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Man as predator : qualitative behaviour of a continuous deterministic model of a fishery system Huberman, Gur 1976

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MAN AS PREDATOR: QUALITATIVE BEHAVIOUR OF A CONTINUOUS DETERMINISTIC MODEL OF A FISHERY SYSTEM by Gur Huberman B.Sc, Tel Aviv University, 1975 . A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE THE FACULTY OF GRADUATE STUDIES (Department of Mathematics) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December, 1976 (c*) Gur Huberman, 1976. in In presenting th is thes is in p a r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary shal l make it f ree ly ava i lab le for reference and study. I fur ther agree that permission for extensive copying of th is thesis for scho la r ly purposes may be granted by the Head of my Department or by his representat ives . It is understood that copying or pub l i ca t ion of th is thes is for f inanc ia l gain sha l l not be allowed without my writ ten permission. Depa rtment The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date f\^V m > i(a) Man as Predator: Q u a l i t a t i v e Behaviour of a Continuous Deterministic Model of a Fishery System ABSTRACT A global p o r t r a i t of the phase plane i s obtained for any acceptable values of the parameters. 3 d i f f e r e n t structures of the phase plane are recovered. The f i r s t predicts an eventual collapse of the f i s h e r y . The second predicts an unstable l i m i t cycle and an eventual s t a b i l i t y of solutions which s t a r t inside the l i m i t c y c l e . The l a s t structure predicts 2 possible stable e q u i l i b r i a , one with high catch rate, and the other one with no catch. Each structure corresponds to a d i f f e r e n t domain i n the parameter space. The boundaries of these domains are found by s o l v i n g the relevant d i f f e r e n t i a l equation for a saddle-to-saddle separatrix i n the phase plane. This procedure u t i l i z e s regular perturbation methods. i(b) TABLE OF CONTENTS PAGE INTRODUCTION "' 2 I. THE MODEL 6 I I . PRELIMINARIES ' . 9 I I I . THE TRAJECTORIES AFFILIATED WITH THE SADDLE POINTS . . . 17 IV. A SOLUTION FOR b . ' 21 V. THE GLOBAL PORTRAIT OF THE PHASE PLANE 31 VI. INTERPRETATION 38 APPENDIX A: POSSIBLE VALUES OF R AND Q . . . . . . . 39 APPENDIX B: THE NUMERICAL SCHEME 41 i i FIGURES PAGE Figure 1. The e q u i l i b r i a i n the phase plane 3 Figure 2. A preliminary d e s c r i p t i o n of the motions i n the phase plane 15 Figure 3. T 2 and T 3 18 Figure 4. T £ > T 3 19 Figure 5. The phase plane p o r t r a i t when b < 32 Figure 6. The phase plane p o r t r a i t f o r x^ < b < b 34 Figure 7. D i v i s i o n of the phase plane by • 35 Figure 8. The phase plane p o r t r a i t f o r b > b . 36 Figure 9. The number of roots of g(x) as a function of various values for R and Q . . . 40 i i i ACKNOWLEDGEMENTS I wish to express my appreciation to my supervisor, Dr. D. Ludwig, for his original suggestion of my thesis topic as well as for his encour-agement and guidance during the preparation of this thesis. I would also like to thank Dr. F. Wan for many helpful discussions considering the final draft of this work. LEAF 1 OMITTED IN PAGE NUMBERING. 2 INTRODUCTION Commercial e x p l o i t a t i o n of animate resources i s one of man's oldest occupations, already mentioned i n Genesis. In recent years an ac c e l e r a t i n g decline i n the p r o d u c t i v i t y of important f i s h e r i e s was observed. Clark [1] mentions the great whale f i s h e r i e s , Grand Banks f i s h e r i e s and the Peruvian anchovy f i s h e r y . A model of the dynamics of animal population and human e f f o r t to harvest i t i s analysed i n t h i s work. The model i s continuous and deter m i n i s t i c . For a non-harvested population i t assumes l o g i s t i c growth perturbed by predation. This implies that there are two possible e q u i l i b r i a f o r the population: a very low one and a high one. Harvesting may drive the population to the low equilibrium, where no harvesting i s worthwhile. For a harvested population we subtract the harvest from the natural growth, and obtain the equation for the dynamics of the population. For the human e f f o r t we assume that the rate of change i n the e f f o r t i s proportional to the net income. This r e f l e c t s the fact that the hunted population i s a common property. Everybody has free access to commercial hunting (or fi s h i n g ) and therefore the t o t a l human e f f o r t increase i s proportional to the t o t a l net income (negative net income and negative increase are not excluded). The net income i s the difference between the t o t a l revenue ( i . e . the harvest) and t o t a l cost. The model i s presented i n Section I. Section II i s a preliminary discussion of the equations obtained i n Section I. They are scaled and brought to the form 3 dx , . — = x g(x) - xy , dt ay(x - b) , g(x) = R(l - ^ ) ^ 1 + x The quantities x and y are proportional to the population density and the human effort respectively, and hence x,y >_ 0. g(x) has three positive zeros x i = 1, 2-, 3 . The possible equilibria of the system are E Q = (0,0); E ^ = (x^,0) i = 1, 2, 3, ; E ^ = (b,g(b)). E ^ is asymptotically stable, We E „ and E » are saddle points and E , is i f (b) 2 3 4 ^ L I • dx _ • unstable > 0 de denote by X q the point where (x) = 0, and then give a preliminary description of the motions in the phase plane. They are summarized in the following figure: Figure 1. The equilibria in the phase plane. 4 Section II concludes with an indication of the main problem of the work: the completion of the phase plane portrait. The main clue for this is the information about T 2 and T^ - the separatrix which goes to E 2 and the one which leaves E^, respectively. In Section III these T^ are defined and the problem arises: for which value of b is a saddle-to-saddle separatrix obtained, i.e. when do the T^ intersect? This b is found in Section IV. Approximate solutions for the saddle-to-saddle separatrix are obtained, and they imply a unique value of b (denoted b) . Of course, b depends on R, Q, a. This b is found with two alternative assumptions: 3/2 (i) a«Q is large, or ( i i ) a*Q^^ is small. 1 The interesting thing i s that always b > x^ . This enables us to draw only three distinct phase plane portraits and eliminate other p o s s i b i l i t i e s . These portraits - as given in Section V - are: (i) For b < X Q a l l the trajectories converge to . ( i i ) For X Q < b < b an asymptotically unstable limit cycle appears. Its existence is proved by the Poincare-Bendixon theorem. A l l the solutions which start inside the limit cycle converge to E^ . Those which start outside the limit cycle converge to E^ . ( i i i ) For b > b, E^ is the attractor for orbits which start under T 2 while .E^ is the attractor for orbits which start above T 2 . From these 3 structurally different portraits we see that given an i n i t i a l point (x(0), y(0)), the solution may or may not converge to E^ . What happens w i l l depend upon the parameters. In certain cases, a modification c 5 the parameters can avoid a collapse of the population (and the harvest). The a n a l y t i c approximations were accompanied and v e r i f i e d by a numerical scheme. It i s described i n the appendix. 6 I." THE MODEL This model is a deterministic continuous model which describes an animal population subject to human harvest. A system of two coupled ordinary differential equations is introduced. One equation describes population changes while the other describes changes in human effort to catch the animals. The equations are simple and therefore one should not expect them to f i t reality in every detail. They do not refer to any particular animal population, but mainly fish populations were in my mind during the work on this paper. The basic equation for the dynamics of the population is 4^  = NG(u) - H . (1.1) dx Where u is the population density, T is time, NG is natural growth rate (i.e. growth rate of the population with the absence of human harvest) and H is the harvest. This equation has been employed in Clark and Munro [2] and Smith [9]. The equation for a population which is not subject to human catch is j 2 ; .. g = NG(u) = r u u ( l - ^ ) - B 2 U 2 (1.2) u a + u The term r u(l - — ) is the right hand side of the logistic U u , . equation, often used by biologists (cf. McNaughton-Wolf [3]). It describes a growth which is exponential initially and then decays due to the finiteness 7 of the environment resources. Ku is the maximal possible population density. It is determined by factors such as limited food supply or space. 2 u The term -8 2 describes.an effect on the growth rate, a + u due to predation. This particular choice of the predation term represents a type III S-shaped functional response (cf. Holling [4]). According to Rolling, the effect of predation saturates at fairly low population densities, i.e. there is an upper limit to the rate of > a mortality due to predation. This implies-that — is small. Another K U feature of predation is a decrease in the effectiveness of predation at very low densities. This is attributed to searching and learning on 2 the predator's part. Finally, we have to remark that -B 2 "—J i s n o t a + u the only way to represent a type III S-shaped response. This particular form is chosen because of mathematical convenience. In order to incorporate the effect of human harvest we follow Clark-Munro [2] who give H(E,u) - the human harvest - the form: Y l Y2 H(E,u) = y u E , (1.3) where E is the human effort and Y. > 0 For reasons of convenience 1 — we set Yx = Y2 = 1 • (1.4) Combining (1.1), (1.2), (1.3), and (1.4) we obtain dT = ru u ( 1 - r> -e V^-T - Y U E • tt-*) u a + u 8 For the economic part we assume that c • Y , (1.6) where Y i s the net income y i e l d . Following Schaeffer [5] we assume that the t o t a l cost i s proportional to the e f f o r t . C = p • E . (1.7) This i s a common assumption among economists (cf. Clark Munro [2]). In general, p = p(E), but here p i s assumed to be independent of E. The t o t a l production has already been given by (1.3), (1.4) and when we combine t h i s with (1.7) and substitute i n (1.6) we obtain g = c YE(u - * > ' • • (1-8) Combining (1.5) and (1.8) we get the system u a + u — = c Y E(u - • The main aim of th i s work i s to study system (1.9). 9 II . PRELIMINARIES This section i s devoted to a s i m p l i f i c a t i o n of the problem and to the der i v a t i o n of some st r a i g h t forward r e s u l t s . S i m p l i f i c a t i o n i s done by s c a l i n g the variables and bringing (1.9) to the form x' = x g(x) - xy , y' = ay(x - b) , X X ' where g(x) = R ( l - —) - . Zeros and a maximum of g(x) are Q 1 + x found and then used i n a discussion on the e q u i l i b r i a and t h e i r asymptotic s t a b i l i t y . The section concludes with a preliminary d e s c r i p t i o n of the motions i n the f i r s t quadrant of the phase plane. The following sections complete t h i s d e s c r i p t i o n . II.1 Scaling We introduce the following parameters and q u a n t i t i e s : ar K c 2 a = - • Y ' a , b - -B- , ya 10 u x = — , a v - ^ F t = — T a In terms of these we have x' = x g(x) - xy » (II.1.1) y' = ay(x - b) , where ' = 4r J g(x) = R(l - £) —j • d t Q 1 + x 2 Since a is the density where predation saturation occurs and K K is the total capacity, Q = — is large. This will be used later, u a II.2 g(x) : its zeros and maximum g(x) may have either one or three zeros. We are interested only in the latter case. This implies certain restrictions on R and Q . They are discussed in the Appendix. We employ algebra to obtain: g(x) = R(I - h ~ — = P ( x ) .t * 1 + x Q(l + x ) where P(x) = x 3 - Qx2 + (1 + |)x - Q . To evaluate x^(i=l,2,3) we set 11 P(x) = 0 . . Hence x ^ - | x ^ + 1 - ± ( x ^ + x j ^ ) 1 * Therefore for 0 < eg £ R < -j w e have : i + A -4 Xl=^-P -rOCQ" 1) , x 2 = R / R + 0(Q _ 1) Since x = - Q + — ^ + = 0 , we obtain 3 R • x_ x- I 3 3 x^ x 3 = Q - \ + 0(Q _ 1) . An approximate value of X Q i s found as follows 2 SLS. = _ R + x ~ 1 , and d x Q ( i + x 2 ) 2 t <*o> - ° • a _ 2 + Q A _ 8R 2 R R / - Q Hence x_ = 0 2 Thus x Q « J\ - f / l + O C Q - 1 ) A In general: (x - x 1 ) ( x - x 2) = e ; x 1 ^ x 2 e . 2, 1 x " X2 1 x, + £ — - x, 1 X l " X2 1 x - x„ 2 12 II.3 Equilibria and their asymptotic stability From the equations: x' = x g(x) - xy , y' = ay(x - b) , we see that the equilibria in the first quadrant are: E Q = (0,0) , E, = E„ = E„ = E, = (x r 0 ) , (x2,0) , (x3,0) , (b,g(b)) E, occurs only i f 0 < b < x, or x 0 < b < x_. The restriction b > x„ means 4 y — — 1 1 — — 3 3 that even i f the biomass were at its maximal possible value, i t would not be worthwhile to make an effort to harvest i t . The restriction b < x 2 means that i t is worthwhile to make a harvesting effort in densities below x 2 - the collapse threshold. Neither case is realistic. We shall concentrate only on the case x 2 < b < x^ • In order to compute the asymptotic stability of the equilibria, we first compute the variational matrix of the system. M(x,y;b) = x gx(x) + g(x) - y ay -x a(x - b) Let >L be' M(x,y;b) evaluated at E.^  . Then we obtain 13 M, R 0 0 -ab -x. a(x 1 - b) M2 = X2 ' gx ( x2 ) 0 -x. a(x 2 - b) M = 3 x 3 • g x(x 3) 0 -x. a(x 3 - b) M4 -b gx(b) a g(b) -b 0 From the assumptions on g(x) and on b (x 2 < b < x3) we can draw the following conclusions: (1) EQ is a saddle point which attracts in the y direction and repels in the x direction. (2) E^ is a stable equilibrium (because § x( xj^ < 0; x^ < b) (3) E 2 is a saddle point (because Sx^x2^ > »^ x£ < that repels in the x direction. (4) E 3 is a saddle point (because S x( x3^ < 0; x 3 > b) which attracts in the x direction. (5) E^ is asymptotically stable (unstable) according to: 14 If g(b) < 0, then is asymptotically stable. If ^ g(b) > 0, then E^ is asymptotically unstable. Graphically, this means that i f b > x^ then E^ Is asymptotically stable; i f b < x^ then E^ is asymptotically unstable. II.4 Phase plane description We consider again the system x' = x g(x) - xy , y' = ay(x - b) . Along the axes: Along the y axis x' = 0 ; y' < 0 . Along the x axis y' = 0 ; sgn(x') = sgn g(x) '. In the interior of the first quadrant we have 5 distinct regi dy ay(x — b) to discuss. Considering -f- = — ,—r we can immediately t e l l the dx x g(x) - xy direction of the motion in each case.. 1. When x < x, and y < g(x) then 4^~ < 0 • 1 dx dy 2. When x 2 < x < b and y < g(x) then < 0 . 3. When b < -x < x. and y < g(x) then 4^  > 0 • 3 dx dy 4. When x < b and y > g(x) then > 0 i . 5. When x > b and y > g(x) then < 0 . At the boundaries of these regions we have (1) y = g(x) implies x'=0; y 1 > 0 (2) x = b implies y'=0; x' > 0 The motions i n the phase plane can 15 i f x > b and y' < 0 i f x < b . i f y < g(x) and x' < 0 i f y > g(x) be i l l u s t r a t e d by the following figure: Figure 2. A preliminary description of the motions in the phase plane. 16 At this stage we are ready to indicate the main problem of the work - the global picture of the phase plane. Given i n i t i a l x and y, we would like to know where the solution of ( I I . 1.1) goes as t -»• 0 0 . Clearly, when is asymptotically stable there are two possible answers E^ and E^ . But even when E^ is asymptotically unstable there is no reason to believe that a l l the solutions converge to E^ . The answer depends not only upon the in i t i a l data but also on the particular value of the parameters. In the next sections the asymptotic behaviour of the solutions will be discussed in the various domains of the parameter space. 17 III. THE TRAJECTORIES AFFILIATED WITH THE SADDLE POINTS A possible way to deal with the problem raised at the end of the last section is a division of the first quadrant into domains bounded by solution trajectories. A solution which starts in such a domain is destined to remain there because two solutions cannot intersect. For the same reason there are only two possible boundaries of this type: periodic solutions and orbits which connect critical points. E^, E 2 and E^ are already known to be connected by one orbit, namely the x axis. There exists another unique orbit which goes to E 2 and another one which goes from E^ . Motivated by this^one is led to investigate properties of these trajectories. Since the right hand side of the system (II.1.1) is twice continuously differentiable we may use the following theorem, which is a slightly modified version of a theorem given by Coddington-Levinson [6]. Theorem. Consider the system d r \ X ' f n(x,y)' dt > y . f 2(x,y) for which the following conditions hold: (i) (x0'^0^ * S a sadd-'-e P°int:' 2 (ii) f , f 2 e C in the neighbourhood of (x 0,y Q) . Then there exist exactly two orbits tending to (xQ,y^) as t -*• °° The angle between these two orbits is 180°, and any orbit starting sufficiently near either of these orbits in the neighbourhood of (xo>vo^ 18 tends away from them as t -*• 0 0 . A corollary of the theorem is that i f (i), (ii) hold, there exist exactly two orbits tending to (X Q , V Q) as t . The angle between the orbits is 180°, and any orbit starting sufficiently close to either of these orbits and to (x ,y ) tend away from them as t •+ -<*> . 0 0 At E 2 = (x2,0) we already know that the latter orbits l i e on the x-axis. We define T 2 as the orbit of the former type which lies above the x-axis. X Figure 3. T 2 and T 3 T 2 must lie above g(x) near E 2 . Indeed, otherwise x' > 0 along T 2 near E 2, and x > x 2 on T 2 for T 2 under g(x) imply that T 2 does not tend to E 2 . Therefore T 2 lies in region 4 near E 2 . At E Q we define T^  - the trajectory leaving E^ . In a 19 similar way to T 2 we can show that lies above g(x) and lies in region 5 near ,. T 2 lies in region 4 near . Therefore, either i t crosses the line x = b or i t converges to E^ as t ->-<», or both happen (if T 2 spirals around E^ in a converging fashion). T^ lies in region 5 near E^ . Hence, either i t crosses the line x = b or it converges to E^ as t -> °° or both. If both T 2 and T 3 cross x = b we say T 2 > T 3 i f T 2 crosses "higher" than T 3 and T 3 > T 2 in the opposite case Figure 4. T 2 > T 3 20 If one T. does not cross the line x = b, then T. > TJ  X0 J h At least one of them must cross x = b . Otherwise both converge to E^, but a simple non saddle equilibrium cannot attract and repel at the same time. There is another possibility yet: both and T^  cross x = b at the same point. Then - by the uniqueness of the in i t i a l value problem - they are identical. 1^ ~ ^ 3* a n c* this is a saddle-to-saddle separatrix. This trajectory connects and E^, so i t can serve us in the way described at the beginning of this section. Hence the motivation to find the value of b for which such a separatrix is obtained. From \ here on we shall refer to this as the solution for b, and denote i t by b . 21 IV. A SADDLE TO SADDLE SEPARATRIX IV.1 Introduction In this section we shall concentrate on the equation dy_ = ay(x - M (IV.1.1) dx x g(x;Q) - xy . We shall find the b for which exists a solution y which satisfies y(x ) =0 i = 2, 3 (IV.1.1) ( j i.e., there is a saddle-to-saddle separatrix. Since (IV.1.1) is too complicated to integrate exactly, we shall use perturbation method. The crucial parameter was discovered in two steps: (i) Clearly, the larger a is, the larger is. Assuming |dx| "large a" we obtained an approximate solution and then compared i t with numerical results. It appeared that the approximation was good also for -2 fairly small values of a such as 10- . (ii) Observing that x„ = is where structural changes occur* we scaled: In fact, i t will be shown later that a Hopf bifurcation takes place near b = x^ . 22 x B = — Now the new form of (IV.1.1) is: (IV.1.3) dY = ai/Q (u - B) • Y du u(G - Y) (IV.1.4) Here Y(u) = y G(u;Q) = R(l - -± ) /Q u/Q 1 + Qu We tried a parameter of the form P = aQm and the following expansions for the saddle-to-saddle separatrix and B: For small P , Y ( s )(u;Q) = Y£ s )(U;Q) + PYJs)(u;Q) + P 2Y^S )(u;Q) + . g(s) = B ( s ) ( Q ) + p B ( s ) ( Q ) + p 2 B ( s ) ( Q ) + ^ ^ (IV.1.5) For large P , Y U )(u;Q) = Y<0(u;Q) + P ^ ^ ( u j Q ) + p" 2Y^ ) (u;Q) + ... , (IV.1.6) B ( i ) = B<A> (Q) + ?-h[l\Q) + P"2B<£)(Q) + ... . ** X i b Following this, u.=-^- j = 0, 1, 2, 3 ; B= — 2 3 (s) BQ (Q) appears to be 0 ( 1 ) as Q -> » and m will be chosen ( s) so that also B^  = 0 ( 1 ) as Q -»• °° . This will indicate that (IV.1 .5 ) is asymptotic. It appears that m = 3 / 2 is the appropriate m . IV.2 An approximate solution for small P We substitute (IV.1 .5 ) in (IV.1 .4 ) and obtain: dY<s) dY<S) . dY<s) -r^-—+ P -r^- + P —— + ... (IV.2 . 1 ) du du du a Q^(Y<s) + PY<s) + P2Y^S) + ...)(u - B<s) - PB<S> - P2B<S) u(G(u) - Y<s)- PY<s) - P2Y^S) - ...) At this point we have to decide i f aQz is "small" or "large". This is k necessary in order to simplify the equation. Certainly, i f m _> \, aQ is small. We assume this, and will show later that m = 3 / 2 . Assuming that aQ2 is small we can balance the two sides of (IV.2 . 1 ) by: (I) setting Y^s)(u;Q) = 0 , or (ii) setting Y^s)(u;Q) = G(u;Q) . (s) We know that Y (u;Q) lies above G(u;Q), so we reject (i) and set Y<s)(u;Q) = G(u;Q) . (IV.2.2) Now the equation has the form: (s) (s) dG + p ^ l + P 2 ^ 2 + du du du (IV.2 .3 ) Q V m(G + PY^ } + P2Y^S) +...)(«- B<s) - PB<s) - ...) u(Y<s) + P Y < 2 ) +P 2Y^ S ) + ...) 24 Formally, we expand the right hand side: ,<s) ^ „2v(s) (s) (s) 2 (s) (G + P Y ^ + P Y^' + ...)(u - B j D / - PB 1 W / - PB ...) u ( Y C s ) + P Y ( S ) + P2Y^S) + ...) Q ^ C G + PY< s ) +P 2Y^ S ) + ...)(«- B<s) PB (s) P 2B^ S ) - ...) Y(s) Y(s) 2 3 ^ — (G + PY1 ( S ) + P M S ) + u Y, 1 I ' ,(s) , ) ( t f - B 0 W -PB^ S ) - ...) ( 1 - P ,(s). _2 ,(s) ,(s) ,(s) ,(s) Q/*-mG(u - B<s)) ,(s) ,(s) ,(s u-Y (s) u Y _ P2 ( Y ( 8 ) ( U . B ( 8 ) ) _ Y ( S ) ( 8 ) _ , ( S ) F v(s) U 2 l u *0 ; Y l B l u Y, ,<s) Br'G - - ^ y ( Y { s ) ( u - B^') ,(s) ,<s) ,(s) ,(s) 21 G(u - BQ) + This expansion is valid only i f — is bounded. 1 Now we substitute this in (IV.2.3) and formally equate the coefficients of the respective powers of P . For P° we have 25 dG du ^ G(u - B<s)) u Y (s) Y^s)(u;Q) C> m • G(u;q) ' [u - B^S)(Q)] u - f (u;Q) (IV.2.4) (s) This Y N (u ;Q) is continuous except - maybe - at u = U Q ( Q ) , where dG — (un;Q) = 0 . To avoid a singularity there we set du u. BoS ) = V°-> ' (IV.2.5) dY For P we have (s) , ,(s) 1 _ " rY(s)r„ R(s)s (s) _2 _ r f „ R ( S M du~~ " ~U~ Y7 [ Y I ( U " B O } " B I G " (s) G ( u " B0 } J 1 ( Y ( S ) ) 2 Y(s) = + 2 G ' U , ( Y1 } d Y l _(s) y(a) G ~ B l Y l u - B (s) (IV.2 Now to avoid singularity we set B^S)(Q) = Qm"S-Yis)<yQ> d Y i r GTu^Q) du~ ( U 0 ; Q ) (IV.2.7) fq") (s) From (IV.2.6) we see that Y^  (u) = H(u) • Y^  (u) where H(u) Y<s)(u) a bounded function on [u.,u,] . Therefore —r~\ = H(u) is bounded 1 Y|S;(u) there. This indicates that the geometric expansion was valid. To get a more explicit form of (IV.2.7) we use l'Hopital's (s, *V rule to obtain Y^  (u^jQ) and — — (u^ ;Q) and substitute in (IV.2.7) to obtain: 26 G(u ;Q)[2 G"(u ;Q) + u G'"(u ;Q)] . B l( s ) = °- ~ ^ ~ Q , (IV.2.8) 1 2 U Q ( G " ( u 0 ) ) J where ( )' - ^ We recall that u Q = ~p+ o(l) , vR G(uQ;Q) = R + o(l) , 3/2 G"(u ;Q) = - 2R_ + 0 ( Q-3/2 } ^ Q2 G'"(u ;Q) - 0(Q- 3 / 2) . Therefore 3/2-m B i 8 )(Q) = a - B ! - + o ( 1 ) • This gives us two things (i) m has to be | in order that B^s) (Q) = 0(1), Q « . This is required to make the expansion (IV.1.5) asymptotic. (ii) B^S)(Q) > 0, and therefore B ( s )(Q) > u Q for large Q . Th will serve us later. The continuation of the expansion is similar to what was done for the 0(P) terms. For 0(P n), n > 1 we have d Y(s) y(s) « * U ' YiS)-^- ±) ' 6 ' [U " B 0 S ^ " V l G + (u - B Q) • 1 y(s) Y(s) Y(s) Y Y H , Y(s) Y(s)._2 3 n-1 ( _2 . -Qzi) , H1 ( Y0' n-1' (s) ' Y ( s ) (s) ) + W n-1' Y ' "' ' ' Y 1 1 1 27 Y2 Yn-1 where H^ ,!^  are polynomials in Yg, Y^  — , —^— which (s) vanish at u^i . From this we calculate the expressions for Y^  , B ( s ) . Y ( s ) will automatically satisfy Y ( s )(u.) =0, 0 = 2, 3, and n n } ' n l . (s) (s) B , is determined so that Y (u) has no singularities in [u„, u ] . n-l n z j IV.3 An approximate solution for large P . (I) For large P we expect y (u) to be large, so we scale i t : (SL), v Let z(u) = y p U J (IV.3.1) Then we assume the following expansions: z(u;B a )) = Z o(u;B^ £ )) + p" 1Z 1(u;B^ ),B^ )) + p- 2Z 2(u;B^,Bf ),B^ )) + B U ) o B < « + > + P" 2B^ + . (IV.3.2) We denote the solution which corresponds to T^ by Z ^ . Substituting (IV.3.1), (IV.3.2) , (IV.1.3) in (IV.1.1) yields: d Z ^ 1 d Z l ( 1 ) ( Z ^ + rh^ + ...)(u - B<» - p"V*> -...)' + P — ± + . . . = — — du du - Qu(Z^l} + P 1Z^ l ) + ... - P 1G) (IV.3.3) Using the geometric series expansion we obtain: 28 dZ du (i) , 7 ( i ) n -1 d Z i + P — - — + - l B du U) • - U K ( U - BQ } -Qu (u - BQ ) u + . . . (IV.3.4) This expansion is valid only i f — is bounded. It will be shown G U) G later that — i s bounded in [ U 2' BQ ^ A N D (3)~ i s bounded i n [B^,u 3] Along with (IV.3.4) we must satify 0 Z j i ) ( u i } = ° 3 = 0, 1, 1 = 1, 2 (IV.3.5) Formally, we equate the coefficients of equal powers of P and solve term by term. Hence Z 0 ± ) ( u ; B 0 O ) = ^ [ B 0 1 } l n u~ - ( u - u i ) ] i = 2, 3 (IV.3.6) i bounded in [U2,BQ is we Q Now we can show that — ^ y G(u) g 1 ( U Z } ° , „U) — = 7—r- < » (we assume Bn < A ) ] . At u = u 2 ) " have lim —-prr j u->u2 Zj (u) u 2 - BQ -Qu > u2) dZ (2) 6 > o . — r — > o du Therefore ? 9 ^ is bounded for u e [u„,6] for some Z 0 < U ) (a) (2) (2~) ( for u E [u2,Bg j] and ZQ '-(u^ ) = 0 implies Z^ ' (u) >_ e for u e [<5,Bg and therefore i - s bounded for u e [6,B^^] . The final conclusion z0 V) ( A ) -29 is that o i ^  * S bounded ^ o r u E tu2>**o^] • ^n a similar way we can Z0 ( u ) treat Z ^ 3 (u) to show that — * s bounded in I B Q ^ >u3^ • Z0 ( u ) Now to find BQ we equate (Bg ;BQ ) Z Q 0 '0 ' and obtain the unique solution of this algebraic equation (A) U3 " U2 B o = i 1 (IV.3.7) 0 In u^ - In u^ The next thing to be shown is that B^  > U Q . This was done numerically and for large Q we have u 3 - u 2 = /Q + 0(1) Q ->- » In -^ T3- = In ^- + 0(1), Q -> «, where x 9 = 0(1) Therefore B, Hence U3 = i„S_ u2 x2 U ) _ Vq 0 x 2 u o = 0(1) u0 for Q While u n , Q -> In order to continue this procedure one has to equate the higher order terms. In general, from (IV.3.4) we shall have: d Znl ) _ R ( G> Z 0 X ) > Z 1 X ) > •••» Zn-1 ; BQ £ ) > ••" "n-l* du Qu QuZQ(u) Where H is a polynomial such that — is continuous in [u„,u ] Z0 30 Since Z(l^(u.) =0 i = 2, 3 we shall have z«>(„> -ru I i— B<° H(G(s), Z^Cs), Z^Cs); B(Q° , B<*>) Qs Q sZ^ }(s) ds i = 2, 3 (2) (3) Now we require (v) = (v) for some v e [ ^ j U ] and obtain the expression for B from n u. 1 Q u„ n H S s • ZQ(s) ds = 0 IV.4 Summary Approximate solutions for the saddle-to-saddle separatrix were obtained in this section. One solution is based on the assumption that 3/2 3/2 aQ is a small parameter and the other assumes that aQ is large. A unique value of b for which the existence of a saddle-to-saddle separatrix is possible was recovered in each case. In the former case "(s) (s) this b has the expansion b = X Q + Pb^ + ... . The expression i As) (s) b l B^  = — — is given in (IV.2.8). In the latter case we have ~ 0 + P b l + ••• W h e r e b0 - ln x 3 - In x 2 ' The important conclusion is that in both cases, b > X Q . This will serve us as the main tool in the next section. * (2) (3) In fact at this stage we already know that Z^  (u) = Z_. (u) 0 <^  j < n 31 V. THE GLOBAL PORTRAIT OF THE PHASE PLANE At this stage we aire ready to answer the question which was asked at the end of Section II. We use the ideas suggested at the beginning of Section III. When T 2 < T^y (which starts at E3) must go to E 1 . As a result we have a domain bounded by solution trajectories. The boundaries are the x-axis and T^ , and any solution which starts inside this domain cannot leave its boundaries. When T 2 > T^ , T 2 serves as a part of the boundary, and similar conclusions will be drawn. Throughout this section we assume b > X Q . This has been shown for "large P" and "small P" . For intermediate values of P i t was verified by a numerical scheme. The details of the numerical scheme are given in the appendix. V.l T 3 > T 2 In this case x 2 < b <_ x Q or x Q < b < b . For x 2 < b < X Q there is only one asymptotically stable equilibrium in the phase plane, namely E^ . E^ is on the boundary of K - the compact set bounded by ' T^  and the x-axis. Therefore every solution (except those starting exactly at the equilibria or on T2) will converge to E^ . This conclusion holds for solutions which start inside K or outside i t in the first quadrant. / Figure 5. The phase plane portrait when b < X Q . Another possiblity, s t i l l in case (1), is that X Q < b < b . Following the formulation of Coddington-Levinson [6] we define L C T J ) as the negative limit set of T2, and T 2 as a negative semi-orbit of * T 2 ' Theorem. L(T 2) is a limit cycle. Proof. We use Poincare-Benedixon theorem as given by Coddington-Levinson. There i t refers to positive semi-orbits but i t can be applied also to negative semi-orbits, which is our case. L(T2)<r K which is a bounded set. The singular points in K are E j . , i = l , 2 , 3 , 4 . E^, E^ £ L(T 2) because they are asymptotically stable. E 2 £ L(T 2) because there are only two trajectories which converge to E.j as t -> and they are on the x-axis. E^ { L(T 2) because E^ = K T ^ ) by definition of T^  and there *i.e. T~ is obtained by starting somewhere on T 2 , and following the solution trajectory as time goes backwards. 33 is only one trajectory in the first quadrant which converges to as t -*- -°° . The conclusion is that L(T 2) contains only regular points and hence either (i) T is a periodic orbit, or 2 (ii) L(T 2) is a periodic orbit (a limit cycle). (i) is excluded because i f T 2 is periodic then T 2 is periodic. But we know that T 2 -* E 2 as t <*> and 4 E 2 . Therefore T 2 is not periodic, and hence a limit cycle exists inside K . E^ is asymptotically stable so the limit cycle is asymptotically unstable. Every solution which starts in the domain bounded by the limit cycle will remain there, and converge to E^ . Outside the closed set bounded by the limit cycle, a l l the solutions converge to E^ . The existence of a limit cycle for b near X Q can be shown also by noticing that at b = x^ a Hopf bifurcation takes place. We shall show this by using the Hopf bifurcation theorem as given by Howard-Koppel [7], Theorem. For b near x Q the system (II.1.1) has a one parameter family of solutions which lie in the neighbourhood of E^, and there are no other periodic solutions wholly in this neighbourhood. Proof. Near b = x Q the eigenvalues at E^ are X^(b) ± iX 2(b) where 34 b g (b) / A2(b) = 4 ag(b) - b g (b) At b = x Q, 8x(b) = 0 and therefore A^b) = 0; A2(b) 4 0 . We also have db b gX x ( b ) < 0 b=x 0 Another condition which is automatically satisfied is that the other eigenvalues are bounded away from the imaginary axis. We see that a l l the conditions of the theorem hold and hence the conclusion. As we can see, a structural change takes place as b becomes bigger than X Q . One feature of this change is the appearance of a limit cycle. By numerical methods i t was observed that its amplitude is small near b = X Q and increases as b gets larger. This increase takes place until T0 = T 0 and the limit cycle disappears. 1 - Figure '6. The phase plane portrait for X Q < b < b 35 is the only asymptotically stable eqilibrium in and E^ is the only asymptotically stable equilibrium in • Therefore a solution which starts in will converge to E^ and a solution which starts in K„ will converge to E. . 36 Figure 8. The phase plane portrait for b > b . Summary. We pointed at 3 structurally different portraits of the phase plane.in the interior of the first quadrant. The first portrait was obtained for b < X Q . Solutions which start at an equilibrium or on T^  stay there, while a l l the other solutions converge to E . . The second portrait was obtained for b e (Xg,b) . An unstable limit cycle appears. Solutions which,start on i t , on o r a t a n equilibrium stay there. Solutions which start in the interior of the domain bounded by the limit cycle converge to E, . All the other 37 solutions converge to E . . The last portrait was obtained for b > b , i.e. when 1^ > T^  Then solutions which start at an equilibrium or on T^  , remain there. Solutions which start under T^  converge to E^, and a l l the other solutions converge to E . A fourth portrait that might have occurred would have E^ as an asymptotically unstable equilibrium and an asymptotocally stable limit cycle surrounding E^. It does not occur because b > x^ . The structural changes were derived from global considerations. However, at b = x^  a Hopf bifurcation - a local phenomenon exists. Its significance is consistent with the portrait derived from a global considerations. r 38 VI. INTERPRETATION As we could see, there is a danger of driving the harvested population to E^ . Clearly, this is an undesirable equilibrium, an equilibrium of a graveyard: low animal density and no harvesting activity. Another feature is the possibility of fluctuations both in the effort and in the population. With b e (xQ,b) these fluctuations may be undesirable but not fatal i f they are inside the limit cycle. But if they are outside the limit cycle they end up at E., - a disaster. If the situation is not too bad a collapse can be avoided by regulating the fishery, i.e. by controlling either a or b . The higher b is the smaller the attraction domain of E^ is . Given (x(0),y(0)) we can determine b such that (x(0),y(0)) are in the interior of the limit cycle (in the second portrait) or under the third portrait) , or we need b > x^  to achieve one of those. Then a collapse is inevitable. The main conclusion is that even from such a simple model i t can be seen that nature is not always forgiving; there is a danger of depleting this type of resource. 39 Appendix A: Possible Values of R and Q g(x) can have either one or three roots. The number of roots is a function of R and Q . To find the domain in the R-Q plain where g(x) has 3 roots we solve simultaneously: g(x) = 0 , Then we obtain R = R(x); Q = Q(x). This is a parametric representation of the curves on which g(x) has one double roots and one simple roots. d 2 -At x where —|- (x) = 0 we have one triple root. These curves are the dx boundary which confines the domain where g(x) has 3 roots. X X The equation g(x) = 0 implies R(l - —) 2 = 0 1 + x 2 The equation ~r~ g = 0 implies X Hence * Q 2,2 2,3 2x3 R(x) = YI i Q00 = — (l + x y x - l By the nature of the problem R and Q are positive. The following properties can be derived from these forms: 40 x 1 + implies Q -* °° and R -»- % + , x -»• implies Q ->• °° and R + 0 + . To find x we consider: dR 2x2(3 - x 2) .; dQ _ 2x 2( X 2 - 3) _ dR _ dQ _ Q a f c ; = / J ( dx = . 2. 3 ' dx . 2 . . 2 dx dx (1 + x ) (x - 1) Figure 9. The number of roots of g(x) as a function of various values for R and Q . 41 Appendix B: The Numerical Scheme A numerical scheme was used in order to verify the results which utilized asymptotic techniques. The main part of the scheme was an integration routine which used a predictor-corrector formula of fourth order. The routine also measured a quantity proportional to the relative error (namely, predicted value - corrected value corrected value ), and changes the time _3 step so that this quantity will be smaller than 10 . When the time step was decreased (or to start the solution) a fourth order Runge-Kutta method was used to generate the next three terms. The system was integrated with i n i t i a l (x,y) near E^ ' or E^, as the case required. Near E. the slopes of T. are x. - b 1 m. = g'(x.) - a — (i = 2, 3) . This observation was used to make x i the integration slightly faster. The in i t i a l y was 10 and the ini t i a l x was taken to be on the line which goes through E^ and has the slope TIK ( i = 2, 3) . To obtain t n e system (II.1.1) was integrated with time going backwards. This was done until the solution crossed the line x = b . The value of T^  at x = b was not necessarily given by the routine because the integration was done with respect to time (rather than integrating 4^  = a ^ / X s — ) . Therefore, the routine took the dx x g(x) - xy ' value at the closest point to x = b, say at x = x,, . Incidentally, the value a^t x„ was either the last or the one next to the last value D of to be computed. The next step was integration of T^  . Now time went forwards 42 and the routine stopped integrating when the solution crossed the line x = x. . This time we were interested in the value of T 0 at x = x , B j B and used a third order Lagrangian interpolation formula to obtain i t . At this point comparison could be made between an& . If y. is the y-value of T. at x„, then F = Y„ - Y„ was the J± J x B 2 3 interesting quantity. As i t was shown in the work, for any (R, Q, a) fixed we had F = F(b) . To find b we had to find the zero of F . This was done by the method of bisection until |F| < 10 ^ . Then the method of false position was used until |F| < 10 . Obviously, each iteration involves an integration of 1^ and T^  • The program was run for R values ranging from 0.014 to 0.5, -5 3 Q values from 50 to 5000 and a from 10 to 10 . For P > 10, the numerical solutions for b was not more than 5% away from the asymptotic "large P" approximation. For P < 1 the numerical solution was not more than 7% away from the asymptotic "small P" approximation. For P between 1 and 10 the results varied. Sometimes they were close to the "large P" approximation, sometimes to the "small P" approximation, sometimes to both and sometimes to none. Yet even in the latter case, the computed b was larger than X Q . The formulas which were used in the scheme were based on Ralston [8]. R Q 3/2 P = aQ ' computed b "small P" b "small P" relative error (%) "large P" b "large P" relative error (%) "small P" is good for P = 1.77 0.5 0.5 0.5 50 50 50 17.67 1.77 0.177 12.57 11.06 10.04 27.37 11.6 10.02 117 4.8 0.1 12.8 12.8 12.8 1.8 15 27 none is good for P = 1.77 0.3 0.3 0.3 50 50 50 17.67 1.77 0.117 15.87 14.03 13.045 37.08 15.22 13.035 133 8.5 0.08 16.25 16.25 16.25 2.3 15 15 both are good for P = 1.77 0.108 0.108 0.108 50 50 50 17.67 1.77 0.117 22.57 22.35 21.74 41.58 23.48 21.67 84 5 0.3 22.58 22.58 22.58 0.04 1 3.9 "large P" is good for P = 5.59 0.0389 0.0389 0.0389 500 500 500 55.90 5.59 0.56 155.80 150.78 131.21 2308.6 332.91 135.35 1381 120.8 3.2 155.99 155.99 155.99 . 0.1 3.5 18.9 Table: A sample of the computer results. LO 44 BIBLIOGRAPHY Clark, C.W., "The Economics of Overexploitation", Science 181, 630-634 (1973). Clark, CW.,and G.R. Munro, "The Economics of Fishing and Modern Capital. Theory: A Simplified Approach", J. Environmental Economics and Management 2., 92-106 (1975) . McNaughton, S.S. and L.L. Wolf, General Ecology, Holt, Rinehart and Winston, Inc. (1973). Holling, C.S., "The Functional Response of Predators to Prey Density and its Role in Mimicry and Population Regulation", Memoirs of the Entomological Society of Canada 45_ (1965) . Schaeffer, M.B., "Some Considerations of Population Dynamics and Economics in Relation to the Management of Marine Fisheries", J. Fish. Res. Bd. Canada 14 (5), 669-681 (1957). Coddington, E.A. and N. Levinsbn, "Theory of Ordinary Differential Equations", McGraw-Hill Book Company, N.Y. (1955). Kop'ell, N. and L.N. Howard, "Plane Wave Solutions to Reaction-Diffusion Equations", Studies in Applied Mathematics _5_2 (4), (1973). Ralston, A., "A First Course in Numerical Analysis", McGraw-Hill Book Company, N.Y. (1965). Smith, V.L., "On Models of Commercial Fisheries", J. Polit. Econ. 7_7, 181-198 (1969) . 

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