γ to (2.19) gives v′(x1) = (u(x1)− γ)/D < 0 contradicting v′(x1) = 0.If x1 = `, then v′(x) < 0 for all x ∈ [x0, `). This statement with v(x0) ≤ 0 in turn impliesv(`) < 0 so that the shooting B.C. v(`) = 0 cannot be satisfied, another contradiction.Therefore, such x1 does not exist and we must have v(x0) > 0 instead. In conclusion, for asolution to (2.19) exists, we must have v(x0) > 0 whenever u(x0) < γ.Next, we substantiate the claim that u(x) < γ for all x < `, which will imply v(x) > 0 aswell by the above argument. We again prove this claim by contradiction.38Since u(0) = α < γ, so suppose, to the contrary that u(x) = γ does happen. Denote x2 ≤ `to be the first such number so that u(x2) = γ, which means we have u(x) < γ for x ∈ [0, x2).Then, v(x) > 0 for x < x2 by the previous claim. Moreover, v(x2) > 0 as well because otherwisev(x2) = 0 together with u(x2) = γ will coincide with the stationary solution u(x) = γ, v(x) = 0to (2.19), which satisfies an incompatible initial condition u(0) = γ and v(0) = 0. Therefore,this is a contradiction to the uniqueness of solutions to ODEs.Now, since v(x2) > 0 and v(`) = 0, we must have x2 < ` and there exists at least one rootto v(x) in the interval (x2, `]. Denote the first root a (if there is exactly one root, then a = `obviously). Thus the number a ∈ (x2, `] satisfies this condition:(∗) v(a) = 0, but v(x) > 0 for all x2 ≤ x < a .Now v(x2) > 0 and v(a) = 0 implies that there exists another number b in (x2, a) such thatv′(b) < 0. But from the equation for v′(x) in (2.19), we observe thatu(b) = v′(b)D + γ < γ .Here, the strict inequality is essential. Now, from the assumption that u(x2) = γ, thisimplies there must exist yet another number c in (x2, b) such that u′(c) < 0. This in turnmeans that v(c) < 0 from the equation for u′(x) in (2.19).By arranging the new numbers in ascending order as x2 < c < b < a, it follows that wehave obtained a number c < a with v(c) < 0, and so contradicting (*).Therefore, the initial assumption that u(x2) = γ for some x2 < ` must then be false, andwe have thus proved that u(x) < γ for all x < `. The first claim then implies v(x) > 0, andsubstituting these results into (2.19), we immediately obtain that v′(x) < 0 and u′(x) > 0,which proves Corollary (2.2) as well. The results above tell us, in particular, that the solution a0 to the nonlinear boundary valueproblem (2.18a) must be strictly increasing, i.e. a0x > 0, and it never exceeds γ. Together with39the solvability restriction that α < a0 < 3α/2, the solution to (2.18a) satisfiesa0(x) < min{γ, 3α/2} ,which holds in the two cases α < γ < 3α/2 and γ > 3α/2, corresponding to where the spatiallyuniform steady-state is Turing stable and unstable, respectively.2.2.2 Reduction to a Quadrature and Existence of a Maximum ThresholdWe may now proceed reduce to reduce the problem (2.18a) to a quadrature.First, we multiply (2.18a) by f(a0)a0x and integrate the resulting expression using a0x(`) = 0to getD2 (f(a0)a0x)2 =ˆ x`f(a0)(a0 − γ)da0dηdη .Since a0x > 0, we have upon labelling µ ≡ a0(`) thatD2 (f(a0)a0x)2 =ˆ a0µf(w)(wγ) dw = G(µ)−G(a0) , (2.20a)where G(u), satisfying G′(u) = f(u)(γ − u), is given explicitly byG(u) ≡ 2u− (2γ + 3α) log u− 3αγu. (2.20b)Notice that G′(u) is also positive on α < u < min{γ, 3α/2} and so the right-hand side of (2.20a)is positive for all x ∈ (0, `). We remark that in [35], similar integration procedures and phaseplane analysis were found for the slow system in the one-dimensional unbounded domain.Next, we take the appropriate square root in (2.20a) and integrate from 0 to x to obtain√2Dx =ˆ a0αf(w) dw√G(µ)−G(w) , (2.21)where we used a0(0) = α and the change of variable w = a0(x), which is valid because a′0(x) > 0.Finally, upon setting x = ` and a0(`) = µ, we obtain an implicit equation for µ given by√2D` = χ(µ) , χ(µ) ≡ˆ µαf(w) dw√G(µ)−G(w) . (2.22)40We now show that the function χ(µ) is well-defined for α < µ < min{γ, 3α/2}.Using f(u) = G′(u)/(γ − u), together with integration by parts, we derive the key formulaχ(µ) =ˆ µαG′(w) dw(γ − w)√G(µ)−G(w) = 2√G(µ)−G(α)γ − α + 2ˆ µα√G(µ)−G(w)(γ − w)2 dw . (2.23)This expression shows that χ(µ) is a well-defined, positive and strictly increasing function onthe range (α,min{γ, 3α/2}). Remarkably, this last expression has important consequences thatdistinguishes the subcritical regime γ < 3α/2 from the supercritical regime γ > 3α/2.For the subcritical regime, γ < 3α/2, χ(µ) diverges to +∞ as µ → γ−. Recall thatα < u(x) < min{γ, 3α/2} = γ for this regime, the implicit relation (2.21) maps the range (α, µ)back to a domain (0, `) of x with unbounded ` as µ→ γ−. Therefore, since the outer problemis always solvable, a single hotspot solution can be constructed regardless of the half-domainlength ` > 0. In Fig. 2.4b, we plot χ(µ) on this range when α = 1 and γ = 1.25.shiteIn contrast, for the supercritical regime where γ > 3α/2, (2.23) shows that χ(µ) attains itsmaximum valueχmax ≡ χ(3α/2) , (2.24)as µ → (3α/2)−, where the upper bound of the range of u is min{γ, 3α/2} = 3α/2 < γ. Inview of (2.22), this translates to a maximum interval length`max ≡√D2 χmax , (2.25)for the outer solution to exist. In Fig. 2.4a, we plot χ(µ) on this range when α = 1 and γ = Crime Hotspot Insertion - the One-Sixth Rule and Fold BifurcationWe now proceed to investigate what is the significance of the following implicit relation in thesupercritical regime γ > 3α/2, especially near its criticality:χ(µ) =√2D` . (2.26)41(a) χ(µ) attains a maximum at µ = 3α/2 when γ > 3α/2,γ = 2.(b) χ(µ) is unbounded on (α, γ) when α < γ < 3α/2, γ =1.25.Figure 2.4: The function χ(µ), from (2.4a), that determines the outer solution for any µ ≡ a0(`)on α < µ < min{γ, 3α/2}. Common parameter values are D = 1 and α = 1.Figure 2.5: χmax against γ for five values of α (1.0, 1.25, 1.5, 1.75, 2.0), given by formulas(2.26) and (2.28). The curves were evaluated on the respective ranges of γ satisfying ((3α/2) ·1.01, (3α/2) · 2). It is evident from the plots that χmax → +∞ as γ → (3α/2)+.42Notice that the left-hand side of this expression depends on γ and α, appearing implic-itly through χ(µ), while the right-hand side depends on the ratio `/√D. Thus, this relationcompares the spatial parameters to those that concerns criminology in a nonlinear fashion.Next, since χ(µ) is monotone, we can invert the one-to-one function χ(µ) to compute a0(`) =µ = χ−1(√2D `) for any ` ∈ (0, `max]. In this way, we identify v0(`) = g (a0(`)) from (2.17). Inparticular, at ` = `max, this gives a maximal criminal density at the endpoint of the interval(0, `) other than which the crime hotspot exists. The value is given simply by substitutingµ = 3α/2 for the values of a and v in ρ = a2v at x = `:limx→`ρ = µ2g(µ) = µ− αµ= 13 , (2.27)which is a surprisingly simple number independent of all model parameters.Combining (2.27) with (2.16), we observe thatρ(`)ρ(0) ∼1/32 =16 .This simple observation gives an interesting one-sixth rule for determining whether a crimehotspot is being born in supercritical regime:One-sixth Rule The ratio of crime density at a nascent hotspot to a preexisting neighbouringhotspot is approximately 1/6.This is the level of criminal density where the outer problem begins to break down and apossibly new boundary layer emerges at this end of the domain, as was suggested earlier by thephase portrait description of the system (2.19).To more accurately determine what actually happens when such a criminal density is reachedat edge of the outer region, we investigate how the solution depends on the parameter D. Thiswill turn out to be an equivalent but more convenient parameter for bifurcation analysis nearthe vicinity where (2.26) begins to break down.For γ = 2 and α = 1, we calculate numerically from (2.23) (see Fig. 2.4a) thatχmax ≡ χ(3α2)≈ 1.0561 . (2.28)43Correspondingly, if we fix `, we can then use χmax to characterize the minimum value Dcritof D for which the outer solution exists asDcrit ≡ 2`2χ2max. (2.29)In most of our numerical simulations below, we consider the domain (−1, 1) containing Kinterior hot-spots. To determine the minimum value of D for this pattern, we simply put` = 1/K to obtain the critical thresholdsDcrit,K =2K2χ2max. (2.30)To compare with our full numerical results in §2.4, we calculate for α = 1 and γ = 2 thatDcrit,1 ≈ 1.7930 , Dcrit,2 ≈ 0.4483 , Dcrit,4 ≈ 0.1121 . (2.31)2.2.4 Determination of Attractiveness Amplitude at the HotspotTo complete the leading-order theory, we must determine the leading-order constant V0 in theinner expansion. By using the outer expansion, we obtain the following limiting behaviour from(2.17) and (2.21):A ∼ a0(0+) = α , v0x(0+) ∼ g′(α)a0x(0+) = 1α2√2D√G(µ)−G(α) . (2.32)To determine V0, we integrate the V -equation of (2.11) over an intermediate region (−δ, δ),with  δ  1, to obtainD(A2Vx) ∣∣x=δx=−δ =ˆ δ−δV A3 dx+O(δ) . (2.33)Since δ  , the outer expansion (2.32) is used to evaluate the left-hand side of (2.33), uponnoting the symmetry condition v0x(0+) = −v0x(0−). In contrast, since V A3 = O(−1) in theinner region, the integral in (2.33) is dominated by contributions from the inner region where44A ∼ w/√V0 and V ∼ 2V0. In this way, using δ   and w(y) =√2 sech y, we calculate thatDα2(v0x(0+)− v0x(0−))= 2Dα2v0x(0+) ∼ 1√V0ˆ ∞−∞w3 dy =√2pi√V0. (2.34)Finally, upon using (2.32) for v0x(0+), we obtainV0 =pi24D [G(µ)−G(α)]−1 . (2.35)Here µ = a0(`) is a root of (2.26) and G(u) is defined in (2.20b).As  is decreased, and for α = 1, γ = 2 and D = 2, in Fig. 2.6 we compare our leading-orderasymptotic results for A(0) (left panel) and for −2V (0) (right panel) with corresponding fullnumerical results computed from (2.11) using the continuation software AUTO-07p (cf. [12]). Theleading-order asymptotic results are A(0) ∼ √2/√V0 and −2V (0) ∼ V0, where V0 is defined in(2.35). These comparisons show that the leading-order asymptotic theory only agrees well withthe full numerical results when  is very small. As such, in order to obtain decent agreementbetween the asymptotic theory and full numerical results when  is only moderately small, wemust provide a higher-order asymptotic theory. This more refined asymptotic theory, done in§2.5, shows that the error in the leading-order prediction is in fact O(− log ), which explainswhy the leading-order theory is rather inaccurate unless  is very small. The asymptotic resultsfrom this improved theory are shown by the thin dotted curves in Fig. NLEP Stability AnalysisIn this section we analyze the stability of steady-state hotspot solutions to (2.10) in the regimewhere D = O(1). We show that all eigenvalues λ, with λ = O(1), of the militarization satisfyRe(λ) < 0 so that steady-state hotspot solutions are stable on an O(1) time-scale.We let ae(x), ve(x) be a hotspot solution to the steady-state problem (2.11). Recall that inthe inner region near the core of the hotspot, we have ae = O(−1) and ve = O(2), while bothae and ve are O(1) in the outer region. Upon introducing the perturbationA = ae + eλtφ , V = ve + eλt3ψ ,450.00 0.02 0.04 0.06 0.08 0.10ǫ0.500.520.540.560.580.600.62ǫA(0), rescaled numerical valuesleading order approximationimproved approximation0.00 0.02 0.04 0.06 0.08 0.10ǫ012345678ǫ−2V(0), rescaled numerical valuesleading order approximationimproved approximationFigure 2.6: Comparison of numerical and asymptotic results for A(0) (left panel) and for−2V (0) (right panel) as  is decreased for a single hotspot centred at the origin. The parametervalues are α = 1, γ = 2, ` = 1, and D = 2. The dashed horizontal lines are the leading-orderprediction of A(0) ≈ √2/√V0 and −2V (0) ≈ V0, where V0 is given in (2.35). The thin dottedcurves are from the improved asymptotic theory with V0 now given by (2.81).into (2.10), we obtain the following singularly perturbed eigenvalue problem on |x| < `:2φxx − φ+ 3vea2eφ+ 3a3eψ = λφ , (2.36a)D(a2e3ψx + 2aevexφ)x= 3a2eveφ+ 3a3eψ + λ(3a2eψ + 2aeveφ). (2.36b)We remark that our choice of O(1) and O(3) perturbations for A and V produce a distinguishedbalance for all terms in the equation for φ in the inner region, where the length scale is O().To examine the stability of a single hotspot solution or a multiple hotspot solution we imposeeither homogeneous Neumann or Floquet-type boundary conditions, respectively, on x = ±`,as was done in [29]. However, as we show below, in the regime   1 and D = O(1), theleading-order stability problem is independent of the specific choice of boundary condition, andthis leading-order problem predicts that hotspot equilibria are unconditionally linearly stableto O(1) time-scale perturbations.We begin by deriving the leading-order stability problem for a hotspot solution centred atthe origin in the domain |x| ≤ `. The eigenfunction component φ is singularly perturbed. Ithas rapid spatial variation near x = 0, but varies on an O(1) scale away from the hotspot core.The leading-order stability problem will consist of a nonlocal eigenvalue problem (NLEP) forφ on the inner-scale centred at x = 0. Alternatively, the eigenfunction component ψ is notsingularly perturbed and varies on an O(1) scale across the entire domain |x| ≤ `.46In the inner region near the core of the hotspot at x = 0 we introduce the local variablesy = x/, φ = Φ(y), and ψ = Ψ(y). Then, from (2.36b), we observe that the leading-orderequation is(w2Ψy)y = 0. The bounded solution to this equation is the constant solutionΨ = ψ(0), which must be determined in the ensuing analysis. With regards to Φ, we recallfrom §2.2 that ae ∼ −1w/√V0 and ve ∼ 2V0. Upon substituting these expressions into (2.36a),we obtain the leading-order inner problem for Φ(y) on −∞ < y <∞, given byL0Φ +1V3/20w3ψ(0) = λΦ , L0Φ ≡ Φyy − Φ + 3w2Φ , (2.37)where Φ→ 0 as |y| → ∞.2.3.1 Derivation of Jump ConditionsNext, we integrate (2.36b) over −δ < x < δ to derive a “jump condition” for ψ, valid as → 0.Here we let δ denote any intermediate scale satisfying  δ  1. This integration yields3D[a2eψx]0+ 2D [aevexφ]0 ∼ 3ˆ δ/−δ/w2Φ dy + ψ(0)V3/20ˆ δ/−δ/w3 dy+ λ[22√V0ˆ δ/−δ/wΦ dy + 2ψ(0)V0ˆ δ/−δ/w2 dy].Here we have defined [f ]0 ≡ limx→0+ f(x)− limx→0− f(x). For → 0, we can then neglect theterms in the square brackets in the expression above, let δ/→ +∞, and use ae ∼ α as x→ 0±from the outer solution. In this way, we obtain the following asymptotic approximation of thejump condition:3Dα2 [ψx]0 + 2Dα [vexφ]0 ∼ 3ˆ ∞−∞w2Φ dy + ψ(0)V3/20ˆ ∞−∞w3 dy . (2.38)Next, we examine the outer expansion for φ and ψ in order to estimate the left-hand sideof (2.38). From the outer approximation of (2.36a), we use v ∼ v0 and a ∼ a0 to obtain that(3v0a20 − 1)φ+ 3a30ψ ∼ λφ .47Then, since v0 = g(a0), where g(a0) is defined in (2.17), we solve for φ0 to obtainφ = 3φ˜ , φ˜ ≡(a30λ− 2 + 3α/a0)ψ . (2.39)Since α ≤ a0 < min{γ, 3α/2} for |x| ≤ `, we conclude by examining the denominator in (2.39)that φ˜ is analytic in Re(λ) ≥ 0. We then let x→ 0± in (2.39) to getφ(0±) = 3(α3λ+ 1)ψ(0) . (2.40)We then use (2.40), together with vex(0±) ∼ g′(α)a0x(0±) and g′(α) = 1/α3, to estimatethe second term on the left-hand side of (2.38) as2Dα [vexφ]0 ∼( 2Dαλ+ 1)3ψ(0) [a0x]0 .In this way, we obtain that the asymptotic jump condition (2.38) for ψ(x) becomes2D(α2 [ψx]0 +2αλ+ 1ψ(0) [a0x]0)∼ 3ˆ ∞−∞w2Φ dy + ψ(0)V3/20ˆ ∞−∞w3 dy , (2.41)where a0(x) satisfies the nonlinear BVP (2.18a) in the outer region.2.3.2 The Outer Problem and Analyticity of CoefficientsNext, we derive the outer problem for ψ(x) on −` < x < 0− and on 0+ < x < ` by consideringthe outer limit for (2.36b), where we have a ∼ a0, v ∼ v0, and φ ∼ 3φ˜. This outer problem isD(a20ψx + 2a0v0xφ˜)x= 3a20v0φ˜+ a30ψ + λ(2a0v0φ˜+ a20ψ), (2.42)where φ˜ depends linearly on ψ from (2.39). We then substitute (2.39), v0 = g(a0), and v0x =g′(a0)a0x into (2.42), where g(a0) is defined in (2.17). After some algebra, we obtain that (2.42)reduces on 0 < |x| < ` to(a20ψx +( 3α− 2a03α+ a0(λ− 2))(a20)xψ)x=((λ+ 1)a30 + λa0(α+ a0λ)λ− 2 + 3α/a0)ψ . (2.43)48The problem for ψ is to solve (2.43) on the intervals 0 < x < ` and −` < x < 0, together withthe continuity of ψ across x = 0, the jump condition (2.41), and subject to either homogeneousNeumann or Floquet-type boundary conditions imposed at x = ±` depending on whether weare considering single or multiple hotspot solutions, respectively. From the solution to thisproblem we can determine ψ(0), which is needed for the inner problem (2.37).The differential equation (2.43) can be written in the general form(b(x)ψx + b1(x, λ)ψ)x = b2(x, λ)ψ . (2.44)where b(x) > 0 on 0 < |x| ≤ `. Since α ≤ a0 < min{γ, 3α/2} for |x| ≤ `, we conclude from(2.43) that b1(x, λ) and b2(x, λ) are analytic in Re(λ) ≥ 0. Since ψ satisfies a linear ODE withanalytic coefficients in Re(λ) ≥ 0, any solution to (2.43) must be analytic in Re(λ) ≥ 0 byclassical ODE theory.Although it is intractable to determine a closed-form for ψ(x), what is essential in thiscontext is that we are still able to determine ψ(0) as  → 0 simply from the leading-orderapproximation (2.41) of the jump condition. Since the outer solution satisfying (2.43) is smooth,we have that ψx(0±) is finite and independent of . Letting → 0 in the jump condition (2.41)we identify thatψ(0) ∼ −3V 3/20(´∞−∞w2Φ dy´∞−∞w3 dy). (2.45)Finally, upon substituting (2.45) into (2.37), we obtain the nonlocal eigenvalue problem(NLEP)L0Φ− 3w3(´∞−∞w2Φ dy´∞−∞w3 dy)= λΦ , −∞ < y <∞ ; Φ→ 0 as |y| → ∞ . (2.46)As shown in Lemma 3.2 of [29], the NLEP (2.46) is explicitly solvable in the sense that its dis-crete spectrum can be found analytically. Lemma 3.2 of [29] shows that any nonzero eigenvalueof (2.46) must satisfy Re(λ) < 0.49−1.0 −0.5 0.0 0.5 1.0x024681012At=0t=10t=1000−1.0 −0.5 0.0 0.5 1.0x1. 2.7: Stability of O(−1)-amplitude and O(1)-amplitude hotspots. Parameter values are = 0.01, D = 1, α = 1, γ = 1.25. Left plot: the O(−1)-amplitude hotspot persists in thissubcritical regime. Right plot: the O(1)-amplitude hotspot is transient and dissipates into theTuring-stable homogeneous state.2.3.3 Conclusions on Stability for Various PatternsSince this NLEP is independent of the boundary conditions for ψ on x = ±`, it applies toboth single and multiple hotspot solutions. It is also independent of whether γ > 3α/2 orα < γ < 3α/2 because the outer solution and its derivatives are the same near x = 0. In otherwords, crime hotspots in both supercritical and subcritical regimes are both stable to linearperturbations.In Fig. 2.7, we confirm numerically the stability of a single hotspot solution and plot theprofile of the attractiveness profile A. We also show a different type of hotspot that is foundfrom numerical computations of the bifurcation diagram. This kind of hotspot has an O(1)amplitude for the attractiveness A, but is unstable. It is an open problem to both constructand show analytically the instability of this kind of hotspot, which we do not consider in Section2.2.Finally, we recall for comparison that, for the regime D = O(−2) studied in [29], the NLEPproblem for O(1) eigenvalues of a K−hotspot solution on an interval of length S is given by:L0Φ− χjw3(´∞−∞w2Φ dy´∞−∞w3 dy)= λΦ , −∞ < y <∞ ; Φ→ 0 as |y| → ∞ . (2.47)50The multiplierχj = 3[1 + D0α2pi2K44(γ − α)3( 2S)4(1− cospij/K)]−1, j = 0, . . . ,K − 1 ,where D0 ≡ D2 corresponds to different “modes” of instability. For a single hotspot solution,i.e. K = 1, we have χ0 = χ1 = 3 which gives exactly the same NLEP (2.46) and, thus, thesolution is stable for allD0. For a multiple hotspot solution, stability is governed by the smallestmultiplier χK−1. This gives a stability thresholdDL0K ≡2(γ − α)3(S/2)4K4α2pi2 [1 + cos(pi/K)] , (2.48)so that a K−hotspot pattern is stable only when D0 < DL0K . In contrast, there is no suchthresholds when D = O(1) and multiple hotspot patterns are stable whenever they exist.2.4 Bifurcation Diagrams of Hotspot Equilibria: Numerical Con-tinuation Computations2.4.1 Supercritical Regime - Fold Points for Spike InsertionIn this section we show full numerical bifurcation results for hotspot equilibria of (2.11) com-puted using the continuation software AUTO-07p (cf. [12]). These computations show that newhotspots are created near the endpoints x = ±l or at the midpoint between two hotspots whenD approaches a saddle-node bifurcation value, which we denote by Dfold. We call this phe-nomena hotspot or spike insertion. Our numerical results show clearly that the fold-point valueDfold tends, as  → 0, to the critical value Dcrit, as derived from the leading-order asymptotictheory of §2.2. However, unless  is very small, our results show that the leading-order theoryfor the critical value of D is quantitatively rather inaccurate. This motivates the need for ahigher-order asymptotic theory in §2.5.By using AUTO-07p we compute from (2.11) the bifurcation diagrams of branches of steady-state solutions starting with either K = 1, K = 2, or K = 4, interior hotspots. Instead ofusing the L2 norm, it is more convenient to use the boundary value A(`) as the vertical axis on51the bifurcation diagram, as this prevents the overlapping of solution branches with boundaryhotspots. The horizontal axis on the bifurcation diagram is D. Our computational results forthe parameter set γ = 2, α = 1, ` = 1, and  = 0.01, displayed in Fig. 2.8(a) show that as Dis varied, the steady-state solution branch with one interior hotspot is connected to a solutionthat has an interior hotspot together with a hotspot at each boundary. Boundary hotspots arecreated at the endpoints x = ±` near the fold point associated with the small norm branch forA(`) versus D.Similar bifurcation results starting from either two or four interior hotspots on the smallnorm A(`) versus D branch are shown in Fig. 2.8(b) and Fig. 2.8(c) to be path-connectedto solutions with mixed boundary and interior hotspots. We observe that new hotspots arenucleated at the endpoints x = ±` and at the midpoint of the interval between adjacenthotspots at the fold point associated with the small norm solution branch.In each set of the four solution profiles in Fig. 2.8 we observe, as expected by symmetry, thatthe right top and bottom figures have the same value of D at the two fold points where D issmallest. In each case, as we follow the small norm solution branch of A(`) versus D towards thefold-point value, we observe a decrease in the amplitude of the hotspot and a gradual bulgingup of the solution at the midpoint between hotspots or at the domain boundaries (i.e. at theboundary of the outer regions). This leads to a large amplitude pattern of hotspots when theboundary value A(`) exceeds a certain value close to 3α/2 = 1.5.As compared to the large O(−1) scale of the hotspot amplitude, the nucleation or formationof hotspots near the boundary, or at the midpoint between hotspots, is not so conspicuous in ourplots, especially when  is small. However, the implication of hotspot nucleation is interestingqualitatively. It predicts that a new crime hotspot can emerge from an essentially quiescentbackground state when the parameters are close to a fold-point value in D.Next, we show how the criminal density changes at the onset of spot insertion. In Fig. 2.10we give various plots of ρ = A2V for values of D at and close to the fold point. We remark thatsince ρ = O(1) in both the inner and outer regions, i.e. globally, the spot insertion phenomenoncan be observed quite clearly in these plots.We now discuss whether the solution branches in Fig. 2.10 corresponding to parametervalues immediately above the fold point, consisting of “small” hotspots between the large am-52(a)0 10 20 30 40 50 60 70D010203040506070A(l)010203040506070A0.0000.0010.0020.0030.0040.0050.0060.0070.008VD = 65.000001020304050A0. = 1.5600−1.0 −0.5 0.0 0.5 1.005101520253035A0.00180.00200.00220.00240.00260.00280.00300.00320.00340.0036VD = 61.8242−1.0 −0.5 0.0 0.5 1.001020304050A0. = 1.5600(b)0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0D05101520253035A(l)05101520253035A0.0000.0050.0100.0150.0200.0250.030VD = 4.00000510152025A0. = 0.3451−1.0 −0.5 0.0 0.5 1.0024681012141618A0.0060.0070.0080.0090.0100.0110.0120.013VD = 3.7617−1.0 −0.5 0.0 0.5 1.00510152025A0. = 0.3451(c)0.05 0.10 0.15 0.20 0.25 0.30D0246810121416A(l)0246810121416A0. = 0.2761024681012A0. = 0.0677−1.0 −0.5 0.0 0.5 1.012345678A0.0240.0260.0280.0300.0320.0340.0360.0380.0400.042VD = 0.2142−1.0 −0.5 0.0 0.5 1.0024681012A0. = 0.0677Figure 2.8: The plots (a), (b) and (c) show the continuation of steady states starting witheither one, two, or four, interior hotspots, respectively, for  = 0.01. The other parameters areγ = 2, α = 1, and ` = 1. The solid and dashed curves in the subplots show the profiles of Aand V , respectively, at various values of D specified on top of the plots. These values of Dcorrespond to the marked points on the bifurcation diagram as shown on the left. Notice thatthe range of A and V are on different scales.53plitude hotspots, are linearly stable or not. An analytical stability theory of the solution branchbeyond the fold point would first require an asymptotic construction of such an alternating pat-tern with new boundary layers for these “small” hotspots (since the previous outer solution isno longer valid), and would be rather difficult to undertake. Some directions are outlined inSection 2.8. Although we do not pursue the analysis here, we conjecture that all such patternsare unstable. Indeed, from a number of numerical experiments, we observe that a solution onthe “upper” branch evolves quickly to its counterpart directly below on the lower branch, i.e.with the same value of D in the bifurcation diagrams (and all other parameters as well) shownin Fig. 2.9. In Fig. 2.11, we show the evolutionary dynamics computed by PDEPE using the“upper” branch solutions computed in AUTO-07p as initial conditions. In all cases shown, thedynamics stabilizes to exactly the pattern of the solution on the corresponding “lower” branch,with no visible difference.Our numerical bifurcation results for the fold-point bifurcation along the small norm solutionbranch for γ = 2, α = 1, ` = 1, and  = 0.01 areDfold,1 = 1.5600 , Dfold,2 = 0.3451 , Dfold,4 = 0.0677 . (2.49)Observe, as expected by symmetry, that these values are quartered as the number of hotspotsdoubles. However, we observe that the quantitative agreement of these fold-point values withthe critical values Dcrit,K of (2.31), as computed from the leading-order theory of Section 2.2,is suggestive, but is not particularly close even when  = 0.01.As a result of this rather poor quantitative agreement at finite , we used AUTO-07p (cf. [12])to perform a codimension-two path-following of the fold point as  is decreased. The compu-tations were done for the case of K = 1, K = 2, or K = 4, interior hotspots. The goal ofperforming this codimension-two continuation to trace the curves Dfold,K() was to establishevidence for the conjecture that lim→0Dfold,K() = Dcrit,K , and to find a range of small  wherethe agreement between Dfold,K and Dcrit,K is close.The results of this codimension-two computation are shown in Fig. 2.9. From this figure,we observe that the “almost” straight solid curves for the numerically computed values ofDfold,K() versus  does seem to extrapolate as → 0 to the leading-order limiting critical value54(a)0.010 0.030 0.050ǫ1.7151.5600.9531.793Dfold,1020406080100120140160180ǫ=0.002801020304050ǫ=0.0100−1.0 −0.5 0.0 0.5 1.012345678910ǫ=0.0500−1.0 −0.5 0.0 0.5ǫ=0.0824(b)0.00 0.01 0.02 0.03ǫ0.41730.34510.20700.4483Dfold,2020406080100120ǫ=0.00240510152025ǫ=0.0100−1.0 −0.5 0.0 0.5 1.012345678ǫ=0.0300−1.0 −0.5 0.0 0.5ǫ=0.0412(c)0.000 0.005 0.010 0.015ǫ0.085780.067730.11210Dfold,40510152025ǫ=0.0051024681012ǫ=0.0100−1.0 −0.5 0.0 0.5ǫ=0.0200−1.0 −0.5 0.0 0.5ǫ=0.0206Figure 2.9: For the parameter set γ = 2, α = 1, and ` = 1, the solid curves on the left of eachof (a), (b), and (c), for K = 1, K = 2, and K = 4, interior hotspots, respectively, show the foldpoint values Dfold,K() associated with the small norm solution branch of A(`) versus D. Thetop tick-mark on the vertical axes in these plots are the approximate values Dcrit,1 ≈ 1.793,Dcrit,2 ≈ 0.448, and Dcrit,4 ≈ 0.112 from the leading-order theory of §2.2. The dashed curvesin each of (a), (b), and (c), are the asymptotic results (2.82) for the fold point value for D,as predicted by the higher-order asymptotic theory of §2.5. For each of the three sets, thenumerically computed A versus x is plotted on |x| ≤ 1 at four values of . At the larger valuesof  the pattern is essentially sinusoidal.55−1.0 −0.5 0.0 0.5 1.0x0.ρLower branchUpper branchFold point−1.0 −0.5 0.0 0.5 1.0x0.ρ−1.0 −0.5 0.0 0.5 1.0x0.ρFigure 2.10: Plots of criminal density ρ near the onset of spot insertion as indicated in Fig.2.9. The parameter values are the same, i.e.  = 0.01, γ = 2, α = 1, ` = 1. The purplecurves correspond to ρ at the numerically computed fold point, while the red and blue curvescorrespond to the upper and lower branch solutions at some identical values of D close to, butrespectively above and below the fold points shown in Fig. 2.9. Such chosen values of D are2.0, 0.5 and 0.1 respectively for one, two and four interior spikes (before the insertion event),corresponding to the sub-figures on the left, centre, and right, respectively. The fold pointvalues of D are given in (2.49).−1.0 −0.5 0.0 0.5 1.0x0.ρ−1.0 −0.5 0.0 0.5 1.0x0.ρ−1.0 −0.5 0.0 0.5 1.0x0.ρFigure 2.11: Time evolution of criminal density ρ for spatial patterns found by continuationbeyond the fold point. Parameter values are the same as in Fig. (2.9), i.e.  = 0.01, γ = 2, α =1, ` = 1. Also, D = 2.0, 0.5, 0.1 for the left, centre and right sub-figures, respectively. Thedotted, dashed and solid blue curves shows the evolution of the upper solution at t = 0, 15, 100,respectively, while the heavy solid red curves shows the lower branch solution. We find thatthe plots at t = 100 all overlap exactly with those of the lower branch counterparts, which arethe linearly stable patterns proved in Section 2.3.of §2.2. Due to numerical resolution difficulties, we were not able to perform computationsfor smaller values of  than shown in Fig. 2.9. However, these computational results do giveclear numerical evidence for the conjecture that lim→0Dfold,K() = Dcrit,K for K = 1, 2, 4.An analytical justification that Dcrit,K does in fact correspond to a fold point is given in theanalysis of hotspot insertion phenomena in § 2.6 below. In Fig. 2.9, we also plot the improvedapproximation for Dcrit,K versus  (dashed curves), as given in (2.82), that will be derived fromthe higher-order asymptotic theory of §2.5.Finally, from the plots of A versus x in Fig. 2.9 at selected values of , we observe that when is only moderately small the steady-state solution more closely resembles a sinusoidal pattern56than a pattern of localized hotspots.2.4.2 Subcritical Regime - Fold Points for Spike Type SwitchingHere, we present the results of analogous studies of the bifurcation diagrams for the case α < γ <3α/2. As discussed previously, this is the subcritical regime studied in [35] on the infinite linewhere homoclinic snaking behaviour observed. In Fig. 2.12, we instead observe an interestingclosed-loop structure, containing crime hotspots solutions of two different order of amplitude:O(1) and O(−1), with the latter type comprehensively developed in this chapter. The foldpoints II and III belong to the supercritical regime γ > 3α/2 and is in agreement with the Fig.2.8 as a reduction in γ leads to an increase of χmax as shown in Fig. 2.5 in the supercriticalregime. Moreover, the same hotspot centre mirroring phenomenon is also observed as we followthe path from point II to point III, analogous to the first plot in Fig. 2.8.In contrast, the fold points I and IV are novel in that they occur for a solution branch withA = O(1). We conjecture that this is the same fold point as shown in Fig. 7 of [29], and thusthey connect the weakly nonlinear regime born from the subcritical Turing bifurcation to thefar-from equilibrium theory in this chapter valid for the solution branch beyond fold points Iand IV. In Fig. 2.7, we took precisely a solution at γ = 1.25 in the branch before reaching pointI from the Turing bifurcation to perform the full numerics. The result was that this solution isunstable as expected.The development of an asymptotic theory for the solution branch with A = O(1) hotspotsis an interesting open problem. However, these solutions are likely all linearly unstable.2.5 Refinements of the Steady State Solution: Higher-OrderTheoryIn this section we present a more refined asymptotic theory than that given in §2.2 to construct asteady-state hotspot solution centred at the origin on the interval |x| ≤ `. The results from thishigher-order theory provide a rather close asymptotic prediction of the saddle-node bifurcationpoint observed in §2.4, as well as providing the error terms associated with the leading-ordertheory. In addition, this more refined asymptotic analysis is essential for the analysis in §2.7,57γ1.||A||−1.0 −0.5 0.0 0.5 Point I: γ=1.16510.0450.0500.0550.0600.0650.0700.0750.080V−1.0 −0.5 0.0 0.5 1.005101520253035AFold Point II: γ=1.76540.−1.0 −0.5 0.0 0.5 1.005101520253035AFold Point III: γ=1.76540.−1.0 −0.5 0.0 0.5 Point IV: γ=1.16510.0450.0500.0550.0600.0650.0700.0750.080VFigure 2.12: A closed homotopy of equilibria with a single hotspot (or two boundary hotspots)from the continuation in γ from the subcritical Turing bifurcation. Model parameters are: = 0.01, D = 1, α = 1. Observe that the amplitude of A at Point I and IV are O(1) and equalbut different to that of Point II and III, which are of O(−1).58where a DAE system for the slow time evolution of a collection of hotspots is derived.In the inner region, where y = −1x and y = O(1), we pose a two-term expansion for A asA ∼ A0/+A1 + . . . , V ∼ 2V .Upon substituting this expansion into (2.11), and retaining the dominant correction terms weobtain(A0yy −A0 + VA30)+ (A1yy −A1 + 3VA20A1 + α)+ · · · = 0 , (2.50a)D[(A20 + 2A0A1)Vy]y− V(A30 + 3A20A1)+ 2 (γ − α) + · · · = 0 . (2.50b)This suggests that we expand V = V0 +V1 + · · · . Upon substituting this expansion into (2.50),and collecting powers of , we obtain our leading-order result that V0 is an unknown constantand A0 = w/√V0, where w(y) =√2 sech y is the homoclinic solution of (2.14). At next order,we obtain that A1(y) and V1(y) on −∞ < y <∞ satisfyA1yy −A1 + 3A20A1V0 = −α− V1A30 , D[A20V1y]y= V0A30 . (2.51)Then, upon substituting A0 = w/√V0 into (2.51), we conclude thatL0A1 ≡ A1yy −A1 + 3w2A1 = −α− V1V3/20w3 ,[w2V1y]y=√V0Dw3 . (2.52)A key step in analyzing the inner region is to determine the far-field asymptotic behaviouras y → +∞ for the solution to (2.52). Since w(y) ∼ 2√2e−y as y →∞, it readily follows uponintegrating the V1-equation that V1 = O(e2y) as y → ∞. Then, since w3V1 → 0 as y → ∞,it follows from the A1-equation in (2.52) that A1 → α as y → ∞. These simple results showthat the V-expansion V = V0 + V1 + · · · becomes disordered when y = O(−(1/2) log ), whilethe far-field as y → ∞ of the A-expansion A = w/ [√V0] + A1 ∼ −12√2e−y + α becomesdisordered when y = O(− log ).As a result, for y > 0, we will need to introduce two additional inner layers before weare finally able to match to the outer solution. For  → 0, we define the mid-inner layer by59y = −(1/2) log  + O(1) and the knee layer by y = − log  + O(1). The asymptotic solution inthe knee layer can then be matched to the outer solution valid on 0 < x < `. By symmetry, asimilar construction can be done for y < 0. In Fig. 2.13 we plot the full numerical steady-statesolution for V , as computed from (2.11), showing the knee behaviour of V for γ = 2, α = 1,D = 2,  = 0.01, and ` = 1.−1.0 −0.5 0.0 0.5 1.0x0.−0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 0.20x0.0000.0050.0100.0150.0200.0250.030VFigure 2.13: Plot of the full numerical solution to the steady-state problem (2.11) showing theknee behaviour of V for γ = 2, α = 1, D = 2,  = 0.01, and ` = 1. The figure on the right is azoom of the one on the left.The asymptotic matching of the expansions of the solution across the mid-inner and kneelayers will specify the appropriate far field behaviour for the solution V1 to (2.52) in the formV1 ∼ b+e2|y| , as |y| → ∞ , (2.53)for a certain constant b+ that is, ultimately, determined by the outer solution. In this way, wecan decompose the inner correction term V1, satisfying (2.52), as V1(y) = V10 + V1p(y), where60V10 is a constant to be determined and where V1p(y) satisfies[w2V1py]y=√V0Dw3 , −∞ < y <∞ ; V1p ∼ b+e2|y| as |y| → ∞ ; V1p(0) = 0 .(2.54)Upon integrating (2.54) on −∞ < y < ∞, we conclude that a solution to (2.54) exists if andonly if √V0Dˆ ∞−∞w3 dy = limy→+∞(w2V1py)− limy→−∞(w2V1py). (2.55)By using the limiting behaviours of w and V1p as |y| → ∞, together with´∞−∞w3 dy =√2pi, weobtain that √V0D= 16√2pib+ . (2.56)Thus, the value of V0 can be found when b+ is known, which is be determined by a far-fieldmatching with the outer solution.Next, we show how to readily determine V1p explicitly. Upon integrating (2.54) from −∞to y and using (2.56), we getw2V1py =√V0Dˆ y−∞w3ds− 16b+ =√V0D(ˆ y−∞w3ds− pi√2). (2.57)Observe thatˆ y−∞w3ds− pi√2= 12ˆ ∞−∞w3ds− pi√2+ˆ y0w3ds =ˆ y0w3ds ,while integrating (2.14) from 0 to y yieldsˆ y0w3 ds =ˆ y0w ds− w′(y) .Therefore, (2.57) implies thatV1py(y) =√V0D(´ y0 w ds− w′(y)w2(y)). (2.58)Then, upon integrating (2.58) from 0 to y, and using V1p(0) = 0 from (2.54), we finally obtain61thatV1p(y) =√V0D(ˆ y0w−2(s)W (s)ds+ 1w(y) −1w(0)), W (s) ≡ˆ s0w(y) dy . (2.59)Note that V1p(y), like w(y), is even in y as well. Also, one can use the explicit formulaw(y) =√2 sech y to computeV1p =√V0D(√2ˆ y0cosh2(s) tan−1[tanh(s2)]ds+ 1√2(cosh y − 1)). (2.60)To determine b+, we now proceed by analyzing the two additional inner layers. We beginwith the knee-layer, defined by y = − log  + O(1), which can be matched to the limitingbehaviour as x → 0+ of the outer solution. In the knee-layer, we introduce the new variablesAˆ, Vˆ , and z byA = Aˆ(z) , V = Vˆ (z) , y = − log + z ,with z = O(1), so that x = − log + z. In terms of these new variables, (2.11) becomesAˆzz − Aˆ+ Vˆ Aˆ3 + α = 0 , D(Aˆ2Vˆz)z− 2Vˆ Aˆ3 + (γ − α) = 0 . (2.61)We substitute the expansion Aˆ = Aˆ0 + · · · and Vˆ = Vˆ0 + · · · into (2.61) to obtain thatAˆ0zz − Aˆ0 = −α. The solution for Aˆ0 that agrees with the far-field behaviour of the innersolution for A isAˆ0 = ce−z + α , c ≡ 2√2√V0. (2.62)In contrast, Vˆ0(z) satisfies[Aˆ20Vˆ0z]z= 0 , −∞ < z <∞ . (2.63)To determine the appropriate far-field behaviour as z → +∞ for (2.63), we expand the leading-order outer solution v0(x) of §2.2 as x → 0, and then rewrite the expression in terms of z to62obtain the matching conditionVˆ ∼ v0x(0+)z + (− log )v0x(0+) + · · · . (2.64)The dominant constant term of order O(− log ) in (2.64) cannot be accounted for by the kneesolution. Instead, we must introduce a switchback term of order O(− log ) into the outerexpansion. More specifically, the outer expansion on 0 < x < ` must have the formA ∼ a0 + (− log ) a1 + a2 + · · · , V ∼ v0 + (− log ) v1 + v2 + · · · , (2.65)with v1(0+) = −v0x(0+), chosen to eliminate the constant term in (2.64)Therefore, to match the knee solution Vˆ0 to the near-field behaviour (2.64) of the outersolution we require that Vˆ0z ∼ v0x(0+) as z → ∞. In contrast, to match the knee solution Vˆ0to the mid-inner solution where V = O(2), we need that Vˆ0 → 0 as z → −∞. Thus, we mustsolve (2.63) subject toVˆ0z ∼ v0x(0+) , as z → +∞ ; Vˆ0 → 0 , as z → −∞ . (2.66)Given the decay as z → −∞ and linear ramp as z → +∞, it is now clear why we refer toVˆ0 as the knee solution. Upon using (2.62) for Aˆ0, a first integral of (2.63), which satisfiesVˆ0z ∼ v0x(0+) as z → +∞, isVˆ0z =α2v0x(0+)(ce−z + α)2. (2.67)Integrating (2.67) and imposing the condition that Vˆ0 → 0 as z → −∞, we findVˆ0 = v0x(0+)F (z), F (z) ≡ˆ z−∞(1 + ce−s/α)−2ds (2.68)This special nonlinear function F (z) furnishes the required knee-shape which allows the tran-sition of scales from the outer to the inner region.63−10 −5 0 5 10z0123456789F(z)Figure 2.14: The knee-shaped function F (z) defined at (2.68). Parameter choices are the sameas in Fig. 2.13 which showed the numerical solution of V .Lemma 2.3. The knee-shaped function F (z) is given explicitly byF (z) = log (c+ αez) + cc+ αez + κ, κ = − log c− 1 , (2.69)and it has the following asymptotic behaviours as z → ±∞:F (z) ∼α22c2 e2z as z → −∞z +(log αc − 1)as z →∞ .Proof: An indefinite integral to (1 + ce−z/α)−2 is given bylog (c+ αez) + cc+ αez + κ ,where κ is a constant to be determined.Now, we let δ = αez/c and rewrite F (z) toF (z) = log(c+ αez) + cc+ αez + κ = log c+ log(1 + δ) +11 + δ + κ .As z → −∞, we observe δ → 0 and so we expand to estimateF (z) ∼ log c+(δ − δ22 + . . .)+(1− δ + δ2 + . . .)+ κ ∼ log c+ 1 + κ+ δ22 + . . . ,64thus we choose κ = − log c− 1 to satisfy limz→−∞ F (z) = 0.As z →∞, we observe δ →∞, and we rewrite F (z) in terms of δ asF (z) = log δ + log(1 + δ−1) + (1 + δ)−1 + log c+ κ ,and then substitute in κ = − log c− 1, δ = αez/c to find the asymptotic behaviourF (z) = log(αez/c)− 1 +O(e−z) ∼ z + log αc− 1 .We conclude from Lemma 2.3 that the knee solution can be given explicitly asVˆ0 = v0x(0+)(log(c+ αez) + c(c+ αez) − log c− 1). (2.70)and that, as z → +∞, we have the asymptotic behaviourVˆ0 ∼ v0x(0+)z +[log(αc)− 1]v0x(0+) . (2.71)This second term in (2.71) provides a matching condition for the outer correction v2 in (2.65)of the formv2(0+) =[log(αc)− 1]v0x(0+) . (2.72)Alternatively, as z → −∞, we conclude from Lemma 2.3 thatVˆ0 ∼ α2v0x(0+)2c2 e2z , as z → −∞ . (2.73)Equation (2.73) yields the far-field behaviour of the mid-inner solution for V , which we nowconstruct.To analyze the mid-inner layer, which is between the inner and knee-layer regions, weintroduce the new variables A˜, V˜ , and η byA = A˜(η) , V = 2V˜ (η) , y = −12 log + η ,65with η = O(1), so that x = −(/2) log  + η. In this layer, where V = O(2), the asymptoticorder of V is the same as that in the inner layer. Therefore, it is the knee layer that has allowedfor the fast transition in V from an O(2) scale to an O() scale, which is then matchable to theouter solution. Upon substituting these new variables into (2.11), we obtain to leading-orderthat A˜ ∼ A˜0 + · · · whereA˜0 =c√e−η + α , c ≡ 2√2√V0,and that V˜ = V˜0 + · · · , where V˜0 satisfies[e−2ηV˜0η]η= 0 , −∞ < η <∞ . (2.74)The solution to (2.74) that matches as η → +∞ to the asymptotics (2.73) of the knee solution,and that satisfies V˜0 ∼ V0 as η → −∞ in order to match to inner layer solution, is simplyV˜0 =α2v0x(0+)2c2 e2η + V0 . (2.75)Finally, we write (2.75) in terms of the inner variable y as η = y + (1/2) log  to obtain thefollowing matching condition as y → +∞ for the inner solution:V ∼ 2V˜0 ∼ 2V0 + 3(α2v0x(0+)2c2)e2y .In this way, we conclude that the solution V1p(y) to (2.54) must satisfyV1p ∼ b+e2y , as y → +∞ , b+ ≡ α2v0x(0+)2c2 , c ≡2√2√V0. (2.76)This rather intricate asymptotic construction has served to identify the constant b+ in (2.54),which can then be used in (2.56) to determine the leading-order constant V0. In fact, uponsubstituting (2.76) into (2.56), and solving for V0, we obtain√V0 =pi√2α2Dv0x(0+). (2.77)Then, by using (2.32) for v0x(0+) in (2.77), we recover the leading order result (2.35) for V0,66as derived previously in §2.2. A higher order approximation for V (0), based on analyzingcorrection terms to V0, is derived in (2.81) below.We remark that although the leading-order theory of §2.2 was also able to determine V0by simply integrating the V -equation in (2.11) over an intermediate, but otherwise unspecified,length-scale δ with  δ  1, as was done in (2.33), the more refined asymptotic approach inthis section provides the gradient information in V that is essential in §2.7 for deriving a DAEsystem for the slow dynamics of a hotspot. Moreover, this asymptotic construction has shownhow the knee-layer solution allows V to make a fast transition between O(2) and O() scales.Next, we proceed to analyze the correction terms in the outer region of the form (2.65). Uponsubstituting the A-expansion of (2.65) into (2.11), we obtain that V = g(A) to both O(− log )and O() terms, where g(A) is defined in (2.17). Therefore, vj = g′(a0)aj for j = 1, 2, and hencethe problems for a1 and a2 can be obtained by replacing a0 in (2.18a) with a0+(− log )a1+a2,and then performing a simple Taylor series expansion. To determine the boundary conditionsfor a1 and a2, we use a0(0+) = α, g′(α) = α−3, and vj = g′(a0)aj for j = 1, 2, together withthe matching conditions v1(0+) = −v0x(0+) and v2(0+) =[log(αc)− 1] v0x(0+), to identifyconditions for a1 and a2 at x = 0+. In this way, we obtain that a0(x) satisfies (2.18a), whilea1(x) and a2(x) satisfyD [f(a0)a1]xx = a1 , 0 < x < ` ; a1(0+) = −a0x(0+) , a1x(`) = 0 , (2.78a)D [f(a0)a2]xx = a2 , 0 < x < ` ; a2(0+) =[log(αc)− 1]a0x(0+) , a2x(`) = 0 .(2.78b)Our key observation is that instead of solving (2.18a) and (2.78) recursively for a0, a1, and a2,these outer approximations are contained in the solution to a renormalized outer problem fora(x), formulated asD [f(a)ax]x = a − γ , x < x < ` ; a(x) = α , ax(`) = 0 , (2.79a)67where x > 0 is defined byx = (− log )x1 + x2 ; x1 = 1 , x2 ≡ 1− log(αc), c ≡ 2√2√V0. (2.79b)The fact that the solutions to (2.79) corresponds correctly to those of (2.78) can be verifiedeasily.First, observe the key nonlinear term in the equation can be formally expanded as follows:D [f(a)ax]x = D[(f(a0) + (− log )f ′(a0)a1 + f ′(a0)a2) · (a0x + (− log )a1x + a2x)]x ,so the O(1), O(− log ) and O() terms are respectivelyD [f(a0)a0x]xD[f ′(a0)a0xa1 + f(a0)a1x]x = D [f(a0)a1]xxD[f ′(a0)a0xa2 + f(a0)a2x]x = D [f(a0)a2]xxwhich gives the L.H.S. of equations in (2.18a) and (2.78).Second, the nonzero matching condition a(x) = α can be formally expanded as:α = a(x) = a0(0+) + a0x(0+) ((− log )x1 + x2) + (− log )a1(0+) + a2(0+) + . . . ,so the O(1), O(− log ) and O() terms are respectivelyα = a0(0+)a1(0+) = −a0x(0+)a2(0+) = a0x(0+)x2 =(log(αc)− 1)a0x(0+)which recovers the matching conditions stated in (2.18a) and (2.78) for a0, a1, and a2.Therefore, the effect on the outer solution of the knee layer is that one needs to accountfor an inner region that is O(− log ) thick. The expression (2.79b) also shows that the outersolution has a weak dependence on the amplitude of the hotspot, mediated by V0. One keyadvantage of using the renormalized problem (2.79) is that the leading-order theory of §2.2 can68still be used provided that we simply replace ` with `−x. This idea of renormalizing the outersolution to account for the switchback term is vaguely similar to the renormalization methodproposed in [8] for analyzing weakly nonlinear oscillators.Next, we use the renormalized problem (2.79) to determine an improved approximationV0 for −2V (0). This new approximation consists of two weakly coupled nonlinear algebraicequations for µ ≡ a(`) and V0. To determine the first relation between µ and V0 we integratethe V -equation in (2.11) from −x < x < x to obtain, in place of (2.33), that2Dα2vx(x) = 2√2D√G(µ)−G(α) ∼ 1√V0ˆ ∞−∞w3 dy − 2x(γ − α) . (2.80)Here we have used vx(x) = g′(α)ax(x), where g′(α) = α−3 and ax(x) is obtained from afirst integral of the renormalized problem (2.79). By solving the expression above for V0, weobtain our first relationV0 =pi22[√2D(√G(µ)−G(α))+ (γ − α)x]2 . (2.81a)The second relation is obtained by replacing µ and ` in (2.26) with µ and `− x, where x isdefined in (2.79b). This yieldsχ(µ) =√2D(`− x) , x = (− log ) + (1 + log(2√2α)− log√V0), (2.81b)where χ(µ) is defined in (2.23).The system (2.81) is a weakly coupled nonlinear algebraic system for V0 and µ ≡ a(`),where the coupling arises through the fact that x depends weakly on V0. By solving thisweakly coupled system using Newton’s method for γ = 2, α = 1, D = 2, and ` = 1, in theright panel of Fig. 2.6 (thin dotted curve) we showed that V0 compares more favourably withthe full numerical result than does the leading-order result V0. Similarly, as was shown in theleft panel of Fig. 2.6, the renormalized approximation√2/√V0 for A(0) compares rather wellwith full numerics when  is small.692.5.1 Improved Approximation of the Fold Point for the Supercritical RegimeFor the supercritical regime γ > 3α/2, (2.81b) implies same maximum threshold χmax =limµ→(3α/2)− χ(µ) as in (2.24) which converges due to the formula (2.23) given in Section2.2. However, the improvement of (2.81b) over (2.26) allows us to determine a more accurateminimum value of D for which a steady-state hotspot solution on |x| ≤ ` exists, which wedenote as Dcrit,.We simply set µ = 3α/2 in (2.81) and solve the resulting system for D = Dcrit, and V0.In particular, for a single hotspot solution on |x| ≤ ` we haveDcrit, ≡ 2(`− x)2χ2max, (2.82a)where χmax ≡ χ (3α/2), and χ(µ) is defined in (2.23). In addition, the minimum value of D fora pattern of K interior hotspots on the domain |x| ≤ 1, is obtained by setting ` = 1/K into(2.82a). This yields the critical thresholdsDcrit,,K =2(K−1 − x)2χ2max. (2.82b)In Fig. 2.9 of §2.4 we showed that the improved approximation Dcrit,,K for the minimumvalue of D for a steady-state pattern of K-interior hotspots on the domain |x| ≤ 1 com-pares rather well with the full numerical results as  is decreased. In comparison with the-independent results of (2.31) of §2.2 for the minimum value of D as obtained from the leading-order theory, our improved theory on |x| ≤ 1 when γ = 2, α = 1, and  = 0.01, yieldsDcrit,,1 ≈ 1.5985 , Dcrit,,2 ≈ 0.3646 , Dcrit,,4 ≈ 0.0771 . (2.83)As seen from Fig. 2.9 and (2.49), these improved approximations for the minimum value ofD compare rather favourably with full numerical results. Moreover, in comparing (2.83) withthe results (2.31) from the leading-order theory of §2.2, it is evident that the effect of the-dependent correction terms is rather significant even at  = 0.01.70For a one-hotspot solution, in §2.6 below we construct a new boundary layer near x = `when D ≈ Dcrit,. For this analysis, we need to determine the local behaviour near x = ` of thesolution to (2.79) when D = Dcrit,, corresponding to when a(`) = a0c ≡ 3α/2. Near x = l, weput a = a0c + a¯(x), where a¯ 1 and a¯(`) = 0. Upon substituting into (2.79), we obtain nearx = ` that [f ′(a0c)a¯a¯x]x ∼ (a0c − γ)/Dcrit,, where f ′(a0c) = −2/a20c. Therefore, near x = `, wehave(a¯a¯x)x ∼a20c(γ − a0c)2Dcrit,.Upon integrating this equation and imposing a¯(`) = 0, we get for x near ` that(12 a¯2)x∼ βa20c(x− `) , β ≡(γ − a0c)2Dcrit,.Integrating once more, and imposing a¯(`) = 0, we obtain that a¯ ∼ √βa0c(x− `) as x→ `−. Weconclude that as x→ `−, the local behaviour of the solution to (2.79) when D = Dcrit, isa(x) ∼ a0c + β1/2a0c(x− `) , as x→ `− , where a0c ≡ 3α2 , β ≡(γ − a0c)2Dcrit,.(2.84)Since, when a(`) = 3α/2, the solution a(x) no longer satisfies the no-flux condition a,x(`) = 0,we need to construct a new boundary layer near x = `. This is done in the next section.2.6 A Normal Form for Hotspot InsertionThe full numerical computations in §2.4.1 for the supercritical regime, and the local analysisin (2.84) of §2.5, motivate the need for constructing a new boundary layer solution near theendpoints x = ±` when D is near the critical value Dcrit,. This boundary layer analysis, whichis shown below to generate multiple solutions in the boundary layer region, characterizes theonset of the peak insertion phenomena at the edges of the domain. We show that the overallmechanism for the creation of new hotspots is markedly similar to the analysis of [28] for theonset of self-replication behaviour of mesa patterns. Indeed, we derive a normal form equation,characterizing the local behaviour of the peak insertion process, that has the same structureas that derived in [28]. However, as an extension of the analysis of [28], we derive a formula,71valid near the critical threshold Dcrit,, that shows analytically how the solution multiplicityassociated with the boundary layer solution near x = ` leads to a fold-point behaviour in thebifurcation diagram of A(`) versus D.Throughout this section, we assume γ > 3α/2 so that α < a0(x) < 3α/2 holds for allx < x < ` and the maximum value χmax = χ(3α/2) exists.To analyze the onset of the peak insertion process, we first write the outer problem for thesteady-state problem (2.11) in the form2Axx +A3 [V − g(A)] = 0 , D[A2Vx]x−A+ γ −A3 [V − g(A)] = 0 , (2.85)on x < x < `, where g(A) is defined in (2.17). We let a(x) and Dcrit, denote the solutionto the renormalized outer problem (2.79) at the critical value where a(`) = 3α/2. In terms ofa(x), v(x) is given by v(x) = g(a(x)), where g(A) is defined in (2.17). In the outer region,away from both the hotspot core and a thin boundary layer to be constructed near x = `, weexpand the outer solution to (2.85), together with Λ ≡ 1/D, asA = a + νa,1 + · · · , V = v + νv,1 + · · · , Λ ≡ 1D= Λ + νΛ,1 + · · · , (2.86)where the gauge function ν  1 and the constant Λ,1 are to be determined. By expandingD = Dcrit, + νD,1 + · · · , and then comparing with the expansion of Λ in (2.86), we identifythatDcrit, =1Λ, D,1 = −Λ,1Λ2. (2.87)We substitute (2.86) into (2.85) and collect powers of ν. Assuming that ν  O(2), we obtainthat a satisfies the renormalized problem (2.79) with a(`) = 3α/2, and that a,1 satisfies[f(a)a,1]xx − Λa,1 = Λ,1a , x < x < ` ; a,1(x) = 0 . (2.88)The asymptotic boundary condition for a,1 as x→ `− will be derived below upon matching tothe boundary layer solution to be constructed near x = `.To construct the thin boundary layer near x = `, we begin by introducing the new variables72A1, V1, and z, byA = a0c + δA1(z) + · · · , V = v0c + δ2V1(z) + · · · , z ≡ σ−1[`− x] , (2.89)where a0c ≡ 3α/2 and v0c ≡ g(a0c) = 4/(27α2). Here the gauge functions σ  1 and δ  1are to be determined. The choice of different scales for A and V is motivated by the fact thatg′(a0c) = 0. We substitute (2.89) into (2.85), and after a Taylor expansion of g(A), we obtainthat2δσ2A1zz + δ2a30c[V1 − 12g′′(a0c)A21]+ · · · = 0 , a20cσ2δ2V1zz = Λ (a0c − γ) + · · · , (2.90)where from (2.17) we calculate thatg′′(a0c) = −2a−40c , a0c ≡3α2 . (2.91)To balance the terms in (2.90) we must relate σ and δ to  by δ = σ and δσ2 = 2, whichyields thatδ = 2/3 , σ = 2/3 . (2.92)With this choice, (2.90) reduces to leading order on 0 < z <∞ toA1zz + a30c(V1 + A21a40c)= 0 , V1zz = − 2βa20c, β ≡ ∆2 (γ − a0c) > 0 . (2.93)In order to satisfy the no-flux boundary conditions for A and V on x = ` we must impose thatA1z(0) = V1z(0) = 0.Upon integrating the V1-equation, and imposing V1z(0) = 0, we get that V1 = V10−βz2/a20c.Then, from the A1-equation in (2.93), we obtainA1zz + a30c(− βa20cz2 + V10 + A21a40c)= 0 , 0 < z <∞ , (2.94)where V10 is an arbitrary constant. To obtain our normal form equation we eliminate as many73parameters as possible in (2.94) by rescaling z and A1 byA1 = bU , z = ξy . (2.95a)By choosingξ = β−1/6 , b = a0cβ1/3 , (2.95b)we obtain that (2.94) transforms to the normal form equationUyy + U2 − y2 + κ = 0 , 0 < y <∞ ; Uy(0) = 0 , (2.96a)where the parameter κ is definedκ ≡ a20cβ−2/3V10 . (2.96b)In order to match solutions to (2.96) with those in the outer region, we need that Uy → −1 asy → +∞, which is consistent with the condition that Ax > 0 as x→ `−.Finally, in terms of the original variables, we obtain from (2.89), (2.92), and (2.95), thatthe boundary layer solution near x = ` is characterized byA ∼ a0c + 2/3a0cβ1/3U (y) , V ∼ v0c + 4/3β2/3a20c(κ− y2), y ≡ β1/6−2/3 (`− x) ,(2.97)where a0c ≡ 3α/2, v0c ≡ g(a0c) = 4/(27α2), and β is defined in (2.93).We observe that −U satisfies exactly the equation (2.26) in [28]. The properties of solutionsto (2.96) were established in Theorem 2 of [28], and we simply restate this result here for theconvenience of the reader.Theorem 1 (From [28]). In the limit κ→ −∞, (2.96) admits exactly two solutions U = U±(y)with U ′ < 0 for y > 0, with the following uniform expansions:U+ ∼ −√y2 − κ , U+ (0) ∼ −√−κ , (2.98a)U− ∼ −√y2 − κ(1− 3 sech 2(√−κy√2)), U− (0) ∼ +√−κ . (2.98b)74−4 −2 0 2 4 6 8s−10−8−6−4−202κ−11−10−9−8−7−6−5−4−3U−1.0−0.8−0.6−0.4−0.20.0s = -3.2094−11−10−9−8−7−6−5−4−3−2−1.0−0.8−0.6−0.4−0.20.0Uys = -2.38320 2 4 6 8 10−10−8−6−4−202U−1.4−1.2−1.0−0.8−0.6−0.4− = 0.62320 2 4 6 8 10−15−10−50510−10−8−6−4−20Uys = 6.5731Figure 2.15: Left: Plot of the bifurcation diagram of κ versus s = U(0) for solutions to thenormal form equation (2.96). Right: the solution U(y) (solid curves) and the derivative U ′(y)(dashed curves) versus y at four values of s on the bifurcation diagram.These two solutions are connected. For any such solution, define s by s ≡ U(0) and considerthe solution branch κ = κ (s). Then, κ(s) has a unique (maximum) critical point at s = smaxand κ = κmax. Numerical computations yield that κmax ≈ 1.46638 and smax ≈ 0.61512.A bifurcation diagram of κ versus s ≡ U(0) for solutions to the normal form equation (2.96)is shown in the left panel of Fig. 2.15. In the right panel of Fig. 2.15 we plot the numericallycomputed solution U(y) (solid curve), and the derivative U ′(y) (dashed curve), at a few selectedvalues of s = U(0).We now proceed to analyze how the solution multiplicity in this boundary layer solutionleads to a fold-point behaviour in the bifurcation diagram of A(`) versus D. This analysis isnew and was not done in [28]. To do so, we first need to determine the far-field behaviour asy →∞ for any solution of (2.96). We let U = −√y2 − κ+ U¯ in (2.96), where U¯  1, to obtainthatU¯yy −(2√y2 − κ)U¯ ∼ − κy3, as y →∞ . (2.99)A homogeneous solution U¯h to this equation has decay U¯h = O(e−2√2|y|3/2/3) as y →∞, whereasthe particular solution U¯p satisfies U¯p = O(y−4) as y → ∞. This shows that any solution to75(2.96) with Uy → −1 as y →∞, has asymptoticsU ∼ −√y2 − κ+O(y−4) ∼ −y + κ2y +O(y−2) , as y →∞ . (2.100)We then substitute (2.100) into (2.97) for A to obtain the following far-field behaviour ofthe boundary-layer solution:A ∼ a0c + 2/3a0cβ1/2(−y + κ2y).Upon recalling that y = −2/3β1/6(` − x), the equation above yields the following matchingcondition for the outer solution:A ∼ a0c + a0cβ1/2(x− `) + 4/3(a0cκβ1/62)1`− x , as x→ `− . (2.101)Since the first two terms in (2.101) agree with the local behaviour (2.84) as x → `− of therenormalized outer solution a(x), we obtain upon comparing (2.86) with (2.101) thatν ≡ 4/3 , (2.102)and that a,1 satisfies (2.88) subject to the singular behavioura,1 ∼(a0cκβ1/62)1`− x , as x→ `− . (2.103)To solve (2.88) subject to (2.103) it is convenient to introduce the new variable a˜,1 definedbya˜,1 ≡ f(a)a,1 .Then, we calculate using (2.84), (2.103), and f ′(a0c) = −2/a20c thata˜,1 ∼ f ′(a0c)a0cβ1/2(x− `)(a0cκβ1/62)1`− x = β2/3κ , as x→ `− .76Therefore, from this limiting behaviour and (2.88), we conclude that a˜,1 satisfies(a˜,1)xx −Λf(a)a˜,1 = Λ,1a , x < x < ` , (2.104a)a˜,1(x) = 0 ; a˜,1 → β2/3κ , as x→ `− . (2.104b)We observe that since f(a) = 0 has a simple zero at x = `, then x = ` is a regular singular pointfor the differential operator in (2.104a). In (2.104) the condition at x = ` is over-determined inthe sense that we are specifying that a˜,1 is bounded as x→ `− and that the limiting behaviourof a˜,1 as x → `− is a per-specified constant. This extra implicit condition in the boundarycondition at x = ` is the condition that determines Λ,1.To determine Λ,1 it is convenient to reformulate (2.104) by introducing the new variable Hby a˜,1 ≡ Λ,1H, so that H satisfiesHxx − Λf(a)H = a , x < x < ` ; H(x) = 0 , H bounded as x→ `− . (2.105)In terms of the solution to (2.105) we identify the constant H` from H` ≡ limx→`− H(x). WithH` now known, we obtain upon comparing (2.105) with (2.104) thatΛ,1 =β2/3κH`. (2.106)Our numerical computations below show that H` > 0.Finally, from (2.87) and (2.97), and the expression for β in (2.93), we obtain a local para-metric description of the bifurcation diagram of A(`) and D in the formD ∼ Dcrit, − 4/3D4/3crit,((γ − α)2/3κ22/3H`), A(`) ∼ a0c(1 + 2/3β1/3U(0)), (2.107)where a0c = 3α/2. Since the graph of κ versus U(0) is multivalued from Fig. 2.15, we concludefrom (2.107) that the graph of A(`) versus D has a fold-point behaviour near Dcrit,. WhenH` > 0, we observe from (2.107) and Fig. 2.15 that D attains its minimum value Dmin, whenκ = κmax ≈ 1.466, corresponding to U(0) ≈ 0.615.To numerically compute the constant H`, we use a shooting method after first formulating77an asymptotic boundary condition to hold as x → `−. Near x = `, we calculate from (2.84)thatΛf(a)∼ rη, r ≡ Λa0c2β1/2, η = `− x ,so that near x = `, (2.105) becomesHηη − rηH = a0c − β1/2a0cη + · · · .The local behaviour of the solution is readily calculated asH(η) ∼ H` [1 + rη log η +O(η)] , as η → 0+ , (2.108)which, after eliminating H`, yields the asymptotic boundary conditionHx ∼ −rH log(`− x) , as x→ `− . (2.109)0.2 0.4 0.6 0.8 1.0x− 2.16: Numerically computed solution (solid curve) to (2.105) with asymptotic boundarycondition (2.109) imposed at x = ` − δ, with δ = 0.000001, and with parameter values γ = 2,α = 1, ` = 1, and  = 0.01. We obtain H` = limx→`− H(x) ≈ 0.303. The dashed curve isthe local Frobenius series approximation for H, valid near x = `, with leading terms given in(2.108).To determine the constant H` we use a shooting method on (2.105), which consists ofiterating on the constant H0 in Hx(x) = H0, and then imposing (2.109) at x = `− δ, where δwith 0 < δ  1 is a regularization parameter. The function a(x) in (2.105) is determined bycalculating a,x from a first integral of (2.79). For γ = 2, α = 1, and  = 0.01, our computations780.90 0.92 0.94 0.96 0.98 1.001.401.451.501.551.60A0.90 0.92 0.94 0.96 0.98 1.000.14700.14750.14800.14850.1490Vǫ=0.00273, Dfold=1.7173, Dcrit=1.72630.90 0.92 0.94 0.96 0.98 1.001.401.451.501.551.60A0.90 0.92 0.94 0.96 0.98 1.000.14700.14750.14800.14850.1490Vǫ=0.005, Dfold=1.6644, Dcrit=1.6823Figure 2.17: The asymptotic results (dashed curves) for A (left) and V (right) in the boundarylayer region near x = ` at the fold point value for D are compared with corresponding fullnumerical results (solid curves). The asymptotic fold point value is given by (2.107). The toprow is for  = 0.00273 and the bottom row is for  = 0.005. The other parameter values areγ = 2, α = 1, and ` = 1.yield H` ≈ 0.303 when δ = 0.000001, which yields Dmin, ≈ 1.598 from (2.107). In Fig. 2.16we plot the numerically computed H(x) versus x when γ = 2, α = 1, ` = 1,  = 0.01, andH` = 0.303. The dashed curve in this plot is a local Frobenius series approximation valid nearx = `, with leading terms given in (2.108).For two values of , in Fig. 2.17 we show a favorable comparison between the asymptoticand full numerical results for A and V in the boundary layer region at the fold point value.The full numerical results are computed using AUTO-07p. Finally, in Fig. 2.18 we compare theasymptotic result (2.107) for A(`) versus D, with H` ≈ 0.303, near the fold point with thecorresponding full numerical result computed using AUTO-07p. The plot is a zoom of the regionnear the fold point.791.56 1.58 1.60 1.62 1.64D1.ℓ)Figure 2.18: The asymptotic result (dashed curve) for A(`) versus D in a narrow interval nearthe fold point, as obtained from (2.107) with H` ≈ 0.303, is compared with the correspondingfull numerical result (solid curve) computed using AUTO-07p. The parameter values are γ = 2,α = 1, ` = 1, and  = Slow Dynamics of Crime HotspotsIn this section we derive a DAE system for the slow dynamics of a collection of hotspots. Inour analysis we assume that a quasi-steady pattern of localized hotspots has emerged frominitial data by way of some transient process for (2.10). As such, we assume that we have“prepared” initial data consisting of a quasi steady-state hotspot pattern. The analysis belowcharacterizing the slow evolution of the quasi steady-state hotspot pattern relies heavily on therefined asymptotic theory of §2.5. We first consider the dynamics of a single hotspot centeredat x0 in the domain |x| ≤ `. In §2.7.2 we extend the analysis to study the dynamics of a multi-hotspot pattern. In §2.7.3 we compare results from the asymptotic theory of slow hotspotdynamics with corresponding full numerical results.2.7.1 The Slow Dynamics of One HotspotTo characterize the slow dynamics of a single hotspot we proceed by adapting the analysis of§2.5. A dominant balance argument shows that the speed of the hotspot is O(2). In the innerregion, where y = −1(x− x0(σ)), with σ = 2t, we pose a two-term expansion for A asA ∼ A0/+A1 + . . . , V ∼ 2V .80Upon substituting this expansion into (2.10), and after retaining the dominant correction terms,we obtain that(A0yy −A0 + VA30)+ (A1yy −A1 + 3VA20A1 + α)+ · · · = −x˙0A0y , (2.110a)D[(A20 + 2A0A1)Vy]y− V(A30 + 3A20A1)+ 2 (γ − α) + · · · = −3(A20V)yx˙0 . (2.110b)As in §2.5, we expand V = V0 + V1 + · · · and substitute this expansion into (2.50) andcollect powers of . This yields the leading-order result that V0 is an unknown constant andthat A0 = w/√V0, where w(y) =√2 sech y. At next order, in place of (2.52), we obtain on−∞ < y <∞ thatL0A1 ≡ A1yy −A1 + 3w2A1 = −α− V1V3/20w3 − wy√V 0x˙0 ,[w2V1y]y=√V0Dw3 . (2.111)Since L0 has a one-dimensional nullspace with L0wy = 0, the solvability condition for theA1-equation in (2.111) providesˆ ∞−∞(αwy +V1V3/20w3wy +w2y√V0x˙0)dy = 0 .Since w is odd with w(±∞) = 0, this condition reduces tox˙0ˆ ∞−∞w2y dy = −14V0ˆ ∞−∞V1(w4)ydy .We integrate this expression by parts and use the fact that w = O(e−|y|) as |y| → ∞, togetherwith V1 = O(e2|y|) as |y| → ∞ (see the discussion below equation (2.52)), to eliminate theboundary term and obtain thatx˙0ˆ ∞−∞w2y dy =14V0ˆ ∞−∞(w2V1y)w2 dy .We then integrate this expression once more by parts to getx˙0ˆ ∞−∞w2y dy =14V0[(w2V1y)I(y)∣∣∞−∞ −ˆ ∞−∞(w2V1y)yI(y) dy], (2.112)81where I(y) ≡ ´ y0 w2 ds. Since I(y) is odd, and(w2V1y)y is even in y from (2.111), it followsthat the integral on the right-hand side of (2.112) vanishes identically. In addition, upon usingw =√2 sech y, we can evaluate the integral ratio´∞−∞w2 dy/´∞−∞w2y dy = 3. In this way,(2.112) reduces tox˙0 =38V0[limy→∞(w2V1y)+ limy→−∞(w2V1y)]. (2.113)The last step in the analysis is to evaluate the two limits in (2.113) by using the gradientinformation on V provided by the knee solution. The analysis of the mid-inner, knee, and outersolutions for A and V proceeds analogously as in §2.5, since these solutions are quasi-steadyon the time-scale of the slow dynamics. As such, in our discussion below, we only highlight theresults of the analysis.In place of (2.79), the renormalized outer problem is now formulated asD [f(a)ax]x = a − γ , on x0+ < x < ` , −` < x < x0− , (2.114a)a(x0+) = α , a(x0−) = α , ax(±`) = 0 , (2.114b)where x0± > 0 is defined by in terms of the hotspot location x0 byx0± ≡ x0 ± x x ≡ (− log ) + (1 + log(2√2α)− log√V0). (2.114c)In terms of a(x), the renormalized outer solution v(x) isv(x) = g (a(x)) , (2.115)where g(a) is defined in (2.17).By matching the renormalized outer solution across the knee and mid-inner solutions onecan obtain, as in §2.5, the gradient information for V1 as y → ±∞. More specifically, in placeof (2.76), we obtain thatV1 ∼ b±e2±y , as y → ±∞ , b± ≡ ±(α22c2)vx(x0±) , c ≡ 2√2√V0. (2.116)82By using this limiting behaviour, together with the asymptotics w ∼ 2√2e±y as y → ±∞, wecan evaluate the two limits in (2.113) aslimy→±∞(w2V1y)= 8 limy→±∞(e∓2yV1y)= α2V0vx(x0±) . (2.117)Then, from (2.115) we calculate vx(x0±) = α−3ax(x0±), where we used g′(α) = α−3. In thisway, (2.113) reduces to an ODE determined in terms of the renormalized outer solution a(x),satisfying (2.114), given byx˙0 =38α [ax(x0+) + ax(x0−)] . (2.118)By calculating a first integral of (2.114), as similar to that in (2.21), we readily derive thatax(x0+) =√2Dα√G(µ+)−G(α) , ax(x0−) = −√2Dα√G(µ−)−G(α) , (2.119)so that (2.118) becomesdx0dσ= 38√2D[√G(µ+)−G(α)−√G(µ−)−G(α)], σ ≡ 2t . (2.120a)Here G(u) is defined in (2.20b), and µ± ≡ a(±l) are determined in terms of the hotspot locationx0 and the constant V0 from the implicit relationsχ(µ+) =√2D(`− x0+) , χ(µ−) =√2D(`+ x0−) , (2.120b)where χ(µ) is defined in (2.23). Finally, to derive the renormalized equation for V0 we proceedas in (2.80) of §2.5, to obtainDα2 [vx(x0+)− vx(x0−)] ∼ 1√V0ˆ ∞−∞w3 dy − 2x(γ − α) .By substituting vx(x0±) = α−3ax(x0±), and using (2.119), we solve the resulting expression83for V0 to get the approximate equationV0 =2pi2[√2D(√G(µ+)−G(α)−√G(µ−)−G(α))+ 2x(γ − α)]2 . (2.120c)For a given x0, (2.120b) and (2.120c) are a weakly nonlinear algebraic system for µ± andV0, where the coupling arises through the weak dependence of x on V0. With µ± determinedin this way for a given x0, the speed of the hotspot when at location x0 is given by (2.120a).In this sense, the system (2.120) is a differential-algebraic ODE system for the evolution of asingle hotspot, starting from some initial value x0(0) with |x0(0)| < `.For the solvability of (2.120b) and (2.120c), corresponding to the existence of an outersolution, we require that the domain lengths for the two outer solutions on either side of x0not exceed a threshold. In particular, as similar to that in (2.25) of §2.2, we require that thefollowing constraint, guaranteeing that no new hotspot is nucleated or created at the domainboundaries, is satisfied:max{`+ x0−, `− x0+} ≤ `max , `max ≡√D2 χmax , χmax ≡ χ (3α/2) . (2.121)We now discuss a key qualitative feature of the dynamics (2.120). Our main observation isthat the dynamics of a single hotspot is symmetrizing in the sense thatx˙0 < 0 if x0 > 0x˙0 > 0 if x0 < 0, (2.122)so that the hotspot is repelled from the domain boundaries. To see this, we note that if x0 < 0,then from (2.120b) and the fact that χ(µ) is monotone increasing in µ, it follows that µ− < µ+.Then, since G(u) is monotone increasing in u, we conclude that G(µ−) < G(µ+), so that x˙0 > 0from (2.120a). As a result of this symmetrizing property of the hotspot dynamics, it follows thatif the constraint (2.121) is satisfied for the initial hotspot location x0(0), then this constraintwill still hold for all time under the DAE evolution (2.120). This implies that no new hotspotscan be nucleated at later times near the domain boundaries under the slow dynamics of a single840 2 4 6 8 10σ=ǫ2t0. location x0Slow ODEǫ=0.005ǫ=0.01ǫ=0.028.0 8.5 9.0 9.5 10.0σ=ǫ2t0. upFigure 2.19: For parameter values γ = 2, α = 1, ` = 1, D = 4, and with initial statex0(0) = 0.3, the asymptotic result (2.120) for slow hotspot dynamics is plotted for three valuesof . These results are compared with the corresponding result when the switchback term− log  is neglected, so that x0± = x0 in (2.120b). The plot on the right is a zoom of that onthe left.hotspot. Since x0e = 0 is the only fixed point of the dynamics, we have that x0 → 0 as σ →∞for any x0(0).In Fig. 2.19 we show the quantitative effect on the slow hotspot dynamics of the switchbackterm − log . Recall that in (2.120b), x0± is defined in terms of this switchback term by(2.114c). With the initial value x0(0) = 0.3, the asymptotic result for the slow dynamics of x0versus σ = 2t is plotted for three values of , and is compared with the corresponding resultwhen the switchback term is not included, so that x0± = x0 in (2.120b).We remark that although the DAE dynamics (2.120) for x0 is highly nonlinear when D =O(1), it simplifies considerably in the limit D  1. For D  1, we can approximate the solutionto the renormalized problem (2.114) bya = α+1Da˜ + · · · ,where a˜ satisfiesa˜xx =(α− γ)f(α) = α(α− γ) , x0+ < x < ` , −` < x < x0− ,a˜(x0±) = 0 , a˜x(±`) = 0 .85From the simple solution to this limiting problem, we calculate for D  1 thatax(x0+) ∼ 1Dα(α− γ)(x0+ − `) , ax(x0−) ∼ 1Dα(α− γ)(x0− + `) . (2.123)By substituting (2.123) into (2.118), we obtain the following simple linear ODE dynamics, withflow towards the origin, for a one-hotspot solution when D  1:x˙0 ∼ − 34D (γ − α)x0 , (2.124)with only exponentially decaying solutions:x0 = C exp(− 34D (γ − α) τ)→ 0 as τ →∞.2.7.2 A DAE System for Repulsive Hotspot DynamicsIn this section we generalize the results for the dynamics of a single hotspot to the case wherethere are N ≥ 1 hotspots on the domain |x| ≤ `. We label the centers of the hotspots by xjfor j = 1, . . . , N , and assume the ordering −` < x1 < x2 < · · · < xN < `. In order to simplifyour analysis, we will use the leading-order result that the spatial extent of the hotspot centeredat xj is xj − x < x < xj + , where x = − log  + O(). By neglecting the O() term inx and using x ∼ − log  the slow hotspot dynamics becomes uncoupled from the heights ofthe hotspots. With this simplification, the adjacent outer problems for the hotspots centeredxj and xj+1 must agree at a common vanishing Neumann boundary point at the midpoint(xj + xj+1)/2. With this observation, and by using translation invariance of (2.10), it is clearthat the one-hotspot results derived above can be readily adapted to determine the dynamicsof a collection of hotspots.More precisely, let `j+ and `j− denote the half-lengths of the outer problems on either sideof the hotspot centered at xj for j = 2, . . . , N − 1. In contrast, for the hotspot adjacent tox = −`, we let `1− be the distance from x1 to x = −`, whereas `N+ is the distance from xN to86x = `. In terms of xj , this yields`j+ ≡ (xj+1 − xj)2 , j = 1, . . . , N − 1 ; `j− ≡(xj − xj−1)2 , j = 2, . . . , N ,`N+ = `− xN , `1− = l + x1 .(2.125a)We then define the constants µj+ and µj− for j = 1, . . . , N by the implicit equationsχ(µj+) =√2D[`j+ − (− log )] , χ(µj−) =√2D[`j− − (− log )] . (2.125b)In terms of the µj±, which depend on the instantaneous locations of the hotspots, the centersof the N hotspots satisfies the slow dynamicsx˙j ∼ 38√2D(√G(µj+)−G(α)−√G(µj−)−G(α)), j = 1, . . . , N. (2.125c)The DAE system (2.125) for the slow evolution of a collection of hotspots is valid providedthat the lengths of the outer regions between adjacent hotspots is below a threshold, i. e.provided thatmaxj=1,...,N{`j−, `j+} ≤ `max , `max ≡√D2 χmax , χmax ≡ χ (3α/2) . (2.126)A steady-state configuration for the N−hotspot pattern is the equally-spaced solutionwherebyxj = −`+ (2j − 1)N` , j = 1, . . . , N .2.7.3 Comparison of Asymptotic and Full Numerical Results for Slow HotspotDynamicsFinally, we compare the asymptotic results (2.120) and (2.125) for slow hotspot dynamics withcorresponding full numerical results computed using the software PDEPE in MATLAB R2013b.For the full numerical computations, we take 10/ + 1 evenly-spaced spatial mesh points inorder to adequately resolve the narrow cores of the hotspots. Initial conditions for the quasi870 2 4 6 8 10σ=ǫ2t0. locations x0ǫ=0.0050 2 4 6 8 10σ=ǫ2tǫ=0.010 2 4 6 8 10σ=ǫ2tǫ=0.02Figure 2.20: The slow dynamics of a single hotspot on the slow time-scale σ, as predicted bythe asymptotic theory (2.120) (dashed curves), are compared with corresponding full numericalresults of the PDE system (2.10) (solid curves) computed using PDEPE of MATLAB R2013b. Thedomain is |x| ≤ 1 and the parameter values are γ = 2, α = 1, and D = 4. Left:  = 0.005.Middle:  = 0.01. Right:  = 0.02.steady-state hotspot patterns are generated by evolving the PDE (2.10) from small initial bumpperturbations, such as shown in the left panel of Fig. 2.2. The resulting transient evolutionleads to the formation of a pattern of hotspots that is essentially stationary on O(1) timeintervals. The locations of the maxima of A for this pattern are then identified numerically.We re-initialize the full numerical computations by using this computed hotspot pattern as theinitial condition for (2.10). Then, the subsequent slow evolution of the maxima of A are trackednumerically over very long time intervals, and compared with corresponding results from theasymptotic theory.In Fig. 2.20 we compare the asymptotic results for the slow dynamics of a single hotspot,as predicted by (2.120), with corresponding full numerical results. The comparisons are donefor three values of . The agreement between the asymptotic and numerical results is very closewhen  = 0.005, but is still decent even when  = 0.02.For the case of multiple hotspots, and with  = 0.01, in Fig. 2.21 we show a very favorablecomparison between the slow dynamics predicted from the asymptotic theory (2.125) and thatcomputed numerically from the full PDE system (2.10) for either a two or four hotspot evolution.2.8 Discussion2.8.1 SummaryIn this chapter, we used the method of matched asymptotic expansions, together with thenumerical bifurcation software AUTO-07p to analyze the bifurcation properties of steady-statehotspot solutions of (2.11) in the limit  → 0 for the regime D = O(1) for any γ > α. It880 2 4 6 8 10σ=ǫ2t−1.0− locations x00 2 4 6 8 10σ=ǫ2tFigure 2.21: Comparison of slow dynamics predicted from the asymptotic theory (2.125) (dashedcurves) and from full numerical simulations (solid curves) of the PDE system (2.10). Thedomain is |x| ≤ 1 and the parameter values are γ = 2, α = 1, and  = 0.01. Left: a two-hotspot evolution with D = 2, with initial locations x0 ≈ −0.300, 0.299. Right: a four-hotspotevolution with D = 0.3, with initial locations are x0 ≈ −0.794, −0.346, 0.151, 0.698.was shown, both analytically and numerically, that new hotspots of criminal activity can benucleated at the domain boundary or in the middle of two adjacent hotspots, and that suchevents are characterized by a saddle-node bifurcation point of the corresponding bifurcationdiagram. Such nucleations are also known as “peak insertion” events, and they occur wheneverthe distance between neighbouring hotspots, or between a hotspot and the domain boundary,increases beyond a critical threshold. This “peak insertion” behaviour effectively determines theminimum number of steady-state hotspots that will occur for a given domain length. Further-more, the peak insertion behaviour for (2.11) is very similar to the mechanism characterizing theonset of the rupture, and ultimately self-replication, of mesa patterns in RD systems (cf. [28]),and the breakup of droplets for a diffusive interface surface tension model under compressibleflow (cf. [34]).Our leading-order-in- asymptotic theory in Section 2.2 features a nonlinear but strictlymonotone outer problem which is at the crux of the distinction of the supercritical and subcrit-ical behaviours of crime hotspots as observed in [48]. Moreover, the asymptotic theory predictsa simple one-sixth rule for the prediction of the insertion of hotspots on page 43. The leadingorder theory was also sufficient for the NLEP theory in Section 2.3 to prove that multi-hotspotsteady-state solutions are unconditionally linearly stable on an O(1) time-scale when D = O(1),regardless of the number of hotspots and the value of D, and as long as the pattern exists as asteady state.89The previous study [29] of the stability of hotspot equilibria for the regime D = O(−2)showed that K-hotspot equilibria with K ≥ 2 are linearly stable on an O(1) time-scale onlywhen D < D0K/−2 for some constant D0K . For other RD activator-inhibitor systems, withoutthe chemotactic term, nonlocal eigenvalue problems characterizing the stability of multi-spikepatterns in 1-D on O(1) time-scales have been derived and analyzed in [13], [20], [24], [52], [53],[54] (see also the references therein).However, our leading-order-in- asymptotic theory in Section 2.2 for the construction ofsteady-state hotspot solutions was found to agree well with full numerical results for (2.11)only when  is quite small. A more refined asymptotic theory in Section 2.5, based on adetailed analysis of a triple-deck structure near the core of the hotspot and the retention of acertain switchback term that is logarithmic in , was shown to provide a significantly betterapproximation of hotspot equilibria at moderately small values of . Switchback terms also arisein the singular perturbation analysis of some other problems, including model problems of lowReynolds number flows (cf. [31], [32], [42]), and the analysis in [33] of singular solutions to a PDEmodel for the deflection of a micro-plate capacitor. In order to include the effect of switchbackcorrection terms, we used a novel procedure, somewhat similar to the renormalization approachin [8], whereby the leading-order-in- asymptotic theory can still be used upon re-defining acertain term with an -dependent quantity.The refined asymptotic theory for the construction of hotspot equilibria was shown to becentral for deriving a differential algebraic system (DAE) characterizing the slow dynamics of acollection of quasi-steady hotspots for the time-dependent problem (2.10) in Section 2.7. Fromthis DAE system, it is shown that the dynamic interactions between neighbouring hotspots arerepulsive. Therefore, due to the geometrical constraint of the 1-D domain, peak insertion eventsthat are triggered dynamically as a result of the distance between two neighbouring hotspots ex-ceeding some critical threshold are not typically possible in 1-D. This behaviour is qualitativelydifferent than the merging-emerging dynamics of localized peaks for the chemotaxis-growthmodel of [39], whereby localized peaks experience attractive, rather than repulsive, dynam-ics. Peak insertion events, together with attractive dynamics between neighbouring peaks, wasshown in [39] to lead to spatio-temporal chaotic behaviour of localized peaks for the chemotaxis-growth model.90We emphasize that our DAE system for the evolution of a single hotspot for the urbancrime model (2.10) has a rather different form to that for the dynamics of a single localizedspike for other singularly perturbed RD systems such as the Gierer-Meinhardt, Schnakenberg,and Gray-Scott systems, studied in [9], [14], [15], [19], [43], [50] (see also the references therein).In these previous studies, the outer problem away from a spike is linear and its solution forthe inhibitor variable can be represented in terms of a Green’s function. This leads to a single,explicit, ODE for the evolution of a spike. In contrast, in our analysis of (2.10), the outerproblem is nonlinear and there is an intricate triple-deck inner layer structure near the hotspotcore for the slow V variable that must be resolved. The resolution of this intricate inner layerstructure leads to the generation of switchback terms characterizing the correction terms forthe outer expansion away from the core of the hotspot. Overall, this analysis leads to a DAEsystem rather than a single ODE characterizing the slow motion of a single hotspot.2.8.2 Open ProblemsWe conclude this chapter by briefly discussing a few possible directions for further research.An open problem evident from the numerical studies in Section 2.4.2 is to construct thepattern observed with O(1) amplitude in both A and V . Our preliminary results show that byexpanding A ∼ A0 + . . . , V ∼ V0 + . . . and substituting to (2.12) gives V0 = const while A0(y)solves the problem.A0yy −A0 + V0A30 + α = 0A0(0) = maxA0, A0y(y)→ 0 as y → ±∞(2.127)is a homoclinic for V0 in the range of 0 < V0 < 427 . This will present a potential solvabilitycondition from the matching process. The fact that A0 = O(1) and V0 = O(1) is numericallyverified by continuation of the solution in the limit of → 0+ as shown in Fig. 2.22.An immediately related question is whether these two types of hotspots can be glued togetherto form a new type of asymmetric pattern. We remark that such a pattern with spiky profilesof different asymptotic order is novel to the best of our knowledge. Fig. 2.23 shows that sucha possibility exists. We also remark that this may be related to the interim states contained in910.000 0.002 0.004 0.006 0.008 0.010ε2.302.352.402.452.502.552.602.652.70maxA0.000 0.002 0.004 0.006 0.008 0.010ε0.1100.1120.1140.1160.1180.1200.1220.1240.1260.128minVFigure 2.22: The continuation of the function values at the core of the unstable crime hotspotshown in Fig. 2.12. Other parameter values are D = 1, α = 1, γ = 1.25. These show thatmaxA = A(0) and minV = V (0) ∼ V0 are indeed O(1) as  → 0+. Note that minV < 427 =0.1˙48˙.0.0 0.5 1.0 1.5 2.0 2.5 3.0l1.ε−1)Figure 2.23: Continuation of O(1) amplitude spike in the A component in domain length `naturally connects to the O(−1)-amplitude spike at a fold bifurcation. We plots the rangeof possible A(`) and check possible overlapping in their range. This is to throw light on thepossibility of asymmetric patterns with neighbouringO(1) andO(−1) spikes. Model parametersare  = 0.01, D = 1, γ = 1.25, α = 1.Fig. 2.8 after the solution goes around the insertion fold point but before the hotspot doublingis complete. When a new spike is being born, it seem plausible that a transitional spike solutionof amplitude O(1) in A exists. However, an asymptotic theory for this kind of spike is currentlylacking.Other open directions include, first and foremost, to consider extensions of the basic 1-D model. For instance, one may analyze hotspot slow dynamics for some extensions of thebasic 1-D model (2.10). Such possible extensions of the basic model include, allowing forspatial variability of the rate at which criminals are re-introduced, so that γ −α depends on x,92accounting for the effect of crime deterrence by police presence (cf. [41], [56]), or allowing fora nonlinear diffusivity of the attractiveness field (cf. [18]). These extensions have a potentialto allow for both heterogeneous patterns and more complex spatio-temporal dynamics notobserved in this chapter.Another key open problem is to extend the preliminary analysis in [29] to analyze theexistence, stability, and dynamics of 2-D localized hotspot patterns to (2.1) in the limit → 0with D = O(1) on bounded 2-D domains. For simpler RD systems, without the chemotacticterm in the inhibitor variable as in (2.1), such as the Gierer-Meinhardt, Schnakenberg, andGray-Scott systems, results in 2-D for the slow dynamics of localized solutions are given in [23],[27], and [10] (see also the references therein).In Section 5 of [29], a formal asymptotic analysis was used in the limit  → 0, and withD  1, to construct quasi steady-state patterns of K ≥ 1 well-separated hotspots for (2.1)in arbitrary, but bounded, 2-D domains. Through the derivation and analysis of a nonlocaleigenvalue problem, it was shown in [29] that, on the D  O(1) regime, a one-hotspot patternis linearly stable on an O(1) time-scale. Additionally, it was shown that K−hotspot patternswith K ≥ 2 are linearly stable on an O(1) time-scale when D < −4D0c/K3, where D0cdepends on γ, α and the area of the domain. Therefore, for multi-hotspot patterns on theregime D  O(1), the stability threshold occurs when D = O(−4). No study of the slowdynamics of 2-D hotspot patterns was made in [29].The mathematical challenge with analyzing the D = O(1) regime in 2-D is that, in contrastto the 1-D case studied herein, the outer problem for the attractiveness field will consist ofa nonlinear elliptic PDE that cannot be reduced to a simple quadrature. However, we doexpect that this nonlinear PDE has a saddle-node bifurcation structure and leads to a peakinsertion phenomena, similar to that studied in 1-D. Furthermore, the characterization of theslow dynamics of a collection of hotspots should depend on the spatial gradient of the solutionto the nonlinear outer problem at the hotspot locations. The goal would be to investigatewhether it is possible in 2-D that new hotspots can be nucleated through peak-insertion eventsthat are triggered, intermittently in time, from the overall collective dynamics of interactinghotspots.Such a mechanism, if it exists, would give rise to highly complex spatio-temporal patterns93of dynamically interacting hotspots, similar to that found for the chemotaxis-growth model of[39] in one spatial dimension.94Chapter 3Police Intervention - a SimpleInteraction ModelIn this chapter, we consider the simple interaction case for the police-criminal dynamics (I(ρ, U) =U) in the three-component reaction-diffusion model introduced in (1.20). The following modelis included as a special case of a general form proposed in [45]. In particular, the simpleinteraction term −U in the ρ-equation (criminal density) represents a criminal removal rateproportional to the number of police present at the same spatial location. The nonlinear policemovement term corresponds to a choice of the function v(A) not explicitly studied in [45], givenbyv(A) = qD∇ logA .We leave the predator-prey interaction case (I(ρ, U) = ρU) for future study, except tomention that our preliminary results suggest that there are three nonlocal terms in the corre-sponding NLEP problem compared to two nonlocal terms for the current system, which we willderive in this chapter in Section 3.2.We begin our study of the simple police interaction model on the finite one-dimensional95domain −` < x < `, formulated asAt = 2Axx −A+ ρA3 + α , (3.1a)ρt = D (ρx − 2ρAx/A)x − ρA+ γ − α− U , (3.1b)τuUt = D (Ux − qUAx/A)x , (3.1c)subject to the no-flux boundary conditionsAx = ρx = Ux = 0 at x = ±` .We first observe that by integrating (3.1c) over the domain, we obtain that the total amountof police´ `−` U(x, t)dx is conserved in time. The parameter q > 0 measures the degree of focusin the police patrol random walk, as was discussed in the paragraphs containing the formula(1.6). The choice of boundary conditions indicates we are studying a closed system, and thismakes sense when the main crime and police deployment are localized in a city sufficientlyisolated from its neighbors.There are two important remarks regarding the police diffusion term (Ux − qUAx/A)x.Firstly, we recall that choosing the factor q = 2 in front of UAx/A in the third equation weare modeling a mimicry police deployment strategy whereby the police concentration diffusesin exactly the same way as the criminals. When q is above or below 2, the police force diffusesin a less or more focused manner, respectively, compared to the movement of the criminals.Secondly, the diffusivity of the police can be regarded as D/τu, and so if τu < 1, the policeare more mobile than the criminals. Conversely, τu > 1 can be interpreted as the police beingcomparatively more “sluggish” in their movements.The key qualitative question we ask is the following. What should the optimum degree offocus be in patrolling? The common overarching question in virtually all the efforts in crimemodeling is to seek strategies to reduce crime, by minimizing the number of crime hotspots ina given region. In our context, we will investigate whether there are optimal values of q andτu, so that the least number of hotspots can be stable. From a mathematical perspective, thisis equivalent to optimizing the stability threshold of the diffusivity D by tuning q and τu.963.1 Asymptotic Construction of a Multiple Hotspot Steady-StateWe now construct steady-state solutions for (3.1) with multiple hotspots. First, we observethat the spatial differential operators on ρ and U are similar, and has the simpler flux form byreversing a product rule∂∂x[(∂∂x− qAx/A)(· · · )]= ∂∂x[Aq∂∂x(A−q (· · · ))] .This suggests that we make the change of variables analogous to (2.9)ρ = V A2 , U = uAq ,so that (3.1) transforms toAt = 2Axx −A+ V A3 + α , (3.2a)(V A2)t= D(A2Vx)x− V A+ γ − α− uAq , (3.2b)τu (uAq)t = D (Aqux)x . (3.2c)The key assumption is that the attractiveness field A is highly localized as compared to boththe criminal and police densities, so that   1. Then, anticipating a homoclinic in A withhotspot amplitude of order O(−1), we will consider the large D regime where D = O(−2).Then, by choosing V = O(2), we obtain a distinguished balance. This motivates the rescalingV = 2v, D = D0/2 ,97so that (3.2) transforms toAt = 2Axx −A+ 2vA3 + α , (3.3a)2(A2v)t= D0(A2vx)x− 2vA3 + γ − α− uAq , (3.3b)τu2 (Aqu)t = D0 (Aqux)x , (3.3c)with Ax = ux = vx = 0 at x = ±`.Observe that, as compared to (2.10), our system will be coupled more weakly as some keynonlinear terms have a different order with respect to . This will facilitate our asymptoticanalysis. The analysis of the stronger coupling regime where D = O(1), as considered for thebasic crime model in the previous chapter, would be the natural next step in future work. Inthis chapter we will focus our analysis on (3.3), pertaining to the D = O(−2) regime.3.1.1 A Symmetric Pattern of Hotspots of Equal AmplitudeTo construct a symmetric steady-state with K hotspots to (3.3), a simple way to do so is toconsider a larger domain of length S, where S = (2`)K, and construct a single hotspot on(−`, `). Owing to the translation-invariance of the equations (3.3), the single hotspot steady-state can be placed side-by-side to give a K−hotspot pattern on the large domain of length S,which we conveniently choose to be (0, S). Consequently, the total number of police, given byU0 ≡ˆ S0U(x, t) dx , (3.4)which is constant in time, can now be expressed asU0 = Kˆ `−`U dx , (3.5)if we integrate (3.1c) on (0, S) and use translation-invariance. We now begin the matchedasymptotic expansions procedure to construct a single hotspot solution on (−`, `)First, we substitute U = uAq into (3.5) to readily obtain the steady-state solution to (3.3c)98asu = U0K´ `−`Aq dx, (3.6)provided the steady-state solution A to (3.3a) is known. Therefore, the steady-state problemfor the 3-component system (3.3) is equivalent to the following two-component system with annonlocal integral term:2Axx −A+ 2vA3 + α = 0 , (3.7a)D0(A2vx)x− 2vA3 + γ − α− U0KAq´ `−`Aq dx= 0 . (3.7b)For the first component, we have A ∼ α + O(2) in the outer region, while in the innerregion, we put y = −1x and A ∼ A0/ to obtainA0yy −A0 + vA30 + α = 0 , D0−4(A20vy)y+O(−1) = 0 .Therefore, to leading order it follows that v ∼ v0 is a constant, and thatA0 =w(y)√v0, (3.8)where w(y) =√2 sech y is the homoclinic solution ofw′′ − w + w3 = 0, −∞ < y <∞ ,w(0) > 0, w′(0) = 0, w → 0 as y → ±∞ .Some particular values of the integrals of w(y) will be needed below. They are collected hereas ˆ ∞−∞w(y)dy =ˆ ∞−∞w3(y)dy =√2pi ,ˆ ∞−∞w2(y)dy = 4,ˆ ∞−∞w4(y)dy = 16/3 ,ˆ ∞−∞w5(y)dy/ˆ ∞−∞w3(y)dy = 32 .99In particular, a general formula for Iq ≡´∞−∞wq(y)dy isIq =ˆ ∞−∞wq(y) dy = 23q/2−1 Γ2(q/2)Γ(q) , (3.9)where Γ(z) is the usual Gamma function. This general formula, along with the analogousformula for a general class of homoclinic solutions is computed in the Appendix at (A.11) and(A.13).We return to (3.6), which is valid uniformly on the whole domain, and estimate the keyintegral ˆ `−`Aq dx ∼ 2`α+ 1−qvq/20ˆ ∞−∞wq(y) dy = O(1−q) ,so that u = O(q−1).Remark 3.1. A key assumption we make is that q > 1 so that the integral´ `−`Aqdx, and thusu depend only on the inner region contribution from Aq, and is decoupled from the O(1) term2`α. With this assumption, the leading order solution to u is given byu ∼ q−1u˜e, where u˜e ≡ U0Kvq/20Iq. (3.10)Next, we determine v0 by integrating (3.7b) on (−`, `) and imposing vx(±`) = 0 to find2ˆ `−`vA3dx = 2` (γ − α)− U0/K .Therefore, since A ∼ α = O(1) in the outer region, while A = O(−1) in the inner region, itfollows that the dominant contribution to the integral arises from the inner region. In this way,we estimate1√v0´w3= 2` (γ − α)− U0/K , (3.11)which determines v0 asv0 = 2pi2 (2` (γ − α)− U0/K)−2 . (3.12)We observe that v0, and thus the amplitude of the hotspot, as determined by A0, is independentof the parameter q.100Remark 3.2. In Chapter 2, the value V0 given in (2.35), which has a role analogous to v0 givenabove at 3.12, are both a limit of the outer solution v(x) at hotspot the spot location, and inboth cases 2/√V0 and 1/√v0 give the leading order approximations of the amplitudes of thecorresponding attractiveness hotspots. However, the strengths of interaction between the innerand outer solutions are significantly different for the D = O(−2) regime, versus the D = O(1)regime studied in Chapter 2), with the stronger interaction case D = O(1) being asymptoticallymore intricate to analyze. Therefore, unlike the leading order approximation for A, which weuse A0 for both, we are distinguishing V0 from v0 by the capitalization.Also, we observe from (3.11) that a necessary condition for a hotspot solution with A =O(−1) to exist is thatU0 < U0,max ≡ 2`K (γ − α) , (3.13)so that the total police deployment must be below some threshold. We will assume such acondition for U0 throughout this section.In the outer region, we expand v ∼ ve(x) + . . . and recalling that A ∼ α + O(2) we findthat ve(x) solves the simple ODE problem on 0 < x < ` given byD0vexx = −(γ − α)D0α2≡ −ζ , 0 < x < ` ; ve(0) = v0, vex(`) = 0 , (3.14)which has the solutionve(x) =ζ2[(`− |x|)2 − `2]+ v0, 0 < |x| ≤ ` , (3.15)with v0 as given in (3.12). Notice that this is a uniformly valid leading order solution for v.We summarize the results for the leading order approximation of a steady-state with a singlehotspot on (−`, `) as follows.Theorem 3.3. For the system (3.3), there exists a symmetric steady-state solution on (0, S)with K hotspots. On each sub-domain with length 2` = S/K, and translated to (−`, `) to contain101exactly one hotspot at x = 0, the steady-state solution, to leading order, is given byA ∼ w(x/)√v0, if x = O(), A ∼ α if x = O(1) ,v ∼ ve = ζ2[(`− |x|)2 − `2]+ v0 ,u ∼ q−1u˜e, u˜e ≡ U0Kvq/20Iq,where v0 = 2pi2 (2` (γ − α)− U0/K)−2 and Iq = 23q/2−1 Γ2(q/2)Γ(q) .Remark 3.4. The leading order behavior is now seen to be entailed predominantly by the localbehavior of the O(−1) spike in the A-component. In contrast, the only term that depends onthe assumption that the the hotspot is located at x = 0 in (−`, `) is ve which has a simpleexplicit formula. Therefore, in the construction of an asymmetric pattern, i.e. with hotspotlocated at 0 6= x0 ∈ (−`, `), only the term ve needs to be changed.We now also summarize the above result in terms of the original variables to facilitate theinterpretation of what it means to the original model (3.3).Corollary 3.5. For the system (3.3), the K-hotspot steady-state that corresponds to that statedin (3.3) is given by:A ∼ w(x/)√v0, if x = O(), A ∼ α , if O() |x| < ` ,ρ ∼ w2(x/) , if x = O(), ρ ∼ 2veα2 , if O() |x| < ` ,U ∼ −1U0Kwq/Iq , if x = O(), U ∼ q−1U0Kαqvq/20 /Iq , if O() |x| < ` .3.2 NLEP Stability of Multiple Spike Steady-State for GeneralPower 1 < q <∞To analyze the linear stability of a K-hotspot steady-state solution, we will first derive the non-local eigenvalue problem (NLEP) by using the method of matched asymptotic expansions. Wefirst derive the NLEP for a one-hotspot solution where the boundary conditions are of Floquettype. From this canonical problem we readily extract the corresponding NLEP correspond-ing to a multi-spike pattern with our desired Neumann boundary conditions by following themethodology described in the Appendix, Section A.1 on page 179). This approach to study the102stability of multi-spike steady-states is a relatively new technique first applied to spike-stabilityproblems for a general class of reaction-diffusion systems [40], for the study of the stability ofmesa patterns [36], and for a spike solution to a competition model with cross-diffusion [30].Our approach is related to the recent work by Kolokolnikov et al. in [29] on the basic crimemodel, but here we extend this analysis to incorporate the third component representing thepolice density. The novelty of the analysis is that it leads to an NLEP that now has two distinctnonlocal terms. Nevertheless, we will show that the spectrum of this NLEP can be reduced tothe study of a simple transcendental equation in the eigenvalue parameter for the case whereq = 3. For this value of q, the equation for the discrete eigenvalue of the NLEP will be sufficientlysimple so as to lead to an explicit characterization of an asynchronous, or anti-phase, oscillationin the hotspot amplitudes for a pattern of two hotspots in a certain parameter regime. Thisoscillation is due to a Hopf bifurcation, whose threshold can be determined analytically. Theexistence of robust asynchronous oscillations in the spike amplitudes is a new phenomena, whichdoes not occur in the study of spike stability for other reaction-diffusion systems such as theGierer-Meinhardt and Gray-Scott models, where synchronous oscillations typically occur.We also mention in passing that such problems with two or more nonlocal terms are relativelynovel in the literature for NLEP stability analysis. Active research is being conducted toinvestigate new ways to treat multiple nonlocal terms in an NLEP, and in this chapter wepresent some new results in this direction for the value q = 3 where the analysis is particularlysimple, and for the more challenging case where q 6= Linearization with Floquet B.C.To study the linear stability problem for a K-hotspot pattern we first introduce a linear per-turbation of the formA = Ae + eλt, v = ve + eλtψ, u = ue + eλtqη , (3.16)where (Ae, ve, ue) represents a steady-state with a single hotspot centered at the origin in−` < x < `. The orders of perturbations (O(1), O() and O(q) for the A, v and u componentsrelatively) were chosen such that φ, ψ, and η are all O(1) in the inner region.103Then, we obtain from (3.3) that2φxx − φ+ 32veA2eφ+ 3A3eψ = λφ , (3.17a)D0(2Aevexφ+ A2eψx)x− 32A2eveφ− 3A3eψ−queAq−1e φ− qηAqe = λ2(2Aeveφ+ A2eψ), (3.17b)D0(qAq−1e φuex + qAqeηx)x= 2τuλ(qAq−1e ueφ+ qAqeη). (3.17c)For (3.17b),(3.17c), we impose the following Floquet boundary conditions on x = ±` η(`)ψ(`) = z η(−`)ψ(−`) , η(`)ψ(`) = z ηx(−`)ψx(−`) , (3.18)where z is a complex-valued parameter. In Section A.1 we discuss how to extract the spectrumfor the Neumann problem for a multi-spike pattern from our initial imposition of Floquetboundary conditions.For (3.17a), in the inner region, we use Ae ∼ w/√ and ψ ∼ ψ0 to obtain thatΦ′′ − Φ + 3w2Φ + ψ(0)v3/20w3 = λΦ , (3.19)where Φ(y) = φ(xj + y) is the leading order inner expansion of φ. In contrast, in the outerregion we obtain from (3.17) to leading-order thatφ ∼ 3α3ψ/[λ+ 1− 32α2ve] = O(3), ψxx = 0, ηxx = 0 .The main goal of the calculation below involves determining the values ψ(0) and η(0). Wewill find that ψ(0) depends on η(0).3.2.2 The Jump ConditionsWe first recall that ue = O(q−1), as shown from our steady-state analysis. As such, we defineue = q−1u˜e and integrate (3.17b) over an intermediate domain (−δ, δ) with 1 δ  . We use104the facts that Ae ∼ w/√v0, φ ∼ Φ(y) and Ae(±δ) ∼ α, and we obtain, upon letting δ/→ +∞,thatD0α2 [ψx]0 + 2D0α [vexφ]0 = 3ˆ ∞−∞w2Φdy + ψ(0)v3/20ˆ ∞−∞w3dy+ qu˜ev(q−1)/20ˆ ∞−∞wq−1Φdy + η(0)vq/20ˆ ∞−∞wqdy +O(2λ) ,where we used the notation [a]0 ≡ a(0+)− a(0−).Since φ = O(3) in the outer region, we can neglect the second term on the left-hand sideof this expression. For eigenvalues for which λ O(−1), we obtain thatD0α2[ψx]0 = 3ˆw2Φ + ψ(0)v3/20ˆw3 + qu˜ev(q−1)/20ˆwq−1Φ + η(0)vq/20ˆwq , (3.20)where we used the convenient shorthand notation that´(. . . ) ≡ ´∞−∞ (. . . ) dy.Now from (3.17b), we use φ = O(3) in the outer region, together the fact with q > 1, sothat the term qηAqe is of lower order. In this way, we obtain the following BVP problem for ψwith jump condition across x = 0:ψxx = 0 , |x| ≤ ` ,e0 [ψx]0 = e1ψ(0) + e2η(0) + e3 , (3.21)ψ(`) = zψ(−`), ψx(`) = zψx(−`) ,where we have defined ej for j = 0, . . . , 3 bye0 = D0α2, e1 =1v3/20ˆw3, e2 =1vq/20ˆwq , e3 = 3ˆw2Φ+ qu˜ev(q−1)/20ˆwq−1Φ . (3.22)This BVP problem for ψ can be solved using the generic procedure outlined in Lemma A.1.This leads to the following expression for the central value of ψ:ψ(0) = − e2η(0) + e3e0 (1− cos(pij/K)) /`+ e1 . (3.23)This expression shows that we need to calculate η(0) as well. To determine this value we first105integrate (3.17c) over the region |x| = O(δ) to obtainD0qαq [ηx]0 +D0qαq−1O(q+2) = 2τuλ[qq−1u˜eq−1v(q−1)/20ˆwq−1Φ + vq/20η(0)ˆwq].If we define τˆu byτˆu ≡ q−3τu , (3.24)we can write the expression above asD0αq [ηx]0 = τˆuλ[qu˜ev(q−1)/20ˆwq−1Φ + 1vq/20η(0)ˆwq]. (3.25)We observe that τˆu = O(1) when q = 3, and we will use this parameter below as a bifurcationparameter.Now in the outer region we obtain from (3.17c) thatD0qαqηxx +O(q+2) = 2τuλ[O(q+2) +O(q)].Therefore, if we assumeτu  O(−2) , (3.26)so that τˆu  O(q−5), then we obtain that ηxx = 0 to a first approximation. In this way, weobtain that the BVP problem for η, with jump condition across x = 0, isηxx = 0 , |x| ≤ ` ,d0 [ηx]0 = d1η(0) + d2 , (3.27)η(`) = zη(−`), ηx(`) = zηx(−`) ,where the constants dj for j = 0, . . . , 2 are defined byd0 = D0αq, d1 =τˆuλvq/20ˆwq, d2 =τˆuλqu˜ev(q−1)/20ˆwq−1Φ . (3.28)106Then, by applying Lemma A.1 again, we determine η(0) asη(0) = − d2d0 (1− cos(pij/K)) /`+ d1 . (3.29)The final step in the analysis is to simplify ψ(0) and express it explicitly in terms of theoriginal parameters, so as to identify the key coefficient ψ(0)/v3/20 in the NLEP (3.19).First, since d0 = D0αq and e0 = D0α2 have similar forms, this suggests that we define akey factor in the denominators of (3.23) and (3.29) asDj,q ≡ D0αq`(1− cos(pij/K)) ≡ Djαq , (3.30)whereDj ≡ D0`(1− cos(pij/K)) . (3.31)We observe that Dj,q < Dj+1,q for any j = 1, 2, . . . ,K − 2 and any q.Then, we substitute (3.30) into both the expressions for ψ(0) and η(0), as given in (3.23)and (3.29), and in this way obtain thatψ(0) = − 1Dj,2 + e1[e3 − e2d2Dj,q + d1]. (3.32)Before continuing, we use the formula for u˜e given in (3.10) to rewrite the expressions fore1, e2, e3, d1, and d2 ase1 =´w3v3/20, e2 =´wqvq/20, e3 = 3ˆw2Φ + U0√v0Kq´wq−1Φ´wq,d1 = τˆuλ´wqvq/20, d2 = τˆuλ(U0√v0Kq´wq−1Φ´wq).With these expressions we can write ψ(0) asψ(0) = − 1e111 +Dj,2/e1[e3 − (e2/d1) d21 +Dj,q/d1]= − v3/20´w311 + v3/20 Dj,2/´w3[e3 − d2/ (τˆuλ)1 + vq/20 Dj,q/´wq/ (τˆuλ)].107We simplify the expression in the bracket by noting that e3 = 3´w2Φ +d2/ (τˆuλ), which yieldse3 − d2/ (τˆuλ)1 + vq/20 Dj,q/´wq/ (τˆuλ).= 3ˆw2Φ +(1− τˆuλτˆuλ+ vq/20 Dj,q/´wq)(U0√v0Kq´wq−1Φ´wq).We then group terms together to define a key quantity χ asχ ≡ −ψ(0)v3/20= 11 + v3/20 Dj,2/´w3[3´w2Φ´w3+vq/20 Dj,q/´wqτˆuλ+ vq/20 Dj,q/´wq(U0√v0K´w3· q´wq−1Φ´wq)]. (3.33)This expression motivates the introduction of a few new quantitiesχ0,j ≡ 11 + v3/20 Dj,2/´w3, χ1,j ≡ U0√v0K´w3χ0,jCq(λ) , Cq(λ) ≡ 1 +τˆuλvq/20 Dj,q/´wq, (3.34)so that χ can be written compactly asχ(λ) ≡ χ0,j 3´w2Φ´w3+ χ1,j(λ)q´wq−1Φ´wq. (3.35)We observe that Cq(0) = 1 and χ1,j(0) = U0√v0K´w3χ0,j .Finally, we substitute these expression back into our NLEP (3.19), and in this way derivethe main result of this section, which we summarize as follows:Proposition 3.6. (Principal Result) The linear stability of a symmetric K−hotspot state givenby Theorem 3.3 is governed by the spectrum of the following NLEP:L0Φ− χ(λ)w3 = λΦ , (3.36)where the local operator is given byL0Φ ≡ Φ′′ − Φ + 3w2Φ , (3.37)108and where the multiplier χ(λ) consists of two nonlocal terms of the formχ(λ) = χ0,j3´w2Φ´w3+ χ1,j(λ)q´wq−1Φ´wq, (3.38)with coefficients defined in The Competition Instability ThresholdWe now turn to determining the competition instability threshold value of the diffusivity D,which is characterized by the zero eigenvalue crossing of the NLEP. Before proceeding, weremind ourselves of the assumptions made on the model parameters during the construction ofthe steady-state solution and the derivation of the NLEP, which are thatq > 1, U0 < U0,max ≡ S(γ − α), τu  O(−2) .We set λ = 0 in (3.36) together with Φ = w. Then, since L0w = 2w3, we have (2− χ(0))w3 = 0,so that χ(0) = 2. We then calculate,2 = χ(0) = χ0,j3´w2 · w´w3+ χ1,j(0)q´wq−1 · w´wq= 3χ0,j + q(U0√v0K´w3χ0,j),which determines χ0,j asχ0,j =23 + qU0√v0/K´w3. (3.39)We then recall the definition of χ0,j given in (3.34), which yields then algebraic equation forDj,2 given by11 + v3/20 Dj,2/´w3= 23 + qU0√v0/K´w3Upon solving for the critical value of Dj,2, we getD∗j,2 =12´w3v3/20(1 + qU0√v0/Kˆw3). (3.40)109For convenience, we define ω byω ≡ U0,max − U0 = S(γ − α)− U0 , (3.41)and rewrite √v0 given in (3.12) in terms of ω as√v0 =K´w3S(γ − α)− U0 =Kωˆw3 . (3.42)In this way, we can further rewrite (3.40) compactly asD∗j,2 =12(´w3)2 ω3K3(1 + qU0ω)= ω34pi2K3(1 + qU0ω). (3.43)Finally, we recall that the outer domain width of a hotspot is given by ` = S/(2K), and sofrom the definition of Dj,2 given in (3.30), we have that the threshold value for the j-th modeis given byD∗j,2 =D0α2 (2K)S(1− cos(pij/K)) . (3.44)Comparing (3.43) and (3.44) we see that there are K−1 zero eigenvalue crossings for the NLEP(3.36), which occur at the following critical values of D0:D∗0,j ≡S8pi2K4α21[1− cos(pij/K)]ω3(1 + qU0ω), j = 1, . . . ,K − 1 , (3.45)where ω = S(γ − α)− U0 > 0.Remark 3.7.1. Note that D0,j is undefined for the j = 0 synchronous mode. We will have to show sepa-rately below that no competition instability threshold, corresponding to a zero eigenvaluecrossing of the NLEP, exists for a single spike steady-state solution.2. The original model parameter in (3.2) is D, and so the zero eigenvalue crossings in termsof the diiffusivity D of criminals are at D = −2D0,j .We conjecture that when D0 < D∗0 ≡ minj(D0,j) = D0,K−1, we will have linear stability forτˆu small enough. However, when D0 > D∗0 we expect instability. This is studied in some detail110later in the chapter. We summarize our main result for the competition instability thresholdderived above asD∗0 ≡S8pi2K4α21[1 + cos(pi/K)]ω3(1 + qU0ω). (3.46)Next, we will analyze the behavior of the competition instability threshold D∗0 with respect tothe degree of patrol focus q and the total number U0 of police.3.2.4 Interpretation of the ThresholdAccording to the results in Corollary (3.5), we can use (3.42) to obtain to leading order thatAmax ∼√2√v0=√2ωK´w3= ωKpi.Therefore, the amplitude of the hotspot in the attractiveness field is directly proportional toω, but inversely proportional to the number of hotspots K. However, the amplitude of thecriminal density ρ at the hotspot locations is ρmax = w2(0) = 2, which is independent of allmodel parameters. In addition, away from the hotspots in the outer region the criminal densityis O(2). Therefore, it is the number of hotspots that is the most important factor in reducingthe total crime in the domain, and we seek to tune the police related parameters q and U0 sothat the range of diffusivity D0 for which a K-hotspot pattern is linearly stable is smallest.By examining (3.46), we observe that D∗0 increases with q in a linear fashion. This predictsthat if the police become increasingly focused on patrolling the more crime-attractive areas,and under the assumption that the police-criminal interaction is of the simple type −U whichdoes not depend on criminal density, then paradoxically the range of D0 where a K-hotspotsteady-state exists and is stable increases. Therefore, for the goal of reducing crime hotspots,a police deployment with intense focus on crime-attractive areas does not offer an advantageover that of a less focused patrol. We remark that the lower limit of q is 1 for which the abovetheory is valid, while q = 0 and q = 2 corresponds to police exhibiting a pure random walk anda movement mimicking the criminals, respectively.For a fixed q > 1, we next examine how the stability of a K-hotspot steady-state changeswith respect to the total police deployment U0. To this end, we substitute U0 = S(γ − α)− ω111into (3.46) and write D∗0 as a function of ω asD∗0 =S8pi2K4α21[1 + cos(pi/K)]g(ω) , where g(ω) ≡ ω3(1− q) + qS(γ − α)ω2 . (3.47)We now analyze the critical points of g(ω).We first observe that dω/dU0 = −1 and that U0 → U0,max as ω → 0. We then differentiateg with respect to U0 to getdgdU0= −3ω2(1− q)− 2qS(γ − α)ω = −ω (3ω(1− q) + 2qS(γ − α)) ,which shows that dg/dU0 ∼ −2ωqS(γ − α) as ω → 0. We conclude that g must have a criticalpoint in 0 < U0 < U0,max, which must be necessarily be a maximum, if and only ifωc =2qS(γ − α)3(q − 1) < S(γ − α) , which implies2q3(q − 1) < 1 so that q > 3 .On the range where q ≤ 3 we have dg/dU0 < 0 for 0 < U0 < U0,max.Our main conclusion from this simple calculus exercise is that if the degree of focus q satisfiesq ≤ 3, then D∗0 is monotonically decreasing in U0, and thus with increasing police deploymentU0 the parameter region where the crime hotspots are stable decreases. However, if q > 3, theninitially as police deployment increases from zero, the stability of hotspots is paradoxicallystrengthened until the critical valueU0,c ≡ S(γ − α)− ωc = S(γ − α) q − 33(q − 1) ,is reached. For U0 > U0,c, the hotspot pattern becomes less stable when more police deploymentis added. We display these results graphically in Fig. Stability of a Single SpikeThe NLEP (3.36) was derived by imposing Floquet boundary conditions to analyze the stabilityof K−hotspot equilibrium with K > 1. For the case of a single hotspot, we can impose theNeumann boundary conditions directly on x = ±`, as the Floquet analysis is not needed. With1120.0 0.5 1.0 1.5 2.0U00246810g(U0)q=1.5q= 3q= 5Figure 3.1: Competition instability threshold nonlinearity g(U0) against police deployment U0at various focus degrees q. Other model parameters are S = 2, γ = 2, α = 1, so U0,max = 2 asshown in the right-most tick of the figure. The competition instability threshold D∗0 is simplya positive scaling of g(U0) according to (3.47).the same procedure as that leading to (3.21) and (3.27) above, we now obtain thatψxx = 0 , |x| ≤ ` ,e0 [ψx]0 = e1ψ(0) + e2η(0) + e3 , (3.48)ψx(±`) = 0 ,together with the BVP for η(x), given byηxx = 0 , |x| ≤ ` ,d0 [ηx]0 = d1η(0) + d2 , (3.49)ηx(±`) = 0 .Here the coefficients e0, e1, e2, e3 and d0, d1 and d2, are as defined in (3.22) and (3.28),respectively.From these two problems it immediately follows that η(x) = η(0) everywhere and thatη(0) = −d2/d1. In addition, we find that ψ(x) = ψ(0) everywhere, with ψ(0) given byψ(0) = − 1e1(e2η(0) + e3)/e1 = − 1e1(e3 − e2d2d1) .113This is precisely the formula given in (3.32) with Dj,2 and Dj,q set to zero.Therefore, by proceeding in the same way as done in the Floquet analysis performed earlier,we simply set Dj,2 and Dj,q to zero in the expression (3.33), and in this way determine χ asχ ≡ −ψ(0)v3/20= 3´w2Φ´w3. (3.50)This leads to an NLEP for a single hotspot solution of the formL0Φ− 3w3´w2Φ´w3= λΦ ,which is independent of all model parameters. Moreover, it is identical to that given in (2.46)in Chapter 2. We conclude from Lemma 3.2 of [29] that any nonzero eigenvalue of (2.46)must satisfy Re(λ) < 0. Therefore, we conclude that a single hotspot steady-state solution isunconditionally stable for any D0 when τu  O(−2).3.3 Analysis of the NLEP - Competition Instability and HopfBifurcation3.3.1 Combining the Nonlocal TermsWe now proceed to analyze the NLEP (3.36), which we first write in the following more explicitform by invoking (3.35):L0Φ− χ0,jw3 3´w2Φ´w3− χ1,j(λ)w3 q´wq−1Φ´wq= λΦ ,χ0,j ≡ 11 + v3/20 Dj,2/´w3, χ1,j ≡ χ0,jCq(λ)U0√v0K´w3,Cq(λ) ≡ 1 + τˆuλvq/20 Dj,q/´wq.(3.51)We now use a special property of the local operator L0, which has the eigenpair L0w2 = 3w2(see Appendix, Section A.2 for a discussion of how we can exploit properties of L0 to help solvethe NLEP). By applying Green’s identity to w2 and Φ, we get´ (w2L0Φ− ΦL0w2)= 0, which114yields ˆw5(χ0,j3´w2Φ´w3+ χ1,jq´wq−1Φ´wq)+ (λ− 3)ˆw2Φ = 0 .We combine terms with´w2Φ to find(3´w5´w3χ0,j + λ− 3) ˆw2Φ = −ˆw5χ1,jq´wq−1Φ´wq.Since the integral ratio´w5/´w3 is 3/2, as seen by using formulas given in (A.12), we canwrite one nonlocal term in terms of the other as3´w2Φ´w3= −99χ0,j + 2(λ− 3)(χ1,jq´wq−1Φ´wq).Therefore, we may combine the nonlocal terms in the multiplier as follows:χ = χ0,j3´w2Φ´w3+ χ1,j(λ)q´wq−1Φ´wq,=(1− 9χ0,j9χ0,j + 2(λ− 3))χ1,jq´wq−1Φ´wq,=[2(λ− 3)9χ0,j + 2(λ− 3)χ1,j]q´wq−1Φ´wq.If we defineC(λ) ≡ 1qχ1,j(1 + 32 ·3χ0,jλ− 3), (3.52)we conclude that 3.51 takes the formL0Φ− χ(λ)w3´wq−1Φ´wq= λΦ, where χ = 1C(λ) . (3.53)Finally, we write C(λ) in terms of the original model parameters and simplify the resultingexpression. First, we substitute √v0 = K´w3/ω into the formula for χ1,j to getχ1,j =χ0,jCq(λ)U0√v0K´w3= χ0,jCq(λ)U0ω.115In this way, we obtain thatC(λ) = ωU0Cq(λ)qχ0,j(1 +(32) 3χ0,jλ− 3). (3.54)We eventually would like to express C(λ) in terms of the parameters q and ω. To this end,we first rewrite Cq(λ) and χ0,j , by using √v0 = K´w3/ω, asCq(λ) ≡ 1 + τ˜jλ , (3.55)whereτ˜j = τˆu/[(√2Kpi/ω)qDj,q/ˆwq], (3.56)andχ0,j =(1 + 2pi2(K/ω)3Dj,2)−1. (3.57)Here we have defined new intermediate parameter τ˜j for subsequent notational convenience.We then defineκq ≡(√2piαKω)q/ˆwq . (3.58)Upon recalling that Dj,q = αqDj , we arrive at simpler expressions for τ˜j and χ0,j given byτ˜j =τˆuκqDj, (3.59)χ−10,j = 1 + ακ3Dj . (3.60)In particular, for the special case where q = 3 we haveκ3 =2√2pi3α3K3ω3(√2pi) = 2pi2α3K3ω3, (3.61)and so for this value of q we have that χ−10,j is related to τ˜j rather explicitly asχ−10,j = 1 + ατˆu/τ˜j . (3.62)116Remark 3.8. In the study of spikes for similar reaction diffusion systems, τ˜j usually is theresult of an “effective” time relaxation constant. Also note that Dj = D0 (1− cos(pij/K)) /`appearing in the formulas above is an important factor resulting from the Floquet boundarycondition. It was found to be crucially related to the competition instability, corresponding tothe zero eigenvalue crossing.We keep here for a quick reference the form of C(λ) which will be central in a subsequentexplicit calculation of eigenvalues:C(λ) = ωqU0(1 + τ˜jλ)(1χ0,j+ 92(λ− 3)). (3.63)Remark 3.9.1. We observe that our current NLEP is of the formL0Φ−(a0 + a1λb0 + b1λ+ b2λ2)w3´wq−1Φ´wq= λΦ .i.e. the multiplier is a proper rational function of degree 2. There have been no priorstudies to our knowledge of NLEP’s arising from other reaction-diffusion systems withthis type of multiplier.2. The key model parameters we will use to analyze the NLEP will be q and ω. The latterbeing a complementary quantity defined in relation to the maximum police presence sothat hotspots can exist, i.e. ω = U0,max − U0, where U0,max = S(γ − α) is defined by themodel parameters.3.3.2 The Zero Eigenvalue Crossing RevisitedWe would like to recover our result for the competition instability threshold given in (3.46) tofurther ascertain that the new form of NLEP with one single nonlocal term is correct.Upon setting λ = 0 in (3.63) we getC(0) = ωqU0(1χ0,j− 32), (3.64)117where we have used the fact that Cq(0) = 1 as noted before. We further set Φ = w in themultiplier to getχ = 1C(0)´wq−1 · w´wq= 1C(0) ,so that our NLEP at (3.53) becomesL0w = w3/C(0) .Then, since L0w = 2w3 implies C(0) = 1/2, we arrive at the equationωqU0(1χ0,j− 32)= 12 (3.65)Upon solving for χ0,j , we get1χ0,j=(3 + qU0/ω2),which is equivalent to (3.39) since ω = √v0/K´w3 by (3.42). Therefore, the exact samecalculations after (3.39) gives that C(0) = 1/2 implies (3.65), which in turn implies thatDj,2 = D∗j,2 ≡ω34pi2K3(1 + qU0ω). (3.66)Now, note that from the definition of χ0,j at (3.57), we obtain that χ0,j , given byχ−10,j = 1 + 2pi2(K/ω)3Dj,2 ,is strictly increasing with Dj,2. Therefore, when Dj,2 is not exactly equal to the j-th modeinstability threshold D∗j,2, we have the following equivalence of inequalities:Dj,2 < D∗j,2 iffωqU0(1χ0,j− 32)= C(0) < 12 , (3.67)with a similar statement for Dj,2 > D∗j,2.If we express using Dj = Dj,2/α2 instead, the statement C(0) = 1/2 can be found to be118equivalent to the inequalityDj < D∗j ≡ω34α2pi2K3(1 + qU0ω). (3.68)3.3.3 Solution to NLEP as Zeros of a Meromorphic FunctionWe now reformulate (3.53) so that we are seeking the eigenvalues λ as the zeros of somemeromorphic function ζ(λ) in the complex plane. We first write (3.53) as(L0 − λ) Φ =(1C(λ)´wq−1Φ´wq)w3 ,so thatΦ = 1C(λ)´wq−1Φ´wq(L0 − λ)−1w3 .We then multiply both sides of this expression by wq−1 and integrate to get(ˆwq−1Φ)(1− F(λ)C(λ))= 0, F(λ) ≡´wq−1 (L0 − λ)−1w3´wq. (3.69)Provided that the eigenfunction satisfies´wq−1Φ 6= 0, the eigenvalue λ solvesζ(λ) ≡ C(λ)−F(λ) = 0 , (3.70)where C(λ) is a proper rational function of degree 2 defined in (3.54).We will proceed to analyze the zeros of the meromorphic function at ζ(λ) = C(λ)−F(λ) intwo cases: q = 3 and q > 1, with the former being explicitly solvable, and the latter requiringthe Nyquist criterion to count the number of zeros in the right half plane. Moreover, we willalso investigate the possibility of Hopf bifurcation, i.e. seeking solutions of the form λ± = ±iλIto (3.70)with λI > 0.Remark 3.10. If´wq−1Φ = 0, then the NLEP becomes simply the local eigenvalue problem:L0Φ = λΦ with an extra condition´wq−1Φ = 0. From Proposition 5.6 of [13], we know thatthere are exactly two discrete eigenvalues, and the only eigenpair with an odd eigenfunction isλ = 0, Φ = w′. This suggests that there should also be discrete spectra of the full problem119that are near zero as → 0. These are the “small” eigenvalues that are related to translationalinstabilities. A relevant treatment for the two-component system without police in the regime  1, D = O(1) was found in Section 2.7. There we observed a mild instability in theslow dynamics of hotspots with an asymmetrical pattern (different outer region lengths). Ouranalysis of the NLEP characterizes only those eigenvalues that are O(1) as  → 0, which canlead to O(1) time-scale instabilities.3.4 Explicitly Solvable Case q = 3 and Asynchronous Oscilla-tionsWe now study the stability of a K-hotspot solution and also illustrate the possibility of a Hopfbifurcation for the explicitly solvable case q = 3, where F(λ) can be written in a closed form.Rather remarkably, we will be able to reduce the problem of determining unstable spectra of theNLEP to finding pure imaginary solutions to a quadratic equation in the eigenvalue parameter.When q = 3, we can calculate F(λ) explicitly by using the principal eigenpair of L0 givenby L0w2 = 3w2. We consider the integral in the numerator of F(λ), and use integration byparts to getI ≡ˆw2 (L0 − λ)−1w3 =ˆw3 (L0 − λ)−1w2 .Since w2 = 13L0w2 =13[(L0 − λ)w2 + λw2], we find thatI = 13[ˆw5 + λˆw3 (L0 − λ)−1w2]= 13(ˆw5 + λI).Upon solving for I we get I = − ´ w5/(λ− 3), and soF(λ) = I´w3=´w5´w313− λ = −(32) 1λ− 3 . (3.71)Upon substituting (3.63) and (3.71) into ζ(λ) = C(λ)−F(λ) = 0, we getωqU0(1 + τ˜jλ)(1χ0,j+(32) 3λ− 3)= −(32) 1λ− 3 .120The resulting equation for λ is simply a quadratic equation, which we write asc2λ2 + c1λ+ c0 = 0 , (3.72)which has a positive leading coefficient c2 = τ˜jχ−10,j/3 > 0. The other two coefficients in thequadratic arec1 = τ˜j(32 −1χ0,j)+ 13χ0,j, c0 =qU02ω +32 −1χ0,j. (3.73)Remark 3.11. If we are only interested in the explicitly solvable case q = 3 where F(λ) has aclosed-form formula, one could have observed that (3.53) takes the formL0[φ]− χwrˆ ∞−∞w(p+1)/2Φdy = λΦ ,where r = 3 and p = 3 is the degree of the nonlinearity in the homoclinic equation (2.14). Onecan then apply the formula in Proposition (A.3) directly to obtainλ = ν0 − χˆ ∞−∞wp+12 +rdy ,where ν0 = 3 is the principal eigenvalue of the local operator L0. This provides an alternativederivation of (3.72) as well. The above treatment highlights how F(λ) in the general formreduces to a simple function when q = 3.Next, we analyze the complex roots of the quadratic equation (3.72) to determine thelocation of eigenvalues. We have that Re(λ) < 0 iff c1 > 0 and c0 > 0. Firstly, we check thatc0 > 0 holds if and only ifqU02ω >1χ0,j− 32which is exactly the inequality (3.67) that in turn is equivalent to Dj,2 < D∗j,2. This confirmsthat the stability of K-hotspot occurs only if Dj,2 is below the competition instability thresholdD∗j,2. Secondly, we check that the condition c1 > 0 holds if and only if13χ0,j> τ˜j(1χ0,j− 32), (3.74)121which is always true if1χ0,j<32 . (3.75)In other words, recalling (3.57), this is equivalent to 1 + 2pi2(K/ω)3Dj,2 < 3/2, which yieldsthatDj,2 <ω34pi2K3 ≡ D∗j,2,min . (3.76)Thus we have arrived at a threshold (3.76) below which we have stability.Remark 3.12. We also note in passing that the same statement can also be expressed using Djand κq from (3.58) and (3.31), which yieldsDj < D∗j,2,min ≡ω34pi2α2K3 =α2κ3. (3.77)This suggests a possibility that D∗j,2,min may not be a uniform bound for all q 6= 3 because itdepends on κ3, which can be the result from choosing q = 3. In contrast, D∗j,2 was derived in(3.66) in a context when q = 3 was not assumed. This observation is related to an eventualdifficulty we will encounter below for proving stability for the range Dj,2 < D∗j,2,min when q 6= 3regardless of τu. This furnishes a possible route of investigation in the algebraic relationshipsbetween the stability thresholds and the problem parameters.Finally, for the parameter range where D∗j,2,min < Dj,2 < D∗j,2, then we find from (3.74) thatthere is a Hopf bifurcation threshold where purely imaginary complex conjugate eigenvaluesexist, given byτ˜j,Hopf ≡(23) 12− 3χ0,j . (3.78)This threshold determines the sign of the coefficient c1 in the quadratic in the following way:0 < τ˜j < τ˜j,Hopf iff c1 > 0 ; τ˜j > τ˜j,Hopf iff c1 < 0 . (3.79)Therefore, in this interval, instability is governed by an “effective” time relaxation constant τ˜j ,which leads to the existence of a Hopf bifurcation.In addition, if we compare the lower threshold D∗j,2,min with the upper threshold D∗j,2 given122in (3.67), we see thatD∗j,2D∗j,2,min= 1 + qU0ω. (3.80)We observe that the case where Dj,2 lies between D∗j,2 and D∗j,2,min does not exist when U0 = 0,i.e. the removal of the police equation. Conversely, if U0 → U−0,max so that ω = U0,max−U0 → 0+,then both D∗j,2 and D∗j,2,min vanish, but D∗j,2/D∗j,2,min → +∞.We now summarize the above findings. The location of the two zeros λ± of the meromorphicfunction ζ(λ) = C(λ)−F(λ), depend on the values of Dj,2 and τ˜j in the following way:Proposition 3.13. Let λ+, λ−, with Reλ+ ≥ Reλ−, denote the two solutions of the quadraticequation (3.72) in the complex plane. Then, their location in the complex plane depends onDj,2 = D0α2(1− cos(pij/K)/` in the following way:1. If Dj,2 > D∗j,2 = ω34pi2K3(1 + qU0ω), then we have a pair of opposite-signed real eigenvaluesλ+ > 0 > λ−, and the K-hotspot steady-state is unstable.2. If Dj,2 < D∗j,2,min = ω34pi2K3 , then Reλ± < 0, and we have stability.3. If D∗j,2,min < Dj,2 < D∗j,2, then if(a) τ˜j = τ˜j,Hopf ≡ 23 12−3χ0,j , we have Reλ± = 0 and a Hopf bifurcation occurs.(b) τ˜j > τ˜j,Hopf , then Reλ± > 0,(c) 0 < τ˜j < τ˜j,Hopf , then Reλ± < 0.Remark 3.14. The multiple hotspot steady-state exhibits an interesting novel phenomenon inthe context of the study of the stability of spike patterns to reaction-diffusion systems. Firstly,the window of oscillatory instability, i.e. the interval (D∗j,2,min, D∗j,2), where a Hopf bifurcationoccurs can occur is the result of the addition of the police equation with a simple coupling term−U in the ρ-equation (criminal density). In particular, for a two-hotspot equilibrium, then j = 1is the only mode of oscillation, and our theory predicts the possibility of an asynchronous Hopfbifurcation so that the amplitudes of the two crime hotspots begin to exhibit temporal anti-phase oscillations. In terms of the urban crime model, this means that when police patrols witha certain specific diffusivity relative to the criminals (determined by τ˜j,Hopf), one observes an123interesting picture that the police concentration is drifting to and fro from the hotspots withoutannihilating any of them. However, if the police patrol diffusivity exceeds such a threshold, thenone of the hotspot will dissipate due to an oscillatory instability. Such a qualitative behaviorin the possible types of detstabilization of localized spike patterns was not observed in otherwell-studied reaction-diffusion systems exhibiting similar concentration phenomena, such as theGray-Scott, Gierer-Meinhardt and Schnakenburg models.3.4.1 Explicit Determination of Hopf Bifurcation and Stability RegionIn this section, we again consider the case q = 3, but we will determine stability thresholds foran arbitrary number K ≥ 2 of hotspots. For K > 2, the number of modes j is K − 1 > 1.To analyze this more complicated situation, we first express our results above in terms of theoriginal model parameters. We first observe that the expressionDj,2 = D0α2 (1− cos(pij/K)) /` ,implies the following ordering relation:D1,2 < D2,2 < · · · < DK−1,2 .Therefore, for a pattern of K-symmetric hotspot on an interval of length S = 2K` with q = 3,it is easy to define a sufficient condition for instability and stability, corresponding to Case 1and 2 of Proposition 3.13 as follows.We first conclude that we have an unstable real eigenvalue due to the K−1-mode wheneverDK−1,2 > D∗j,2. This motivates introducing D0,upper byD0,upper ≡ D0(D∗j,2/DK−1,2)= Sω38pi2α2K4 (1 + cos(pi/K))(1 + 3U0ω),so that we have instability if D0 > D0,upper.Similarly, for the eigenvalues with respect to all j = 1, 2, . . . ,K − 1 modes to be in the left124half-plane Re(λ) < 0, we need that DK−1,2 < D∗j,2,min. This motivates the introduction of asecond threshold D0,lower byD0,lower ≡ D0(D∗j,2,min/DK−1,2),= Sω38pi2α2K4 (1 + cos(pi/K)) .We conclude that the K-spike pattern is linearly stable if D0 < D0,lower.We observe that the ratio of these two thresholds, given byD0,upperD0,lower=D∗j,2D∗j,2,min= 1 + 3U0ω,holds regardless of the value of τ˜j = τˆu/(κ3Dj). Therefore, this ratio is independent of thevalue of τˆu = τu as well (since q = 3, see (3.24)).However, if D0,lower < D0 < D0,upper, then we certainly have D∗j,2,min < DK−1,2 < D∗j,2. As aresult, there will be a Hopf bifurcation associated with the K−1-mode when τ˜K−1 = τ˜K−1,Hopf ,as defined in Proposition 3.13, i.e. at τu = κ3DK−1τ˜K−1,Hopf . Moreover, it can also happenthat D∗j,2,min < Dj0,2 for some minimal 1 ≤ j0 < K − 1 as well.This indicates that there can be multiple Hopf bifurcation curves of the formτu = Hj(D0) for j = j0, j0 + 1, . . . ,K − 1 ,whose domains may or may not overlap. If they do not overlap, then all the Hopf curves Hjfor j < K − 1 exist in domains to the right of D0,upper, and consequently do not affect stabilitysince D0 > D0,upper is already sufficient to conclude instability due to a real unstable eigenvaluein the K − 1 mode. However, if they do overlap, then one needs to determine if they haveintersection points.To analyze this more complicated case, we need to first discuss, the domain of Hj , i.e. therange of D0 so that Hopf bifurcation is possible for each mode j separately.125First, we define oscillatory instability thresholds D−0,j and D+0,j for the j-th mode as follows:D−0,j ≡ D0(D∗j,2,min/Dj,2)= Sω38pi2α2K4(1− cos jpiK) ,D+0,j ≡ D0(D∗j,2/Dj,2)= D−0,j(1 + 3U0ω)= Sω38pi2α2K4(1− cos jpiK) (1 + 3U0ω).Then, with respect to a specific mode j only, the steady-state is stable if D0 < D−0,j , is unstableif D0 > D+0,j , and a Hopf bifurcation occurs at some τu = τj,Hopf when D0 is on the intervalD−0,j < D0 < D+0,j .Notice that D0,lower = D−0,K−1 and D0,upper = D+0,K−1. Moreover, we have the followingordering for the thresholds for each mode of oscillation:D−0,j < D+0,j for j = 1, 2, . . . ,K − 1 , (3.81)owing to the fact that D+0,j/D−0,j = 1 + 3U0/ω > 1. Moreover, if K ≥ 3, we haveD±0,j+1 < D±0,j for j = 1, 2, . . . ,K − 2 , (3.82)as a result of the fact that Dj+1,2 > Dj,2 for j = 1, 2, . . . ,K − 2.An issue of interest is then to determine whether D+0,j+1 ≤ D−0,j for all j, so that the domainsof the Hopf curves Hj do not overlap and a complete ordering is possible:D−0,K−1 < D+0,K−1 ≤ D−0,K−2 < D+0,K−2 ≤ · · · ≤ D−0,1 < D+0,1 . (3.83)To investigate this issue, we first note that 1 ≥ D+0,j+1/D−0,j =(D−0,j+1/D−0,j)(1 + 3U0/ω)gives the inequality1 + 3U0ω≤ 1− cosj+1K pi1− cos jKpi,which implies thatωU0 ≥3(1− cos jpiK)cos jpiK − cos j+1piK.Next, recall that since ω = U0,max − U0, the expression above implies the following condition126on the ratio U0/U0,max depending on j and K:U0U0,max=(ωU0+ 1)−1≤ cosjpiK − cos j+1piK3− 2 cos jpiK − cos j+1piK≡ rj,K . (3.84)To maintain a weaker ordering compared to (3.83) so that no other j-th thresholds appear inthe interval(D0,lower, D0,upper) =(D−0,K−1, D+0,K−1),that isD−0,K−1 < D+0,K−1 ≤ D−0,K−2 < · · · , (3.85)where terms in the ellipsis is of unknown order (though still subject to the restrictions (3.81)and (3.82)), then, it is sufficient to require that D+0,K−1 ≤ D−0,K−2 holds, and so we applyj = K − 2 to (3.84). We would then conclude that (3.85) holds if and only ifU0 ≤ U0,max rK−2,K , (3.86)where rj,K is defined in (3.84).In such a case, one can conclude that there is only one Hopf curve τu = H(D0) due to theK − 1 mode, with the domainD0,lower = D−0,K−1 < D0 < D+0,K−1 = D0,upper .There would then be stability with respect to the remaining modes j = 1, . . . ,K − 2, becauseD0 < D+0,K−1 ≤ D−0,K−2 < D−0,K−3 < · · · < D−0,1 .Although the condition (3.86) on the total amount of police guarantees this ordering, the actualdetermination of the threshold values D+0,j ’s for j = 1, . . . ,K − 2 would require further work.From the numerically computed values in Table 3.1, one sees that (3.86) is quite restrictiveand actually implies a complete ordering of the thresholds for the given values of K. For127j \ K 3 4 5 61 0.4000 0.4459 0.4660 0.47662 0.1907 0.2297 0.25003 0.1129 0.14294 0.0752Table 3.1: Values of the constant rj,K defined in (3.84).example, stability can occur without a complete ordering ifminjD−0,j = D−0,K−1 .In general, it is possible that there exists a minimal j0 so thatD−0,j0 < D+0,K−1 .A closer examination of this would require a detailed analysis of the rj,K coefficients, which wedo not attempt here.Instead, we only note that 0 < rj,K < 1 for any K ≥ 3 and j = 1, . . .K − 2, and we leaveopen the possibility that D+0,j1 > D−0,j2 , with j1 < j2, may occur in general. In particular, werecognize the possibility that the adjacent Hopf curves (i.e j1 = j2 − 1) may have overlappingregions on their domains. See Fig. 3.2 for an illustration of some possible cases where bothoverlapping and disjoint domains are possible.Bearing in mind that the domains of the Hopf curves may overlap, our key question nowis to determine whether the graphs Hj(D0) could intersect in the τu − D0 plane, leading tothe possibility of different modes oscillations depending on the location of D0 in the interval(D0,lower, D0,upper).To this end, we first express χ0,j in terms of D−0,j asχ−10,j = 1 + 2pi2(K/ω)3Dj,2 = 1 +Dj,22(4pi2K3ω3).128(a)(b)(c)Figure 3.2: Regions of stability (shaded) and Hopf curves as function of D0 for K = 2, 3, 4according to (3.92). Model parameters are S = 4, γ = 2, α = 1, U0 = 1. The vertical dottedlines denote D0,lower and D0,upper respectively.129Then, by using the definition of D∗j,2,min in (3.76), and recalling (3.81), we getχ−10,j = 1 +Dj,22D∗j,2,min= 1 + D02D−0,j. (3.87)Then, we can rewrite τ˜j,Hopf asτ˜j,Hopf =2312− 3χ0,j =23(χ−10,j2χ−10,j − 3)= 23(D0/(2D−0,j) + 1D0/D−0,j − 1)= 13 +(D0D−0,j− 1)−1. (3.88)Remark 3.15. The simplicity of the final expressions in (3.87) and (3.88) are interesting in theirown right, but are not strictly necessary for the following results to hold.Since we have the ordering D−0,j+1 < D−0,j , we conclude that, for any fixed D0, it impliesthat the ordering of the (rescaled) thresholds isτ˜j+1,Hopf < τ˜j,Hopf ,for all j = 1, . . . ,K − 2 when K ≥ 3.To obtain some qualitative properties for τ˜j,Hopf as a function of Dj,2, we first observe thatfor each j = 1, 2, . . .K − 1, we haveD0D−0,j= Dj,2D∗j,2,min→1+ as Dj,2 →(D∗j,2,min)+,1 + 3U0ω as Dj,2 → D∗j,2 .This yields the following limiting behaviors for τ˜j,Hopf as Dj,2 approach the lower and upperthresholds:τ˜j,Hopf →+∞ as Dj,2 →(D∗j,2,min)+,13(1 + ωU0)as Dj,2 → D∗j,2 .(3.89)Finally, we unpack the definition of τ˜j,Hopf to define the Hopf threshold with respect to theoriginal parameter τu. We recall from (3.59) that when q = 3 we have τu = τ˜jDjκ3, and so wedefineτj,Hopf ≡ τ˜j,HopfDjκ3 . (3.90)130Therefore, apart from a positive scaling factor Djκ3, the limiting behavior (3.89) still holdsqualitatively for τj,Hopf .Finally, according to (3.31), we have Dj+1 > Dj as well. Therefore, if D−0,j < D+0,j+1, thenfor any D0 ∈ DomHj+1⋂DomHj = (D−0,j , D+0,j+1), we have following majorizing property forthe Hopf curves:τj+1,Hopf = Hj+1(D0) < Hj(D0) = τj,Hopf , (3.91)with this holding for all j = 1, . . .K − 2 if K ≥ 3 whenever there is an intersection of domains.In addition, from (3.89) and (3.90), we also see that the Hopf curves τj,Hopf = Hj(D0) tend to+∞ as D0 → D−0,j , and to some finite positive limit if D0 → D+0,j . This verifies the shape ofthe numerically computed Hopf bifurcation curves shown in Fig. 3.2.In conclusion, (3.91) gives a negative answer to the possibility of Hopf curve intersection.More precisely, if Hj1 and Hj2 are any two Hopf curves with j1 < j2, then their graphs donot intersect even within the overlap of their domains. Instead, the lower mode curve alwaysmajorizes the higher mode curve wherever they overlap in their domains, according toHj2(D0) < Hj1(D0) for any D0 ∈(D−0,j1 , D+0,j2).In other words, for any D0 ∈ (D0,lower, D0,upper), as τu increases from 0 to infinity, the firstHopf bifurcation that is triggered is always the K − 1-mode, i.e. the mode associated with theamplitudes of neighboring spikes oscillating in opposite directions.Finally, we would like to determine the formula of τj,Hopf more explicitly in terms of D−0,j .To this end, we first note thatD−0,j =S2K ·ω34pi2α2K3(1− cos jpiK ),= α2D0 ·`D0(1− cos jpiK )· ω32pi2α3K3 ,= α2D0 (Djκ3)−1 .By using this expression, we can thus rewrite the scaling factor τj,Hopf/τ˜j,Hopf = Djκ3 (see131(3.90)) asτj,Hopfτ˜j,Hopf= Djκ3 =α2D0D−0,j.With this expression for τ˜j,Hopf in terms of D−0,j in (3.88), we conclude thatτj,Hopf = Hj(D0) ≡ α2D0D−0,j13 +(D0D−0,j− 1)−1 , (3.92)which is in agreement with the asymptotic behavior (3.89) as D0 →(D−0,j)−derived above,without appealing to this formula. Moreover, we haveHj(D+0,j) =α2(13 +U0ω)(1 + ωU0).We summarize all these findings in the following main result.Proposition 3.16. For D0,lower < D0 < D0,upper, a Hopf bifurcation occurs at some uniqueτu = τHopf > 0 defined as a function of D0 byτu = τHopf = HK−1(D0) = α2D0D0,lower13 +(D0D0,lower− 1)−1 , (3.93)where the upper and lower thresholds are defined byD0,lower ≡ D−0,K−1 =Sω38pi2α2K4(1 + cos piK) , (3.94a)D0,upper ≡ D+0,K−1 = D−0,K−1(1 + 3U0ω). (3.94b)The oscillation at τu = τHopf corresponds to the highest mode j = K − 1, i.e. the amplitudes ofneighboring hotspots oscillate in opposite directions (anti-phase oscillatory instability).If τu < τHopf , we have stability, while if τu > τHopf we have instability, due to complexeigenvalues with negative and positive real parts respectively.Moreover, ifτu < τmin = limD0→(D0,upper)−H(D0) = α2(13 +U0ω)(1 + ωU0),132the no Hopf bifurcation occurs and the pattern is stable on this interval of D0.3.5 General case q 6= 3In this section we analyze the NLEP for the general case where q 6= Determining the Number of Unstable Eigenvalues by the ArgumentPrincipleWe will return to the expression given in (3.70) where the discrete eigenvalues of the NLEP arethe complex zeros the function ζ(λ) = C(λ)−F(λ). We recall that C(λ) can be written asC(λ) = a(1 + τ˜jλ)(1− b3− λ), (3.95a)where the coefficients are given bya = ωqU0χ0,j, b = 9χ0,j2 , χ0,j = (1 +2pi2K3ω3)Dj,2 = 1 + ακ3Dj , (3.95b)τ˜j = τˆu/[(√2Kpi/ω)qDj,q/ˆwq]= τˆuκqDj. (3.95c)We observe that C(λ) is a meromorphic function with a simple pole at λ = 3, and with coefficientsdefined by all the model parameters.The latter function F(λ) is also meromorphic with a simple pole at λ = 3, but is insteaddefined by integrals involving the local operator L0 and the ground-state solution w asF(λ) =´wq−1(L0 − λ)−1w3´wq. (3.96)It has a closed form given in (3.71) when q = 3. Since closed forms for F(λ) are not availablefor q 6= 3, in this section we must appeal to the argument principle to count the number ofzeros of the meromorphic function C(λ)−F(λ) in the unstable right half-plane Re(λ) > 0.133By using the identity w3 = 12L0w, as given in Lemma A.2, we calculateF(λ) = 12´wq−1 (L0 − λ)−1 [(L0 − λ) + λ]w´wq,= 12 +λ2´wqˆwq−1(L0 − λ)−1w , (3.97)which will be in a form more amenable for analysis below. Next, we derive a formula, based onthe argument principle, that can be used to determine the number of zeros of ζ(λ) in the righthalf plane. First we observe that the simple poles of C(λ) and F(λ) do not cancel as λ → 3−,since when restricted to the real line we getF(λ)→ +∞ while C(λ)→ −∞ as λ→ 3− .We then introduce the anticlockwise “Nyquist” contour that covers the right half plane asR→∞ consisting of the union of the segmentsΓI+ = segment from iR to 0, ΓI− = segment from 0 to − iR,CR = half circle from − iR to iR ,where R > 0. A plot of this contour is shown in Fig. 3.3.Let N denote the number of roots of ζ(λ) = 0 in Re(λ) > 0, corresponding to the number ofunstable eigenvalues of the NLEP. Then, since the contour encloses the simple pole wheneverR is large enough, we have by the argument principle thatlimR→∞([arg ζ]|CR + [arg ζ]|ΓI+ + [arg ζ]|ΓI−)= 2pi(N − 1) . (3.98)To determine the change in the argument of ζ over the semi-circle, we use the asymptoticsof F and C as |λ| → ∞, given byF(λ) = O(1/|λ|) and C(λ) ∼ aτ˜λ ,134 Re0 5 10 Im-10-8-6-4-20246810A Nyquist contour in the complex planeΓI−ΓI+CRFigure 3.3: Schematic plot of the Nyquist contour used for determining the number of unstableeigenvalues of the NLEP in Re(λ) > 0.Re-1 0 1 2 3 4Im-2-1.5-1-0.500.511.52ζ(ΓI+)ζ(ΓI-)Figure 3.4: An image of the Nyquist contour transformed by ζ. Notice that ζ(CR) shrinks tothe complex infinity ∞ as R → ∞, and both ζ(Γ+) and ζ(Γ−) are asymptotically parallel tothe imaginary axis due to (3.101).to derive thatlimR→∞[arg ζ]|CR = pi, for any τ˜ > 0 .Therefore the problem of determining the number or unstable eigenvalues is is reduced tocomputing the change of argument of ζ = C −F as we go down the imaginary axis. In Fig. 3.4we show a typical picture of how ζ transforms the Nyquist contour. The key question is whetherthe origin (marked by “x” in the figure) is inside or outside of the fish-shaped loop.135We further observe that since ζ(λ) = ζ(λ), it readily follows that [arg ζ]|ΓI+ = [arg ζ]|ΓI− .In this way, we conclude that 2pi(N − 1) = pi + 2 [arg ζ]|ΓI+ , which determines N asN = 32 +1pi[arg ζ]|ΓI+ . (3.99)Due to the symmetry, the visual interpretation of whether the “fish loop swallows the origin”(as shown in Fig. (3.4)) then depends crucially on the distinction of whether ζ(0) = C(0)−F(0)is to the left or right of the origin, and then whether the loop is anticlockwise or clockwise withrespect to the origin. To this end, we will need to examine the properties of C and F on thepositive imaginary axis in detail. More explicitly, our strategy in the following sections is to letλ = iλI and let λI decrease from ∞ to 0 to determine the behavior of the functions C and Fin the complex plane.Before we proceed, we would like to state a typical choice of model parameters, which wewill use repeatedly in the subsequent figures for illustrating various stages of the analysis:α = 1, γ = 2, q = 2, U0 = 1, ` = 1, K = 2, j = 1, S = 2K` = 4 . (3.100)Our value q = 2 corresponds to the “cops on the dots” strategy, whereby the police patrolmimics that of the criminals.3.5.2 The Starting and Ending Point of the Path and the Two Main CasesThe location of the end of the path at λI = 0 is easily determined. We have F(0) = 1/2 from(3.97), and soζ(0) = ζR(0) = C(0)− 12 = a(1−b3)−12 ∈ R .Thus, the equation ζ(0) = 0 corresponds to exactly C(0) = 1/2, which we derived in (3.65) inSection 3.3.2 where we studied the competition instability threshold. Moreover, we concludefrom (3.95) thatζ(0) > 0 when Dj,2 > D∗j,2 ; ζ(0) < 0 when Dj,2 < D∗j,2 .136Next, we consider the limiting behavior as λI → +∞. We begin by decomposing ζ(iλI)into real and imaginary parts as ζ(iλI) ≡ ζR(λI) + iζI(λI). Then, we readily have the limitingbehavior for ζ(iλI) as λI →∞, given byζ(iλI) ∼ a (1 + τ˜jb) + iaτ˜jλI as λI → +∞ ,so thatζR(λI) ∼ a(1 + τ˜jb), ζI(λI) ∼ aτ˜jλI , as λI →∞ . (3.101)This means, with respect to the origin, that the path begins asymptotically close to the positiveinfinity of the imaginary axis.Therefore, this reduces the problem to determining whether the path crosses the imaginaryaxis and real axis on the range 0 < λI <∞.The easiest case to consider is for Dj,2 > D∗j,2 when the end point is on the positive realaxis. Then, in order to prove[arg ζ]|ΓI+ = −pi2 ,it suffices to show that the path never crosses the imaginary axis. When this occurs we obtainthat N = 1, which gives one unique unstable eigenvalue for the NLEP. Fig. 3.5 shows the pathζ(ΓI+) in two distinctive cases.The next natural step would be to consider Dj,2 < D∗j,2, so that the end point is on thenegative real axis. Then, by continuity, the imaginary axis must be crossed. If one can provemonotonicity in the real component it follows that there must be exactly one such crossing,and it would remain to determine whether it occurs on the positive or the negative imaginaryaxis. If it occurs on the positive imaginary axis, then it is clear that[arg ζ]|ΓI+ =pi2 ,so that N = 2, and we have two unstable eigenvalues. Otherwise, if it occurs on the negativeimaginary axis, then[arg ζ]|ΓI+ = −3pi2 .137Re-0.6 -0.4 -0.2 0 0.2 0.4Im-0.5-0.4-0.3-0.2-ζ(0) < 0ζ(0) > 0Figure 3.5: Close-up of two distinctive cases of the Nyquist plots for K = 2. For the modelparameters stated in (3.100), and for τu = 1, we show the Nyquist plot for two values of D0.The upper threshold, corresponding to the competition instability, is given by D0,upper = D+0,1 ≈3.103. The curve with ζ(0) < 0 and ζ(0) > 0 correspond to the choice D0 = 3.0 and D0 = 3.2,respectively. The number of unstable eigenvalues are respectively N = 0 and N = 1.Re-0.6 -0.4 -0.2 0 0.2 0.4Im-0.5-0.4-0.3-0.2- 3.6: Close-up of two distinctive cases of Nyquist plots for K = 2. Parameter valuescommon to both are α = 1, γ = 2, U0 = 1, D0 = 3 and ` = 1 (but S = 2K` = 4). The curvesthat are clockwise and anticlockwise with respect to the origin correspond to the choices τu = 1and τu = 4, respectively. The number of unstable eigenvalues are numerically determined to beN = 0 and N = 2 for these particular cases.In this case we would have N = 0, and no unstable eigenvalues. See Fig. 3.6 for examples ofboth paths.In order to distinguish between these two cases, we must determine detailed properties ofthe component functions C(λ) and F(λ), and especially for the latter, as it has no closed form138representation.3.5.3 Key Global and Asymptotic Properties of C(λ) and F(λ)We will analyze the global and asymptotic properties of C(λ) and F(λ) restricted on the positiveimaginary axis. We first consider F(iλI) = FR(λI) + iFI(λI) whereF(iλI) =´wq−1(L0 − iλI)−1w3´wq.If we rewrite the operator as(L0 − iλI)−1 = (L0 + iλI)[(L0 + iλI)−1 (L0 − iλI)−1],= L0[L20 + λ2I]−1+ iλI[L20 + λ2I]−1,we readily obtain upon separating the real and imaginary parts thatFR(λI) =´wq−1L0[L20 + λ2I]−1w3´wq, FI(λI) = λI´wq−1[L20 + λ2I]−1w3´wq. (3.102)Now, we call some rigorous results for FR(λI) and FI(λI), as proved in [52].Fact 3.17.The real functions FR(λI) and FI(λI) have the following properties:(i) FR(λI) = O(λ−2I ) as λI → +∞, FR(0) = 1/2.(ii) F ′R(λI) < 0 when q = 2 [Proposition 3.1 of [52]].(iii) FI(λI) = O(λ−1I ) as λI → +∞.(iv) FI(λI) ∼ λI4(1− 1q)> 0 as λI → 0+ when q > 1 [Proposition 3.2 of [52]].(v) FI(λI) > 0 for q = 2 and q = 4.For q = 3, we have the explicit formula F(λ) =(32)13−λ from 3.71, and so for this case it iseasy to calculate thatF(iλI) = 32( 13− iλI)= 32(3 + iλI9 + λ2I)= 32(39 + λ2I+ i λI9 + λ2I).139Therefore, for q = 3, we have the explicit formulasFR(λI) = 3239 + λ2I, FI(λI) = 32λI9 + λ2I, (3.103)which show clearly that property (v) is also true for q = 3.Moreover, we can then further computeF ′R(λI) = −9λI(9 + λ2I)2 , F ′I(λI) = 3(3− λ)(3 + λ)2 (9 + λ2I)2 , (3.104)andF ′′R(λI) =27(λ2I − 3)(9 + λ2I)3 . (3.105)Therefore, it follows that (ii) is true for q = 3, and furthermore, we observe that there is aunique maximum of FI occurring at the principal eigenvalue of L0 given byλI = 3 = ν0 ,and that the inflection point of FR occurs atλI =√3 = √ν0 .Apart for the asymptotic behaviors (i) , (iii) and (v) which is known to be true for all q > 1,we now have some sound reasons to further assume for (ii) and (iv) that the following conjectureshould hold:Conjecture 3.18. The properties (ii) and (iv) in Fact 3.17 still hold for all integers q > 1,i.e.F ′R(λI) < 0, FI(λI) > 0 . (3.106)We already have rigorous justifications that F ′R(λI) < 0 when q = 2, 3 and that FI(λI) > 0when q = 2, 3, 4. In Fig. 3.7 we plot the numerically computed functions FR(λI) and FI(λI)for various values of q.Next we consider C(λ) restricted to the positive imaginary axis. We first separate C(iλI)140λI0 2 4 6 8 10FR00.λI0 2 4 6 8 10FI-0.0500. 3.7: Plots of FR and FI for q = 2, 3, 4, 5. Note that FR(0) = 1/2 and FI(0) = 0, andthat the maximum of FI occurs near λI = 3. In fact, the maximum does occur exactly atλI = 3 when q = 3.into its real and imaginary parts asC(iλI) = a (1 + iτ˜jλI)(1− b3− iλI)= a (1 + iτ˜jλI)(1− b(3 + iλI)9 + λ2I)Therefore, if we denote C(iλI) = CR(λI) + iCI(λI), we have thatCR(λI) = a(1 + τ˜jb− 3b9 + λ2I(1 + 3τ˜j)), (3.107a)CI(λI) = aλI9 + λ2I(3τ˜j (3− b)− b+ τ˜jλ2I). (3.107b)We now list several elementary properties of CR(λI) and CI(λI).Properties of CR(λI)(i) CR(0) = a(1 − b/3). Therefore, CR(0) > 0 if and only if b < 3 (also,CR(0) < 0 if and only if b > 3).We also had earlier a similar condition that CR(0) > 1/2 if and only ifb− 3 + 3/(2a) < 0, which occurs if and only if Dj,2 > D∗j,2.(ii) C′R(λI) > 0 for λI > 0, and CR(λI)→ a(1 + τ˜jb) > 0 as λI →∞.(iii) Suppose b > 3, then CR(λI) < 0 on the interval0 < λI <√3(b− 3)1 + τ˜jb.141Properties of CI(λI)(iv) CI(λI) ∼ aλI [3τ˜j(3− b)− b] as λI → 0+, and CI(λI) ∼ aτ˜jλI as λI →+∞.(v) CI(λI) < 0 on 0 < λ2I < 3(b− 3) + b/τ˜j . Otherwise, we have CI(λI) > 0.Therefore, if b > 3/(1 + 13τ˜j ), it follows that there is a range of λI , given by0 < λI <√3(b− 3) + bτ˜j≡ λII ,for which CI(λI) < 0.(vi) Alternatively if b < 3/(1 + 13τ˜j ), we can then show that CI(λI) > 0 forall λI > 0.To establish property (vi) above we begin by calculatingC′I(λI) =a(9 + λ2I)2[τ˜jλ4I + λ2I (18τ˜j + b+ 3bτ˜j) + 9 (τ˜j(9− 3b)− b)]. (3.108)The expression in the brackets of 3.108 is a quadratic in x = λ2I , and upon setting C′I(λI) = 0,we see that the two roots x+ ≥ x− satisfyx+ + x− = − [18τ˜j + b+ 3bτ˜j ] /τ˜j , x+x− = 9 (τ˜j(9− 3b)− b) /τ˜j .Since x+ + x− < 0 always holds, there is a root x+ = λ2I > 0 if and only if x+x− < 0. Itfollows that there exists λ2I > 0 for which C′I(λI) = 0 if and only if τ˜j(9 − 3b) − b < 0, whichimplies that b > 3/(1 + 13τ˜j ) must hold. Therefore, C′I(λI) > 0 for any λI > 0, and with the factthat CI(0) = 0, we conclude CI(λI) > 0 always holds as well. This establishes property (v) forCI(λI).In Fig. 3.8 we show various plots of CR(λI) and CI(λI) for τu = 1 and for the modelparameters given in (3.100). These plots illustrate our established properties for CR(λI) andCI(λI).From property (ii) of CR(λI) we have C′R(λI) > 0, while from property (ii) of FR(λI) we142λI0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2CR-0.500.511.522.533.54D0 = 0.4D0 = 0.6D0 = 1.0λI0 0.5 1 1.5 λII 2CI-0.500.511.522.5D0 = 1.0D0 = 0.5Figure 3.8: Plots of CR and CI for τu = 1 and the model parameters given in (3.100). The valuefor D0 was chosen as follows for each curve: Left figure: three cases of CR(λI) are plotted: (i)b > 3 (D0 = 0.4), (ii) 3 − 3/(2a) < b < 3 (D0 = 0.6), (iii) b < 3 − 3/(2a) (D0 = 1.0). Rightfigure: two cases of CI(λI) are plotted: (i) b < 3/ (1 + 1/(3τ˜j)) (D0 = 1.0), b > 3/ (1 + 1/(3τ˜j))(D0 = 0.5), and the right end point of the interval of negative values for CI is at λII ≈ 1.6204,indicated by a heavy dot.conclude that F ′R(λI) < 0. Therefore, we have that ζR(λI) = Re [ζ(iλI)] satisfiesζ ′R(λI) = C′R(λI)−F ′R(λI) > 0 for all λI > 0 . (3.109)This leads to a key result that the path ζ(ΓI+) can only intersect the imaginary axis exactlyone or zero times. The only remaining issue will be to determine whether any such intersectionpoint occurs on the positive or negative imaginary axis.3.5.4 Above Competition Instability Threshold Dj,2 > D∗j,2; A Unique Unsta-ble Real eigenvalueSuppose that b < 3 − 3/(2a) so that CR(0) = C(0) > 1/2. Recall from (3.67) that this impliesthat Dj,2 > D∗j,2, and so we are above the competition instability threshold.Since FR(0) = 1/2, it follows that the endpoint of the path is at ζ(0) = CR(0)−FR(0) > 0.Then, since the path ζ(ΓI+) begins “at positive infinity” in the imaginary axis, stays withinthe right half-plane and ends at the positive real axis, owing to the key monotonicity property(3.109), it follows that we must have [arg ζ] |ΓI+ = −pi2 . Therefore, we conclude from (3.99) thatN = 3/2 + 1pi(−pi2 ) = 1 for all τ˜j > 0 .143Our main conclusion regarding stability for the regime Dj,2 > D∗j,2 is that for any j =1, . . . ,K − 1, a K-hotspot steady-state solution is unstable due to a unique positive real eigen-value that occurs for any τ˜j > Below Competition Instability Threshold Dj,2 < D∗j,2Now suppose that b > 3 − 3/(2a), so that CR(0) = C(0) < 1/2. From (3.67) this means thatDj,2 < D∗j,2 and so we are below the competition instability threshold.For this case, since ζ(0) = CR(0) − FR(0) < 0, while C′R(λI) > 0 for any λI > 0 andFR(λI) = 1/2 at λI = 0 but decreases monotonically to 0 as λI → ∞ (by Conjecture 3.18),there must be a unique root λ∗I to ζR(λ∗I) = 0. Since ζI(0) = 0, the endpoint is on the negativereal axis, with exactly one crossing at λI = λ∗I when traveling from λI = +∞. It is then clearthat there are two distinctly different cases:(I) If the crossing occurs on the positive imaginary axis, i.e. ζI(λ∗I) > 0, then [arg ζ] |ΓI+ =pi/2, and so N = 2.(II) If the crossing occurs on the negative imaginary axis, i.e. ζI(λ∗I) < 0, then [arg ζ] |ΓI+ =−3pi/2, and so N = 0.These two distinguishing cases were presented in Fig. 3.6, where we also noticed that case(I) and (II) implies that the curve approaches ζ(0) < 0 as λI → 0+ in an anticlockwise andclockwise direction, respectively, with respect to the origin. Therefore, the final key step in theanalysis is to distinguish path (I) from path (II).We first establish that when b > 3 we obtain path (II) if either τ˜j  1 or τ˜j  1. Recallthat b > 3 means 9χ0,j/2 > 3, or equivalently1χ0,j<32 .This is precisely the condition (3.75) which was applied to (3.74) to show that the eigenvaluesare stable regardless of τ˜j in the q = 3 explicitly solvable case. This condition (3.75) implies alower thresholdDj,2 < D∗j,2,min < D∗j,2, Dj,2,min =ω34pi2K3 ,144which for q = 3 case guarantees stability regardless of τ˜j . However, in the general case q 6= 3,the lack of an explicit form for F(λ) prevents us from obtaining the same conclusion as easily.We must then appeal to the local and far-field asymptotic properties of C and F .Firstly, if τ˜j  1, we obtain from (3.107a) thatCR(λI) ∼ a[τ˜jb− (3b)(3τ˜j)9 + λ2I]= aτ˜jbλ2I9 + λ2I.Then, upon setting CR(λI) = FR(λI) and using FR(0) = 1/2, we obtain that ζR(λI) = 0 hasa root with λI = O(τ˜−1/2) when τ˜  1. In particular, if we write λ∗I = λ∗I,0τ˜j−1/2 so thatλ∗I,0 = O(1), then we see that ab(λ∗I,0)2/9 ∼ 1/2, which yieldsλ∗I,0 ∼1√ab3√2.Therefore, ζR(λ∗I) = 0 has a unique root when τ˜j  1, and that this root is located near thethe origin with asymptoticsλ∗I ∼3√2abτ˜j−1/2 . (3.110)Now since CI(λI) < 0 on the range 0 < λI <√3(b− 3) + b/τ˜j ∼√3(b− 3) for τ˜j  1, wehave CI(λ∗I) < 0 and so ζI(λ∗I) < 0. We then conclude that the crossing occurs on the negativeimaginary axis as stipulated, which yields path (II).Next, if we instead consider τ˜j  1, then we have from (3.107a) thatCR(λI) ∼ a[1− 3b9 + λ2I]= a[3(3− b) + λ2I9 + λ2I], (3.111)which is independent of τ˜j . Thus, the intersection point where CR = FR must occur at some λ∗I =O(1). However, since CI(λI) < 0 on 0 < λI <√3(b− 3) + b/τ˜j = O(τ˜j−1/2) as τ˜j → 0, whichis now asymptotically large as τ˜j → 0, we conclude that 0 < λ∗I = O(1)√3(b− 3) + b/τ˜j . Itfollows that ζI(λ∗I) < 0, and so the crossing occurs on the negative imaginary axis as well. Thisagain yields path (II).In both cases, since the point ζI(λ∗I) < 0 is the only place the curve ζ(ΓI+) crosses theimaginary axis, we have path (II) so that N = 0 by the formula (3.99). As a consequence, there145are no unstable eigenvalues of the NLEP.The remaining issue is to consider whether the statement above regarding N is still true forany τ˜j > 0 for the general case q > 1. We now consider three possible strategies for exploringthis question.One way to begin an analysis is to observe that FR is independent of τ˜j and thatCR(λ∗I)−FR(λ∗I) = 0 , (3.112)gives us an implicit relation for λ∗I as a function of τ˜j . If we differentiate (3.112) with respectto τ˜j we get∂∂λ∗ICR(λ∗I)dλ∗Idτ˜j+ ∂∂τ˜jCR(λ∗I)−∂∂λ∗IFR(λ∗I)dλ∗Idτ˜j= 0 .By letting (. . . )′ denote derivatives with respect to λ∗I , and by using (3.107a), we obtain that[C′R(λ∗I)−F ′R(λ∗I)] dλ∗Idτ˜j = − ∂∂τ˜j CR(λ∗I) = −ab(1− 99 + λ2I). (3.113)Since C′R(λ∗I) > 0 and F ′R(λ∗I) < 0 by (3.107a) and Conjecture (3.18), the expression aboveyields thatdλ∗Idτ˜j< 0 , for all τ˜j .Next, we observe that the upper end point of the interval of negativity of CI(λI) is a functionof τ˜j as well. We will denote this endpoint as h(τ˜j). It satisfiesdhdτ˜j≡ ddτ˜j√3(b− 3) + b/τ˜j = − b2τ˜2j√3(b− 3) + b/τ˜j< 0 .In this way, we would conclude that the root λ∗I stays within the interval (0, h(τ˜j)) if we canprove that the following inequality holds for all τ˜j > 0:dλ∗Idτ˜j<dhdτ˜j. (3.114)Then, since λ∗I(τ˜j) ∈ (0, h(τ˜j)) has been shown above to be true for τ˜j  1, the inequality above146τu0 2 4 6 8 10λ I0.ζR(λI) = 0λI0 0.5 1 1.5ζ I(λI)-0.7-0.6-0.5-0.4-0.3-0.2-0.100.1Figure 3.9: Numerical verification of a Nyquist path that predicts stability. Left: a plot of theunique roots to ζR(λI) = 0 as τu varies. For very large τu, λI can be quite close to zero. Right:a plot of the possible range of values of ζI(λI) when the Nyquist contour hits the imaginaryaxis, where λI was chosen to vary from zero to the maximum value plotted on the left. Themodel parameters chosen were D0 = 0.4 and the parameters specified in (3.100).would imply thatλ∗I(τ˜j) ∈ (0, h(τ˜j)) for all τ˜j > 0 .However, it seems analytically intractable to prove that (3.114) holds.A second route of resolution would be to explore this issue numerically. For instance, inFig. 3.9 we fix D0 = 0.4 so that b ≈ 3.1279 > 3. Then, we find the unique root to ζR(λI) = 0for each τu from 0 to a large number to determine the range of possible λI values so that theeigenvalue is on the imaginary axis. Then, we compute ζI(λI) for λI on this range to determineits sign. For q = 2, where the NLEP is not explicitly solvable, we are able to confirm thatζI(λI) < 0. Therefore, since path (II) applies, we conclude that the steady-state spike patternis linearly stable for this parameter configuration. Nonetheless, it is still desirable to providean analytical proof to show that the eigenvalues are in Re(λ) < 0 for all choices of q > 1 and0 < τu <∞.The final route that we pursue to resolve this final case where Dj,2 < D∗j,2,min (b > 3) isto assume an extra condition, as we will derive below. We first observe that CR(λI) = 0 atλI = λIR ≡√3(b−3)1+τ˜jb , and we recall that CI(λI) = 0 at λI = λII =√(3(b− 3) + b/τ˜j . Then,147λI0 λIR0.5 λIM1 1.5 λII2-1-0.500.511.522.53CRCIFigure 3.10: Locations of the three special points λIR , λII and λIM on the graphs of CR and CIversus λI , as discussed in the text. The model parameter values τu = 1 and D0 = 0.4, togetherwith those stated in (3.100), were used. The value b ≈ 3.1279 > 3 was also numericallyconfirmed.since b/τ˜j > 1/(1 + τ˜jb) for b > 1, it follows that the inequalityλIR < λII ,holds, because we have b > 3. For a typical picture of this scenario, we refer to Fig. 3.10 wherewe present full numerical computations for a particular parameter set.We now determine the value λIM for which CR(λIM ) = 1/2. We conclude that the root λ∗Ito ζR(λI) = 0 must occur in λIR < λ∗I < λIM . If we can further show that λIM < λII , so thatCI(λIM ) < 0, it follows also that CI(λ∗I) < 0. This would imply that ζI(λ∗I) < 0 and N = 0.Therefore, the desired order of these special values of λI , which will guarantee that path (II)holds and so N = 0, is thatλIR < λ∗I < λIM < λII . (3.115)For a particular set of parameter values this is illustrated in the numerical results shown inFig. 3.11 .We now show that the ordering (3.115) does in fact hold under an additional assumption.To show this, we consider the real part of (3.107a), and solveCR(λIM ) =12 = a[1 + τ˜jb− 3b9 + λ2IM(1 + 3τ˜j)].148λI0 λIR0.5 λIM1Re-λI0 1 λI* λII2Im-0.6-0.4- 3.11: Numerical results for the location of λ∗I (indicated by a star) relative to λIR , λIM ,and λII (indicated by heavy dots), as stated in (3.115). The model parameters, τu = 1 andD0 = 0.4, together with the parameter values stated in (3.100), were used.This readily yields, after a little algebra, thatb(1 + 3τ˜j)9 + λ2IM= 13(1 + τ˜jb− 12a)if 1 + τ˜jb− 1/(2a) > 0 .Next, we consider the imaginary part of (3.107a). Upon substituting the expression aboveinto it, we calculate thatCI(λIM ) = aλIM[τ˜j − 3bτ˜j + b9 + λ2IM]= aλIM[13(1 + τ˜jb−12a)].This expression can be written asCI(λIM ) = −a3λIM[τ˜j(b− 3) +(1− 12a)].From this last expression, we conclude that if b > 3 and the extra condition a > 1/2 holds,then λIM exists and CI(λIM ) < 0. It follows that N = 0 since CI(λ∗I) < CI(λIM ) < 0, and thenζI(λ∗I) < 0 because FI > 0.We summarize this result in the following statement:Proposition 3.19. Suppose thatDj,2 < D∗j,2,min =ω34pi2K3 (i.e. b > 3) ,149and thatωqU0χ0,j>12 (i.e. a >12),or equivalently,Dj,2 >(qU02ω − 1)(ω32pi2K3). (3.116)Then we have three different possibilities:(I) If qU02ω − 1 < 0, i.e. qU0 < 2ω = 2(U0,max − U0), or equivalentlyω = S(γ − α)− U0(1 + q/2) > U0q/2 , (3.117)then the lower bound is negative and the inequality (3.116) is automatically true. Therefore,(3.117) together with Dj,2, < D∗j,2,min implies N = 0 and thus stability, or:(II) If qU02ω − 1 > 1/2, then this conflicts with Dj,2 < D∗2,jmin and no conclusion regardingstability can be made, or:(III) If 0 < qU02ω − 1 < 1/2 , then if(qU02ω − 1)(ω32pi2K3)< Dj,2 < D∗j,2,min ,we have N = 0 for all τ˜ > 0. This gap condition holds with an analogous gap condition in U0.U0q/3 < ω = S [γ − α]− U0 < U0q/2 .It is an open problem to close the gap and drop the condition (3.116), and prove that N = 0for all τ˜j > 0 and 0 < Dj,2 < D∗j,2,min, in agreement with what we discovered for the specialcase q = 3. However, Remark 3.12 also suggests that the threshold D∗j,2,min could possibly berevised, as it may depend on q.3.5.6 A Gap Between the Lower and Upper Thresholds: Dj,2,min < Dj,2 < D∗j,2,Existence of Hopf BifurcationHaving considered Dj,2 above D∗j,2, where there is instability due to one unstable eigenvalue,and below D∗j,2,min, where we have stability subject to the extra condition, we now consider the150λI0 λIR0.5 λIM1Re-τu1 λI* λII2λI-1012345ζR = 0ζ I = 0Figure 3.12: Locating the Hopf bifurcation point. The two curves shows the locus of the rootsof ζR(λI) and ζI(λI) as functions of τu. Hopf bifurcation is determined numerically to occurat τu ≈ 3.33 and λI ≈ 0.307, at the intersection of these two curves. The model parametersare D0 = 0.6 and those stated in (3.100). For this parameter set we confirm numerically that3− 3/(2a) ≈ 2.397 < b ≈ 2.714 < 3, which satisfies (3.118).gap region defined byD∗j,2,min < Dj,2 < D∗j,2 .In this gap region we will argue that a Hopf bifurcation exists with respect to the parameterτ˜j . An example of a such a Hopf bifurcation point is illustrated in Fig. 3.12.Unlike in the explicitly solvable case, q = 3, where a Hopf bifurcation point can be locatedby a closed-form formula τ˜j = τ˜j,Hopf = O(1) using (3.92) (but only the mode j = K − 1 willactually occur), corresponding to the critical case ζI(λ∗I) = 0 (and by definition ζR(λ∗I) = 0 aswell, so λ = ±iλ∗I are a pair of purely imaginary eigenvalues corresponding to Hopf bifurcation),we seek only to confirm that if τ˜j  1, we have path (II), i.e. stability, while if τ˜j  1, thenwe have path (I), i.e. instability.We first observe that when3− 32a < b < 3 , (3.118)then Dj,2 is on the gap rangeD∗j,2,min < Dj,2 < D∗j,2 .For the typical model parameter set given in (3.100) with two hotspots, these bounds are0.4559 ≈ D∗0,min < D0 < D∗0 ≈ 0.7599151Since purely imaginary eigenvalue implies λ = ±iλ∗I , we still seek to find λ∗I such thatζR(λ∗I) = ζI(λ∗I) = 0 directly. In other words, we are seeking the boundary case where theimage of the positive imaginary axis is neither away from nor enclosing the origin, but exactlycrosses through the origin.Now on the range τ˜j > b3(3−b) > 0, we have CI(λI) > 0 for all λI > 0. We can again letτ˜j →∞, as in the derivation of (3.110), to conclude that ζR(λ∗I) = 0 whenλ∗I ∼3√2abτ˜−1/2j  1 .Next, by using λ∗I = O(τ˜−1/2j ), together with CI(λ∗I) from (3.107a), we estimate thatCI(λ∗I) ∼aτ˜−1/2j9 [3τ˜j(3− b) +O(1)] ∼aτ˜1/2j9 (3− b) > 0 ,since b < 3. Then, since CI(λ∗I) = O(τ˜1/2j )  1, but FI(λ∗I) = O(τ˜−1/2j ) when λ∗I = O(τ˜−1/2j ),we conclude that ζI(λ∗I) > 0 as τ˜j → +∞. This implies that path (I) holds and N = 2.On the other hand, when τ˜j → 0+ we can repeat a previous calculation given in (3.111) toobtain that the intersection of CR(λI) and FR(λI) must occur at some λ∗I = O(1) > 0. Then,since for τ˜j → 0 we have from (3.107a) that CI(λ∗I) < 0 because b > 0, we must conclude thatζI(λ∗I) < 0 when τ˜j  1. This implies that in the limit τ˜j  1, path (II) holds and N = 0.We then prove the existence of a Hopf bifurcation by appealing to a continuous path ar-gument, namely that λ∗I = λ∗I(τ˜j) as a continuous function. Since N = 0 for τ˜j  1, whileN = 2 for τ˜j  1, there must exist a minimum value of τ˜j,Hopf > 0 such that ζI(λ∗I(τ˜j,Hopf)) = 0exactly with λ∗I > 0. However, it is not clear whether there are other values of τ˜j at which otherHopf bifurcations occur for τ˜j > τ˜j,Hopf > 0.One way to attempt to prove uniqueness of the Hopf bifurcation value would be to provea condition of one-way traversality at the onset of a purely imaginary eigenvalue. To do so,we may again let λ = λR + iλI and consider both real part and imaginary part as a functionof τ˜j , so that the Hopf bifurcation occurs at λI(τ˜j) = λ∗I where λR = 0. If one can furthershow that dλR/dτ˜j > 0 whenever we have a root on the imaginary axis, it would follow thatthe locus of any eigenvalue in the complex plane can cross the imaginary axis only once, and152in only the direction from left to right. It appears to be analytically rather difficult to provesuch a one-way transversal crossing condition.3.5.7 Conclusions on the Stability of a Symmetric K−Hotspot Steady-StateFor q = 2, 3, 4, suppose that Conjecture 3.18 holds and that K ≥ 2. Let N denote thenumber of eigenvalues with positive real part. Then, for any mode of competition instabilityj = 1, . . . ,K − 1, we have established the following statement:(i) If Dj,2 > D∗j,2, then N = 1 for all τ˜j > 0. The steady-state K-spike pattern isunstable for any q.(ii) If D∗j,2,min < Dj,2, < D∗j,2, then there exists a Hopf bifurcation threshold τ˜H whichis possibly non-unique when q 6= 3. Moreover, we have N = 2 when τ˜j  1, andhence instability. Otherwise if τ˜j  1, then we have N = 0, and hence stability.When q = 3, the threshold τ˜H is unique with formula given in (3.88), and we haveN = 0 and N = 2 for τ˜j below and above the threshold τ˜H , respectively.(iii) If Dj,2 < D∗j,2,min, then for q 6= 3 we have N = 0 if τ˜j  1 and τ˜j  1. If wesuppose further the additional condition that ω = S(γ − α)− U0 > U0q/2, then wehave N = 0 for any τ˜j > 0. For q = 3, we have that N = 0 for any τ˜j > 0 with noextra condition needed.Remark 3.20.1. A reason for considering integral values of q only in the range {2, 3, 4} is that we assumedin the course of deriving the NLEP 3.36 that τu  O(−2) (see the calculations leadingto 3.26). Then, for the parameterτ˜j = τuO(q−3) ,to be O(1), we must have that τu = O(3−q). This establishes that q < 5 is required tosatisfy the assumption that τu  O(−2). If q = 5, then the ODE for the perturbation ηat (3.27) will be changed toD0αqηxx = 2τuλαqη ,153where τu = O(−2). This will result in a different problem to be solved for η(x), andconsequently a different value for η(0). Ultimately, this leads to a different NLEP thatrequires a separate analysis.2. Conjecture (3.18) is rigorously established in the literature except for the statementF ′R(λI) < 0 for q = 4. However, this monotonicity condition for q = 4 was readilyestablished numerically in Fig. Asymptotic Determination of Hopf Bifurcation ThresholdFinally, we would like to see if we can determine λ∗I and τ˜j,Hopf asymptotically as b → 3 frombelow. We recall that for the explicitly solvable case q = 3 that we have τ˜j,Hopf →∞ as b→ 3−(Dj,2 →(D∗j,2,min)−) and, correspondingly, that λI,H → 0+ . This motivates us to consider thegeneral case q 6= 3 to examine whether only the known local behavior of CR, CI , FR, and FI asλI → 0, is involved in estimate the Hopf bifurcation point.From [52], it is well known for any q > 1 thatFR(λI) ∼ 12 − kcλ2I + · · · , FI(λI) ∼λI4 (1− 1/q) as λI → 0 ,where kc > 0 is a constant depending on q. We then recall that CR(λI) can be written asCR(λI) = a9 + λ2I[3(3− b) + (1 + τ˜jb)λ2I].To examine the region near b = 3, we introducing a detuning parameter δ by δ = 3 − b,where 0 < δ  1. We then look for a root to CR(λI) = FR(λI) near λI = 0 when τ˜j  1. Tothis end, we expand CR(λI) near λI = 0 asCR(λI) = a9 + λ2I[3δ + (1 + 3τ˜j)λ2I +O(τ˜jδλ2I)].This suggests that we must have the dominant balance τ˜jλ2I = O(1) so that CR → 1/2 = FR(0)as λI → 0. This motivates the introduction of the rescalingλI ∼ τ˜−1/2j λI,0 ,154which must be chosen in such a way thata9[3λ2I,0 +O(δ)]∼ 12 .This yields thatλI,0 =√32a .Therefore, for τ˜j  1 and b = 3 − δ with 0 < δ  1, the unique root of ζR(λI) = 0 is locatedasymptotically atλI ∼√32aτ˜−1/2j .Next, we relate τ˜j to δ by enforcing that CI(λI) ∼ FI(λI) as λI → 0+. This latter conditionyields thataλI(3τ˜jδ − 3 +O(δ) + τ˜jλ2I9 + λ2I)∼ λI4 (1− 1/q) .Upon cancelling λI from both sides of this expression, and putting τ˜jλ2I = 3/(2a), we geta9[3τ˜jδ − 3 + 32a]= 14(1− 1/q) .Upon solving this equation for τ˜j , we getτ˜j ∼ δ−1(1 + 14a −34aq).In summary, provided that1 + 14a(1− 3/q) > 0, (3.119)we have that as b→ 3−, the Hopf bifurcation occurs atτ˜j = τ˜j,Hopf ∼ 1δ(1 + 14a (1− 3/q)), (3.120)with corresponding frequencyλ∗I ∼ δ1/2√32a(1 + 14a (1− 3/q))−1/2, (3.121)155where δ = 3− b→ 0+.Finally, let us compare this result for general q to the explicit result that we obtained earlierfor the exactly solvable case q = 3.Firstly, we observe that the quadratic equation (3.73) for λ that occurs when q = 3 isequivalent toλ2 +(1 + τ˜j(b− 3)τ˜j)λ+ [b− 3 + 3/(2a)]τ˜j= 0 .We conclude that a Hopf bifurcation occurs if the coefficient of λ vanishes, i.e. if τ˜j,Hopf =(3 − b)−1 = δ−1 in agreement with setting q = 3 in (3.120). In addition, since at the Hopfbifurcation we have(λ∗I)2 = b− 3 + 3/(2a)τ˜j= 1τ˜j( 32a − δ),we conclude thatλ∗I ∼ δ1/2√32a .This expression agrees with (3.121) upon setting q = 3.When q = 4, we readily observe that the assumption (3.119) for (3.120) and (3.121) isalways satisfied. However for q = 2, we have to require that a > 1/8 in order for (3.119) tohold. This means for a = ω/(U0χ0,j), b = 9χ0,j/2 = 3, which gives χ0,j = 2/3, that we musthave the conditiona = ω43U0>18 ,which is equivalent to the requirement that ω > U0/6. Therefore, we conclude that a Hopfbifurcation occurs when Dj,2 → (Dj,2,min)+ with asymptotics given by (3.120) and (3.121) ifthe conditionω = S(γ − α)− U0 > U0/6 , (3.122)holds. In other words, the asymptotic conclusions hold provided we are not too close to theexistence threshold ω = 0 of the hotspot pattern.Remark 3.21. The fact that, for q = 2, the Hopf bifurcation with the asymptotics (3.120) and(3.121) does not occur for0 < S(γ − α)− U0 < U0/6 ,156when b → 3−, i.e. as Dj,2 →(D∗j,2,min)+does not contradict the fact that there exists a Hopfbifurcation whenD∗j,2,min < Dj,2 < D∗j,2 .It only shows that the limiting asymptotics is not of the form where τ˜j → ∞ and Dj,2 →(D∗j,2,min)+simultaneously.3.6 Stability of a Stripe Pattern, Explicitly Solvable Case.In this section we extend our 1-D spike stability analysis to study the transverse stabilityof a homoclinic stripe for our three-component crime model with simple police interaction.A homoclinic stripe occurs when the attractivness concentration A concentrates on a planarcurve in the 2-D domain. The simplest type of homoclinic stripe solution is a stripe of zerocurvature, which results when a 1-D homoclinic spike solution is trivially extended along themidline of a rectangular domain. The main goal of this section is to analyze the linear stabilityof this type of stripe solution to transverse perturbations for our three-component model. Inour stability analysis we will assume that the patrol focusing parameter is q = 3, which will leadto an explicitly solvable NLEP spectral problem, which is then highly tractable analytically.In this way, we are able to explicitly identify a band of unstable transverse wavenumbers, andcalculate both the growth rate and most unstable mode within this band. This instabilityis shown to lead to the breakup of the stripe into a localized hotspot of criminal activity.For two-component reaction-diffusion systems, such as the Gierer-Meinhardt and Gray-Scottmodels, there have been several prior studies of the stability of homoclinic stripes to transverseperturbations ([16], [25], [26]). As far as we are aware there have been no prior studies forthree-component systems.We formulate our three-component model in the rectangular domain defined byΩ = {(x1, x2)| − ` < x1 < `, 0 < x2 < d} . (3.123)157The three-component RD system (3.3) in 2-D now takes the formAt = 2∆A−A+ ρA+ α , x ∈ Ω ; ∂nA = 0 , x ≡ (x1, x2) ∈ ∂Ω , (3.124a)ρt = D∇ ·(A2∇(ρ/A2))− ρA+ γ − α− U , x ∈ Ω ; ∂nρ = 0 , x ∈ ∂Ω , (3.124b)τuUt = D∇ · (Aq∇ (U/Aq)) , x ∈ Ω ; ∂nU = 0 , x ∈ ∂Ω , (3.124c)where Ω is the rectangular domain of (3.123).3.6.1 Extension of a 1-D Spike Solution to a 2-D Stripe SolutionIn the construction of the steady-state stripe solution below, we will need to assume that q > 1.It will be shown that for the case q = 3, the NLEP governing the transverse stability of thisstripe is explicitly solvable.We will analyze the transverse stability properties of a steady-state stripe solution for (3.124)for the regime O(1)  D  O(−2). As motivated by the scalings in [29], we introduce thenew variables v, D0, and u byρ = 2vA2 , U = uAq , D = D0/2 , (3.125)where we will assume that D0  O(2) so that D  1. In terms of these new variables, (3.124)becomesAt = 2∆A−A+ 2vA3 + α , x ∈ Ω ; ∂nA = 0 , x ≡ (x1, x2) ∈ ∂Ω , (3.126a)2(vA2)t= D0∇ ·(A2∇v)− 2vA3 + γ − α− uAq , x ∈ Ω ; ∂nv = 0 , x ∈ ∂Ω , (3.126b)2τu (Aqu)t = D0∇ · (Aq∇u) , x ∈ Ω ; ∂nu = 0 , x ∈ ∂Ω . (3.126c)We now construct a steady-state stripe solution consisting of a localized region of highattractiveness, which we center along the midline x1 = 0 of the rectangle. To do so, we simplyconstruct a steady-state 1-D spike A = A(x1), v = v(x1), and u = u(x1) as in Section 3.1,and extend it trivially in the x2 direction. Since the total number of police is conserved due158to (3.124c), we have´Ω U(x, t) dx = U for all time, where U > 0 is the initial number of policedeployed. As such, we have for the steady-state stripe solution thatˆ `−`U(x1) dx1 = U0 ≡ U/d, (3.127)where d is the width of Ω. It follows from (3.126c) that the steady-state 1-D solution u(x1) isa constant given byu(x1) = U0/ˆ `−`[A(x1)]q dx1 , (3.128)and that the steady-state 1-D problem for A(x1) and v(x1), from (3.126a) and (3.126b), is2Ax1x1 −A+ 2vA3 + α = 0 , |x1| ≤ ` ; Ax1(±`) = 0 , (3.129a)D0(A2vx1)x1− 2vA3 + γ − α− U0Aq´ `−`Aq dx1= 0 , |x1| ≤ ` ; vx1(±`) = 0 . (3.129b)In the inner region |x1| ≤ O() near the pulse we set y = x1/ and expandA = A0/+ · · · , v = v0 + · · · ,as in [29] and also Section 3.1. Note that we are reusing the notations A0 and v0 for the leadingorder expansions for this context, because the procedures and structures of the asymptotics arevery similar.We readily obtain that v0 is a constant and A0yy−A0 + v0A30 = 0 as before. This yields theleading order inner solutionA(x1) ∼ 1√v0w (x1/) , v ∼ v0 , (3.130)where w(y) =√2 sech y satisfies (2.14), and where the constant v0 is to be determined. Todetermine the constant v0, we integrate (3.129b) over |x1| ≤ ` to get−2´ `−` vA3 dx1+2`(γ−α)−U0 = 0. Since A = O(−1) in the inner region, while A = O(1) in the outer region, the integralabove can be calculated asymptotically as 2´ `−` vA3 dx1 ∼ v−1/20´∞−∞w3 dy = v−1/20√2pi. In159this way, provided that 0 < U0 < 2`(γ − α), we calculate v0 asv0 =2pi2[2`(γ − α)− U0]2, (3.131)which looks very similar to (3.12) with K = 1, except thatU0 = U/d ,now carries a different meaning. Therefore a stripe solution exists only if the total policedeployment U0 per cross-sectional length satisfiesU0 < 2`(γ − α).In the inner region, the police concentration U(x1), given by U(x1) = uAq, becomesU = U0Aq´ `−`Aq dx1∼ U0wq´∞−∞wq dy, |x1| ≤ O() . (3.132)In contrast, in the outer region we have from (3.129a) that Aout = α + O(2). To determinethe leading-order outer problem for v, we first need to estimate the integral´ `−`Aq dx1 in(3.129b). Since A = O(−1) in the inner region |x1| ≤ O(), while A = O(1) in the outer regionO()  |x1| ≤ 1, it follows that when q > 1 the contribution to the integral´ `−`Aq dx1 fromthe inner region is dominant, with the estimate´ `−`Aq dx1 = O(1−q) 1. We will henceforthassume that q > 1, so that the nonlocal term in (3.129b) can be neglected to leading-order inthe outer region. Then, from (3.129b) we obtain that the leading-order outer problem for v isv ∼ v˜0 + o(1), where v˜0 satisfiesv˜0x1x1 = −(γ − α)D0α2, 0 < |x1| < ` ; v˜0x1(±`) = 0 , v˜0(0) = v0 . (3.133)This yields the leading-order outer solution for v˜0 as given below in (3.136).Finally, we calculate u. Since q > 1, we use A ∼ −1w/√v0 to estimate the integral in160(3.128). This yields thatu = U0´ `−`Aq dx1∼ q−1u˜e , where u˜e ≡ U0vq/20´∞−∞wq dy. (3.134)We summarize our result in the following statement.Proposition 3.22. For  1, D  1, and U < 2d`(γ −α), the steady-state spike solution for(3.126) is given to leading order in the inner region byA(x1) ∼ 1√v0w (x1/) , v ∼ v0 , U(x1) ∼ U0wq´∞−∞wq dy, |x1| ≤ O() , (3.135)where v0 ≡ 2pi2 [2`(γ − α)− U0]−2, U0 = U/d, and w =√2 sech(x1/). In the outer region,O() |x1| ≤ `, we haveA ∼ α , v ∼ (γ − α)2D0α2[`2 − (`− |x1|)2]+ v0 , U ∼ q−1U0αq vq/20´∞−∞wq dy. (3.136)The criminal density in the inner and outer regions, as obtained from (3.125), isρ(x1) ∼ [w (x1/)]2 , |x1| = O() ; (3.137a)ρ(x1) ∼ 2α2[v0 +(γ − α)2D0α2(`2 − (`− |x1|)2)], O() |x1| ≤ ` . (3.137b)In Fig. 3.13, we plot the above asymptotic results. We observe that the criminal density ρis accompanied by a large quantity A, the attractiveness, at the vicinity of the crime hotspot.Moreover, the degree of patrol focus q, can be reflected by the shape of the police concentration.For the larger value of q we observe a more narrow profile near the crime hotspots.3.6.2 The Stability of a StripeNext, we derive an NLEP governing the stability of the stripe solution to transverse pertur-bations that can lead to the breakup of the stripe into localized hotspots. Let Ae, ve, and161−0.3 −0.2 −0.1 0.0 0.1 0.2 0.3012345678Aρ−0.3 −0.2 −0.1 0.0 0.1 0.2 0.30246810121416q=2q=3Figure 3.13: Plot of the steady-state spike from Proposition 3.22 in the inner region on thex1 axis for  = 0.05, ` = 1.0, γ = 2, and α = 1. The x2 direction is omitted because itextends trivially. Left plot: The attractiveness A ∼ w (x1/) /(√v0) + α (heavy solid curve)and criminal density ρ ∼ w (x1/) (solid curve), these curve do not depend on q. Right plot:The police density U from (3.136) for q = 2 (heavy solid curve) and for q = 3 (solid curve).ue denote the steady-state solution constructed in the previous subsection and summarized inProposition 3.22. We then extend it trivially in the x2 direction to make a stripe. To determinethe stability of this stripe with respect to transverse perturbations we introduceA = Ae(x1) + eλt+imx2φ(x1) , v = ve(x1) + eλt+imx2ψ(x1) , u = ue + eλt+imx2qη(x1) .(3.138)Here m = kpi/d where d is the width of the rectangle and k > 0 is an integer. The relative sizesin  in (3.138) are such that φ, ψ, and η are all O(1) in the inner region. Upon substituting(3.138) into (3.126), we obtain on |x1| ≤ l that2φx1x1 − (1 + 2m2)φ+ 32veA2eφ+ 3A3eψ = λφ , (3.139a)D0[A2eψx1 + 2Aevex1φ]x1− m2D0A2eψ − 32veA2eφ− 3ψA3e−qAqeη − qAq−1e ueφ = λ2(A2eψ + 2Aeveφ),(3.139b)D0[qAqeηx1 + qAq−1e uex1φ]x1− qm2D0Aqeη = 2τuλ(qAqeη + qAq−1e ueφ). (3.139c)In the analysis of (3.139), we must allow for spatial perturbations of high frequency as → 0because the hotspot O() width is small relative to the O(1) domain diameter of Ω. As such,162we consider the range 0 < m ≤ O(−1). Below, we show that the upper stability thresholdoccurs when m = O(−1).In (3.139b) and (3.139c), we note that uex1 = 0 and ue ∼ q−1u˜e, where u˜e is given in(3.134). In the outer region where Ae ∼ α we obtain from (3.139b) that φout = O(3ψout),when 0 < m ≤ O(−1). Next, we estimate the terms in (3.139b) in the outer region. We obtainfrom (3.134), and our estimate of φout, that qAq−1e ueφ = O(q+2ψout). Moreover, since q > 1,we have qAqeη  O(). In this way, we obtain in the outer region that (3.139b) reduces toψx1x1 −m2ψ = 0 , O() < |x1| ≤ l ; ψx1(±l) = 0 . (3.140)Similarly, since τu = O(1) and uex1 = 0, we obtain from (3.139c) that to leading orderηx1x1 −m2η = 0 , O() < |x1| ≤ l ; ηx1(±l) = 0 . (3.141)In the inner region, we look for a localized eigenfunction for φ in the form φ = Φ (x1/),i.e. which is constant in x2 direction. Since the equations for ψ and η are not singularlyperturbed, we obtain that ψ ∼ ψ(0) and η ∼ η(0) to leading order in the inner region. Then,since Ae ∼ −1w/√v0 and ve ∼ v0 in the inner region, as obtained from (3.135), we find from(3.139b) that Φ(y) satisfiesΦ′′ − Φ + 3w2Φ + 1v3/20w3ψ(0) =(λ+ 2m2)Φ , −∞ < y <∞ . (3.142)Next, we derive the jump conditions for η and ψ across x = 0. To do so, we introduce anintermediate length-scale δ with O()  δ  1 and integrate (3.139b) from −δ < x1 < δ anduse Ae ∼ α at x1 = ±δ. This yields thate0 [ψx1 ]0 = e1ψ(0) + e2η(0) + e3 , (3.143a)163where we have defined [ψx1 ]0 ≡ ψx1(0+)− ψx1(0−). Here ej for j = 0, . . . , 3 are defined bye0 = D0α2 , e1 =D0m2v0ˆ ∞−∞w2 dy + 1v3/20ˆ ∞−∞w3 dy ,e2 =1vq/20ˆ ∞−∞wq dy , e3 = 3ˆ ∞−∞w2Φ dy + qu˜ev(q−1)/20ˆ ∞−∞wq−1Φ dy .(3.143b)In a similar way, we integrate (3.139c) across −δ < x1 < δ and use Ae ∼ α at x1 = ±δ.This yields thatf0 [ηx1 ]0 = f1η(0) + f2 , (3.144a)where fj for j = 0, . . . , 2 are defined byf0 = D0αq , f1 =D0m21−qvq/20ˆ ∞−∞wq dy + 3−qτuλvq/20ˆ ∞−∞wq dy ,f2 =3−qqu˜eτuλv(q−1)/20ˆ ∞−∞wq−1Φ dy .(3.144b)In (3.143b) and (3.144b), u˜e is given in (3.134).Next, we must solve for ψ(x1) and η(x1) from the solution to (3.140) and (3.141) subject tothe jump conditions (3.143) and (3.144), and the boundary conditions ψx1(±`) = ηx1(±`) = 0.From this solution, we calculate ψ(0), which then determines the NLEP for Φ(y) from (3.142).To solve for η, we introduce the Green’s function Gm(x1) satisfyingGmx1x1 −m2Gm = −δ(x1) , |x1| ≤ ` ; Gmx1(±`) = 0 . (3.145)The explicit solution to this problem isGm(x1) =cosh [m(`− |x1|)]2m sinh(m`) , (3.146)for m > 0. In terms of Gm(x1), and using [Gmx1 ]0 = −1, the solution to (3.141) with (3.144) isη(x1) = η(0)Gm(x1)Gm(0), η(0) = − f2f1 + f0/Gm(0). (3.147)164Similarly, for m > 0, the solution to (3.140) subject to (3.143) isψ(x1) = ψ(0)Gm(x1)Gm(0), ψ(0) = − e2η(0) + e3e1 + e0/Gm(0). (3.148)We estimate the asymptotic order of the terms in (3.144b) as f0/Gm(0) = m tanh(m`)·O(1),f1 = m21−q ·O(1) + 3−qτu ·O(1), and f2 = 3−qτu ·O(1). As such, when τu = O(1) and q > 1,we conclude for any m > 0 with m O() thatf1 +f0Gm(0)∼ D0m21−qvq/20ˆ ∞−∞wq dy ,η(0) = − f2f1 + f0/Gm(0)∼ O(3−q)O(1−qm2) = O(2/m2) 1 .Since η(0)  1 when q > 1, τu = O(1), and m  O(), we conclude from (3.148) that, in thisparameter regime,ψ(0) ∼ − e3e1 + e0/Gm(0). (3.149a)Upon using (3.134) for u˜e, the coefficients in (3.149a) aree0 = D0α2 , e1 =D0m2v0ˆ ∞−∞w2 dy + 1v3/20ˆ ∞−∞w3 dy ,e3 = 3ˆ ∞−∞w2Φ dy + qv1/20 U0´∞−∞wq−1Φ dy´∞−∞wq dy,(3.149b)Upon substituting (3.149a) into (3.142), and by using (3.131) for v0, together with´∞−∞w2 dy =4 and´∞−∞w3 dy =√2pi, we obtain the following NLEP with two nonlocal terms:L0Φ− χ0w3´∞−∞w3 dy(3ˆ ∞−∞w2Φ dy + qv1/20 U0´∞−∞wq−1Φ dy´∞−∞wq dy)=(λ+ 2m2)Φ , (3.150a)χ0 ≡(1 + 4D0m2−1[2`(γ − α)− U0] +4D0α2pi2m tanh(m`)[2`(γ − α)− U0]3)−1. (3.150b)1653.6.3 Analysis of the NLEP - Stripe Breakup InstabilityThe analysis of the spectrum of (3.150) is challenging for general q > 1 owing to the presenceof the two nonlocal terms. In our analysis below, we will focus on the special case q = 3, as wehave done before for our 1-D hotspot analysis, for which this NLEP with two nonlocal termscan be transformed to the following NLEP with only one nonlocal term:L0Φ− χw3´∞−∞w2Φ dy´∞−∞w3 dy=(λ+ 2m2)Φ , χ ≡ χ0[ 6`(γ − α)2`(γ − α)− U0], (3.151)which is explicitly solvable. Here χ0 is defined by (3.150b). It is an open problem to analyze(3.150) for arbitrary q > 1.The NLEP (3.151) for q = 3 is a special case of the class of explicitly solvable NLEP’s of Prin-cipal Result 2.2 in [37]. Upon replacing λ, σ, g(w), and h(w) with λ+ 2m2, 3, w2/´∞−∞w3 dy,and w3 respectively in the formula (2.2) of [37], we getλ = σ − χ(λ)ˆ ∞−∞g(w)h(w) dy. (3.152)By using the definition of χ at (3.151), we obtain the following explicit formula for any unstableeigenvalue of (3.151):Proposition 3.23. Let → 0, q = 3, τu = O(1), U0 < 2`(γ − α), with m > 0 and m O().Then, the transverse stability of a stripe solution for (3.124) on an O(1) time-scale is determinedby the sign of the discrete eigenvalueλ = 3− 2m2 − 9`(γ − α)[2`(γ − α)− U0][1 + 4D0m2−1[2`(γ − α)− U0] +4D0α2pi2m tanh(m`)[2`(γ − α)− U0]3]−1. (3.153)To determine the edges of the instability band for a stripe, we set λ = 0 in (3.153) and solvefor m. For   1, the upper edge m+ of the instability band is m+ ∼√3/, with λ < 0 form > m+. In contrast, the lower edge m− of the instability band satisfies m− ∼ 1/2m0− wherem0− satisfies3 ∼ 9`(γ − α)[2`(γ − α)− U0](1 + 4D0m20[2`(γ − α)− U0])−1.166Upon solving for m0−, we conclude for  1 that λ > 0 when1/2m0− < m <√3, m0− ≡√`(γ − α) + U04D0. (3.154)We remark that the lower O(1/2) edge of the band is consistent with the assumption m O()used to derive (3.153). In addition, we note that the lower edge of the band increases with thelevel U0 of police effort. This shows that as U0 increases, less transverse modes become unstable.Finally, we estimate the mode mdom within the instability band that has the largest growthrate. To do so, we set dλ/dm = 0 in (3.153), and obtain that mdom is the root of22m ∼ 9`(γ − α)[2`(γ − α)− U0](4D0m2−1[2`(γ − α)− U0] + · · ·)−2 ( 8D0m [2`(γ − α)− U0] + · · ·).For   1, this reduces to 16D0m4 ∼ 36`−1(γ − α). For   1, this yields the most unstablemode asmdom ∼ −1/4[ 94D0`(γ − α)]1/4, (3.155)which is independent of U0. We predict that a stripe for the urban crime model (3.124) on arectangular domain of width d will break up into N localized hot-spots, where N is the closestinteger to mdomd/2pi.In Fig. 3.14 (left plot), we use (3.153) to plot λ versus m for the parameter set  = 0.05,D0 = 1, γ = 2, α = 1, ` = 1, and U0 = 1. In the caption of the figure, the asymptotic predictionsfor the edges of the instability band, as obtained from (3.154), are compared with results from(3.153). From (3.155), the asymptotic prediction for the most unstable mode is mdom ∼ 2.59,which compares well with the numerically computed result mdom ≈ 2.44 as computed from(3.153). In Fig. 3.14 (right plot), we compare λ versus m near the lower threshold m− forU0 = 1 and for U0 = 1.5.For the same parameter values as used in Fig. 3.14, we further put d = 2 to computemdomd/2pi = 2.59/pi ≈ 0.82. (3.156)This suggests that a stripe on a square domain of side-length two should break up into only1670 5 10 15 20 25 30 35m−λ0.0 0.2 0.4 0.6 0.8 1.0m− 3.14: Principal eigenvalue λ as a function of frequency m. Left: plot of λ versusm, as given in (3.153), for  = 0.05, D0 = 1, γ = 2, α = 1, l = 1, and U0 = 1. Theasymptotic prediction as → 0 for the instability band from (3.154) is 0.158 < m < 34.64. Thecorresponding numerical result is 0.131 < m < 34.56. Right: plot of λ versus m near the lowerthreshold m− for U0 = 1 (solid curve) and U0 = 1.5 (heavy solid curve). The lower edge of theinstability band increases as U0 increases.one spot, which is relatively few as compared to, say, the case of the Gierer-Meinhardt modelstudied also in [37] for an explicitly solvable case. To validate this claim we computed fullnumerical solutions to (3.126) for the parameter set  = 0.05, D0 = 1, γ = 2, α = 1, U0 = 1, onΩ = [−1, 1]× [−1, 1], i.e. ` = 1, d = 2.The computations were done using the adaptive grid finite difference solver VLUGR2 [3].The initial conditions were taken to be the leading order steady-state stripe solution of Propo-sition 3.22. The results for A at different times, as shown in the gray-scale plot of Fig. 3.15,confirm the theoretical prediction that the stripe breaks up into only one spot.3.7 DiscussionsThe modelling aspect of this chapter is relatively preliminary in nature, and we have considereda police-criminal interaction of the simplest type. Nonetheless, what insights can we still drawabout how a chief of police should instruct the patrolling policemen?We discovered that if there are sufficiently many policemen, it would preclude the math-ematical existence of crime hotspots. Since this is usually not the case, when crime hotspotsdo exist, we consider how the policemen should patrol in order to destabilize multiple crimehotspots (single crime hotspot is unconditionally linearly stable). In other words, we want to168Figure 3.15: Spontaneous breakup of a stripe into one spot, obtained from a full numericalsimulations of (3.124) using VLUGR2. The model parameter values were  = 0.05, D0 = 1, γ =2, α = 1, U0 = 1, d = 2 and ` = 1. The time instants chosen were t = 0.000, 0.7299, 8.220, 9.985from top to bottom respectively.increase the range of criminal diffusivity so that a crime hotspot is unstable (i.e. lowering thecompetition threshold). This is exactly the main thrust of our mathematical analysis.Firstly, we found that the competition threshold does not go down necessarily as the num-ber of policemen increase. Paradoxically, if the degree of focus of police patrol is too high,the threshold goes up to a maximum before going down as the number of policemen furtherincreases. This suggests that "hotspot policing" which focuses policemen too much on policehotspots could be counter-productive. The optimal strategy, thus, lie in the middle ground169between "random policing" and "hotspot policing", which suggests some sort of "containmentapproach".Secondly, we found that below the competition threshold, the relative speed of policemenalso play a role in destabilizing crime hotspots through a novel asynchronous oscillatory insta-bility. The policemen need to travel fast enough relative to criminals so as to trigger instabilityof crime hotspots through this mechanism. This suggests that upgrades of mobility of patrollingpolicemen could be potentially valuable to decrease the total number of crime hotspots in acity.As in many modelling endeavours, both the pros and cons of a model may serve as inspi-rations for more accurate and applicable models in the future. To this end, here we brieflysummarize our main mathematical results, and we suggest a few directions that warrant fur-ther investigation, both for the sake of completeness for the analysis of this model, and also forimproving it to make it more realistic.3.7.1 SummaryIn this chapter we used the method of matched asymptotic expansions to construct a steady-state hotspot solution to (3.3) having K hotspots of a common amplitude in the limit  → 0for the regime D = O(−2). We then studied the spectrum characterizing the linear stabilityproperties of this steady-state solution by analyzing an NLEP with two nonlocal terms. Westudied the NLEP by first considering a special case with patrol focus degree q = 3, whichresults in an explicitly solvable NLEP and, consequently, an explicit formula for the principaleigenvalue. Explicitly solvable NLEP problems also appear in [29, 37, 38]. The general casewas then studied using the argument principle to count the number of unstable eigenvalues inthe right half plane. This procedure was first developed to study the stability of steady-statespike patterns for the Gierer-Meinhardt model (cf. [52]) and now has a rather large body ofliterature (see [29] and the references therein). Our conclusions from the explicitly solvablecase q = 3 are considerably stronger than those for the non-explicitly solvable case q 6= 3. Inparticular, when q = 3, two thresholds D0,lower and D0,upper given in (3.94) were determinedso that the a multiple-hotspot pattern is stable when D0 < D0,lower and unstable due to acompetition instability when D0 > D0,upper. Moreover, an explicit formula for the existence of170Hopf bifurcation τu = τHopf when D0,lower < D0 < D0,upper was given in (3.93). In contrastto the absence of a Hopf bifurcation for the basic crime model with no police intervention, asdiscovered in [29], the window of existence for a Hopf bifurcation given by (D0,lower, D0,upper)vanishes exactly when U0 = 0. In other words, the third component of the PDE system,modeling the police interaction, is essential to inducing the possibility of oscillations. Moreover,unlike the case of the Gray-Scott and Gierer-Meinhardt models studied in [9, 52, 50], wheresynchronous oscillatory instabilities of the spike amplitudes robustly occur and are the dominantinstability, our three-component system exhibits asynchronous oscillatory instabilities. Theseasynchronous, anti-phase, oscillations of the spike amplitudes have the qualitative interpretationthat, for a range of police diffusivities, the police presence is only able to mitigate the amplitudeof certain hotspots at the expense of the growth of other hotspots in different spatial regions.However, when q 6= 3, we had difficulty in analytically proving results as strong as for thecase q = 3. In particular, we were not able to prove, without assuming further conditions,that a multiple hotspot pattern is stable when the rescaled criminal diffusivity D0 is below thesame lower threshold defined earlier in the q = 3 case. One possibility is that the definitionof the lower threshold should be revised and should change with q. When D0 is between thelower and upper thresholds, we were able to prove the existence of a Hopf bifurcation, but wecannot show uniqueness of the critical Hopf bifurcation value in τu. These are interesting openproblems that warrant further study. Most importantly, we would like to investigate what arethe mathematical relationships between the explicitly solvable case q = 3 and the non-explicitlysolvable case q 6= 3, so that the strong results from the explicitly solvable case can potentiallycarry over to the general case.3.7.2 Open Problems and Future DirectionsWith regards to our police model, with simple police interaction, studied in this chapter, itwould be interesting to consider the more challenging D = O(1) regime. One key questionwould be to investigate whether the police presence can eliminate the peak insertion behaviourthat was found for the basic crime model to lead to the nucleation of new spikes of criminalactivity. In this direction it would be interesting to determine the influence of the police presenceon the global bifurcation of multiple hotspot steady-state solutions.171A second interesting direction would be to study the effect of police presence on crimepatterns when the police interaction is modelled by the predator-prey dynamics case I(U, ρ) =Uρ for (1.20). Preliminary results suggest that the NLEP will now have three non-local terms,which makes a detailed stability analysis very challenging. However, the determination ofthe competition instability threshold, corresponding to the zero eigenvalue crossing, should bereadily amenable to analysis.A third direction would be to consider spatial patterns in more than a simple 1-D spatialcontext. In Section 3.6 we studied (3.3) on a closed and bounded two-dimensional domain,where we observed that a homoclinic stripe can undergo a breakup into a localized hotspot intwo spatial dimensions. 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Russell (2005), The slow dynamics of two-spike solutions forthe Gray-Scott and Gierer-Meinhardt systems: competition and oscillatory instabilities,SIAM J. Appl. Dyn. Syst., 4(4), pp. 904–953.[51] W.-H. Tse and M. J. Ward (2015), Hotspot formation and dynamics for a con-tinuum model of urban crime, Europ. J. Appl. Math., available on CJO2015.doi:10.1017/S0956792515000376.177[52] M. J. Ward and J. Wei (2003), Hopf bifurcations and oscillatory instabilities of spike solu-tions for the one-dimensional Gierer-Meinhardt model, J. Nonlinear Sci., 13(2), pp. 209–264.[53] J. Wei (1999), On single interior spike solutions of the Gierer–Meinhardt system: unique-ness and spectrum estimates, Europ. J. Appl. Math., 10(4), pp. 353–378.[54] J. Wei (2008), Existence and stability of spikes for the Gierer-Meinhardt system, bookchapter in Handbook of Differential Equations, Stationary Partial Differential Equations,Vol. 5 (M. Chipot ed.), Elsevier, pp. 489–581.[55] J. Q. Wilson and G. L. Kelling (1998), Broken windows and police and neighborhood safety,Atlantic Mon., 249, pp. 29–38.[56] J. R. Zipkin, M. B. Short and A. L. Bertozzi (2014), Cops on the dots in a mathematicalmodel of urban crime and police response, DCDS-B, 19(5), pp. 1479–1506.178Appendix ALemmas and General FormulasA.1 A Floquet Boundary Condition Approach to NeumannNLEP Problems on a Bounded IntervalWhen considering the stability of a symmetric, equally-spaced, K−spike pattern on a onedimensional interval, we must impose homogeneous Neumann boundary conditions. We observetwo special properties for this class of problem. If we denote the spectral problem as Lφ = λφ,we see that:1. Translation-invariance: If φ(x) is a Neumann eigenfunction on the interval [a, b], i.e.φx(a) = φx(b) = 0, then the translated eigenfunction φ˜(x) = φ(x − x0) is a Neumanneigenfunction with the translated boundary condition: φ˜x(a+ x0) = φ˜x(b+ x0) = 0.2. Even symmetry of the linear operator L: The function φ˜(x) = φ(−x) solves Lφ˜ = λφ˜whenever φ solves the same problem. A sufficient condition for this is that L containsonly even derivatives and the nonlocal term is an integral on the whole domain with aneven kernel.The spectral problems on one spatial dimension we considered in this thesis satisfy the aboveproperties. We now show how these simple considerations can be used to reduce a spectralproblem with Neumann boundary conditions for a multiple spike pattern to one for a singlespike with a Floquet-type boundary condition.The first step is to recognize the Neumann problem as a restriction of a periodic problem.179A.1.1 Converting a Neumann problem to a Periodic Problem with Twicethe Domain LengthBecause of translation invariance, we may translate our domain to wherever we feel convenient.We describe here a bijective correspondence of Neumann eigenfunctions on a domain of lengthL to periodic eigenfunctions on a domain of length 2L.Suppose φ(x) is a Neumann eigenfunction on [0, L]. Then, if we apply an even extensionφ˜(x) =φ(x) for 0 ≤ x ≤ Lφ(−x) for − L ≤ x ≤ 0 ,it follows that φ˜(x) satisfies periodic boundary conditions on [−L,L], i.e. thatφ˜(L) = φ˜(−L), φ˜x(L) = φ˜x(−L) .Conversely, if φ(x) is a eigenfunction on [−L,L] satisfying periodic boundary conditions,then since the domain is symmetric about the origin, the functionφ˜(x) = φ(x) + φ(−x) ,restricted to [0, L] solves Lφ = λφ and it satisfies φ˜x(L) = φx(L)−φx(−L) = 0 by 2L-periodicityand φ˜x(0) = φx(0)− φx(0) = 0 by construction. Thus, φ˜(x) is a Neumann eigenfunction.Therefore, we conclude that the spectrum of the Neumann problem on a domain of length Lis exactly the same as the spectrum of the periodic problem on a domain length of 2L (whetherthe domain is actually symmetric is irrelevant, because of translation-invariance)A.1.2 Converting a Periodic Problem to a Floquet ProblemNow, if we have an eigenvalue problem with periodic boundary conditions on a domain [a, b]with length S, then we can partition the domain into N subintervals of equal length as[a, b] =N−1⋃j=0[xj , xj+1] ,180and consider the following complex-valued problem with the following Floquet-type boundarycondition on each subinterval:z φφx∣∣∣∣∣∣∣x=xj= φφx∣∣∣∣∣∣∣x=xj+1, for j = 0, 1, . . . , N − 1 ,where z is a complex number to be determined.Then, by collapsing the N conditions, we find thatzN φφx∣∣∣∣∣∣∣x=x0=a= φφx∣∣∣∣∣∣∣x=xN=b.Thus, periodicity requires zN = 1, and hence z is determined by the N -th roots of unity aszj = e2piij/N , j = 0, 1, . . . , N − 1 ,which yields N possible choices for the multiplier z.Finally, if the problems restricted on each subinterval are exactly the same (i.e. the functionsinvolved defining L are identical on each subinterval), which occurs when we consider thestability of a symmetric, equally-spaced multiple spike pattern subject to a S-periodic boundarycondition, then we only need consider the Floquet eigenfunction that satisfiesLφ = λφ , −` < x < ` ; φ(l) = zjφ(−l), φx(l) = zjφx(−l) ,where 2lN = S, so that l = S/(2N).A.1.3 The Floquet Eigenvalue Problem for the Stability of a K-spike Sym-metric Pattern with Neumann Boundary ConditionsFor our problem defined on the domain [−L,L] with length 2L with K spikes, we first applyan even reflection as described above, which yields the periodic problem with S = 4L having181N = 2K spikes. Therefore, we look for eigenfunctions that satisfyLφ = λφ , −` < x < ` ; φ(l) = zjφ(−l), φx(l) = zjφx(−l) ,with l = S/(2N) = 4L/(4K) = L/K, andzj = e2piij/N = epiij/K , j = 0, . . . , N − 1 ,which when solved, gives N different modes depending on j to the original problem.We end this discussion with a lemma which we use frequently to solve a BVP problemresulting from imposing Floquet boundary conditions.Lemma A.1. The following BVP with a jump condition at x = 0 and, subject to Floquetboundary conditions on the interval [−`, `], formulated asηxx = 0 , −` < x < ` ; d0 [ηx]0 = d1η(0) + d2 ,η(`) = zη(−`), ηx(`) = zηx(−`) ,is solvable with the central value η0 = η(0) given byη0 = d2[d02`(z − 1)2z− d1]−1.In particular, if z = zj = epiij/K , where 0 ≤ j < K is an integer, thenη0 = − d2d0 (1− cos(pij/K)) /`+ d1 .Proof: Let η0 = η(0). The solution of the ODE is continuous but not differentiable at x = 0,with the general formη(x) =η0 +A+x if 0 < x < ` ,η0 +A−x if − ` < x < 0 .182Upon imposing the Floquet boundary conditions we obtainA+ = zA− , η0 +A+` = z (η0 −A−`) = zη0 −A+` ,so that A+ = (z−1)2` η0. Then, upon imposing the jump condition we getd1η0 + d2 = d0 [ηx]0 = d0 (A+ −A−) =d0η02` (z − 1)(1− 1z),which determines η(0) = η0 asη0 = d2[d02`(z − 1)2z− d1]−1, (A.1)as was claimed. If we set z = epiij/K , we obtain(z − 1)2z= (z − 1)(1− 1z)= −(1− z)(1− z¯) = −2− (z + z¯) = −2 (1− cos(pij/K)) .A.2 Properties of the Local Operator L0 in One Spatial Dimen-sionConsider positive solutions on −∞ < y <∞ tow′′ − w + wp = 0 , (A.2)which vanish as y → ±∞. It is well-known that the unique positive solution to this problem isw(y) =((p+ 1)2 sech2[(p− 1)y2])1/(p−1). (A.3)In particular, for p = 2 we havew(y) = 32sech2(y2), (A.4)183while for p = 3, we getw(y) =√2sech(y) . (A.5)We define the linear operator obtained from linearizing around this solution byL0[φ] := φ′′ − φ+ pwp−1φ . (A.6)We refer to L0 as the local operator in the context of our NLEP stability analysis.First, we recall a few well-known results for the discrete spectrum of L0. By convertingthe differential operator to a hypergeometric equation, a more precise statement for the localeigenvalue problem L0ψ = νψ is that it has exactly two discrete eigenvalues when p ≥ 3 is aninteger, given byν0 =(1 + p2)2− 1, ψ0 = w2 > 0 ; ν1 = 0, ψ1 = w′ , (A.7)and there are no other discrete eigenvalues in −1 < ν < 0. When p = 2, then ν2 = −3/4 is alsoan eigenvalue. The proof of these results is given in Proposition 5.6 of [13].Secondly, we list several algebraic properties when L0 acts on functions of w.Lemma A.2. The local operator L0 satisfies the following identities(i) L0[w] = (p− 1)wp, so L−10 [wp] = wp−1 ,(ii) L0[yw′] = 2(w − wp), so L−10 [w] = 12yw′ + 1p−1wp,(iii) L0[ws] = (s2 − 1)ws , iff s = p+12 ,(iv) L0[ws] =(s2 − 1)ws + (p+ s) (1− 2sp+1)ws+p−1 , for any s > 1.(v) For s > p− 1 and s 6= p− 1 + p+12 , we have the reduction formulaL−10 [ws] =p+ 1(s+ 1)(2s− 3p+ 1)[(s− p)(s− p+ 2)L−10 [ws−p+1]− ws−p+1].Proof: (i) and (ii) can be verified readily by calculatingL0[w] = w′′ − w + pwp = (p− 1)wp ,184as well asL0[yw′] = 2w′′ + yw′′′ − yw′ + y(pwp−1w′) ,= 2(w − wp) + y(w′′ − w + wp)′ ,= 2(w − wp) .Now we show (iii). We multiply A.2 by w′ and integrate to getw′22 −w22 +wp+1p+ 1 = 0 ,which yields that (w′)2 = w2 − 2p+ 1wp+1 .Thus, we obtain thatL0[ws] =(sws−1w′)′ − ws + pwp+s−1 ,= s(s− 1)ws−2 (w′)2 + sws−1w′′ − ws + pwp+s−1 ,= ws−2(s(s− 1) (w′)2 + sww′′ − w2 + pwp+1) ,, ws−2Tp,s[w] . (A.8)Then, using w′′ = w − wp and (w′)2 = w2 − 2p+1wp+1, we getTp,s[w] = [s(s− 1) + s− 1]w2 + [−2s(s− 1)/(p+ 1)− s+ p]wp+1 ,=(s2 − 1)w2 + 1p+ 1(p2 − (s− 1) p− 2s2 + s)wp+1 ,=(s2 − 1)w2 + (p+ s)(1− 2sp+ 1)wp+1 . (A.9)This shows (iv), and it is now obvious that the second term in (A.9) vanishes iff s = (p+ 1)/2,which establishes (iii) as claimed.Finally for (v), we proceed by a direct computation to getL0[ws−p+1] =[(s− p+ 1)2 − 1]ws−p+1 − s+ 1p+ 1(2s− 3p+ 1)ws .185So taking L−10 on both sides and rearranging yields the desired result. Observe that 2s− 3p+1 = 0 exactly when s = p − 1 + (p+ 1)/2. Moreover, when s = p, we recover (i), given byL0[w] = (p− 1)wp. In other words, (iii) means that for s > 1, the principal eigenvalue for L0 isν0 = s2 − 1 > 0 ,with ws being the corresponding eigenfunction. Since we have neither assumed p and q tobe integers, in fact, for any p > 1, we have s = (p+ 1)/2 and w(p+1)/2 is an eigenfunctioncorresponding to the principal eigenvalue ν0 =(1+p2)2 − 1 > 0. In particular, for the casesp = 2 and p = 3 we get ν0 = 54 and ν0 = 3, respectively.The identities (i) and (ii) are useful in the analysis of a class of nonlinear function definedimplicitly by L0, given byF(λ) =´wm−1(L0 − λ)−1wp´wm,when m−1 = s. This function appears frequently in the NLEP analysis of spike stability wherethe eigenvalues can be found to satisfy the equationg(λ) = C(λ)−F(λ) = 0 ,where C(λ) is usually some rational or transcendental function of λ defined in terms of themodel parameters of the reaction-diffusion system.The identity (iii) turns out to be very powerful and gives tremendous simplifications to theNLEP analysis, which effectively allows F(λ) to be rewritten explicitly in a closed form. Inthis way the study of the roots of g(λ) = 0 will only require finding roots of some explicittranscendental equation in the eigenvalue parameter.186A.2.1 Applications to Explicitly Solvable NLEPProposition A.3. Consider the NLEPL0[Φ]− χwrˆ ∞−∞wsΦdy = λΦ , −∞ < y <∞ , Φ→ 0 as |y| → ∞ ,where s = (p+ 1)/2 and r > −s is arbitrary. Then, any unstable eigenvalue must be root ofλ = ν0 − χˆ ∞−∞ws+rdy =[(p+ 12)2− 1]− χˆ ∞−∞wp+12 +rdy . (A.10)Proof: We apply Green’s identity to ws and Φ, using decay properties of ws and Φ as |y| → ∞together with integration by parts. This yields0 =ˆ ∞−∞wsL0[Φ]− ΦL0[ws]dy ,=ˆ ∞−∞(ws[λΦ + χwrˆ ∞−∞wsΦdy]− Φν0ws)dy ,=(ˆ ∞−∞wsΦdy)(λ− ν0 + χˆ ∞−∞ws+rdy).Therefore, when´∞−∞wsΦdy 6= 0, we obtain λ = ν0 − χ´∞−∞ws+rdy as claimed.Moreover, the condition´∞−∞wsΦdy 6= 0 fails only if the NLEP was in fact the local eigen-value problem, given by L0[Φ] = λΦ. Hence´∞−∞wsΦdy = 0 implies that Φ is an eigenfunctionof L0. However, since the principal eigenfunction of L0 is one-signed, the only eigenpairs of L0that satisfy´∞−∞wsΦdy = 0 are either the translation mode λ = 0 where Φ = w′ or possibly aneigenfunction whose eigenvalue is in −1 < λ < 0. Therefore, the transcendental equation forthe eigenvalue given above will characterize any unstable eigenvalue of the NLEP. A.3 Miscellaneous FormulasA.3.1 Formulas for the Lq-Norm of the Ground State w(y): p = 3 CaseLemma A.4. For p = 3, w(y) =√2sechy. We have the general formula for its q-th integral:187Iq =ˆ ∞−∞wq(y)dy = 23q/2−1B(q2 ,q2) = 23q/2−1 Γ2(q/2)Γ(q) , (A.11)where B(s, t) is a beta function. In addition, we have the reduction formulaIq+2 =2qq + 1Iq . (A.12)Proof: Iq/2q/2 = 2q´∞−∞dy(e−y+ey)q = 2q´∞−∞(e−2y)q/2(e−2y+1)q dy. So we substitute t = e−2y to obtainIq/2q/2 = 2q−1ˆ ∞0tq/2−1(1 + t)q dt = 2q−1B(q2 ,q2) ,where the last identity for the Beta function can by shown from the definition by the substitutions = 11−t − 1 = t1−t , ds = 1(1−t)2dt = (s+ 1)2 dt. In this way, we calculate thatB(x, y) =ˆ 10tx−1 (1− t)y−1 dt ,=ˆ 10(t1− t)x−1(1− t)x+y−2 dt ,=ˆ ∞0sx−1 (s+ 1)−(x+y−2) (s+ 1)−2 ds ,=ˆ ∞0sx−1(1 + s)x+yds .The reduction formula can then be seen as a result of the general formula, but can also beeasily obtained from a simple integration by parts. The general formula is most useful for real numbers 0 < q < 2 and for general q we canapply the reduction formula first. For integers q = 1, 2, 3, 4, one can readily recover the followingvalues, which were frequently used in this thesis: I1 =√2pi, I2 = 4, I3 =√2pi, I4 = 16/3.A.3.2 Formulas for the Lq-Norm of the Ground State w(y): General CaseFor general p, the formula of the ground state is:w(y) =((p+ 1)2 sech2[(p− 1)y2])1/(p−1).188Therefore, we computeIp,q ≡ˆ ∞−∞wq(y)dy =(p+ 12)q/(p−1) ˆ ∞−∞(sech2[(p− 1)y2])q/(p−1)dy .Let u = tanh (p−1)y2 , then du =p−12 sech2[(p−1)y2]dy. We calculate thatIp,q =(p+ 12)q/(p−1) (p− 12)−1 ˆ 1−1(1− u2)q/(p−1)−1du .With the substitution t = (1 + u)/2, we further obtain thatIp,q =(p2 − 1)2(p+ 12) qp−1−1 ˆ 10[4t(1− t)] qp−1−1 dt ,= (p2 − 1)2 (2p+ 2)qp−1−1ˆ 10tqp−1−1(1− t) qp−1−1dt ,= (p2 − 1)2 (2p+ 2)qp−1−1B( qp− 1 ,qp− 1) ,= (p2 − 1)2 (2p+ 2)qp−1−1Γ2( qp−1)Γ(2qp−1) . (A.13)This expression also implies a reduction formula:Ip,q =p2 − 12 (2p+ 2)( q−(p−1)p−1 −1)+1Γ2( q−(p−1)p−1 + 1)Γ(2(q−(p−1))p−1 + 2) ,= Iq−(p−1)(2p+ 2)(q−(p−1)p−1)2(2(q−(p−1))p−1 + 1) (2(q−(p−1))p−1) ,= Iq−(p−1)(p+ 1)q − p+ 12q − p+ 1 . (A.14)For p = 3, the expression (A.13) simplifies toI3,q = 4 · 8q/2−1Γ2( q2)Γ(q) = 23q/2−1 Γ2(q2)Γ(q) ,while (A.14) yieldsI3,q = I3,q−24(q − 2)2q − 2 =2(q − 2)q − 1 I3,q−2 .189in full agreement to (A.11) and (A.12).We remark that the integral Ip,q has a closed form with simple surds and factorials whenqp−1 is an integer, and also whenqp−1 = n+ 1/2 for some integer n, but with an extra factor ofpi (because Γ(12) =√pi). In all cases, Ip,q can be reduced to a formula of surds and factorialsexcept for a factor of Γ(qc) where qc = mod (q, 1) is the non-integral part of the number q.190


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