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Approximate solutions of some predator-prey ecological models Brearley, John R. 1978

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APPROXIMATE SOLUTIONS OF SOME PREDATOR-PREY ECOLOGICAL MODELS by John R. Brearley B.Eng. (Honours) McGill University, 1976 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES in the Department of El e c t r i c a l Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June, 1978 © John R. Brearley, 1978 In presenting t h i s thesis in p a r t i a l f ulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shall make i t fre e l y available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for fi n a n c i a l gain shall not be allowed without my written permission. Department of £^ &CTI*-ICtfl 9hf £///^gjg,Qfl/ 6-The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 ABSTRACT The purpose of this thesis i s to develop and evaluate approximate analytical solutions of nonlinear differential equations that purport to model physical ecological systems of a predator-prey nature. A variety of analytical techniques, some well known and some new are used to obtain analytical solutions for these models. Solutions are obtained for five different predator-prey systems, and simulations show that these solutions are accurate over wide ranges of i n i t i a l conditions and system parameters. These l i t e r a l parameter solutions allow the user to readily determine which components of the solutions are sensitive to which parameters, and thus avoid the traditional heuristic computer sensitivity studies. i i TABLE OF CONTENTS Page ABSTRACT . . . . . i i TABLE OF CONTENTS i i i LIST OF ILLUSTRATIONS wi ACKNOWLEDGEMENTS x i i INTRODUCTION . . . . 1 CHAPTER ONE LOTKA-VOLTERRA SYSTEM PROTOTYPE 1.1 Introduction • 4 1.2 Analytical Technique . . . . . . . . . . 4 1.3 Application to the Lotka-Volterra System . . . . . . 6 1.4 Simulation Study of Ritz Solutions . . . . . . . 8 1.5 Improved Approximate Amplitude Solution 21 1.6 Simulation Study of Approximate Amplitude Solution 23 1.7 Summary 24 CHAPTER TWO ADVANCED MODELS AND LINEAR SOLUTIONS 2.1 The Linear Solution 35 2.2 The Volterra-Gause-Witt Model . . . 37 2.3 Linear Solution of the Volterra-Gause-Witt Model 37 2.4 Simulation Study of the Volterra-Gause-Witt Model 3 8 2.5 Holling's Model 5 3 2.6 Linear Solution of Holling's Model 54 2.7 Simulation Study of Holling's Model 56 2.8 Rosenzweig's Model 56 2.9 Linear Solution of Rosenzweig's Model 66 i i i 2.10 Simulation Study of Rosenzweig's Model . 68 2.11 O'Brien's Model 72 2.12 Linear Solution of O'Brien's Model s . . 72 2.13 Simulation Study of O'Brien's Model 74 2.14 Summary 74 CHAPTER THREE EQUIVALENT LINEARISATION SOLUITON OF ADVANCED MODELS 3.1 Introduction 79 3.2 The Method of Equivalent L i n e a r i z a t i o n . , 80 3.3 Equivalent L i n e a r i z a t i o n Solution of the Volterra-Gause-Witt--Model 84 3.4 Equivalent L i n e a r i z a t i o n Solution of Holling's Model 84 3.5 Simulation Study of Holling's Model 8 6 3.6 Equivalent L i n e a r i z a t i o n Solution of Rosenzweig's Model 87 3.7 Simulation Study of Rosenzweig's Model 98 3.8 Equivalent L i n e a r i z a t i o n Solution of O'Brien's Model 98 3.9 Simulation Study of O'Brien's Model 1 0 2 3.10 Summary 1 0 6 CHAPTER FOUR REFINEMENT OF THE METHOD OF EQUIVALENT LINEARIZATION 4.1 Introduction 108 4.2 The Additive Correction Factor 108 4.3 A n a l y t i c a l Determination of the Additive Correction Factor . . . . 109 4.4 Additive Correction Factor f o r the VGW Model 114 4.5 Additive Correction Factor f o r Holling's Model 120 4.6 Additive Correction Factor for Rosenzweig's Model 120 4.7 Additive Correction Factor f o r O'Brien's Model 136 4.8 Summary 141 i v CHAPTER FIVE CONCLUSIONS .142 APPENDIX A E l l i p t i c Functions as Approximate Solutions . . 147 APPENDIX B Preliminary Developments of the Additive Gorrection Factor 151 REFERENCES 155 LIST OF ILLUSTRATIONS Figure Eage 1.1 L o t k a - V o l t e r r a Model R i t z S o l u t i o n f o r (x »y Q) = (35,10) 9 1.2 L o t k a - V o l t e r r a Model R i t z S o l u t i o n f o r ( x 0>y o) = (35,10) 10 1.3 L o t k a - V o l t e r r a Model R i t z S o l u t i o n f o r (x ,y ) = (35,10) . . . . . 11 1.4 L o t k a - V o l t e r r a Model R i t z S o l u t i o n f o r (x ,y ) = (60,35) 12 o o 1.5 L o t k a - V o l t e r r a Model R i t z S o l u t i o n f o r ( x Q , y ) = (60,35) 13 1.6 L o t k a - V o l t e r r a Model R i t z S o l u t i o n f o r '(x ,y Q) = (60,35) 14 1.7 L o t k a - V o l t e r r a Model R i t z S o l u t i o n f o r (x ,y ) = (85,35) . . . . • 15 o o 1.8 L o t k a - V o l t e r r a Model R i t z S o l u t i o n f o r (x »y ) = (85,35) 16 1.9 L o t k a - V o l t e r r a Model R i t z S o l u t i o n f o r (xr,,y ) = (85,35) 17 ; o o 1.10 L o t k a - V o l t e r r a R i t z S o l u t i o n Amplitude E r r o r L o c i 19 1.11 L o t k a - V o l t e r r a R i t z S o l u t i o n Frequency E r r o r L o c i • 20 1.12 L o t k a - V o l t e r r a Model Approximate Amplitude S o l u t i o n f o r (x ,y ) = (35,10) • 2 4 o o 1.13 L o t k a - V o l t e r r a Model Approximate Amplitude S o l u t i o n f o r (x ,y ) = (35,10) 2 5 o o 1.14 L o t k a - V o l t e r r a Model Approximate Amplitude S o l u t i o n ' f o r ( x Q , y o ) = (35,10) 26 1.15 L o t k a - V o l t e r r a Model Approximate Amplitude S o l u t i o n f o r :(x o,y o) = (60,35) 2 7 1.16 L o t k a - V o l t e r r a Model Approximate Amplitude S o l u t i o n f o r ( x Q , y o ) = (60,35) 2 8 1.17 L o t k a - V o l t e r r a Model Approximate Amplitude S o l u t i o n f o r (x ,y ) = (60,35) 2 9 o o v i 1.18 Lotka-Volterra Model Approximate ••Amplitude Solution for (x .y )=•= (85,35) . 30 1.19 Lotka-Volterra Model Approximate Amplitude Solution for (x o,y Q) = (85,35) . . . 31 1.20 Lotka-Volterra Model Approximate Amplitude Solution for (x ,y ) = (85,35) 3 2 1.21 Lotka-Volterra Model.Approximate Amplitude Solution Amplitude Error Loci . , 3 3 2.1 Possible Singularity Types and Positions for the VGW Model 3 9 2.2 VGW Model Linear Solutions for <x ,y ) = (85,60) 41 2.3 VGW Model Linear Solutions for (x.o y Q) = (85,60) 42 2.4 VGW Model Linear Solutions for (x ,y ) = (85,60) . . . . 43 2.5 VGW Model Linear Solutions Amplitude Error Loci 44 2.6 VGW Model Linear Solutions Amplitude Error Loci 45 2.7 VGW Model Linear Solution for (x >y ) = (3.5,1) 46 2.8 VGW Model Linear Solution for" (x ,yQ) = (3.5,1) 47 2.9 VGW Model Linear Solution for (x ,y ) = (3.5,1) . 48 o o 2.10 VGW Model Linear Solution for (x ,y ) = (6,3.5) 49 o o 2.11 VGW Model Linear Solution for (x ,y ) = (6,3.5) 50 o o 2.12 VGW Model Linear Solution for (x ,y ) = (6,3.5) 51 o o 2.13 VGW Model Linear Solution Amplitude Error Loci 52 2.14 Possible Singularity Types and Positions for Holling's Model 55 2.15 Holling's Model Linear Solution for (x »y ) = (17.5, 42.5) 57 v i i * 2.16 Holling's Model Linear Solution for (x o,y o) = (17.5, 42.5) 58 2.17 Holling's Model Linear Solution for (x Q,y o) = (17.5, 42.5) . 59 2.18 Holling's Model Linear Solution for (x o,y Q) = (30,30) 60 2.19 Holling's Model Linear Solution for 2.20 Holling's Model Linear Solution for (x Q,y o) = (30,30) 62 2.21 Holling's Model Linear Solution Amplitude Error Loci 63 2.22 Holling's Model Linear Solution Frequency.Error Loci 64 2.23 Holling's Model Linear Solution Amplitude Error Loci 65 2.24 Possible Singularity Types and Positions for Rosenzweig's Model 67 2.25 Rosenzweig's Model Linear Solution Amplitude Error Loci 69 2.26 Rosenzweig's Model Linear Solution Frequency Error Loci 70 2.27 Rosenzweig's Model Linear Solution Amplitude Error Loci 71 2.28 Possible Singularity Types and Positions for O'Brien's Model 73 2.29 O'Brien's Model Linear Solution Amplitude Error Loci 75 2.30 O'Brien's Model Linear Solution Frequency Error Loci 76 2.31 O'Brien's Model Linear Solution Amplitude Error Loci 77 3.1 Holling's Model Equivalent Linearization Solution for (x Q,y o) = (17.5, 42.5) 88 v i i i 3.2 Holling's Model Equivalent Linearization Solution for (x ,y ) = (17.5, 42.5) 89 o o 3.3 Holling's Model Equivalent Linearization Solution for (x ,y ) = (17.5, 42.5) 90 o o 3.4 Holling's Model Equivalent Linearization Solution for (x ,y ) = (30,30) 91 o o 3.5 Holling's Model Equivalent Linearization Solution for (x o,y Q) = (30,30) . . 92 3.6 Holling's Model Equivalent Linearization Solution for (x Q,y o) = (30,30) . . 93 3.7 Holling's Model Equivalent Linearization Solution Amplitude Error Loci 94 3.8 Holling's Model Equivalent Linearization Solution Frequency Error Loci 95 3.9 Holling's Model Equivalent Linearization Solution Amplitude Error Loci 96 3.10 Rosenzweig's Model Equivalent Linearization Solution Amplitude Error Loci 99 3.11 Rosenzweig's Model Equivalent Linearization Solution Frequency Error Loci « . . . 100 3112 Rosenzweig's Model Equivalent Linearization Solution Amplitude Error Loci 101 3.13 O'Brien's Model Equivalent. Linearization Solution Amplitude Error Loci 103 3.14 O'Brien's Model Equivalent Linearization Solution Frequency Error Loci . 104 ix 3.15 O'Brien's Model Equivalent Linearization Solution Amplitude Error Loci 105 4.1 VGW Model Corrected Equivalent Linearization Solution for (x ,y ) = (85,60) 116 o o 4.2 VGW Model Corrected Equivalent Linearization Solution for (x ,y ) = (85,60) . 117 o o 4.3 VGW Model Corrected Equivalent Linearization Solution for (x Q,y o) = (85,60) 118 4.4 VGW Model Corrected Equivalent Linearization Solution Amplitude Error Loci 119 4.5 VGW Model Corrected Equivalent Linearization Solution for (x o,y Q) = (3.5,1) 121 4.6 VGW Model Corrected Equivalent Linearization Solution for (x o,y Q) = (3.5,1) , . . . 122 4.7 VGW Model Corrected Equivalent Linearization Solution for (x o,y Q) = (3.5,1) 123 4.8 VGW Model Corrected Equivalent Linearization Solution 124 4.9 VGW Model Corrected Equivalent Linearization Solution for (x Q,y o) = (6, 3.5) 125 4.10 VGW Model Corrected Equivalent Linearization Solution for (x o,y Q) = (6, 3.5) for (x ,y ) = (6, 3.5) o o 126 4.11 VGW Model Corrected Equivalent Linearization Solution Amplitude Error Loci 127 4.12 Holling's Model Corrected Equivalent Linearization Solution for (x Q,y o) = (17.5, 42.5) 128 x 4.13 Holling's Model Corrected Equivalent Linearization Solution for (x Q,y ) = (17.5, 42.5) 4.14 Holling's Model Corrected Equivalent Linearization Solution for (x ,y ) •=• (17.5, 42.5) o o 4.15 Holling's Model Corrected Equivalent Linearization Solution for (x ,y ) = (30,30) o o 4.16 Holling's Model Corrected Equivalent Linearization Solution for (x Q,y ) = (30,30) 4.17 Holling's Model Corrected Equivalent Linearization Solution for (x »y Q) = (30,30) 4.18 Holling's Model Corrected Equivalent Linearization Solution Amplitude Error Loci . 4.19 Holling's Model Corrected Equivalent Linearization Solution Amplitude Error Loci 4.20 Rosenzweig's Model Corrected Equivalent Lineariza-tion Solution Amplitude Error Loci . . . .... . 4.21 Rosenzweig's Model Corrected Equivalent Lineariza-tion Solution Amplitude Error Loci ... .• 4.22 O'Brien's Model Corrected Equivalent Linearization Solution Amplitude Error Loci ' . 4.23 O'Brien's Model Corrected Equivalent Linearization Solution Amplitude Error Loci 5.1 Comparison of the 10% Amplitude Error Loci for Rosenzweig's Model for three Solution Techniques xi ACKNOWLEDGEMENTS I t i s with great pleasure that I acknowledge the invaluable advice of Dr. A.C. Soudack, whose weekly i n t e r a c t i o n provided a continuing source of drive and i n s p i r a t i o n , without: which t h i s undertaking would not have been completed. This research was supported by National Research Council Grant A-3138, as well as a U.B.C. Fellowship. This support i s g r a t e f u l l y . • acknowledged. x i i 1. INTRODUCTION The purpose of this thesis is to develop and evaluate approx-imate analytical solutions of systems of nonlinear differential equations that purport to model physical ecological systems of a predator prey nature. These analytical solutions w i l l give insight into the system behaviour and the effect of system parameter variations without resorting to extensive computer simulation studies. A variety of analytical techniques, some well known and some new, are used to obtain the analytical solutions and are outlined in the following chapters. The ecological systems of interest usually consist of one or more interacting species whose population dynamics are influenced by many factors, such as carrying capacity of the environment (see Gilpen 21), age and spatial distributions (see Hoppenstead 27 & Sanchez 50), natural evolution (see Zwanzig 57), prey nutrient control (see Rosenzweig 47), harvesting (see Brauer 7,8 & San Goh 51), and time delays (see Murray 41 & Mazanov 38) to name just a few. There are numerous methods for mathematically describing the various observed biological phenomena. To begin with, the time variable may be modelled as continuous or discrete. The actual population variables are of the integer type in reality, but for large populations these are approximated by continuous variables. The model w i l l consist of a system of difference, integral (see Greenberg 24), or differential equations. The differential equations may be either partial or ordinary, and either deterministic or stochastic in formulation. In this thesis, we w i l l be considering only pairs of f i r s t order coupled, continuous, time invariant, ordinary differential equations 2. modelling predator-prey systems. For those interested in discrete time models, the papers by May (36) and Oster (45) are recommended as introductory reading. Diamond (14) discusses some st a b i l i t y theory and results, while Comins (12) discusses the effects of a predator which can choose between several prey species. The mathematical phenomenon of bifurcation and the so-called "chaos" problem is expounded upon by Hoppenstead (26 & 28), May (35 & 37), and Smale (52). Some results concerning linear time varying systems' are given by Mulholland (40). The joys and complications of third and higher order coupled systems are elaborated on by McGehee (39) , Oaten (43), Rapport (46), and Rosenzweig (49). The major current problem in ecological modelling l i e s in that the more elegent the biologically observed phenomena becomes, the less tractable the associated mathematics becomes, and vice versa. In an attempt to comprehend the dynamics of a given ecological system, the researchers have often neglected the second order effects when formulating their mathematical models. In practice, i t is usually very d i f f i c u l t to backtrack and experimentally verify the validity of the complete model. The individual terms in the models are experimentally j u s t i f i e d , however the combination given in the model may s t i l l not be enough to adequately describe the system at hand. In the models that w i l l be presented in the following chapters, application, data, and results for the models w i l l be given when available. A single predator, single prey biological system i s modelled (see Brauer)7)by a pair of f i r s t order, coupled, ordinary differential equations of the form x = xf(x,y) x ^ x ( Q ) with ° Y - yg(x,y) y Q k y(0) 3. where x(t) and y(t) are, respectively, the prey and the predator populations. It should be noted that with this form of model, populations i n i t i a l l y at zero remain there. Here i t is assumed that the growth rates of the populations are dependant only on the present population sizes. Thus the model neglects the numerous factors which may influence real populations as previously mentioned. In a more general model, the expression for y(t) might contain a second term to include the effects of harvesting the predator (see Brauer 7, or Rapport, 46). These effects are not included in the models considered in this thesis, and permits the mathematics to remain tractable. For the system of differential equations to represent a predator prey system, we must impose the following conditions f (x,y) < 0 g x(x,y) > 0 g y(x,y) <. 0 and x L 0 y L 0 where the subscripts indicate partial derivatives (see Brauer 7). The last two conditions are characteristic of a l l ecological systems, as negative populations are meaningless. The f i r s t two conditions imply that the prey population is decreased by increasing predator population and that the predator growth i s dependant of the existence of the prey. The third condition expresses the possibility that competition amongst the predators for the prey may exist. Often g(x,y) i s independant of y, implying that there is no competition amongst the predators. In the following chapters, a number of ecological models w i l l be introduced. Several different analytical techniques w i l l be discussed and applied to the models to yield analytic approximate solutions, and these w i l l be evaluated in detail. 4. CHAPTER ONE LOTKA-VOLTERRA SYSTEM PROTOTYPE 1. Introduction The Lotka-Volterra predator-prey system was selected for a proto-type study. Predator-prey systems have been the subject of much mathematical study since the original work of Lotka (33) and Volterra (55) , but to our knowledge, no one has developed approximate solutions that show the functional relation of the system parameters in the solution. While there are many valid objections to the use of the Lotka-Volterra equations for describing physical predator prey systems (see May 34, or Ayala 1), i t has been found that these equations do characterize some natural systems, such as the lynx and the Arctic hare populations, and some fish populations. With some modification, the equations can be used to model multi-species predator prey systems. It i s this use of the Lotka-Volterra equations as the basic building block of a number of more complicated models (see Rapport 46) that motivates the analytical study of the system behaviour. 1.2 Analytical Technique The Equivalent Ritz Method, also known as the Principle of Harmonic Balance, was developed about 60 years ago by W. Ritz (see Cunningham, 13), and w i l l be used to determine approximate solutions of the Lotka-Volterra equations. The Ritz method finds an approximate solution, x(t) to the equation *x + f(x) = 0 such that the functional t, J = / D F(x,x,t) dt a 5. i s a minimum, and F(x,x,t) i s chosen such that the s o l u t i o n f o r the minimum of J, the Euler-Lagrange equation, i s the d i f f e r e n t i a l equation we wish to solve, that i s Oj OJ r« The approximate s o l u t i o n x(t) i s of the form x(t) =? >) a . T . ( t ) , where the i <f^(t) form a l i n e a r l y independant set, and are chosen from an a p r i o r knowledge of the d i f f e r e n t i a l equation's behaviour. To obtain x ( t ) , the fun c t i o n a l J must be minimized with respect to the n c o e f f i c i e n t s a^. I f we specify that <|>.(t ) = 4>.(t, ) = 0, or i>. i s p e r i o d i c i n t -t, , we obtain the x a 1 b l a b conditions t / a<p. ( t ) E ( x ( t ) ) d t = 0, i = 1,..., n b 1 For o s c i l l a t o r y systems the Equivalent R i t z Method eliminates the OJ necessity of performing the above n i n t e g r a t i o n s . I f we take x(t) = A cos u>t as an approximate s o l u t i o n to x + f(x) =0, i t i s found by evaluating the above i n t e g r a l that the R i t z method i s equivalent to choosing UJ and A such that x(t) s a t i s f i e s the d i f f e r e n t i a l equation with the harmonic terms generated by f(x) being neglected (see Cunningham 13). That i s to say that the sum of the c o e f f i c i e n t s of cos cot i n E(x(t)) i s i d e n t i c a l l y zero. N S i m i l a r l y , i f we take x(t) = l a cos nut, we f i n d that we must have the 0 n Oj c o e f f i c i e n t s of cos nwt, for n = 0 , N, i n E(x(t)) must balance to zero (hence the name P r i n c i p l e of Harmonic Balance). 6. 1.3 A p p l i c a t i o n to the Lotka-Volterra Equations Consider the Lotka-Volterra predator prey system 4 x = xf(x,y) f(x,y) = a - gy y = yg(x,y) g(x,y) = -y + <5x where a, B, y, 6 > 0. This model was formulated by r e t a i n i n g the f i r s t terms of the double Taylor series expansion of f(x,y) and g(x,y). The c o e f f i c i e n t y represents the predator's natural mortality rate. The c o e f f i c i e n t s g and 6 represent the magnitude of the predator prey i n t e r a c t i o n s . The system has two s i g n u l a r i t i e s , at (x . , y . ) = (0,0) and ( y / 6 , a / 3 ) . L i n e a r i z a t i o n sing J sin?/ of the d i f f e r e n t i a l equation about the s i n g u l a r i t i e s shows the former to be a saddle point and the l a t t e r to be a centre. We w i l l apply the equivalent R i t z method to obtain approximate solutions i n the neighbourhood of the centre. The saddle point i s of l i t t l e i n t e r e s t to the discussion, the associated separatrices being the x and y axes. Thus we are guaranteed that i f we s t a r t with both x and y greater than zero, they w i l l always remain so. An approximate s o l u t i o n of the form x(t) = A + B cos tot + C s i n tot y(t) = D + E cos wt + F s i n cot i s postulated, where A, B, C, D, E, F and co have yet to be found. Substituting the above solutions i n t o the d i f f e r e n t i a l equations and equating the c o e f f i c i e n t s of cos cot, s i n cot and the constant terms to zero, we obtain the following equations 7. aC - gAF - 3 CD + OJB = 0 Cw - aB + gAE + gBD = 0 i . O A _ gBE gCF _ aA - gAD ^ 2~ = u)E - yF + 5 A F + 6 CD = 0 wF + yE - 6 A E - 6 B D = 0 , . N < 5 B E ,: 6 C F _ 6AD - yD H ^ 1 2~ = By matching i n i t i a l values, we obtain x(0) = A + B y(0) = D + E We a p r i o r i assume that A= y/6 and D = a/g i n order to obtain closed t r a j e c t o r i e s about the desired equilibrium point i n the x-y plane, (x . , y . ). Solving the above set of equations, we obtain s x n g • ' s x n g - - ° n ' B = x(0) - y / 6 C = ±(y(0) - a/g)(g/6)/y7^ E = y(0) - a/g F = *(x(0) - y / 6 ) ( 6 / g ) A 7 7 ai = +v/ya" Thus the approximate solutions are given by x(t) = y / 6 + (x(0) - y / 6 ) cos/yat -(y(0) - a/g) (g/6')v/y7a sin/yat y(t ) = a/g + (y(0) - a/g) cos/rat + (x(0) - y/6) (6/g) /a/y sin/yat These solutions describe an e l l i p s e i n the x-y plane, and i f the technique used i s adequate, should compare favourably with numerical solutions of the Lotka-Volterra equations. Subsequent analysis reveals that 8. these solutions are i d e n t i c a l to those obtained by l i n e a r i z i n g about the singular point, even"though-this was not the method used to obtain them. As a general r u l e , the R i t z method w i l l not normally generate l i n e a r i z e d solutions as i t did i n t h i s case. 1.4 Simulation Study of R i t z Solutions To test the v a l i d i t y of the approximate solutions developed i n the previous section, the Lotka-Volterra equations were simulated on the University of B r i t i s h Columbia's IBM 370/168 Computer using a v a r i a b l e step Runge-Kutta in t e g r a t i o n routine. The numerical and approximate solutions v.. are p l o t t e d i n the phase (x-y) plane, and also as functions of time. In the simulation, the c a l c u l a t i o n s were perfomed with the centre s i n g u l a r i t y located at (50,50) using the a r b i t r a r y parameter choice a = y = 500, and 3 = 6 = 10. V a l i d data for t h i s model i s d i f f i c u l t to locate. The t r a j e c t o r i e s were computed for i n i t i a l conditions i n the ranges 10 <_ x(0) <_ y(0) <_ 90 i n increments of 5. The t r a j e c t o r i e s and phase plane curves are shown i n Figures 1.1 to 1.9 for the i n i t i a l conditions (x(0),y(0)) = (35,10), (60,35) and (85,35). I t i s seen that there i s close agreement between the two solutions for i n i t i a l conditions l o c a l to the center s i n g u l a r i t y . From the time plots of the numerical s o l u t i o n , the actual frequency of the o s c i l l a t i o n i s computed, and the frequency er r o r i s given as the percentage difference between the R i t z frequency and the actual frequency. Also from the time pl o t s of the s o l u t i o n s , the amplitude error i s computed, being the worst case percentage difference between the R i t z s o l u t i o n amplitude peaks and the actual s o l u t i o n amplitude peaks. This method of computing amplitude errors neglects any phase err o r between the peaks and thus gives a measure of error of the s o l u t i o n amplitude envelope. o.o to', coo ifl'.OflO HO.010 103.OOO 120.000 mo.000 160.000 K (T > 160.000 200.000 • „ , . . . , « i « , e . i » u u T . i « . . " » I C K "..uTio. »• <"*" , 0 8 I " " s ^ " 0 " 5  ^.»M«.W ... ..1.M.»;«»-«l«»*»-lt, »t. ..,»).(-0.5....F.M. «.*•••"•«> „ R A U E N C ,-.,.S..I..I «..« « " • > • • .-.i«M»f..i« . » n n u o t « L>i . W""t» t « i . . . . « « w » »ouitlo»-nf-i«F.-iorr<«-»'iL't.««* »»»tr«— "Lotka-Volterra Model R i t z s o l u t i o n f o r ( x o > y o ) - (35,'10) 10. 6.0 loe'.oooi O'.MO o . o i s o.n?o O.OM o'.ojo 0.035 o.ono , o,o«5 o.oso ' I .' I ..*•" I . l ' . t I !'.,'..»•... .... I .... I .... • 200,000 O.<)?0 . I t ' n . . t .*. , . 1 I » # *• I . . f r + T i T f » »rrr»i . 0 , 0 « 0.050 0,031 0.04S 0.050 T i n t SEC. «SS>S a * h « L » M C » l SntUTtDN ••••• * MIIHEBIC*L SOLUTION « POINTS CO^ON TO BOTH SOLUTIONS -Soa-0 al3o00C+0? Vfl«-ft,tOaflnE*Oa-»IWKUt*RtTV-AT--(*,»)•I-0,50oOOF*02, 8,S0000t*O«) • A|.pHt« O'.fftQOOOflt hfT*» O.loOCnf*B? G«HA« 0.50fnOE«n3 OfLTA« n,toooOE»o? - » ! « SntttHO»».flf.-TMf;..tnT*A-WOLtfM*A--*¥*UH...~ 1.2 Lotka-Volterra Model; Ri t z s o l u t i o n f o r .(xQ,y ) = (35,10) 11. 8.0 o'.oos °.pj°. ."•".I5 . .'.t'.'A.I " •1 • • • •' • -UO'.OQO--UHE01CAI SOLUTION • 'OWS Cl)M«QN TO BOTH SOLUTIONS UStft a »N*L*TICH SnLKTTON _.,.-«,tt.0O(.O! V..-..IO.0.»i«-St«6UlMH». «-<».»>•«•••.«••"'•"• •.«»'"«•"> ; f»OOR. o'.iatSM.Oit «H.LITUO. E«»0«l »RCL».0.3T»I*£.H» tT«a»..0.,«6».C.0» fsroi'tNCvi **• o*?.snef..n* MUM. n,h.a*«>r..oi t , LPN». e'.Moeor.M «ft.. o.iooo-i.o? <.•««• o.Monnr.M orLt.. B.IOOOOE.OJ -«|tl »nulIIO»-(tf-tMf-ll>I4»-wnLtf''»» S<»If» F i g . 1.3 L o t k a - V o l t e r r a Model R i t z S o l u t i o n f o r ( x Q , y o ) - (35,10) ioo.aooi....(.>•. I.... I.... I. , 40.ooo •V-.50.000 , . 1 . . . . 60.000_ "."..I... . TO.000 „eo. ooo _90.000_.._100.i 70.000- 5^ * y -4-M *4J N 70.000 f/t* \^*\#- NUW3&\CftL f / * . \ \ SoLuT/OfJ 60.000- - 60.000 f S OLUr/orJ \ \ * 4 50.000- 50.000 v\ f% v 40.000 - 40.000 %L AY* 30.000 - 30.000 20.000 - 20.000 to.ooo - 10.000 o.o 1 1-. — i- — t 1 1 1 1 1 |....i ....I o.o 0^ 0 LP«°J>°_ ?0*ot>°_ _ 30.000 43.QO0_ _50.000 60.030 _ 70.000 60.000 90.000 100.QQQ %%%%% • AN* LY T | C At, SOLUTION »»» • NilUgRICAL SOLUTION POINTS _C0HM0N TO,.BOT H SOI.UTI O^ jJL-X0- 0.60000E*02 YO- 0.35000E*02 SINGULARITY AT IX.Yl-( 0.500001*02, 0.50000E+02I ^fteK"tTft'YT"lN-"^7»95 7PF»6T NUM™0i'7863 TUO E ~ E R R 0 RT "Ef U>"^ 0~;53792E*6Vt"~E *81^6738635E"• 01'"' ALPHA- 0.50000E»03 BETA" 0.|0000E*02 GAMA" 0.5000QE»03 DELTA- 0.100006*02  BIT! SOLUTION OF THE LOTKA-VOLTERRA SYSTEM 1.4 Lotka-Volterra Model Ritz Solution for (x ,y ) = (60,35) - . o o 13. XIII 0.0 ToSTTool." - 40.000 90.000- .  - 80.000 60.000- — I S 2X. — r— ~ . - 30.000 10.000- - J J0.000 0.0 I....I....)....I. 0.0 0.005 Ij'r»SI_»-»>V-lV.i.»«LJ.'_o.oi? P.OJO.... 0.035 .0.0*0 ._a».<»» o.osg . . . . . . .N.LTT.r. .! SntUT.OW NUME.IC.L SOLUT I (IN . . . . . • P<H»Tj COMMON TO BOTH SOWlQH . XO- O.60OO0E.O2 YO- 0.350O3E.O* SINGULARITY »T IX.YI.I O.50OO0E.02. O.500OOE.OJI ^ \ AI.FHA. 0.500000") BfTA. 0.10000E.0? GAMA. 0.50000E.fH DELTA, O.IOOOOE.O; . : . RITt SOLUTION. OF THE LOTKA-VOLTERRA SYSTEM Fig. 1.5 Lotka-Volterra Model Ritz Solution for (x Q,y o) = (60,35) 14. 100.0001 100.000 90.000- 90.000 U-t) \ 80.000- 80.000 70.000-i r ^ Vk A i\ '• 70.000 80.000 60.000 \ \ 1 \ I \ ' { 50.000 1 1 50.000 1 \ \ 1 I • W \ ' \ \ : \ / \ // \ \ 40.000 1 \ / W I F I - 60.000 t \ J \ it y \ 30.000 - 30.000 20.000 - 70.000 0 .0 I . . . .| . . . . I 1 I I....I 1 I I.... I.... I....I 1 I I 1 . . . . | . . . . | .<••!*•*.I o . o 0.0 9>°'° _ 0.015 _0.020 0.025 0.030_ _ 0.035 _ _ 0.0*0 _ 0.0*5 0.050 ... - - ...... TINE SEC. _*i»«t • ANALYTICAL SOLUTION • NUMERICAL SOLUTION - POINTS COMMON TO BOTH SOLUTIONS  *0- 0.60000E*02 YO- 0.35000E»02 SINGULARITY AT (X.Yl-l 0.50000E»02. 0.5000QE*021 nFWEQU£l<CYr"TN^ Cl9r7HEV0^  TUOE ERROlfi EYREL--0.53792E*0l* EY*BS--0.38635E*0l ALPHA- 0.50000E»03 SETA- O.IOOOOE*02 GA *<A. 0.S0300E«-03 PEL T A" 0.^ 00O0E»Q2 IITI SOLUTION OF THE LOTKA-VOLTERRA SYSTEl Fig. 1.6 Lotka-Volterra Model Ritz Solution for (x o>y o) = (60,35) 15. Till 0.0 IQ.30ft 20.000 JO.OOO 40,000 50.000 £0*0?J> JO-OOP 60.000 90*000 103-000 T3o7obo 1 1 1; ;;7T77:7r.T.TT7:. 7r.~...1...  rr.. .TTTT:,.... i . . . . r.... i . . . . j.... i . . . . i.TTTT. . I ... .TT. .. I I OO. OOO 0 . 0 | . . . . | . . . . | . . . . t 1 . . . . | . . . . 1.1.. I.... I 1 . . . . 1 1 . . . - 1 1 . . . . . . . . . . . . . . I 1 1 1 0 . 0 0^0 10_.100 20._0_00_ 30.000 40.000 50.000 60.000 70.000 80.000 _ _ 90,000 JOO.OQQ • " " ~ XIT1 ttttt • aNALVTKAL SH1.IJT10N »'«'« • NJ^ Et 1CAL SOLUTION M<it • POINTS COHHQN TO BOTH SOl.UjMP'iS '. X0« O.S5000E»02 TO- 0.35003E*02 SINGULARITY AT IX.TIM O.5QO0OE*O2« 0.50000E*02J TSI oTETiC TT"^ N^ ~0T7^  5 K € ^07" N ^ *L»H»" 0.30Q00E»03 OFT*- p. 100QQC*02 GAHA* 0.500QOE»03 OFLTA- O',Q0.?P£.tP2 HIT* SOLUTION OF THE LOTKA-VOLTERRA SYSTEM Fig. 1.7 Lotka-Volterra Model Ritz Solution for (x o,y Q) = (85,35) 16. x m 0.0 0.005 0.010 0.015 0.020 0.025_ 0.030 0.035 '_ 0.040 „ 0.0*5 0.050 ~.\i...ri77.%i...rir^ ,00.000 il»t • ANALYTICAL SOLUTION M « I > • NUNf 0 | CAL SOLUTION M»It • POINTS .C.OHWnN JO ,J>OT.H..SOLUT I ONS  X0« O.SSO0')E»0? YO- 0.35003E*02 SINGULARITY AT I X , Y » »! ' 0. 50000E *02 » O.5O0OOE*O21 HFR~EWFT£TTTN^H)77«^ ERROR t~EXREL«-0l?6999^ ALPHA- 0.53nOOE»03 HEX A' O.IQ00nE»0? GAM A- 0.,500006 »03. DFLTA^O. IQOff 3E «02  RITZ SOLUTION OF THE LGTKA-VOLTERPA SYSTEH Fig. 1.. Lotka-Volterra Model Ritz Solution for (x ,y ) = (85,35) • o o 17. VIT1 o.o o.ons 0 - o i 0 _ i do .a oo i . . . . ] . . . . i rr: V I I . . ; . _0.015 0.020 0.025 0.030 0.035 0.0*0_ _ 0.045 0.050 .: i . : i — i - —T—11 - ...i — i — i . . . . i . .v. i . . . : i:.. . T . 1 . . . . i — 1 100.600 - ANALYTICAL SniUTIOl ••••• - NJMF.R |C 4 L _ SOLUT1 ON ' POINTS COMMON TO BOT H SOtUTlONS XO« 0.85000^*02 VO* 0.35003E*02 SlNf.ULAPtTV AT (X.Vl- l o;50000E*02. 0.50000E*02I ~>h'EQuTNCTi~*N-"07795 7i'EVa>'NUN»~ 0.766 76E*02~ERBOR"'6/37 799E + 011 XMPLI TUOE E RRORr EVREU~0T*9^^ A9S^~-0.il536E*02 | 6t»HA« O,50000F»03 RET A" 0. |0000E«02 GAHA« 0.30300E»03m DELTA" 0.130Q0E»02 \ RI 7 2 SOLUTION OF THE LOTKA-VOLTFRRA SYSTEM F i g . 1.9 Lotka-Volterra Model Ritz Solution f o r (x ,y ).= (85,35) o J o 18. In the simulation, amplitude and frequency errors were stored for each set of i n i t i a l conditions, and from this datajloci of constant error, as a function of i n i t i a l conditions, are plotted in Figures' 1.10 and 1.11. From these curves i t i s seen that Ritz frequency is quite accurate for a large region of i n i t i a l conditions, while the amplitude of the Ritz solution is accurate over a smaller region of i n i t i a l conditions. In this case (and future ones similar to i t ) where the data i s either arbitrary of of questionable accuracy, we shall be arbitrarily assuming that certain stated percentage accuracies of the approximate solutions are acceptable for use with the model. It i s seen that in the region of acceptable amplitude error (say 20%) , the corresponding frequency error is much less (about 5%). We observe from additional runs with different parameters that the general shapes of the error l o c i remain the same for different centre singularity locations. We also note that the farther the centre singularity from the origin, the larger the regions enclosed by the error l o c i become, thus extending the region of i n i t i a l conditions for which the approximate solutions are useful. It i s essential to remember that both the amplitude and frequency error l o c i plots should be studied when deciding on the useful ranges of i n i t i a l conditions of the solutions, as information in the frequency error l o c i i s not reflected in the amplitude error l o c i , and vice versa. The actual relation between the amplitude and frequency error plots i s , at present, unknown and may be pursued at a later date. The preceding results have been summarized in a paper by Brearley and Soudack (9). At this point i t should be mentioned that a number of other researchers have developed formulae for the period of the solution of the Lotka-Volterra system. Frame (17) has given the exact formula for the period 19. CONSTANT RELATIVE AMPL1TOUE ERRCP LOCI GRAPH t k R O R X tO.OOOl SINGULARITY AT IX .V I - I O . S O O O j E . 0 2 . O.50O00E.O21 ' ALPHA. 0.5OO0UE.03 bETA" 0 .10300E.O2 &AHA- C .S0000E.03 UfcLTA. O . W O U U t . u 2 J | T t J s * S t t ^ J E t J f i l - ^ 9 l ! ^ * l M M J B t H j l ! l — 1.10 Lotka-Volterra R i t z Solution Amplitude Err o r L o c i 20. 0.0 &O..Q00 20,. OOO. 30.000., *0.000_ 50.00Q ,_60.00p _70»pOO 60.000. .VO.OOP 100.000 100.0001 100.000 • • 1 ^ ^ j 90.000- 9 7 / *> * * \5 6 7 - 90.000 / \ : 80.000- / * 3 2 2 2 3 3 * \ 4 80.000 / " ^ '• 7 5 2 2 1 . 3 4 V / • • ' \ : 70.000- * 3 2 • 1 2 3 • V 70.000 9 6 / 1 V 2 * \ / ' \ \ * 60.000- 60.000 2 30.000* 3 Y 2 50.000 __L X41MA,¥$M6\ i \ 1 1 ' \ \ / / : 60.000- * *\ s V V 2 1 * / 60.000 \* 3 2 I s , 1 ' ' ' * / \ ^ / 30.000- 7 3 ^ * 3 2 2 2 2 2 3 * 30.000 \ y V • 3 • * Si » 7 20.000- 7 ^^V— « 7 8 9 • 20.000 I 9 ~ 9 9 9 10.000- • 10.000 0.0 0 .0 10.000 20.0C0 30.000 *0.000 50.000 60.000 7J.000 80.000 90.000 100 0.0 .000 CONSTANT RELATIVE FREQUENCY ERRCR LOCI GRAPH ERROR X 1-0001 SINGULARITY AT (X-VI-C 0.5000dE>02. 0.50000E»021 ALPHA* 0.5000GE+01 BETA* 0.10000fc*02 CAMA« 0.5CC&OE*03 DELTA" 0.10000E*02 .MTL.iOWiIICN_Qf_ lME_J._CTrU.-yQLTERRA_ SYSTEM .' 1.11 Lotka-Volterra Model Ri t z Solution Frequency Err o r L o c i in the form,of an i n f i n i t e series, which he then approximates by a series of Bessel functions. Grassman (23) develops a complicated asymptotic formula which works under certain i n i t i a l conditions. Both these methods of determining the frequency are rather long and tedious, and do not yield much insight into the system behaviour. The extra accuracy gained for the labor involved is of dubious value. For these reasons, the simple, concise Ritz frequency formula seems to be the most practical solution. We now proceed to another method of obtaining solutions for the Lotka-Volterra system. 1.5 Improved Approximate Amplitude Solution Being encouraged by the results of the previous section, we:: attempted to improve the amplitude of the Ritz solution. It was desired to be able to more accurately estimate the peaks of the solutions of the Lotka-Volterra. system. The differential equations may be integrated directly in the phase plane to give the exact solution as follows: x = ax - gxy y = yy + ^ xy -• djc _ ax - gxy dy -yy + 6xy /. dx(- - + 6) = dy(- - B) x J y which upon integration, becomes - yln(x) + 6x = aln(y) -By + C where C = 6x + By - yln(x ) - aln(y ) o Jo o o This is a well known result which gives the exact phase plane solution. Graphical techniques (see Clark 11) have been developed which permit the sketching of the solutions only in the phase plane. 22. To obtain the peaks of the sol u t i o n s , we proceed as follows. At the peaks of x ( t ) , we have x(t) = 0, which implies from the d i f f e r e n t i a l equation that y = a/3. Substituting this value of y i n the phase plane s o l u t i o n we obtain - yln(x) + 6x = a(ln(a/B)-l) + C This equation has two roots which correspond to the maxima and minima of the x(t) o s c i l l a t i o n . This equation could be solved by any number of standard numerical analysis methods. To provide a comparitively simple estimate of the actual roots, we approximate ln(x) by i t s truncated Taylor s e r i e s ln(x) ='i.ln(a) -1 + x/a where a i s some sui t a b l e point to develop a Taylor s e r i e s about. This y i e l d s x = a ( l n ( a / g ) - l ) + C + y(ln(a) -1) P e a l c 7 ; ~ ~ 6 - y/a A f i r s t approximation to the peak deviation from the singular point i s given by the i n i t i a l displacement from the singular point r = /(x - x . ) 2 + (y - y . ) 2 o o sing o sing Thus we choose a as follows x . - r for . x . sing o mm x . + r for x sing o max subject to the constraint a > 0. Having estimated the peaks of the s o l u t i o n , we construct a s o l u t i o n that o s c i l l a t e s between the peak points at the R i t z frequency. The s o l u t i o n i s given by x(t) = x ^ + cos (cot + <j> ) where x 0 = (x + x . ) 12 L)C v max mm' A = (x - x . )/2 x max min Note: i f i ( t ) > 0, put A = -A so that the i n i t i a l x x derivative has the correct sign, co = Ritz frequency = /cxy ** = C O S " 1 ( x o " X D C ) A x The y(t) solution i s constructed in exactly the same manner. This completes the second approximate solution of the Lotka-Volterra system. 1.6 Simulation Study of Approximate Amplitude Solution We now wish to evaluate the second approximate solution of the Lotka-Volterra system. In the simulation study, the calculations were performed with the same parameters as the Ritz solution, and over the same ranges of i n i t i a l conditions. The trajectories and phase plane curves for the same i n i t i a l conditions are shown in figures 1.12 to 1.20. The corresponding amplitude error plots are shown in figure 1.21. The frequency error l o c i plots are the same as the Ritz solution plots, and need not be shown here. It i s seen that the 10% and 20% amplitude error l o c i cover a much wider range of i n i t i a l conditions than the Ritz case, and the improvements are readily seen in the given time plots. Eventually, the true solution departs from a sinusoidal shape, and the approximate solution has l i t t l e value. This completes our evaluation of the second approximate solution of the Lotka-Volterra system. 24. I6fl'.0o0 • 160,000 \ UO'.OoO- me.ooo I I — loo'.ono- 1 7 v r 100,000 '] 7 \ 1 AO.OOO If \ f eo.ooo 40'. 000 , 60f000 ^ / 40'.OOO * - ao.ooo * 'o'.ooo » 20,000 0,0io'.ooo" "on.o^o* "fcS'.ooo "fto'.ooo* ioo.ooo" iio.ooo iio.ooo iio.oco IBO.OOO zoo.ooo X(T) I S I * S • ANALYTICAL SOLUM0*4 • NUMtOIC*t SOLUTION •»•#* • POINTS COMMON TO BOTH SOLUTIONS - t « * - 6 a \ S « 0 0 r * 0 » VOa-6.10600i»0?-MNfiULA»ITv-.AT-<ii rV)a< -0.50000r*02r- o,sooooc*oa> PfifQUtNC*! AN* 0*.T9«»75E«02 Nu«a (l.fto23of*02 EHP09* n*. t09?«E*02t AfpLlTUOF f«BOW| E»El» 0.10563E»©2x E*83« 0,15J77E*02 IMt l f t a n'.|flT3RE«01 t"kttAm 0,iMrtlE»f>3 S>TN«« l ' . •» ^ Pfl'lE * n 1 «MIN»» 0,903PSE*Qt --fHANAa O'. l«7 fl, t M *vi t*n\ v«!N9»-tl9^QHuf *11 V«INAa*«,9016SE*0I : :  ALPNAa O'.S0<l01F*0J PETA» 0 . 1 00 o n t X I ? fi» " » • 0 , 500 A Of *fl3 D E L T A S fl,IOflOOE*02 APPROXIMATE A ^ P L I T H ^ t SOttJflON f\f THF | CTfc A-V O L T F RH A 3*STtM Fig. .1.12 Lotka-Volterra Model Approximate Amplitude Solution for (xQ,y<)=(35 10) 25. • . NUHC«lr.L JOU'llOM ..... • POINTS CONHON TO BOTH SOLUTIONS IStSS • A N A L Y T I C A L S O L U T I O N _l..-o.»s.oo«.o» Y A . O . I O « O O £ . . O ^ B I » G U L . S M Y A T . . ( « , Y > . < O . S O O O O F . O J , O . S O O O O E . O J I -F P F O U C N C Y I A N « o'.7«575Et»J N U M . n.t.JJor.o? E B B O B » » I I B . O'.IATiar.lH » H A » A . O.IM/tlt.Ol ««t«. o.«V>««F«»l K N I N A . 0,»0!»5t.01 -ALPHA.-I'.IHMIMVMIU-A.IUUE. 1». G » « A . 0 .SOOdOf •(! I. Of L 1 A « 0 . I 0«0«E .0! tppBOIIHaTC A«PLTTIPHt SOLUTION 0' T«C I 0T«A-YOLTF»«A SYSTEM • o'.ll«J«t'.»J* AMPLITUDE E B P O B I E « O E L . O . 1 0 5 S 5 E . 0 J X E Y A 9 S " O . H I T T C . O t .1.13 Lotka-Volterra Model Approximate Amplitude Solution for (x ,yQ) = (35,10) 26. • • f l . . . . ° ' - o f l ' . o l o i S O,Oj»0 0 . 075 0".0)0 o'.055 0 .000 0.0H5 0.050 o.o(ioi...'.iv.;..i.-..,.r..'..i.;.-.i... i . . . . i i . . . : I . . . . i . . . . i i . „ ' . r . . m . o O.OIO 0.015 0.0..0 Q.0?5 * 0.050 t • • a I «n»4 4-.-»-, • *-»• • • I 4> 0.035 O.OttO 0.0*15 0.050 TIWE SCC. i»t» m ANALYTICAL SOLUTION . win-fRItAc SOLUTION «*««• > fOlMTS COMMON TO BOTH SOLUTIONS . • - * < 3 * ' " ' * F L * * 0 ? V0* 0 , IOfl<-o£*'»>-8lNr.ll,*()ITY -AT (K, V)«<a,5000OF.•('2, • 0,50000E*O2>- —-F Q U C K C Y I AN» Q'.TQST5F*r.a N I I W A , fro?S«F E R R O R * t>*. t ««a«£*0?*, AMPLITUDE E R R O R t E Y R E I B 0, 100 S«C •02X EVApSR 0,!«6eiE*02 ANIla 0.)aT38E»fll YNAITAB 0,l6.nif..o3 tMINR* 0'.•M«»«i'F*01 VKI*»« 0,*»0Sfl5t»01 • H A » - ft'.500DOf Of r »• -0. I AOA***A? G*u«»-0,SrtOinF*i>i OFLT»«. A # | 0000k•OJ" -•mOlfATC A H P L I T I I O E SOLUTION flF f M E |"oTH A-VflL Tf HB A SYSTEM 1.14 Lotka-Volterra Model Approximate Amplitude Solution for (x o,y o) = (35,10.) 27. •/JL" in n n n in.nnn *n.ann Af,. (100 50.000 60.000 70.000 80.000 90.000 100.000 100.000 | T.TTTrrrir...rr7:".']7:.vr:...r; /.TTTTTTI.- . .rr;-. -. i — ( — i — i — i — i — J — i — i — * — ^ IUU,ouu 90.000- 90.000 80.000- ao.ooo 70.000- W — S O L U T I O N - 70.000 f . W I 60.000- 60.000 : 50.000- 50.000 1 k — ••' V;—^ ?//"«^ 1 •>•'•' J I 40.000 • ; j X "' y* 1 • ' . ' 30.000 - 30.000 20.0Of - 20.000 10.00C : 0.0 |'o" 10.000 ^0.000 30.000 40.000 50.000 60.000 70.000 BO. 000 90.000 10 '\ 0.0 O.OOQ ,„„ . tMM.tTICM. SOLUTION' »'»•' •• NJMCRICAL SULUTI ON »«.» - PL INT S COMMON TO BOTH SOLUTIONS HO- O.60O00E.O2 YO- 0.35000E.02 SlNiULARIIY AT IX.YI-C O.50000E«02. O.5OO0OE.02I renufNCY. ,N;-d^ S»ST7oTTO^ nCT»S32^  MHAXR. O.TH2aF.O? X«»«A. 0.7;I?0E'Q2 XM1NR- 0.33I70E.Q2 MM- 0.330!.aE.02 YNAXR. 0.Tl«»t.02 YMA XA. O.72120MO2 Y Ml NR. 0.3J170L.O2 YMINL 0.33058E.O2 . APPRUXIMAIE AMPLITuqr SOLUTION OF THf LO T< ft - VOL T ERR A SYSTEM F i g . 1.15 Lotka-Volterra Model Approximate Amplitude Solution f o r ( x Q , y o ) - ( 6 0 , 28. XITI 0.0 lo'o.oool.... 0.005 0.010 0.515 0.020 0.02 5 '0.030 , 0.0J5 0.040 «.J4J 0 . 0 » 0 _ _ rrrnr.rrrrrr:j.:."rrirrrrrr:.Trrrrri;.~r.x:.rr.i.Trrrrr:..i...rr..-.i — r r r r r i . . . . i . . . . r . . . . i — i ioo.ooo »»!»» - ANALYTICAL SOLUTION - NJMEHlCAL SULUTION t « » - POINTS COMMON TO BOTH SOLUTIONS 10- 0.HO00OE.02 VO- 0.550001.02 SINGUL4RITY AT CX.YI-I 0.5O000E.02. 0.5OO00E.O2I —W—JSHC,, AN. c.7<ii7rtTo2-iiua.ni;ian->.t-oj-t«-RB«—071 i « 7 t SOW-JSPI;nMC-tw>Mi ExRETr-KSTVTTFrdorTxrts-. O.4»639E.OO JMAIR- 0. TI925E.02 XMA X A- 0.72U0E.Q2 XM1NR- 0.>il70t>02 AM1 NA- 0.51056002 ; ; _ ALPHA- 0.50001E.03 SET A- 0.100O0C.02 CAMA- 0.5O0O0E.O3 DELTA- O.1000OE.02 "~A»R'0"xfMTATE~A'MPinTu5E^0TurTuN lit 7HE LOUA-VOLIERKA SYSTTM Fig. 1.16 Lotka-Volterra Model Approximate Amplitude Solution for (xo,yo>-(60,:35) 29. _o.oo» 0.010 o.oii 0.020 o.Q2» . ."j; . 1 "., . . .! jHfTTTr^TTTT^i ~ fooloobt. A . A . Y M C A C S O L U T E • » « - ^ - 6 R , » L SOLOT.ON ™ » ™ TO BO, H SOIUT.OKS ,0- 0 . 6 0 0 0 0 E . 0 2 Y O . 0 . 3 M O 0 E . O 2 S • N G U L A K I , , AT . X . Y I - . 0 . 5 0 0 0 0 E . 0 2 . 0 . 5 0 0 0 0 E . 0 2 . - A W O E K Y . ^ - K w ^ v o r w ^ w m ^ W ^ o T i T ^ r ^ ^ n t ^ w . m a - I ^ I M * ! » » , o . u . « i ^ ~ o . T H 2 « f . Q 2 Y H A X A - O . T 2 1 2 0 f . Q 2 Y H I M R . O . 3 3 1 7 0 t . Q 2 Y M 1 N A . 0 . 3 3 0 > a f 0 2 ! \ • U . P H A . 0 . 5 0 0 0 0 1 . 0 3 B E T A . 0 . I 0000E .02 " C A M . O .SOOOOE.Oi O C l T A - 0 . 1 0 0 0 0 E . 0 2 —"IPPAUXIHATE AMRLfTOoE^SOCufTON"Df^ -THE LTJT^A-VOl.~fERRA~iY*iTL'S 1.17 Lotka-Volterra Model Approximate Amplitude Solution .for (x Q,y o) (60, 30. V (T ) 0 . 0 10.000 20 . 000 ".".".Tl 30.000 40.000 50.000 aO.000 70.000 80.000 90.000 100.000 l o o . o o o T ...-[..:.I.;..I77.-.-J . . . T i . . . T r . . . ; i . . . . i . . . . . . . . . . . . . . . — i . . . . i . . . . i . . . - 1 - . . . t iuu.uuu - • " _ . m 90 .000 - 90.000 s \ x / \ Y • • 80 .000- - 80.000 // \ \ if \ V 70 .000-II w 70.000 ti w 60 .000- r i« t V * \ * - 60.000 50.000 - 50.000 V " tr m-*f/ I st' § j . . . .ri . ••• \ 60.000 • — X - 60 .000 \\ J 30.000 - 30.000 • \ \ . TV' ' .-• 20.000 - 20.000 . „ ^ _ ^ Art AM TIC Al-- V inUuTld^f Yo.000 10.000 0.0 . 1 0 . 0 tl\%% m ANALYTIC AL SOLUTIU't - NUMERICAL SULUIION • POINTS COMMUN TO BOTH SOLUTIONS XO- 0.65000E.02 YO. 0.35000E.02 SINGULARITY XT I X , Y l - l 0.500006.02. 0.5JOOOE.O2I TSTTUENCY. x„ioV79???r.orXOS—OT7a76lV02'ERSo«—0-.178*6 XH«<R. 0 . 9 | 7 6 9 t * 0 2 0 . ^1627E '02 X HINR" 0.236Q9E»02 AMINA. 0.1b90<6»02 YrtAXR* 0 .91769F.02 YMAXA- 0 .91627E.02 YB IHR. 0.23t>09E«02 YHINA- 0 . U904E«02 T^OiTTTnTOOT.Tl^ fflTAV-oTroTlW^ 'J GAHA- 6.S0OTOT5O3 O E L I A - O.IOOOSE .02 APPROXIMATE A MR LIT UO E SOLUTION OF THf LOTXA-VOLTERR A SYSTEM F i g . 1.18. Lotka-Volterra Model Approximate Amplitude Solution for (x o,y o)=(85,35) 31. XITI 0.0 0"005_. 0.010 0 .015 1.020 0.025 0.030 0.035 0.0*0 0.0*5 0.050 „ — — i i i i i i....rr777TTTr7r7Tr7rr7rrT 1 0 0 . 0 0 0 nil» • ANALYTICAL SOLUTION • > " « • NUMERICAL SOLUTION 1 IME S E C . tA.fl . POINTS CUHMON TO BUT H SOLUTIONS «0" O.B5000E.02 YO- 0.350OOL.O2 SINGULARITY AT U.YI-I 0.50000E.02. 0.500001.021 XHAXR. Q.«l7l,<f»02 XHAXA. 0.91621002 XHINR. u..>jOTE.Q2 XN I NA • 0.1590*002 , , ALPHA. 0.50000003 ui T A . 0.100011 • 02 GAMA. o.«-"'"l»03 UELI A" O.10OOOC.02 —lPPRU"xlN~AtE-ANPLllunE StlLU f f C^TlIF I HE LUfXA-VOl TEMMA SYSTCM Fig. 1.19 Lotka-Volterra'Model Approximate Amplitude Solution for ,yQ) = (85,35) 32. • — • — • — * — » — » 0.005 0.010 0.015 . . I . . . . I . . . . I . . . . I . . . . I . . . . I . • • • • • — I 0.020 0.02, 0.030 0.035 » l t t t • ANALYTICAL SOLUTION • NJHERTCAL SOLUTION Tim: sirr: • POINTS COMMON TO BOTH SUL'JTIONS ..I . . . . I . . . . I 1 1 o . o 0.0*0 0.0A5 0.050 »0- 0.85000E.02 YO- 0.35000L.02 SI NGULA R11Y AT I..YLI O.5OO00E.02. 0.50000E.02I • 1HUK. 0.<176«.O2 YMA.A. O.^H,2TH02 YMINR. 0.2 31,0^.02 Y HI NA • 0. IB10AE.07 . _ _ . ALPHA. 0.50000E.03 BETA" O.IOOOOE'02 GA MA. 0.50000003 DEL t A" O.10000E.02 T»pW6«i«TE~AMP"Lnio"jE"Sua;nG"rruf" Fig. 1.20 Lotka-Volterra Model Approximate Amplitude Solution for (x o >y o)-(85,35) 33. Vf 0 1 0.0 10.000 20.000 . 30.000... .40.000 _.50.CQQ_._ 60.000. ,.,7.1.000 60.000 90.000 100.000 • 9 0 . 0 0 0 ^ 6 4 3 3 3 3 3 6 9 0 . 0 0 0 3^fS' 2 2 2 " ~ - » ^ ^ ^ 3 9 : / \ . * o°/ff B O . 0 0 0 B O . O O O - Sf 1 . - . 1 1 1 L 1 ^ . 1 — — - S\ 1 \ \ 6 -7 0 . 0 0 0 - 7 0 . 0 0 0 * / \ 1 6 0 . 0 0 0 - 10 " A ' 1 3 • 6 0 . 0 0 0 ^r** T • 40.000- 5 0 . 0 0 0 1 »>— — • (*V/YT T ^  "BT*  f *" 1 ^" 4 0 . 0 0 0 - 1 3 - 4 0 . 0 0 0 * \ 11 3 0 . 0 0 0 - l\ P I * • 3 0 . 0 0 0 - 3 \l / / 3 6 70.000- \ l lA 77 - 2 0 . 0 0 0 j 1 0 . 0 0 0 - 1 1 - 1 . .1 I- 1 *1 1 / • 1 0 . 0 0 0 ' — - - • • 0 . 0 0.0 10.000 20.000 30.000 40.000 50.000 60.000 To.000 80.000 90.000 100 000 X I O I C O N S T A N T R E L A T I V E A M P L I T U D E E R R C R L O C I C R A P H E R R O R X 1 0 . 0 0 0 1 S I N G U L A R I T Y A T I X . Y I - I 0 . 5 0 0 0 J E » 0 2 , 0 . 5 0 0 0 0 6 . 0 2 1 ALPHA- O.50O0OE.03 BETA- 0.10000E.02 GAMA- 0.5COOUE.U3 DELTA- 0.10000E.02 _APPS0XIMATE _APIPlITV0e_S0lUTI.CN CF _TH£_ LOTKArVQLTCHRA. SYSTEM F i g . 1.21 Lotka-Volterra Model Approximate Amplitude Solution Amplitude Error L o c i 34. 1.7 Summary We have developed two different analytical solutions to the Lotka-Volterra predator-prey system, using f i r s t of a l l , , the classical Ritz method (which for the case at hand yielded the linearized solution), and secondly, an approximate solution which uses a combination of two different approximating techniques. The solutions (see page 7") permit great insight into the system behaviour and the effects of each system parameter without resorting to computer studies. We should note here that in general, the Ritz method w i l l not generate linearized solutions as i t did with this example. As a rule of thumb, the Ritz solution i s valid (< 20% amplitude error and 5% frequency error) in the set of i n i t i a l conditions enclosed by an ellipse centred on the centre singularity with major and minor axes equal to 50% ofthe perpendicular distance between the singularity and the x & y axes. The amplitude error"is the limiting factor in this case. For the case of the approximate amplitude solution, the limiting factor is the frequency. The corresponding e l l i p s e has axes that are 70% of the distance between the singularity and the axes. These results raise the possiblity of developing approximate solutions for more complex (and more realistic) models. Such models and techniques for determining the approximate solutions are considered in the following chapters. 35. CHAPTER TWO ADVANCED MODELS AND LINEAR SOLUTIONS 2.1 The Linear Solution The f i r s t and simplest technique of obtaining solutions to the nonlinear models i s to l i n e a r i z e the d i f f e r e n t i a l equations about the singular point of i n t e r e s t , (x ,'y )• sing' sing We s h a l l write the d i f f e r e n t i a l equations as follows:-x = F(x,y), x(0) = X Q y = G(x,y), y(0) = y Q where F(x,y) = xf(x,y) G(x,y) = yg(x,y) The matrix of p a r t i a l derivatives i s computed to y i e l d the c o e f f i c i e n t s a^ '3-x 9y A = 3G 3G 8x 9y -The d i f f e r e n t i a l equation i s reduced to the l i n e a r form(where th p a r t i a l s are evaluated at the equilibrium point)• x = a l ; Lx - a 1 2 y y = a 2 i x + a22 y where x and y are the variables translated to the singular point of i n t e r e s t . The eigenvalues, s^ of the system are computed from det(A-sI) = 0. These are used to determine the s t a b i l i t y of the singular points, and thus determine the phase plane behaviour. a l l a12 a21 a22 36. We may use the well known transition matrix technique of linear, time invariant differential equation theory to solve the above equations. If the eigenvalues of A are complex, the complete solution i s given by: x(t) = x . + (x - x . ) e a t(cosc-t L-T :(a. --o)sincot) sxng o sing' _J__L__ • CO at, , . - e (y - y . )a,-smut Jo Jsing 12 y(t) = y . +(y - y . )e° (coscjt+(a„0 - a )sinwt) J -'sing K Jo Jsing 22 to + e t(x - x . ).a-- sincot o sing 21 where a = + a 2 2 ^ 2 co = % A ( a 1 2 a 2 1 + a 1 ; La 2 2) - (a^ + a ^ If the eigenvalues are real, the solution is given by: a l t X ( t ) = Xsing + ( xo " x 8 l n 8 > « a l l " °2)e + ^°1 ~ a n ) e a 2 t ) / ( a 1 - a 2 ^ a l t a 2 t " a12 <yD " 7sluJ ( e " e >/<°i- C T2 ) °l t y ( t ) = ysing + ( yo " y s i n g ) ( ( a 2 2 " °2)e + (a± - a 2 2 ) e ) J(a1 ~ a2' + a (x - x . )(e - e - a ) 21 o sing 1 2 where 01,2 = % ( a l l + a 2 2 ± / ( a n + a 2 2 ) 2 - 4 ( a 1 2 a 2 1 + a^a^) ) For very large i n i t i a l conditions, these local solutions depart from the true solutions. The regions of i n i t i a l conditions where these solution are valid w i l l be determined in the following sections. We now have another 37 solution which explicitly relates the system parameters and i n i t i a l conditions to the system behaviour. We w i l l how discuss some models and apply this method to obtain some specific solutions. 2.2 The Volterra-Gause-Witt Model The Volterra-Gause-Witt (VGW) (See Dutt 15) model i s the f i r s t model to be considered and is a modification of the Lotka-Volterra model discussed in chapter one. A quadratic term is added to the prey equation, which w i l l limit the prey population in the absence of the predator, and reflects the capacity of the environment to support the prey (see Gilpen 21). The model i s : 2 x = ax - Xx - 3xy c.,-f3,Y,<5,A.>.0 y =-yy + 6xy The model has three singularities which are located at (0,0), (a/X,0), (y/6, a - Ay/6) and these w i l l be discussed in greater detail in the next section. Gause (19&20) applied this model to two species of yeast with a good measure of success and also to two species of protozoa with some success. 2.3 Linear Solution of the Volterra-Gause-Witt Model We start our evaluation of the linear solution by applying this method to the Volterra-Gause-Witt model. We compute the A matrix of the VGW model to obtain: ' a -. 2Ax - ,(3y -Bx + 6y -6 + <5x 38. We now wish to examine the phase plane behaviour of the system (see f i g . 2.1). For the singularity at (0,0), we obtain the eigenvalues s 1 = a > 0-, s 2 = -y^O from which we conclude that this is a saddle point. The eigenvalues of the signularity at (a/A,0) are s = -a<0 and s = -y + ~- . ;If we have 6 C6 — < A (see figure 2.1A), the singularity is a stable node. Also, in this case, the third singular point has moved into the fourth quadrant of the phase plane, where i t is of no interest to us, as we are only interested nonnegative populations, which occur only in the f i r s t quadrant of the phase plane. If we have > A, the singular point at (a/A,0) is a saddle point. The third singular point ( x s i n g > y s l t l g ) = a - ^ Y / 6 ) ' 1 s a s t a b l e focus (see figure 2.IB) i f we have 0 < A < 26(-l + / l + a/y) and has eigenvalues s i n = ~ 1 y^ay - Ayz - A2y2" , 1,2 icj . ^2 If we have 26(-1 + / l + a/y < A < 6a/y , the third singular point is a -stable node (see Figure 2.1C), and is the singularity about which we wish to obtain solutions. The coefficients of the linearized differential equations at the third singularity, (x . ,v . ) are: sing Jsing a l l = " ¥• a12 " 3 Y / 6 a6 - Ay „ a21 " f3 a22 - 0 The system is always stable. The question of s t a b i l i t y in the general n'th order VGW model i s explored by Gilpen ( 22 ). This completes the determination of the linear solution of the VGW model. 2.4 Simulation Study of the Volterra-Gause-Witt Model We now wish to evaluate the linear solution of the VGW model. The solutions were computed for the arbitrary parameter choice a = y = 500, g = 6 = 10, A = l ( f o r which (x . , y . ) i s arbitrary a stable focus) over • sing sing 39. 3 4_S T/)&L£ y A ( O : ^> < — %Pit>OL£ POINT'S NO OB X F i g . 2:1 Possible S i n g u l a r i t y Types and Positions f o r the VGW Model 40. the i n i t i a l conditions ranges of 10 < x^ < 90, and 10 < y Q < 90 i n increments of 5. The time plots and phase plane curves for the i n i t i a l conditions (85,60) are given in figures 2.2 to 2.4. The corresponding amplitude and frequency error l o c i are shown in figures 2.5 and 2.6. As was the case with Ritz solution, the frequency was found to be substantially more accurate th an the amplitude for given i n i t i a l conditions. We shall assume that a 10% or 20% e r r o r i i n the solution i s acceptable. Another arbitrary set of parameters was used for which (x . , y . ) is a stable node. These were a = 275, 3 = 6 = 10, y = 50, sxng' •'sxng ' A = 45. The computations were done over the ranges of i n i t i a l conditions 1 < x < 9, and 1 < y < 9 in increments of %. The time plots and phase o o r r plane curves for the i n i t i a l conditions (3.5, 1.0) and (6.0, 3.5) are given in figures 2.7 to 2.12. The corresponding amplitude error l o c i curves, which are the only ones needed for solution evaluation, are given in figure 2.13. For the case of the stable node, the amplitude error i s taken as the worst case percentage difference, on a point by point basis between the actual and approximate solutions. It should be noted here that with the stable node, i f there i s a slight error in the i n i t i a l derivative, there may be a large difference between successive time plot points, which results in a large solution error even though the solution i s visibly quite good. As a result, we may consider solutions with up to 40% amplitude error.-as acceptable. For the case of the stable focus, this "problem" in solution evaluation does not occur, as the amplitude and frequency errors are treated separately, and do not influence each other. We have evaluated the linear solution of the VGW model and found that i t is a good approximate solution (see error l o c i figures) over a wide range of i n i t i a l conditions. We now proceed to study another ecological model. 41. 1 Q . _ 0 _ 2 0 - 0 0 0 3 0 . 0 0 0 A O . 0 0 0 5 0 . 0 0 0 6 U . 0 0 O . 7 0 . 0 0 0 . . 8 0 . 0 0 0 9 0 . 0 0 0 1 0 0 . 3 0 0 . n.iMF«ir.ti. tmiiT mw - P M N T S mum., T C B O T H S O I U T I O N S ;o- 0 . B 5 0 0 0 E . 0 2 V O - O.fc-OOOOE.02 S I N G U L A R I T Y A T lx .TI-1 0 . 5 0 0 U 0 E . 0 2 , C . . 5 0 0 0 E . 0 2 I F K E - . U E N - . V 1 «l»."oTfii'7lt.O. Nj7.V~o772 7-iT V-2" 0««'i»- 0 . 3 4 5 2 5 E . 0 1 « "AlPLiruuE" E R U O R T E H EI..-6 . 3 0 9 3 - E . O 2 I E A B S — O . 1 0 1 4 2 E . 0 2 iim>. o.5moif.r<5 p.riA. o.ioonnF.02 G A » A . 0.500001.03 ariri- O . I O O O O E . 0 2 n m M - O . I O P O O E » I I : '. E Q U I V A L E N T ' L I N E A R I S A T I O N S 3 L U I I U N OF LOT . .A-VOL TE R R A 5 Y S I E H « I T H P R E Y C A R R Y I NO C A P A C I T Y F i g . 2.2 VGW Model Linear Soltuions for (x >y.0) = (85,60) 42. a m ...0,005 0.010 „ 0.015 U.020 .. 0.025 . . 0.030 . 0.035 0 .0*0 0 .045 0.350 . . .._ I . . . . I . . . . i . . . . I . • I 100.000 n.nn^ O . H O 0.015 0.020 0 .025 0.030 33.035 — I • • • - — • o.o _Q.0*0___j n . n . 5 n-nsn U m , - /m.iYTir.i SOIIITIIIN HIKFRICAI SDIUIHIH TIME SEC. I IH I • PUlHTS COMMON TO PUT H SOIIITIQMS X0- 0.850001*02 YO" 0.60003E.02 SINGULARITY AT IX .Y1 - I 0.50UOOE.02, 0.A5000E.021 ~~FREyTENC Yi »Ni"oT7527!f»02 NUM- 0."/'2TI,TE'«02 ERRJHV ' t , M ) ! ! E t l l l AMPLITUDE ERRCR1 bxREL.-0~.3»9iaT»6"2 «_uT8S.-0.10l*2E»02 AIPHA- u.sniOOf.03 nrTf 0.100011.OJ OAHA. U.5QOOOL.Q3 DELIA. 0.1000UE.U2 L AM ADA* 0.10000E.Q1 ; — EQUIVALENT LINEARIZATION SULJTION UF LUTKA—VOL TERRA SYSTEM . ITH PREY C AKk Yl NS CAPACITY Fig. 2.3 VGW Model Linear Solution for (x »y Q) = (85,60) 43. T i l l . _ O . Q . _ 1 0 0 . 1 0 0 1 . . . . I. J . 1 0 S 0 . 0 1 0 . . . . . . . I . . . . . . . . . . . 0 . 0 1 5 . . . . ...I I. 0.025 .......... 0 . 0 . 0 . . I . . . . 0 . ( 1 3 5 . . I . . . . . 0 . 0 * 0 ... 1 I 0 . 0 * 5 . . . I I.. 0 . 0 5 U - _ . I 1 0 0 . 0 0 0 ...r.. 5,11,.HON  11 H E S C O . „ . 0 . . 5 0 0 0 E . 0 2 . 0 . O . . O 0 O O E . O 2 S l h E U L . M I . « I...... O . 5 0 O 0 0 E - 0 2 . O . * , , 0 0 E . O 2 ^ - , ; r H n < 1 B . 1 „ , „ . , „ »»>lltU..€ E . R O R T C T K E L . - O . I S ^ E . O J X « S _ . - 0 . . » f i . - . . l " f l l E 0 J E " N C < l » N - 3 . T 5 2 7 3 F . 0 2 N U M - 0 . 7 2 7 6 1 E O 2 E R R O R - 0 . 1 * 5 2 5 1 . u i m r i n - • — ' . n ' P n . - . " ' ^ " f " ' « " t - l " W M 1 ; . . • \ t Q U , V U E N , . . . . . . . . . . . C N M L U . . O H W U . 7 K . - V 0 L , . - . . - « . » - T H F R E , . « . ' • « . Fig. 2.4 VGW Model Linear Solution for (x o >y o) - (85,60) 2.5 VGW Model Linear.Solution Amplitude Err o r L o c i 2.6 VGW Model Linear Solution Frequency E r r o r Loci 46. . . I . . . . 2.000 ».ooo-P ^ - ^ T ^ 5 ^ r ) -o.o i . . " ! - - - - ' - — 1 " " ; ! ^ — 1 0 - " 1 .300 _ *" XITI 1,00!! 2.C.0O 3,005. ...wrDir.i sniuTION • POIN.T? ™H»nH TO BOTH SILUTIONS. ,0. 0.1S000E.01 TO" 0.100O0E.01 S.NGU...AMY .T I».»"•-1 O.5C0OOE.01. 0.500006.011 "ERROR" 0.0 t AHPII TUBE' ERROR!'""mi.- O.8«102e.02t E.BS- .0.15»9*E»01 fREOUENCYi' "»N" 0.0 "UH- 0.0 £„U,V.LE»T UkE.RI2.nOH SCIUMCN OF L CTK.-VOLTERF.. S,STEH -ITH RREY »....« « - « l T < _ F i g . 2.7 VGW Model Linear Solution f o r (x o , y Q ) = (3-5.1) , 47. i . . . . ..... i . . . , ( . . . , j . . . . ( . . . . I.... I . . . . . . . . I 10.00 2*000' TIME SEC ...» • ANALYTICAL SULUTIOH UHFR . f A. 4n.HT.n- . „,M„ r A m m -„ „-•„ <„„•,-,.,•  X^O- O.J5.00E.01 TO. O..O0O0E.OI SINGULARITY AT {,,„.( 0.50000E.01. 0.50000E.OU .ALPHA. O..TSCO_.0) BETA. 0. |0r,00f .c, ..... 0..0-aot.„ 0.lr,00-r.-, „.„•„,..„ ' EOUIVALfNT LI.1EAMZATI0N SCLOTIPN OF U'TKA-VOl'FRAA SYSTEM -ITH P«E» CAR.TIN. CAPACITY F i g . 2.8 VGW Model Linear Solution for (x ,y ) = (1-5 1) o o ' 48. V | T | J 3 . 0 — 9-030 p«Q*Q. _P - ,Q90 Of 120.. 0 . 1 5 _ 0 0.. 180. 0 . 2 1 0 9 , 2 * 0 0 . . 2 7 0 0 . 3 0 0 1Q.3JU 1 10.000 9 . 0 0 0 - 9.000 6 . 0 0 0 - - 6 .000 7 , 0 0 0 - 7.000 6 . 0 0 0 - 6.000 5.000 5 . 0 0 C -J>€%^ -v.oco-AAlAMTlcM . j r • .000 3 . 0 0 0 - 3.000 / S^* * SQLoTloti 2 . 0 0 0 • / >- - 2 .000 l.OOOl 1.000 -0 .0 0 1 0 . 0 .0 0.C30 0 .0 *0 0 .040 G.120 0.150 0.193 3.210 0 .2 *0 0.27O TIME SEC. » H » • ANALYTICAL S n f T I Q H •NUMERICAL SOLUTION • P31NTS COMMON TO BOTH SJIUTIONS X O " 0.350000*01 YO- O.lCOOOfi+Ol SINGULARITY AT I X . Y I - l 0.S0G00E»Ol* 0.50000E >>01J "PREQJENCY t" AN» ( . 0 "~ N U M - 0 ^ ~ E P P O k * 0 .0 " ~ " "f AMTPLITUOF ERPO *VEVPEL»~6 . 8 *90 iE»62* E Y A B S " 6.l5994E*6l A L P H A . Q . 2 7 5 0 G E + 0 3 BETA- 0.100Qfl(»C2 C A M A » 0 . S O P P O , . *02 DELTA* 0 .10003E+J2 LAMBPA* 0 . * 5 0 0 0 E » 0 2 ; \ EQUIVALENT L I ti 6 Af- I CAT I C N S I .L I 'MCt . t-F LnTKA-VtUTFfc*A S Y S T E M WITH PRl V C A R R Y I N G CAPACITY Fig. 2.9 VGW Model Linear Solutions for (x ,y ) = (3.5,1) 49. Y ( T » 0 .0 1.000 2 .000 3 .000 4.000 . 5.00Q „. *...QOtt 7-000 B.000 9__00Q 10,303 lo.uouj 9.000- 9.000 \ 8.000- a.ooo 7.000- 7.000 6.000- 6.000 5.000- Y 1.000 4.000-\^ " 56 L I) Tl A A* 6.000 3.000 - 5.000 2, 000-- 7 .000 1.000 - 1.000 : . • • ' 1 0.0 0.0 1.000 2.000 2-000 4.000 5.000 6.003 7.000 8.000 9.000 ftlitl - ANALYTICAL SOLUTIfW • - NUMFRICAL SnuiTiPN ***** - POINTS C3.H.M0N TO BOTH SJLUT1QNS «0- 0 .60000E.01 7 0 - 0 .35000E -01 SINGULARITY AT IX.VI•< 0 . 5 0 0 0 0 E . 0 I . 0 -50300E.011 Ti^NCYi'Tw-'ora T."UM."O.O'~ .wot.- 6:0-—— , :AM>LITUOE" ERROR, E R F O o : - T 2 6 0 - . 6 i i . - M ^ - I M I T ^ O O ALPHA- Q.2T500E.03 P-FTA- Q. | QOf.nf .f 2 GAMA. 0.500001.07 OtlTA- 0.10000_»_? LAHPOA- n.4iP(H.f OZ 60UIVALf:.T L INEAF IJA'inN S(luT!t,N IF I nTK A- V.LTFRH A SYSTEM MI TH PREY CARRYING CAPACITY Fig. 2.10 VGW Model Linear Solution for (x ,V ) = (6,3.5) 50. .0,1.50 ..I 1. 3.210. _ 0.2AQ.„. . I . . . . I . . 0.270.... . . I . . . . I . 0.303 . I 10.000 • • • 9.000- . 9.000 * — /\ \_ ' • .nnn • • 7.000- • T.000 6 . 0004 6.000 •wiiilHifn* "^  jj^ u^.^ ^^ aawBTTirrrM*^ ^ * A • - • • . • . • • • • • ^ • ^ • • • • • • • • • • • • • • i j j l . l t l j t . t l l l .. nnn 5.0OQ- - T • v ftNfllSiriCAL ... ... !__ %.ooo-! • 3.000- 3.000 2.000- - — . ; — '- 7.000 - — ; - — : • 1.000-• 0.0 0.0 Oloio o loeO. WtO_ .0.120 ..0.130 0.183 3. 210...:_0.2*0. _ „ . 0.H«_ .1 0.0 0.350 . . . . . a A M Al VT If A V »«P«ini sniliTinN ••»•« • PUNTS CJHHON TO BOTH SJLU.I10NJ S*.3>t • ANfll, T 1 i l *> L, <0" 0.«0000£*01 YO FREQUENCY t AN* CO 0.3500GE.01 SINGULARITY AT IXiYLl 0 . 50 0 0 0E«01, 0.50C.00E «01l s HUM 0.~0 ERRORV-6.0 flHPLITUOf EA«OR~l EXPEL—0.6A891E.00I 6XABS.-6.136A7E-01 ' " « • - " «nn..i,PWi» otiTA. 0.100r,OE.02 LAHUOA. O.ASOOOS'O.J EQUIVALENT LINEARIZATION SOLUTION Of LOTKA-VOLTERRA SVSTEH HlTH PREY CARRYING CAPACITY 2.11 VGW Model Linear Solutions for (x T,y Q) = (6,3.5) 51. Yl 0.0 10.0001 . T ' 0.030 o . 0 6 0 _ o . o . o . 0.120 o.iH o-iM ^^-^^rxrr^T^;^^ ...\....\. ' . ' ..\ 1 1 1 I....I....I....I....I • 9.000-9 .000 • 1 11.000 • -. 7.000^ • -• • 6.000-6 .000 • • ! 4.000-./^ SOLUTION -• . 3 .000 3 .000 : • • ' " ... « — • — Z j.noo • • • -1.00C • - l . n n a -• 0.0 ).o" C.030 0.060 0.090 „ .0 .1J .__ . 0.150 ... 0.180 0.210 0.240 . - - « ' " . 0 M M ... . ANALYTIC! SPL UT 1 ON . NUMFPI-AL SOLUTION ..... . POINTS CJHHON TQ BgTH SuLUTlOTS I. 0.60000t«01 TO- 0.35000F.01 SINOULAPITY AT IX.YIM 0.50000E.01. 0.500C0E.0U ^ i R O f W A - i M T T S fM.Sji Y « * * ~ K S « AMPLITUDE ERROR. EYPEL" 0.3724-».»0l 1 -YABS" 0V.5317.V0O .PHA. 0.773CCE.03 BE TA. O.lQPOCr.02 GAMA. O.5COOOE.02 OF I T A • 0.13003E.Q2 L AM BOA* 0 .45 000E»» . : E4UIVALEN7 IINEATUATION SOLUTION CF LOTFA-VOLTERRA SYSTEM WITH PREY CARRYING CA»4CII» . 2.12 VGW Model Linear Solution for (x 1,y Q) = (6,3.-5) 52. ° n i p I . c op .i....i....i....i.-"i---:i;--'""l.;;-'"-,!ooo',""'6!ooo"L-'^ •' j.QQQ > . 0 0 0 5 . 0 0 0 6 . 0 0 0 ' . " O O _ | 0 | CONST-NT B E l t T I V E • M P f I T I T E E R P O P L O C I " W APM EKFOR » 1 0 . 3 0 0 X S I N O U I M I T V . T <».VI-« 0 . 5 0 0 0 - 6 . 0 1 , O . S O O O O E . O l l ^ r o T T T ^ ^ c r T B T r l ^ ^ ^ M.PH* ^ q o j X t t l a l - U . 1 F i g . 2.13 VGW Model Linear Solution Amplitude Err o r L o c i 53. 2.5 Holling's Model The next model we wish to study is Holling's model (see Holling 25). This model is one in which another physical characteristic of real ecological systems i s incorporated, that of f i n i t e k i l l i n g capacity of the predator, which is also dependant on the quantity of prey available. The model i s : x = r x ( l - x/K) - xy/(A + x) r,K,A,s,J > 0 v = sAy(x - J) y (J + A) (A + x) In the prey equation, r governs the growth rate for small populations, while K is the carrying capacity of the environment. For large prey populations, the predator-prey interaction term, xy/(A +x), (known as a Michaelis - :Menton type interaction) indicates that the predators k i l l prey at the maximum rate of one prey per unit time per predator, which i s the upper limit of the saturating term. For small populations, the prey are more d i f f i c u l t to locate and thus the k i l l i n g capacity of the predator drops in a nonlinear manner. The parameter A controls the prey population level at which this effect becomes noticable. Note that for A » x, the equations reduce to the VGW model. A complete derivation of this effect i s given by Frederikson ( 18. ) and Holling (25 ). In the predator equation, the parameters s & J control the natural mortality rate of the predator and the magnitude of the predator-prey interaction. This model has three singularities, which are located at (0,0), (K,0), (J,(J + A)r(1-J/K)), and are discussed in the next section. Holling (25) has numerous examples of ecological systems-that obey the Michaelis-Menten type interaction. Two of these are short tailed shrew (predator) and sawflies in cocoons (prey); and the mantidae and adult 54. housefly. Validation of the complete model is. s t i l l pending. 2.6 Linear Solution of Holling's Model We now proceed to determine the linear solution of Holling's model. The A matrix corresponding to Holling's model is A = r - 2rx/K - Ay/(A + x ) 2 - x/(A + x) sAy/(A + x ) 2 sA(x ~ J) (J + A) ;(A + x) and For the singularity at (0,0), we have the eigenvalues s^ = r > 0 s 2 = -sJ/(J + A) < 0 which indicates that this is a saddle point. The eigenvalues of the singularity at (K,0) are s^ = -r < 0 and - (J+A)—(K+A)" If K > J, we have a saddle point. If J > K, we have a stable node, (see figure 2.14A), with the third singularity having moved down into the fourth quadrant of the phase plane, where i t i s of no interest. For K > J, the third singularity, ( x s i ng>y s i n g) = (J,(J + A)r(l - J/K)) is the one of interest to us. If K < J < K (where K £ can be determined from the system eigenvalues with some d i f f i c u l t y , see Brauer 7 ). we have a stable focus (see figure 2.14B). If %(K-A)<J we have a stable node (see figure 2.14C). If we have 0<J<%(K-A), the singularity becomes unstable, iand Kolmogorov's theorem (see Brauer) reveals that this gives rise to a limit cycle (see figure 2.14D.). The major difference between Holling's model and the VGW model i s that the prey isocline in Hollings model is humped and allows for the possiblity of a limit cycle, while the prey isocline of the VGW model is linear. The linearized system coefficients at the third singularity (x . , y . ) are: J 6 J sing' •'sing a n = r ( l - 2J/K) - Ar(l - J/K)/(J + A) a 1 2=J/(J + A) a 2 1 = sAr(l - J/K)/(J + A) a 2 2 = 0 55. V T 7 /< \ STABLE T / K F i g . 2.14 Possible S i n g u l a r i t y Types and Positions for Holling's Model The problem of determining solutions for the limit cycle have not been considered in this thesis. This completes the determination of the linear solution of Holling's model. 2.7 Simulation Study of Holling's Model We now wish to revaluate the linear solution of Holling's model. The solutions were computed for the arbitrary parameter choice of r=2, K=45, s=l, A=10, J=20 (for which (x . ,y . ) is a stable focus), over sing; sing 7 ' . the ranges of i n i t i a l conditions 5<x <45 and 5<y <45 in increments of 2.5. o o The time plots and phase plane curves for the i n i t i a l conditions (17.5, 42.5) and (30.0, 30.0) are shown in figures 2.15 to 2.20. The corresponding amplitude and frequency error l o c i are shown in figures 2.21 and 2.22. As before, the frequency i s much more accurate then the amplitude for given i n i t i a l conditions. The above computations were repeated for K=25, for which (x . ,y . ) is a stable node, and the corresponding amplitude sing'-'smg ' t- a t-error l o c i are shown in figure 2.23. We have evaluated the linear solution of Holling's model and found that i t i s a good approximate solution (see error l o c i curves). We now proceed onwards to study a third ecological model. 2.8 Rosenzweig's Model The third model we wish to study i s Rosenzweig's model (see Rosenzweig 48). The same physical phenomenon found in Holling's model are incorporated i n Rosenzweig's model. The model i s : x = rx(l - x/K) - b ( l - e )y r,K,b,c,s,J>0 , -cJ -ex. y = sy(e - e ) 57. ~>ttlll—,r7?j™~l~ia1VM , li;'>00 , -• , « . 0 0 0 - _ JO.OOO.-_JS.OOO_..._*0.000 _.«.o i . . . . i . . . . i . . . . i . . . : r . : . . i . . . r i T : : : i OQ 50. 000 ...I....I 50.000 * M 1> \ —V— Ay»*.«.« jl — . • T ^ * \ * \ : 30.000- \» - 30.000 • 0.0 l....|....|....|....|.„.| I....I....I 5jjiafl ijjjaao is.ooo ... zo. , . H I I t • RNALTIKAI • in i l lT lUN MiaiirtL s m i r i m . * * * i.noa IO.OOOIT .is. o"oo '''"ii!ooo\L\^'o!ooo!.N llll 10. 0.17500E.OJ TO. 0.4P500E.02 SINGULARITY AT l>,v|.| U..IOOOOC «02, 0 .15U3E.02I -FRE*JENCY. ttiV^TTX-a7*Y*M.ox ERROR. O.ITT ? ,E .O ? < AMPLITUDE e,ia>ri,^.o.2i^^c^ro^i\i7oT B- -Vl «,• o.Ayinn.07 » • O.IOQJOF.,-)? <. o.i.ianofwi i. ii.i>n»lf.|)f ' _ ^ LINEAR SOUTH]* OF HOLIING'S HJ'ltL ' Fig. 2.15 Rollings Model Linear Solution for (x Q,y ) = (17.5,42.5) 58. 50.000 III 0. _5.000.„. 10.000 ._15,300 20.000 . 25.000 30.000. „..35.000„ ... .40.000 ...45.033. .50. 000.. . ; 45,000- 45.000 40.nno- •_n .nnn 35.060- 35.000 30.000 30.000 1 \* N 0/V) CQU 7S.010 A \ - .nnn ' j - \ • ^ 20.000 20.000 _ J / \ \ // • 13.000 \ 1 ^ frHM-IJ \CfrL, • 15.000 tf in.nnn \ I - V . in.nnn 5.000- 5.000 0.0 o 0.0 .nnn .0 5.0,00 in.nnn IS. Of)*. 20.000 Z5.000 30.000 35.030 40.000 ...45.000 SQ TIME sec. \\%\\ • 4HrVt,YTict,' <nniTinM •«»«< - .i-jurn;CAL sunn ION •., pgip.TS.CfJH^ N^.TO MITH yimrmis X0- 0.1T*>0£*02 ¥0- C.42500E*02 SINGULARITY AT ix.V I* I 0.200005*02, 0.333336*02) fREQJENC Yl AN* b~794.6E-01 NiJ"V»'"o^  67462E-')1 FRHOH" O.I7721E*0?t AMPLI TU-H t RH Oil fX RCI--0.12411E*02« EX AtfS»-0. 3 7265E *01 • • o.go:inor»oi n« n.*,^ nnnF»»7 A- P. i-Juanf *n? t- a. ia»nrt£ mi J - o.2noQat=»Q? ; : tl NF Aft S0LJT10N f.F HOL I [NG* i HtiOLL Fig. 2.16 Holling's Model Linear Solution for (x »yQ) = (17.5,,42.5) 59. V 0 . )T_* 5 . 0 0 0 . 10.000 . 1 5 . 0 0 0 20 .000 2 5 . 0 0 0 .. 30 . 000 _. 35.000 ^ 40 . 000 ^ . .45.000 ......50. 000 50 .000 SO* 0001 45 .000^ 4 5 . 0 0 0 . L i • 4 0 . 1 0 0 -\ ANALYTICAL - 4n.nnn \\ K * ~ 3 5 . 0 0 0 -••- y t ; • ~ L I L I V \ t t i M i t t • 11 / / \ \ fi /iTS^LM > 3 3 . 0 0 0 j r i l V A / / * i ^ * * * ^ ^ A j M ^ r f T r ? * * * . \ \ / A \ \ A / ^ t W i f c w r ^ * -30 .000 - 1 \* / J 1 30.000 24 .100 \ 1 »4.onn » U N\ BO.) CftL j'o.oo'o '. V S — SOLO Tl* - 20 .000 1 5 . 0 0 0 - I S . 0 0 0 Io.oon - lo.noo j .000 5 .000 0 .0 1 0 . 0 -ft s . n n o I0.OJO 15 .000 20.000 . 25 .000 30 .000 35 .000 .40.000 45-.QOQ 51] .ftfttl T I M E S E C . m i l . « N A I Y T i r « i v i ' N T f r . i . . . . . . M.IHFPICAL 1ULIJT10N •<<<> - P L - I M S . C I U H U H IL) OniM S tnUTlOHS XO" O . 1 7 5 0 0 E » O 2 Y 0 - 0.4250111. •02 SINGULARITY AT ( X , Y l « ( 0 . 2 O 0 O 0 E » O 2 , 0 . 3 3 3 3 3 E * 0 2 I FREQJCNCYi AN-" 3~.79Vlif-0l NUMV 'yr6 7462r ' -0l lERR0R - t 0 . 1 7 7 2 l E * 0 2 * AMPLITUDE t R P O R T E V R E t - O . 2 l 4 » 9 E » 0 2 t EY ABS • 0.4861 IE i b l R i Q . 2 0 0 0 Q f » 0 1 4Sf1^Q^.g? t>- O . t O O 0 0 E » Q 2 S» O.10000t*-01 J ' <) .20nOOE»0? . LINE AR SOLUTION OF H U L L l N G ' S NUOLL Fig. 2.17 Holling's Model Linear Solution for (X q,y o) = (17.5,-42.5) 60. YIT> _ 3.0 '- 5.000. ' 50.0001 .... I..--I. •• • . 10.000 ...I.... 15.000 — 20.000 ...|....I ....I .... 5.OOO - 30.000 ......I....I... 35.000 I....I.... AO.000 ...I... SS.OOO.... . 1 I. I. . 50.000 ...I 50.000 65.300- • - A5.000 .... _ • 1 .n. nnn tn.nnn- — — • % %. * — ^ 35.000- I - 35.000 —• ' \ I \ \ \ \ . *\ \ wl""" I \ \ \ " — 30.000-- 30.000 • \\ut\i\ea\cAL $ 0 / . 0 TJ <LtL - 1 • '. ?5.nnn 75.000-~~ " : 20.000 -• ""7" • 20.000 • * -• 15.000- . - 15.000 ' » . O.30100E.02 YO. 0.300031.02 SINGULARITY AT -I 0.20000E.02. 0.33333E.021 . . - f t ^ C S:TS-yiVE~V0VT^5^b, E»«0«.-O.l12nE.OU-ANRL.tU3E |K.... EREL-" 0^33T.3E.02. EASS. 3.TS.5.E.01 . . n n n o r . m fl- o u n n o r . . . ; n.inooor.Q? s- n.mmiiiFHii i - n.?(intwt • LINEAR SOLUTION OE HOLLING'S MODEL •' , —• F i g . 2.18 Holling's Model Linear Solution for ' ( x ^ y ^ - (30, 61. 0.0. 5.000 10.000 15.000 20.000 25.000 30.000 35.000 40.000 J5.000 ... 50.000 . . . . 50.0001.... i . . . . I.... I •••• I.... I. ...I ....I . . . . I . . . . I • ...I I. ... S O L U T I O N s.nio 10.000 15.000 . 20 | , .| t | | | | | I 1 1 1 0-0 ,000° "25.000 ...30.000. 35.0UO ....40.000 .45.000. 50.0.00 ..... . i»,i.nf.i 4.11 .IT.nv, ..UT.!!-!. sn.uiioN iHti - poihn rnmnN ill will inninaNS »0" O.3300OE.02 YO. n.300006.02 SII.aoi.4PI IY 41 II.YI'I 0.2OO0OE.02, 0.31333S.02I ^ PRE 0, U E Ni Y» AN* iT76»3f=o7 hUN."o.7%5.F-Ol ERRUK-U. 1121 IE .01« ANPLITUUt ERRORT EXREL* 0.33 263E*02I E A ABS" 0.738S6E.01 p. n.MiMfni n.nminni A. n. innooF.n? s. Q.looooE.QI •• o.?l)OOOE'0.2 _ — LINEAR SOLUTION OF HOLLING'S NOOEL F i g . 2.19 Holling's Model Linear Solution for ( x »y o) = (30,30) 62. . 0 . 0 s . 0 0 0 10.000 . 15 .000 . . .20 .000 25.030 . 30.000 . 35.000 * 0 . 0 0 0 . . - * 5 . 0 0 0 50 .000 _. • 5 . 0 0 0 - - 45.000 l AO.OOO- - 40.0(10 '- J \ S0LL>r/»A/ : '• 1 /** i \ i • • '35".00d-" \j • 35.000 ' V OLJMH>><i'»ll»IHb|J»» •J 3 0 . 0 0 0 f - 30.000 : A <*or U T/OA/ • 7S.nnn- - 7<5.onn -Vb."ooo- - 20.000 • * * 1 5 . 0 0 0 - - 1 3 . 0 0 0 i n.ifio- lo.nno : -• --5.000- 9 . 0 0 0 1 " ' 0 . 0 5.1-100 10. 000 15.000 20.000 25.30 0 .30 .000 _. 35.030 <.n.nno fcs.nnn *o.non T IME SEC. A l t 11 - AMAl YT1rM ninT iMN »#•••• - MiMFHir.Ai sniiirmfi - PCINTS C M M M 'TN in RO- 0.33000E»02 YO- 0 . 30()0>t>02 SINGULARITY AT (X,Y)-{ 0 .200006*02, 0.33333F»02) FREQJfNCVI AN- OW0763t-6l NUrt- 0 . 7S65&E.-01 f RRUH • - 0 . II 2 HE* 014 AMPLITUDE EPRLRl EYRCL- 0.76627F*0U EYAdS- 0 .278*0£*0l R- 0.20100?*01 K- 0 4«,on1-iF*ii.'* A - (i.innnnr»n? s« o. ioonof»oi J - o.?noooE»02 L I N E A R SOVJTION Of HOLLINu'S MODEL Fig. 2.20 Holling's Model Linear Solution for (x ,y ) = (30,30) 63. TIOI 0 .0 5 0 . 0 0 0 ! . . . . _ 5 . 0 0 0 10 ( 000 15 .000 . 2 0 . 0 0 0 iS-GOo!. ,0.00 0 3 5 . 00 0 _.»0 j O O C ^ 5 . 0 0 0 . ^ 5 0 . 0 0 0 ^ J . . . . . . - . ; j ; ; - - » - - ; ; ! S ^ S ! ^ CONSTANT RELATIVE AMPL1TU0E ERROR LOCI GRAPH ERROR X 10 ,0001 S INGULAUTY AT I X . T I M O . 2 0 0 0 J E . O 2 , 0 .33333E.02 I » . 0 .20000E .01 » • O.A5000E.02 A - 0 .10000E .02 S- 0.1O0O0E.O1 J - 0.2COOOE.02 _ U N £ A P . S O L U T I O N .Of -HOLLING-S BOOEL -F i g . 2.21 Hollingf'.s- Model Li n e a r S o l u t i o n Amplitude E r r o r L o c i 64. 1 n 5 0 * 0 0 0 101 0.. . 5 . 0 0 0 1 0 . 0 0 0 - 1 5 . 0 0 0 . 2 0 . 0 0 0 - 2 5 . 0 0 0 3 0 . 0 C 0 3 5 . 0 0 0 . „ . * 0 . 0 0 0 - 4 5 . 0 0 0 . , 3 3 . 0 0 0 . _ 1 1 l l l l I i • I i I I I i I 1 I I I c n n n n 4 5 . 0 0 0 - 7 5 4 3 2 2 3 3 4 4 - 4 5 . 0 0 0 8 4 3 2 2 • 1 2 2 3 3 7 \ 7 ' " ^ ^ ^ ^ ^ 7 7 7 - .40-OrtO 6 * 2 l / 1 l ^ V l 1 I 3 5 . 0 0 0 - 3 - 3 5 . 0 0 0 i 3 0 . 0 0 0 -/ ' r - 3 0 . 0 0 0 \ 1 \ 1 * M . n n n - 9 \ 1 1 - ? * . n o n 3 2 2 ^^N^ ; 2 0 . 0 0 0 - 5 4 3 3 3 3 >v / - 2 0 . 0 0 0 1 5 . 0 0 0 - 6 5 5 5 5 1 1 - 1 5 . 0 0 0 i n. nnrv 1 l o . r o o 5 . 0 0 0 5 . 0 0 0 0 . 0 0 1 0 . 0 . 0 3 . 0 0 0 l Q ^ O O " 15 . 0 0 0 2 0 . 0 0 0 2 5 . 0 0 0 3 0 . 0 0 0 3 5 . 0 0 0 4 0 . 0 0 0 . 4 5 . 0 0 0 5 - n n n X I 0 I C O N S T A N T R E L A T I V E FREQUENCY ERROR L O C I GRAPH ERROR X 1 0 . 0 0 0 1 , S I N G U L A R I T Y AT I X . V I M 0 . 2 0 0 0 J E . 0 2 . 0 . 3 3 3 3 3 E . 0 2 I 0 . 2 0 0 0 0 E + 0 1 K « O . A 5 0 O 0 E . O 2 A - 0 . 1 0 0 0 0 E . 0 2 S> 0 . 1 0 0 0 0 E . 0 1 J * O . 2 0 0 0 C E . 0 2 L I N E A R . SOLUT ION .DF_.H0LL I N G ' S M00EL . . _ - . — — 2.22 Holling's Model Linear Solution Frequency Error L o c i 65 v J . O O Q . : 1 0 . 0 0 0 . , . . . -15.300 2 0 . 0 0 0 2 5 . 0 0 0 30 .000 _ . 35 .030 S 3 . 0 0 3 ..".5.000 5 0 . C O N S T A N T R E L A T I V E A M P L I T U D E i R R O K L O C I G R A P H ERROR X 1 0 . 0 0 0 1 S I N G U L A R I T Y AT I X . Y L I 0 . 2 0 0 0 0 E . 0 2 , O . I 2 0 0 0 E . O 2 I « . 0 .200006*01 O . 2 5 0 O 0 L - O 2 A . 0 . 100001i.02 S - 0 .10OTUE.01 J - O .2000OE.02 ^l»ULS.OJLUJJD!l_QI_Jl(;L.LU.aiS_.'lC.1tL -F i g . 2.23 Holling's Model Linear Solution Amplitude Error Loci In this model, the saturation of the predator appetite is expressed in terms of exponentials (known as the Ivlev type interaction),; instead of a ratio of polynomials as in Holling's model. Here, the parameter b represents the maximum k i l l i n g capacity of the predator, while c governs the prey level at which the predator appetite saturation effect occurs. Rosenzweig's model behaves similarly to Holling's model, as the same physical phenomenon are modelled in each. As was the case with Holling's model, the prey isocline is humped. The model has three singularities which are located at (0,0), (K,0), (J, r4r^  J / j ^ ) — a n c j these are discussed in the next section. b(1 - exp(-cJ)) Ivlev (29) has several examples of ecological systems that obey the above type of predator-prey interaction. Some of these are carp fed on nonliving food; roaches fed on chironomid larvae; and bleak fed on Daphnia pulex. Again, the validation of the complete model for specific ecological systems is pending. 2.9 Linear Solution of Rosenzweig's Model We now proceed to determine the linear solution of Rosenzweig's model. The A matrix corresponding to Rosenzweig's model i s : A = r - 2rx/K - bcye c x -b(l - e C X) -ex „ / - „ - " C J ~ C X N scye s(,e -e ) For the singularity at (0,0), we have the eigenvalues -cJ s^ = r > 0 and = s(e -1) < 0, which indicates that this i s a saddle point. The eigenvalues at the singularity (K,0) are = -r < 0 and s 2 = se c J ( l - e c ^ ~ ~ J j ) , i f J<K, we have a saddle point. If J> K , we have a stable node, with the third singularity having moved down into the fourth quadrant of the^phase plane, where i t is of no interest (see Figure 2.24A). 67. ° c Ala ' ^ T f ; < X ' > " : F i g . 2.24 Possible S i n g u l a r i t y Types and Positions f o r Rosenzweig's Model 68. The t h i r d s i n g u l a r i t y , (x . y . ) = ( J , r J ( l - J/K) i s normally s i n g , s i n g b ( 1 _ e x p ( _ c J ) ) (for K>J) the one of i n t e r e s t . I f K < J < K, (where K ean be determined • c c from the system eigevnalues we have e i t h e r a stable node or a stable focus (see figure 2.24 B & C). If J < -K , the s i n g u l a r i t y becomes unstable (see figure 2.24D), and Kolmogorov's theorem (see Brauer 7 ) reveals that this gives r i s e to a l i m i t cycle. The l i n e a r i z e d system c o e f f i c i e n t s at the t h i r d s i n g u l a r i t y , (x . , y . ) are: & sing' J s i n g a±1 = r ( l - 2 J / K ) - r c J ( l - J / K ) / ( e C j -1) a±2 = b ( l - e" c J) a 2 1 = s c r J ( l - J/K)/b/(e C J -1) a £ 2 = 0 The problem of determining solutions for the l i m i t cycle has not been considered i n this t h e s i s . This completes the determination of the l i n e a r s o l u t i o n of Rosenweig's model. 2.10 Simulation Study of Rosenzweig's Model We now wish to evaluate the l i n e a r s o l u t i o n of Rosenzweig's model. The solutions -were computed f o r the a r b i t r a r y parameter choice r = 2, K = 45, b=l, c=0.1, s=l, & J=20, (for-which x ' ,y . ) i s a stable focus) over the ' s i n g ' 7 sing range of i n i t i a l conditions 5<x <45 and 5<y <45 i n increments of 2.5 The o J o amplitude and frequency e r r o r c l o c i are shown i n figures 2,25 & 2.26. As before, the frequency i s more accurate than the amplitude f or given i n i t i a l . conditions. The above computations were repeated for K=25, for which (x . ,y . ,) i s a stable node, and the corresponding amplitude error l o c i sing •'sing ' v b v are shown.if figure 2.27. We have evaluated the l i n e a r s o l u t i o n of Rosenzweig's model and found i t to be a good approximate s o l u t i o n (see error l o c i figures) over a wide range of i n i t i a l conditions. We now proceed onward to study one f i n a l e c o l o g i c a l model. 69. • ' —.,— -» — — — 0 . 0 1 • • • • ( • • , , 1 — 1 — . . . . . I * " 0 0 0 1 0 . 0 0 0 . 5 . 0 0 0 2 0 . 0 0 0 2 5 . 0 0 0 . 3 0 . 0 0 0 . . . | . . . . . . . . . 1 9 : 0 3 5 . 0 0 0 4 0 . 0 0 0 4 5 . 0 0 0 5 0 . 0 0 0 0 . 0 CONSTANT R E L A T I V E AHRL I TUUE ERROR L O C I GRAPH ERROR X l .O .OOOt S I N G U L A R I T Y AT i X t Y I " ( 0 . 2 0 0 0 3 E * 0 2 . 0 . 2 5 T 0 0 E * 0 2 I _________ R* Q . 2 0 0 0 0 E * 0 . * • - Q . 4 5 0 0 0 £ » 0 2 B " 0 . 1 0 0 0 U E » 0 I C - 0 . 1 0 0 0 0 E » 0 0 S- 0 . 1 0 0 0 0 C » 0 1 J • 0 . 2 o o o a t * o < : tiMF»R_soLUTicw OF aosrNZwtii;'S_Moo{L. F i g . 2.25 Rosenzweig's Model Linear Solution Amplitude 70. 5 . 0 0 0 -• - J — : — I....I....I 1 1 *>•" 3 5 . 0 0 0 4 0 . 0 0 0 4 4 . 0 0 0 5 0 . 0 0 0 ; 0 . o ">»uuii . i u • v w .———- ••—— CONS..NT H U t H I " E 3 U E S C Y ERROR L u e i - — - » « « ERROR X , 0 . 0 0 0 , S I N G U L A R I T Y AT U . Y I - I 0 . 2 0 0 0 3 E . 0 2 . 0 . 2 3 T 0 0 E O 2 I - . . 0 . 2 0 0 0 0 E . O I K . o . v . o o o t . u . » • 0 . 1 0 0 0 0 E . 0 1 C- O . i O O O O l . O U 5- U . I U U U U E . O I J L I N E A R ^ n . i i T i n N OF R O S E N Z W E I G ' S NGOEL. — • • o . z o o u u t . u . 26 Rosenzweig's Model Linear Solution Frequency 0 . 0 5 . 0 0 Q 1 0 , 0 0 0 _ _ . . . I ' .OOO J 9 . 0 0 0 . _ J 5 . 0 C 0 1 0 . 0 0 . J » . 0 O 0 « „ -5oTooo|.TrruT..I...-l 1 1 1 — > — ' — • — 1 — 1  2 5 . 0 0 0 -9 9 9 9 9 9 7 T 7 7 7_ 1 I 1_ 5 " 5 5 5 5 » I » - 2 0 . 0 0 0 2 0 . 0 0 0 - 9 • • ^ ^ — 7 / — ' ' ' ' ' ' '-JO/, 1 5 . 0 0 0 - / 2 1 J - 1 _ i _ _ Z I - 1 3 . 0 0 0 ! 6 f — L ^ - l ^ / o •• ' • - 1 0 .OOO 1 0 . 0 0 0 \ ^ ( XSYKS , y<^ A) y.. • ' 6 5 . 0 0 0 i 2 — — , - . - - 1 5 - 5 . 0 0 0 : x : , 1 1 1 1 1 . 1 . . . . I . . . . I 0 . 0 " • " n l ; - 1 — i l ; : ; - ' - ; ; ! ; ; " 1 " ; ; ! ; ; ; • ' • " O ! O O O , V _ » . O O Q J _ » P . O ° O — » , m — < & . m — ± ^ < m — » * a a — E L X T I V E A - P L I T U O c ERROR L O C I GRXPH ERROR X 1 0 . 0 0 0 1 S l N O U L . R U t XT I X . T L I 0 . 2 0 0 0 J E . 0 2 . . . 0 . 2 0 0 0 0 E . 0 1 « - 0 . 2 5 0 0 C E . 0 2 » • 0 . 1 0 0 0 0 E . 0 1 C 0 . 1 0 0 0 0 E . 0 0 S - 0 . 1 0 0 0 C E . 0 1 J - O . 2 0 O O 0 E » 0 2 ^ B E « ^ 0 L U U C N . J J f ^ 5 | N 2 i E L 0 1 l J ! 6 Q - f . l . • 2.27 Rosenzweig's Model Linear Solution Amplitude E r r o r L o c i 72. 2.11 O'Brien's Model The l a s t e c o l o g i c a l model to be studied i s O'Brien's model (see O'Brien 44). This model has been s u c c e s s f u l l y used i n the study of phytoplankton dynamics. The model i s : x = r -xy/ (A + x) r,A,S,J > 0 j = sAy(x - J) (J + A) (A + x) In the model, r i s the rate at which prey (nutrient) i s added to the system. As with the previous two models, the predator has a f i n i t e appetite. The predator equation i s i d e n t i c a l to Holling's predator equation. There i s only one s i n g u l a r i t y i n this model, which i s located at ( J , r ( J + A)/J), and i t w i l l be discussed i n the next s e c t i o n . 2.12 Linear Solution of O'Brien's Model We now proceed to determine the l i n e a r s o l u t i o n of O'Brien's model. The A matrix corresponding to O'Brien's model i s : -yA/(A + x ) 2 -x/(A + x) sAy/(A + x ) 2 sA(x - J) (J + A) (A + x) For the only s i n g u l a r i t y (x . y . ) =. ( J , r ( J + A)/J), we J ° J sing,-'sing have the eigenvalues s, „ = -Ar ± % / A 2 r 2 1 , Z 2(J + A)J / 7f~ + -4sAr (J + A) 2 j 2 (J + A) 2 2 The system i s stable for a l l combinations of parameters. I f Ar > 4sJ , then 2 we have a stable node (see figure 2.28A). I f we have Ar < 4sJ , then we:have a stable focus (see figure 2.28B). The l i n e a r i z e d system c o e f f i c i e n t s at the s i n g u l a r i t y are: a.. 1 = -Ar a, „ = J / ( J + A) (J + A)J a 9 1 = sAr a~„ = 0 J ( J + A) 73. A \ (A) \ 4 ST fi&LE O y A ( f t ) O F i g . 2.28 Possible S i n g u l a r i t y Types and Positions for O'Brien's Model 74. This completes the determinationoof the linear solution of O'Brien's model. 2.13 Simulation Study of O'Brien's Model We now wish to evaluate the linear solution of O'Brien's model. The solutions were computed for the parameter choice r=2, A=10, s=2, J=5 (for which (x . ,y . ) is a stable focus) over the ranges of i n i t i a l sing'-'sing & conditions 1 < x < 9, and 1 < y < 9 with increments of %. These parameter o ' Jo r values are scaled versions of those used by O'Brien in his model. The amplitude and frequency error l o c i are shown in figures 2.29 & 2.30. As before the frequency i s more accurate then the amplitude for given i n i t i a l conditions. The above computations were repeated for s = 0.1, for which (x . ,y . ) is a stable node, and the corresponding amplitude error loci Sing ''sing ' - r . b r are shown in figure 2.31. We have evaluated the linear solution of O'Brien's model and found i t to be a good approximate solution (see error l o c i figures) over a wide range of i n i t i a l conditions. This completes our study of the method of linearizing nonlinear differential equation ecology models. 2.14 Summary In this chapter we have started our study of advanced models by applying the classical principle of linearization to four nonlinear systems of differential equations that model predator-prey systems. These solutions accurately describe the behaviour of the systems for i n i t i a l conditions that are more than a distance e away from the singular point (see figures of error l o c i ) . The value of these solutions i s two fold. First of a l l , the complete l i t e r a l parameter solution i s made available. Secondly, one can quickly ascertain the effects of variations of a given parameter on the 75. , . .°L_. .0<>0_ 3 . ° 0 0 . . ^ 0 0 ^ 0 0 ^ e.ooo 9 . 0 3 0 1 0 . 0 3 0 ' - • ' I 1 0 . 0 0 0 a.o 1.1111" 7.000 J.ooo,. «r.000. " » » » — 8 ( o i CONSTANT R E L A T I V E A N P L I T U O E ERRCR L O C I GRAPH ERROR « 1 0 . 0 0 0 . S I N G U L A R 1 T T AT M . V I - I 0 . 5 0 C C . l E . G l . O . G G O O O E . C 1 1 » . 0 . 2 0 0 0 0 E . 0 1 A . 0 . 1 0 0 0 0 E . O Z S - C . i O O O O E . O l J - O . S O O O O E . O l _ L I N E A ( l _ S 0 l U T 1 0 N J J _ O i > » I E N L S _ H C P i ; i . • F i g . 2.29 O'Brien's Model Linear Solution Amplitude Err o r L o c i 76. TIOI 0 ,0 1 .000 2.000 3,000 4.000 J . 000 6..000 Z..O0Q 8.002 i . M a L 8 .S0S_ 10 .0001 I I....I I I I I I.... I I....I I I I I.... I.... I . . . . I . . . . I.... I 10 4 3 2 1 1 2 CONSTANT R E L A T I V E FREQUENCY ERROR L O C I GRAPH ERROR X l . O Q G X S I N G U L A R I T Y AT ( X . Y J . l O . S O O O j E . O l , 0 . 6 0 0 0 0 E . 0 1 I 8> 0 . 2 0 0 0 0 E . 0 1 A - 0 . 1 O O Q L E . 0 2 S - 0 . 2 0 0 0 0 E . 0 1 J > 0 . 5 G 0 Q 0 E . 0 1 2.30 CL'Brien's Model Linear Solution Frequency E r r o r L o c i 77. 2.31 O'Brien's Model Linear Solution Amplitude 78. singular points, time constant and frequency of the s o l u t i o n . For some combinations of parameters, some terms i n the i n d i v i d u a l a..'s. may be n e g l i g i b l e , or may be dominant. Thus, the user may r e a d i l y determine which components of the s o l u t i o n w i l l be s e n s i t i v e or i n s e n s i t i v e to v a r i a t i o n s i n a given parameter without re s o r t i n g to guesswork i n computer simulations. These solutions provide great i n s i g h t i n t o the system behaviour, and y i e l d a basis on which more refined solutions are constructed i n the following chapters. 79. CHAPTER THREE EQUIVALENT LINEARIZATION SOLUTION OF ADVANCED MODELS 3.1 Introduction A number of techniques were used in an attempt to develop solutions that were acceptable over a wider range of i n i t i a l conditions than the linear solutions discussed in Chapter two. A brief discussion of the general formulation of nonlinear differential equations is necessary for comprehension of the nature of the problems encountered here. Perturbation methods for approximating the solution of nonlinear differential equations have been studied (see Nayfeh 42 & Boguluibov 5) often 2 took a form (such as x + to x + ef(x,x) = 0) in which the magnitude of the nonlinearity was controlled by a small parameter e. Thus when £ -> 0, the differential equation degenerated to one that was solvable in terms of standard trigonometric functions or some other special functions, such as the v t e l l i p t i c functions (see Barkham 2 & 3) . The frequency of the soltt^ion was usually readily discernable. The techniques used to approximate the solution when e ^ 0 were such that the solution would degenerate to the exact solution when e was set to zero. Turning to the ecological models at hand, several important points should be emphasized. First of a l l , in the absence of nonlinear terms, the differential equation's behaviour usually becomes exponential in nature (often unstable), instead of oscillatory, due to a lack of cross coupling terms. Secondly, there is no small parameter e that governs the magnitude of the nonlinearity in both equations. Third, the nonlinearities involved are not small perturbations of a linear system, but large ones that determine 80. the qualitative behaviour of the systems. It i s these factors that caused many of the previously developed techniques for nonlinear differential equations to be of l i t t l e use for :the cases a t hand. The above information was obtained by much t r i a l and error while working with the VGW model as a prototype. A brief summary of the techniques that were applied to l i t t l e avail is given below. The well known perturbation series techniques with x ( t ) = Ee'Sc.Ct) was tried (see also Dutt 15 & 16). A variety of ideas for a r t i f i c i a l l y inserting a parameter e ( and then setting e = 1) were tried, as well as means of adding in linear cross coupling terms to produce oscillations in the case of negligible nonlinearities. These methods yielded f i r s t order solutions that were several orders o f magnitude t oo large. The techniques of Krylov-Boguluibov (KB) were modified to handle pairs of f i r s t order differential equations. This method failed due to a lack of a parameter e, and due to a lack of oscillations in the absence of nonlinearities. The averaging techniques presented by Lin(32) are a modification of the KB method, and suffer the same problems as the KB method. A slight variation of Van der Pol's method was applied to the Volterra-Gause-Witt model. The results were not as good as those obtained for the linear solution, and thus do not jus t i f y a lengthy description of the convoluted method of solution. 3.2 The Method of Equivalent Linearization We now turn our attention to the next method of solution which provided good results. The method of equivalent linearization i s a means of approximating the solutions of non-linear differential equations, and was originally developed for second order systems by Krylov and Boguluibov. The extension to f i r s t order coupled systems i s sketchily described by Lin (32) . 81. A complete explanation and derivation i s given below. We s t a r t with the system x = F(x,y) Y = G(x,y) I f the s i n g u l a r i t y of i n t e r e s t i s nonzero, we translate the variables to the singular point so that the o r i g i n of the phase plane i s now the singular point of i n t e r e s t . This i s done to reduce the complexity of the algebra which follows. We wish to approximate the o r i g i n a l d i f f e r e n t i a l equation by a l i n e a r d i f f e r e n t i a l equation of the form: x = a u x - a 1 2 y y = a 2 1 x + a 2 2 y whose c o e f f i c i e n t s a., w i l l be determined s h o r t l y . The s o l u t i o n of the l i n e a r d i f f e r e n t i a l equation i s given i n section 2.2. As a c r i t e r i o n for choosing the a^y w e decide to minimize the mean square der i v a t i v e error (as only derivatives are available) over one period between the o r i g i n a l system and the l i n e a r system. For the one period the l i n e a r system i s equivalent to the o r i g i n a l nonlinear d i f f e r e n t i a l equation, and hence the name of the method. For q u a s l l i n e a r d i f f e r e n t i a l equations, the l i n e a r d e s c r i p t i o n of the o r i g i n a l system w i l l continue to be v a l i d a f t e r the i n i t i a l period. Thus we minimize the functional J , with respect to the a „ , where J i s given by J = — J2* ((F(x,y)-a 1 ; Lx + a ^ y ) 2 + ( G ( x , y ) - a ^ x i - a ^ y ) 2)d6 For the minimum of J, we require that 3J 3J 3J 8J ) a l l 8 a12 9 a21 9 a22 = 0 Upon computing the above p a r t i a l d e r i v a t i v e s , we obtain: 82. 9J 8 a l l 1 2TT f2"-0 9J 3 a 1 2 1 2lT 0 9J 8 a 2 1 1 2TT 2TT / 0 9J 3 a 2 2 1 2TT 0 * i r ' ~i2 j x21~ ~22 J x21~ °22 J The above i n t e g r a l s w i l l give two sets of two equations i n two unknowns, which must be solved f or the a..'s. The above i n t e g r a l s cannot be solved exactly as the solutions x(t) & y(t) are not know. Hence we must resort to approxmimations that w i l l allow the i n t e g r a l s to be evaluated. We make the assumption that over one period of the s o l u t i o n , the envelope of the so l u t i o n i s slowly changing and may be taken as a constant. The solutions may thus be approximated by x = r ^ cosG & y = r 2 sin9, over one period. Inserting these approximate solutions i n the f i r s t of the four above:r . i n t e g r a l s , we obtain: — = h- (F(r-.cos8,r„sine) - a,., r, cosG+a.. 0r„sin6) r. cos6 d0=O 9 a ^ 2IT o 1 2 11 1 12 2 1 a r 2 1 2TT 11 1 .*. -z— S F(r.,cos6, r„sin6)r.. cosG dO = 2TT o 1 ' 2 1 2 1 2TT or a,., = / F(r,cos8, r„sin6)cos6 dG 11 nr^ o 1 2 S i m i l a r l y , we obtain: 12 -1 w r 2 ,2* 0 F(r^cos6 ),r 2sin0)sinG dG 21 1 j.2"n 0 G(r^cos6 ),r 2 sinG)cosG dG l22 1 i r r 2 j.2ll 0 G(r^cos6 ), r 2 sinG)sinG dG 83. We now- turn our attention to the constants r ^ & r 2 • The choice of the approximate s o l u t i o n says that we are assuming the t r a n s l a t e d equivalent l i n e a r d i f f e r e n t i a l equation to be of the form x = ~a-^^y a n (* y = a 2 1 x o v e r o n e P e ri°d. This has a s o l u t i o n x = r^cos(cot + (j)^ ) y = r 2 s i n ( c t + ^ ^ e ^ = ^ ^ - / \^ , a12 , s2 r n = /(x -x •. ) + (y -y . ) 1 o sxng . a 9 j -° sing • X~ = v — - (x -x . ) + (y -y . ) 2 a ^ o sxng w o •'sxng * - t - ^ " 1 r ( Y o " y s i n g ) a12. *1 " t a n ( ( x - x . ) 0) } o sxng , „ -1 ,( yo ~ y s i n g ) a21 4>0 = tan (7 ^ — -2 (x - x . ) co o sxng We now have a l l the equations necessary for the computation of the a „ . We note that r ^ and r 2 are dependant only on a ^ a n d a2j_ • Thus, i n the equations for a^ 2 and a2i> the same variables w i l l probably appear on the r i g h t handside i n the i n t e g r a l s , to give two nonlinear algebraic equations i n two unknowns, a^ 2 and a ^ which must be solved. Once the c o e f f i c i e n t s a^ 2 and a^ have been computed, the c o e f f i c i e n t s a^ and a 2 2 are quickly determined, as the r i g h t hand sides of the equations for these c o e f f i c i e n t s w i l l contain at most, only a^ 2 and ' ^ n S e n e r a i > w e s e e that the c o e f f i c i e n t s a „ depend on the i n i t i a l conditions of the d i f f e r e n t i a l equation, which i s an e f f e c t that does not occur i n l i n e a r d i f f e r e n t i a l equations. I t should be noted that the minimum mean square error c r i t e r i o n w i l l correct large d e r i v a t i v e errors (which occur near the time o r i g i n for dampled o s c i l l a t i o n s ) while neglecting smaller e r r o r s . Thus we may expect the s o l u t i o n to be best near the time o r i g i n . 84. We now have a second technique f or obtaining a l i b e r a l parameter dependant s o l u t i o n (which w i l l probably depend nonlinearly on the i n i t i a l conditions of the system)„of nonlinear d i f f e r e n t i a l equation models of e c o l o g i c a l systems. We now proceed to apply this method to the models discussed i n chapter two. 3.3 Equivalent Linearized Solution of the Volterra-Gause-Witt Model The f i r s t : model we wish to apply the technique of equivalent l i n e a r i z a t i o n to, i s the VGW model. We begin by t r a n s l a t i n g the singular point (x . , y . ) = (Y/6, (a - Xv/6)/0) to (0,0). The d i f f e r e n t i a l sxng Jsxng equations become: x = x -y -jr - Bxy - X x = F(x,y) y = 6xy + ( a < s ~ AY), x = Q.(x,y) We compute the a „ to obtain: a l l = - X y / / < 5 a12 = e Y , / < S a 2 1 = ( a 6 "• Xy^$ a22 = ° which are exactly the same as those f or the l i n e a r s o l u t i o n obtained i n Chapter Two. The equivalent l i n e a r °f this very simple model are independant of i n i t i a l conditions. Thus we may consult the r e s u l t s of section 2.4 for the evaluation of t h i s s o l u t i o n . We now proceed onwards to the second model discussed i n Chapter Two. 3.4 Equivalent L i n e a r i z a t i o n Solution of Holling's Model The next model we wish to study i s Hollings (see section 2.5) model. We s t a r t by t r a n s l a t i n g the s i n g u l a r i t y ( x s ^ n g ys±ng) = (J»(J+A)r(l-J/K)) to (0,0) to obtain: 85. x = xr + Jr - (x^+jN^Jx)^ - (xy + x(J+A)r (1-J/K)+Jy+(J+A)r (1-J/K) A + J + x y = sA (xy + x (J+A)r(1-J/K)) (J+A)(J+A+x) Before we can evaluate the a... we must pause to obtain some results in calculus. Using the method of contour integration of complex variables (see Wayland 56), we evaluate ah integral needed in the computation of the a. . r2iT e J n x dx _ _2TT i—2 n o - ( p + q cos x) f ( * ~ 1 ~ q " ^ ' P > q p - q q , -2TT sin nx dx „ . . / - r — r- =0, p > q o ( p + q c o s x ) and f2iT cos nx dx _ 2ir , / p 2 , ~ _ I P s 1 1 > . J o ( p + q cos x) 2 / & Q ' p - q We may now compute the a „ as follows: a n = r(l-2J/K) - (J+A) r(1-J/K) { i + ( / X ^ £ H - 1 - 1 ^ * 1 . ) } / (J+A)2 - r x 2 2J (J+A)r(1-J/K). , /(J+A) 2: -^/(J+A) 2 - r x 2 /' r l r l a I l t / ^ - ! - -(J+A) ) 3} r l 1 / r l r l 2 + J{1 - ( / ( J + A ) -1 - (J+A) 2] r l r l / ( J + A ) 2 - r 1 2 86. (J+A) y } a = S A r l r /(J+A) 2 (J+A) a22 y ,. Z y J " ~ 2 (J+A) / ( J + A ) 2 - ^ 2 / r l 1 _ / / (J+A) 2 , _ (J+A) . 3 , r l 2 ^ The equations for a & a are not yet i n usable form. To solve for *12 " "21 a i ? & a 9 i » w e define 5= a 1 0 / a 0 1 . Thus we have r. = / £ x -x . ) 2 _ r ( y N2" x / Z J- 1 / 2 1 1 s o sxng y •'sing By taking the r a t i o of the equations for & a21' w e O D t a i n a n equation of the form 5 = h ( 5 ) , which we solve for £. We then back substitute to obtain the four parameters a... We observe that as the i n i t i a l conditions of the system tend to the singular point of i n t e r e s t , the c o e f f i c i e n t s a „ tend to the values obtained for the l i n e a r s o l u t i o n s . Computer simulation has shown that we may approximate the s o l u t i o n of the equation for 5 by the r a t i o of the l i n e a r solution's a,„ & a„,, i e : £ = — ? — / •,„. This does not 12 21' sAr(rJ/K) a f f e c t the s o l u t i o n accuracy to any noticable degree. Thus we have been able to s i m p l i f y the s o l u t i o n somewhat, and s t i l l r e t a i n the e f f e c t of the a.. being influenced by the i n i t i a l conditions of the d i f f e r e n t i a l equations. This completes the equivalent l i n e a r i z a t i o n s o l u t i o n of Holling's model. 3.5 Simulation Study of H o l l i n g ' s Model We now wish to evaluate the equivalent l i n e a r i z a t i o n s o l u t i o n of Holling's model. The equivalent l i n e a r i z a t i o n solutions were computed for the same parameters and i n i t i a l conditions as the l i n e a r s o l u t i o n (see section 2.7). The same samples of time p l o t s and phase plane curves are 87. given i n f i g u r e s 3.1 to 3.6. The corresponding amplitude and frequency e r r o r l o c i are given i n f i g u r e s 3.7 to 3.9. We note that we have an improvement i n s i z e of the amplitude e r r o r l o c i , but t h i s i s at the expense of the frequency e r r o r l o c i . Thus, as a s o l u t i o n to t h i s apparent t r a d e - o f f between amplitude and frequency e r r o r , we use the amplitude of the equivalent l i n e a r i z e d s o l u t i o n , and before computing the s o l u t i o n , we s u b s t i t u t e i n the frequency of the l i n e a r s o l u t i o n , so as to o b t a i n the maximum b e n e f i t of both methods. Hence we obt a i n a s o l u t i o n that i s an improvement over--the c l a s s i c a l l i n e a r s o l u t i o n . We now proceed onwards to study the next e c o l o g i c a l model discussed i n Chapter Two. 3.6 Equ i v a l e n t L i n e a r i z a t i o n S o l u t i o n of Rosenzweig's Model The t h i r d model we wish:to study i s Rosenzweig's (see s e c t i o n 2.8) model. We begin by t r a n s l a t i n g the s i n g u l a r i t y (x . ,y . ) = & J b e J sing'•'sing ( J , r J ( l - J / K ) ) to (0,0) to o b t a i n : b ( l - e x p ( - c J ) ) x = r J ( l - J / K ) + r x ( l - 2 J / K ) - x 2 r / K -b(y + r J ( 1 ~ J / K ) ) ( 1 _ e ( ^ b ( l - e C J ) j = s(y + J r ( l - J / K ) ) ( e ~ C j -e~C^X)) b ( l - e " c J ) Before we can evaluate the a „ , we must pause to evaluate the f o l l o w i n g i n t e r g r a l s , / 2 i r sin(nx) e a c o s x dx = 0 (by symmetry) T ,2TT , . a cosx , - 2 7 T , , , !? a 2 K r 1 ,2K . I = f cos (nx) e dx = J cos(nx){ E T ^ - , . L —ZZ? ( „ ) n o o K=l • 2 -i ov- oo „2K-1 + 1 ( c o s 2 K x + . . . + ( K ^ ) c o s 2x)] + Z ^ , [ c o s ( 2 K - l ) x + 2 K—1 / . . .+(^_^) cosx] } dx 88. o'oTl 5 . 0 O O . 1 0 . 0 0 0 . .15.000 2 0 . 0 0 0 i 5 . 0 0 0 .. 3 0 . 0 3 0 . ... S i . 0 0 0 4 0 . 0 0 0 ... *5.0Q0 ...SO. 000 5 0 . 0 0 0 45.000*- 4 5 . 0 0 0 . • i n . n n n • ; / s-Af/AlrVTICti-L SOLUTION : / 1 3 5 . 0 0 0 - / f tf,... \. : k la.1.1.* \ ^..J ~ : > • 3 3 . 0 0 0 • 1 \ T K . * , *** /H V \ • " I A 4* \ m»*if I* i A« , ' -\ \>» vr» [' \ ; 3 0 . 0 0 C K iv y \ —— • 3 0 . 0 0 0 _ m\ 1 „ „„„; \ ^-m^Blt\Cf)L f ' »4.nnn \ / -SOLUTION JC • - ib.'ooo" 2 0 . 0 0 0 -. '. n.ooo"- - 1 5 . 0 0 0 . 1 -1 0 . 0 0 0 -3 . 0 0 0 - S . 0 0 0 • 0,0 4.nnn i n . n o n 1 3 . 0 0 0 2 0 . 0 0 0 2 5 . 0 0 0 3 0 . 0 0 0 ... 3 5 . 0 0 0 4 0 . 0 0 0 4 5 . 0 0 0 5 i i 0 . 0 .nnn . . . » . i M M V T i r . 1 smiiTihk » » . » • MIIMCHICAL SOLUTION _ J « J • • • PniNTS f i i M l f l N Tn ftflTH Sd.UTIQNS. : XQm 0.17500E*02 Y0« 0.425CCE*02 S I N G U L A R I T Y AT (X|Yt"« 0.20000E*02. 0.333336*02) 7liVQ~UENCY~I *K- '^ 84~972l^ U^M-'b76T321E-oi ERROR" 0.2622OE«02» AMPLITUOE ERROR* E R E L " 0.25*0fle*02* EABS- 0.S7->6SE*01 i . n . j o n n n F . n i K , r . ^ s o n n F f n ? A . P.. maooF«02 0-i.OQQQFtOl J " Q.2nOOQft02 : • E Q U I V A L E N T L ItlEAR I Z A T I O N SOLUTION OF HO L L I N G " S MODEL . F i g . 3.1 Holling's Model Equivalent L i n e a r i z a t i o n Solution for' (x o,y o)H17.5, 89. o!oTI 4.000 -.8.000 1 2 .COO . 1 6 . 0 0 0 2 0 . 0 0 0 - 24.000. 2 C . 0 0 0 32.000 )M00 40.000^..^.-. *s.oo<>: / (•£ ) - 45.000 . • * • 4 0 . 0 0 0 • : 35 .000- 35.000 • • 30.000 • 1 \ L »».nnn ~ I J \ \ ±**ry+4m _________________ I 20.000- ly \ Jr t " S _ . 7^..._i^^---' J > /^^^ f r >"' l 1 v~' ! [/ X* \ \ / / JVW~*1K____-«^ » 2 0 . 0 0 0 ; If \ i ( J -- 13.000 ' n. J r ' lo-nnn- V / - 1 0 . 0 0 0 -V 5.000- 5.000 I 0.0 0 - 0 4 - C Q n ft.OOO 1 2 . 0 0 0 1 6 . 0 0 0 2 0 . 0 0 0 2 * . O O O . . 2 d . 0 0 0 3 2 . 0 0 0 3 6 . 0 0 0 41 i.oon T I N E S E C . » » l • AMfllVTICAl SniliTtnN -+«-» • NUMFft It* AL SOLUTION ••.HIM • POINTS CJHHQN TQ BOTH -JLUTI Qt-S . XO- 0.17500E*02 V O " 0.42500E*02 SINGULARITY AT lX,Y,-l O.20OQ0E»O2t 0.33333E*02t VREQUENCV. AN" b~.849'72l-6rNUM""o.6T321E-01 ERROR- 0 . 2 6 2 2 0 E * 0 2 l AMPLITUDE ERROR* EXREL- 0 . 2082«VEVQ2 * EX*BS«-b."3383lE*01 fl- Q. ?OOQQFtP- -O 0-*snnOP.tQ? 0. 10000F»02 S- O . l Q D O Q E t f l l — I " Q.2Q0QQEI-2 • EQUIVALENT LINEARIZATION SOLUTION OF HOLLING'S MOOEL Fig. 3.2 Holling's Model Equivalent Linearization Solution for.- (x ,yQ) = (17 .5,42.5) 90. oVo"! *.ooo. . . o o o . . ii-ooo. .....ooo 2 o . o o o ».o.o w.ooo »»-wo.l;:;j;D1«;-1--t!i0!.rooo— "ToToooi. — I — I — i — I — i.-.-i — — i 1  - 1 0 . 0 0 0 1"- *- — • ; # : t s.ooc- • S . 0 0 0 • • 0.0 o . o fc-onn n - n n n 1 2 . 0 0 0 . . .. 1 6 . 0 0 0 . 2 0 . C O O __..2A.O0O T IME S E C . 1 0 0 . . . . . • A m i y T l f A l S n i l l T i n N ! • • • • _ i NuMFftirn s m u i i n N . . . . . . » n » N T < riMMON r n R U T H ^.II i i T i n n s « 0 - 0 . 1 7 5 0 0 E . 0 2 T O . 0 .42500E .02 SINGULARITY «T U . r i ' l 0 . 2 0 0 0 0 E . 0 2 . 0 .33333E .02 I ~ » £ Q U E N : V 1 A N . ' 6^.'ei972E-bTNUM." 0 .67321E -01 E R R O R . 0 . 2 6 2 2 0 E . i ) 2 « AMPLITUDE E R R O R T E ' T R E L - 0 . 2 5 . 0 O E . 0 2 J S T A B S - 0 . ! A . n . ? n n n n c . n i « • n . t s n n n f . n ; A . n . m n n n f . n ? S. 0 . l O O O O E - O I J- Q . 2 0 0 C 0 E f 0 2 • E Q U I V A L E N T L INEARIZATION SOLUTION OF H O L L I N G ' S MOOEL . F i g . 3.3 Holling's Model Equivalent L i n e a r i z a t i o n Solution f o r (x^,yo)-(17.5,42.5) \ 91. m i 0.0 S.000 . 1 0 . 0 0 0 15.000 20.000 25.000 30.000 55.000 4 0 . 0 0 0 .45.000 5 0 . 0 0 0 • • 4 5 . 0 0 0 - - 4 5 . 0 0 0 • -4 0 . n n n -: / — MA-LYTICAL : / \__S.OL\JTI 0>M 3 5 . 0 0 0 - 7 Y >. \ - 3 5 . 0 0 0 3 0 . 0 0 0 - f \ - 30 .000 • • S OL-UTlOtf J 5 . n n n -~ : . — — 2 0 . 0 0 0 - ' " . " 2*o.~6ob • • is.ooo^ ; 1 5 . 0 0 0 • : • ' • . : m . n n n l ; • : 3 . 0 0 0 - : 3 . 0 0 0 • : 0 * 0 S^flflfl LO^QQO— 15 .000 .. .20 ,000 .. 2 5 . 0 0 0 ., 30 .030 3 5 .000 4 0 . 0 0 0 45 .000 5 0 . 0 . 0 nnn XITI " " ' - A N A L T T I f t l Sni l lT l f lN . MlHFRtrAI Sfll IITTHN «•••. . POINTS r ,«nn« Tn BHTH <,.iiiTin«i«  / XO- 0.3OOOOE.O2 TO* 0 . 3 0 0 0 0 E . 0 2 SINGULAR I TV AT IX,V|.( 0 . 2 0 0 0 0 E . 0 2 , 0 .33333E.02 I FREOUENCVI AN. 0 . S23S8F.-bi NUN. 0 .80026E -01 ERROR. 0 .31642E .01X AHPLITUOE ERROP.7~EREL.-6.^ 1 2 0 8 4 ^ MTYllB^iljT^mEioT —n-7onr\nF*m * - n . n f l n n p . n ? A . n. m. innpf f i ? s . n . i n o a o F . n i j . n . j n n n n p . T ? '  EQUIVALENT LINEARIZATION SOLUTION OF HOLLING'S NOOEL. 3.4 Holling's Model Equivalent Linearization Solution for (x ,y )=(30, 92. oT' 4 000 8.000 12.000 . 1 6 . 0 0 0 . 2 0 . 0 0 0 . . 2 4 . 0 0 0 2 a . 000 32 .000 . 3 6 . 0 0 0 . 4 0 . 000 3 0 . 0 0 0 • -4 5 . 0 0 0 . -• 4 0 . 0 0 0 • $ 5 , 0 0 0 -• 35 .000 3 0 . 0 0 0 t A ' ' - 30 .000 • \ \ * w '. 75.nnn 1 20 .000 ls.ooo^  \j* S0i-(j y• j o yv/ - 13 .000 : A. ANftLHTlCflL I 10 .000 ...it . ANAiYTtrAi s n u - T i n N _ 1 2 . 0 0 0 . . M I I M F B i r » i SniUTION l.-.-l t....I . . . . I I t . . . . I . . . . 1 0 . 0 16.000 20.000 _ 24.030 2d .000 3 2 . 0 0 0 36.000 40.000 T I M E S E C . • P U N T S f.^MMON T O W I T H S J L U T I O N S X O - 0 . 3 0 0 0 0 E . 0 2 T O - O . 3 O 0 O 0 E . 0 2 S I N G U L A R ! T Y AT « X , V 1 « 1 0 . 2 0 0 O Q E . 0 2 . 0 . 3 3 3 3 3 E . 0 2 1 " T R E O U E N C Y I AN- 0~.825 58E -or N U M - ~ C . 8 0 0 2 6 E - 0 1 E R ROR- ~0.31642E .018 A N P L I T U O E E R R « X E X R E L - - 0 . 1 2 6 8 » E . 0 2 « E X A B S . - 0 . 2 0 0 8 2 E . 0 1 ft. n.yonnnF.oi K . n.45nnoFt07 A- n.innnnp.n? s- n.innnnF.oi j . 0 . 7 0 Q O O E . Q 2 _ — — E Q U I V A L E N T L I N E A R I Z A T I O N S O L U T I O N O F H O L L I N G ' S MODEL Fig. 3.5 Holling's Model Equivalent Linearization Solution for"(X q,y o)=(30,30) 93. SO.OOO • W T . V V V . B . V W V . . . W W W c g . u u u ( . . U W C O . W V 9 ( . V U U P O . W U U . _ I U 5 0 . 0 0 0 4 5 . 0 0 0 • 4 5 . 0 0 0 4 0 . 0 0 0 - - 4 0 . 0 0 0 '• !/ \ SOLUTION 3 5 . 0 0 0 -\ f* >* T_f*-l* ^ tH A»l-f ~ » » • 3 5 . 0 0 0 3 0 . 0 0 0 } srrw 3 0 . 0 0 0 SOLUTION • J4.<>nn. 2 4 . 0 0 0 : - • • : 2 0 . 0 0 0 ~ " ~ 2 0 . 0 0 0 -1 3 . 0 0 0 - 1 3 . 0 0 0 i n . n o n . 1 0 . 0 0 0 5 . 0 0 0 - 9 . 0 0 0 0 . 0 n 0 4 . 0 0 0 fl.noo 1 7 . 0 0 0 | A , n o o 2 0 . 0 0 0 2 4 , 0 3 0 ? t t . 0 0 0 1 2 . 0 0 0 3 4 . 0 0 0 4 0 . 0 . 0 0 0 0 _ a i i > • mat vTtrAr <m iiTinn ..--NUHERIC-L -QLU.T I ON I I I a * - P-H^TS, rinjM.n_N TO B O T H uTinNt X 0 - 0 . 3 0 0 0 0 E * 0 2 V O " O . 3 0 0 O 0 E * 0 2 S I N G U L A R I T Y AT I X . V I - t 0 . 2 0 0 0 0 6 * 0 2 , 0 . 3 3 3 3 3 E + 0 2 i F R E Q U E N C Y . A N - 0 . 8 2 5 5 8 F - 0 1 NUM- 0 . 8 0 0 2 6 E - O I E R R O R - 0 , 3 1 6 « W E * 0 i X AMPL ITUDE E R R O R i E Y R E L - O . T 2 1 3 / E » O H E Y A 8 S - 0 . 2 6 2 1 7 E * 0 l „P 0-300QOf»oi nm iio..v.soQar,tO'; A - P-lononF+oj s- O . I O Q O Q F » O I J- O.7QOQQF*O?  E Q U I V A L E N T L I N E A R I Z A T I O N S O L U T I O N OF H O L L I N G ' S MODEL Fig. 3.6 Holling's Model Equivalent Linearization Solution for (X Q > Y Q ) =(30,30) 94. V t O I , . 0 00 3 5 . 0 0 0 . 4 0 . 0 0 0 . 4 5 . 0 0 0 S O . 0 0 0 _ 3 . 0 0 0 1 0 . 0 0 0 . . 1 5 . 0 0 0 ._. . . .2Q.000 . " ' • ' • • • • ^ ' ^ ^ " ' " " T i T . ' I ' K ' t T i T ^ 5 0 . 0 0 0 5 6 7 6 •> 1 6 1 1 1 9 2 2 2 ' o.o . ^ • • . • • • . " • l : ; : - ' " \ - , ; . : r ' " ^ — :'.V.;"'T.V.o;J H . M O . - - . 2 0 . 0 0 0 . . . . . 2 5 . 0 O O . _ _ J 3 . O 0 O _ . . 3 CONSTANT K E 1 4 T I V E 4 H » 1 1 T J J F ERKUH l U t l -,S4PH t , , U » » 1 0 . 0 0 0 . S 1 N S U L . . I T Y «T ! « . » . • < 0 . 2 0 0 0 0 E . 0 2 . 0 . 3 3 3 3 3 E . 0 2 ) 1 . . 0 . 2 0 0 0 0 E . 0 1 K . 0 . 4 5 O 0 0 E . O 2 4 - O . 1 O 0 0 O T . O 2 S- 0 . 1 0 0 0 0 E . 0 1 J - U . 2 0 0 C 0 E . 0 2 _ £2U IV . * L f.Nl . .U!.aWli -U10 ! t . . SM . J t [ i l N t f H u l U V i ' S « W U F i g . 3.7 Holling's Model Equivalent L i n e a r i z a t i o n E r r o r L o c i 95. CONSTANT R E L A T I V E F R E i ' J F N C V ERROR L U C l GRAPH ERROR X 1 0 . 0 0 0 1 S I N G U L A R I T Y AT ( X * V I « ( 0 . 2 0 3 0 3 E * 0 2 t 0 . 3 3 3 3 3 E * 0 2 ) « • 0 . 2 0 Q 0 0 E * 0 l K-> 0 . - . 5 0 0 0 E » 0 2 * • 0 . 1 0 0 U 0 E * 0 2 S> 0 . I 0 0 0 0 £ * 0 l J " 0 . 2 0 0 0 J E * O i 3.8 Holling's Model Equivalent L i n e a r i z a t i o n Solution Frequency Error L o c i 96. y IO I J...0 5.100 10.000 .5.000 _li3.?J_ 25.01", 30.:m .3 5.. 3.10 A.Q,..00.1 15.000 50.000 _ - ? 4 5 . 0 0 0 - 4 5 . 0 0 0 4 0 . 0 0 0 - 4 0 . 0 0 0 3 5 . 0 0 0 - 9 8 3 5 . 0 0 0 7 6 ) 6 s e 3 0 . 3 0 3 - 5 4 B 5 3 b 8 - 3 0 . 0 0 0 • ": » 5 / 3 3 3 3 3 4 4 ~ ^ » ^ S 7 3 . 3 3 3 - 6 / 2 2 ? 2 2 2 2 3 3 / - 2 5 . 0 0 0 1 A6% ; 3 ^ 1 1 _1 1 1 1 ^ 2 2 2 -2 0 . 3 0 3 - 1 * 2 1 / 1 1 I I I NJ " 2 0 . 0 0 0 q / \ • 1 3 . 3 0 0 - 7 f ' 1 5 . 0 0 0 1 0 . 0 0 0 \ * 1 l ^ f 2 2 3 '- 1 0 . 0 0 0 \ >^ - 6 j^ 3 2 l _ 1 - T—~T~~~ 2 2 3 ^ f S ^ : 5 . 0 0 0 5 . 0 0 0 0 . 3 i 3 . 0 °* 0 5 . 0 0 0 * 10.000 15.300 23.000 25.3-10 30.330 35.030 40.000 45.000 50.000 xtot CONSTANT RELATIVE AHPLTTJOE E4RUR LOCI 3 U H E**0* * 10.030X SINGULAR.T» AT U ,V» - ( 0.20303 E *-02 * 0.12303EO2* km 072533JE*0l K.- .0.2iO1>3FO2 A- 0.100QOF*02 S" G.10000E»01 J« O.2OOO0E»O2 E3JlV*l6Nf LlVEftftl^ATlCN S3LUTI0N Gf H O L L I N J ' S M03EL WITH 11 NEAR fKEqjENCY. Fig. 3.9 Holling's Model Equivalent Linearization Solution Amplitude Error Loci We w i l l need the values of I for n = 0,1,2. We obtain: n ' ' n = 0 I = 2TT.I1 + E ^ — Q ) ] ° K=l (2K)!2 2 K K i x v a 2 K - 1 ,2K-1 n = 1 I = TT E —j ( ) K=l (2K-1)!2 n = 2 I = TT E £ — _ ( 2 * ) Z K=l (2K) !2 2 K 1 K 1 where (^ ) = ^ n' n!(K-n!) We may now compute the a „ as follows: 2K a = r(l-2J/K) - " J ( 1-J/K) { ± + £ ] ( e J C - l ) K=l 2 / K(K+l)!K! -Jc 0 0 ( " - l ) a = b - be J C [ 1 + E -y-± ] K=l 2 (K+1)!K! a - BrJc(l-J/K) " ^ " l ' , 21 Tr L ~9T? J b ( e J - l ) K=l 2 Z R(K+1)! K! , . 2K " J c -Jc M , " ( ~ C r l } . a = se - se [ 1 + E — ] K=l 2 (K+l)! K! Simulation of the equations show that we may truncate the i n f i n i t e series to one term i f we have |c| < 1. As with Holling's model, we define 5= a1_2'/a2i' '^n^s 8 l v e s us a quadratic equation for E, whose solution i s given J~T~ 2 a2 , srJ(l-J/K)c 3 (Y - Y . ) 2 where a_ = - o sxng L 8 b ( e J C -1) 98. s r J ( l - J / K ) , . . c 3 , ,2, , b e J ° c 2 , v „ ,2 a = ( c + — ( x -x . ) ) + ——5 (Y -Y . ) 1 b ( e J c _ 1 ^ 8 o smg' 8 o sing' _ T " • 2 ? a n = b e J C [ 1 + £ - (X -X . ) ] - b 0 8 o sing Thus we compute E,, and then the a^^, which completes our s o l u t i o n . As before, when the i n i t i a l conditions tend to the s i n g u l a r i t y , the c o e f f i c i e n t s a.. tend to the values obtained for the l i n e a r s o l u t i o n . This completes the equivalent l i n e a r i z a t i o n s o l u t i o n of Rosenzweig's model. 3.7 Simulation Study of Rosenzweig's Model We now wish to evaluate the equivalent l i n e a r i z a t i o n s o l u t i o n of Rosenzweig's model. The equivalent l i n e a r i z a t i o n solutions were computed for the same parameters and i n i t i a l conditions as the l i n e a r s o l u t i o n (see section 2.10). The corresponding amplitude and frequency error l o c i are shown i n figures 3.10 to 3.12. We note that we have a s u b s t a n t i a l improvement i n the s i z e of the amplitude err o r l o c i , but this i s again at the expense of the frequency er r o r l o c i . Thus, for the complete s o l u t i o n , we proceed as with the case of H o l l i n g ' s model, by using the equivalent l i n e a r i z a t i o n s o l u t i o n amplitude and the l i n e a r s o l u t i o n frequency, and obtain the best of both worlds. Again, we have an improvement over the l i n e a r s o l u t i o n . We now proceed onwards to the fourth e c o l o g i c a l model discussed i n Chapter Two. 3.8 Equivalent L i n e a r i z a t i o n Solution of O'Brien's Model The l a s t model to be studied\is.0'Brien's (see section 2.11) model. We begin by t r a n s l a t i n g the s i n g u l a r i t y (xs-j_ng>ys-;ng) = (J »r(J+A)/J) to (0,0) to obtain: 99. 3.10 Rosenzweig's Model Equivalent L i n e a r i z a t i o n Solution Amplitude Err o r L o c i 100. Zl" 5 . 0 0 0 1 0 . 0 0 0 . . 1 5 . 0 0 0 2 0 . 0 0 0 2 5 . 0 0 0 3 0 . 0 0 0 3 5 . 0 0 0 4 0 . 0 0 0 . . 4 5 . 0 0 0 5 0 . 0 0 0 -w:oooi .^7rr.T.i.. . . v . . . . i . . . . i . . . . 1—*—*—*—'—I ' " " I M P lolooo I3.00.0.__20.0PO 25.C00_30.000. . 35.000 .._4.0.000 45.000 5^000 •-R E U T 1 V E FREQUENCY ERRCR L O C I Gft APM ERROR » 1 0 . U 0 M S I N G U L A R I T Y AT I X . Y l - l 0 . 2 0 0 0 J E . 0 2 . 0 . 2 3 7 0 0 E . 0 2 1 >• 0 . 2 0 0 0 C t > 0 1 » • 0 . 4 5 0 0 0 E . 0 2 a - 0 . 1 0 0 0 0 E . 0 1 C - 0 . 1 0 0 0 0 E . O O S - O . 1 0 0 0 0 E « 0 1 J " 0 . 2 0 0 0 0 E . 0 2 _J<»«t f ( . SS l -V I I s t—1 * «T !» l_Sp i )> t l » Of. K O S 6 N 2 W . I C S MOOEL - . - 0 U A 0 R A T 1 C EON EOR 21 3.11 Rosenzweig's Model Equivalent Linearization Solution Frequency Error Loci 102. 5 0 T w ! ° j 5 J 0 n 0 - - j — 1 0 . 0 0 0 . 1 5 . 0 0 0 . . . . 2 0 . 0 0 0 2 5 . 0 0 0 . . . 3 0 . 0 0 0 .... 3 5 , 0 0 0 . 4 0 . 0 0 0 . _ . . 4 5 . 0 0 0 . . 8 8 3 9 0 4 6 6 7 7 7 2 0 . 0 0 0 9 \T i~ •""»"". " • " j " j " V i ^ ^ f r f ) • 20 .000 8 * 2 1 1 1 1 1 1 1 1 1 . 1 1 1 5 . 3 0 0 7 - 13 .000 3 I 1 0 . 0 0 0 * i o . n o n Y X / 6 3 > v I ^ ^ ^ ^ 2 2 3 . ^ - ^ \ " 5 . 0 0 0 - 1 s \ 3 2 2 2 3 ' • 5 .000 0 . 0 0 o 5 , 1 0 3 1 0 , 0 0 0 1 5 , 0 0 0 7 0 . 0 0 0 7 5 . n o n » n . n n n ^ 5 , ' > i o 4 0 . 0 0 0 4 5 . 0 0 Q i Q j 0 . 0 o n o CONSTANT R E L A T I V E AMPL I TUDE ERROR L U C 1 GRAPH ERROR X 1 0 . 0 0 0 1 S I N G U L A R I T Y AT t X , Y ) - t 0 . 2 0 0 0 3 E » 0 2 • 0 . 9 2 5 2 1 E * 0 1 ) ' , R- 0.20000£*01 K- 0.25Q0UE*Q2 0* O.IOOJOE+Ol C- O.IUOOUE»00 S- 0. lOOOOE^O I J - 0.200008*02 ^QULMltSiNLUHU*^ FREQUENCY — Q U A O R A I I C EQN f O R Z l _ „ „ F i g . 3.12 Rosenzweig's Model Equivalent L i n e a r i z a t i o n Solution Amplitude Err o r L o c i 102. (x+J)(y + r(J+A)/J) x — r "'" 1, 1 . (J+A+x) • = sAx(y + r(J+A)/J)  7 (J+A) (J+A+x) We may now compute the a „ as follows: -2r(J+A) //(J+A)2 \ , T X . / , T , ^ r-, , / /(J+A) 2 -. ..../^ a 1 ; L =. ~~"^~7~:— — — - 1 -(J+A) ^ - r(J+A){l+(/ - — ~ -1 -(J+A)^ j r 1/(J+A) 2-r 1 2 r l r l r l / (J+A) 2-r 1 2 The coefficients a-^' a21 a22 a r e t* i e s a m e a s ^ o r Holling's model. Again we may approximate the solution of equations for a^ & a 2 l ' a s P r e v i o u s ± y described in section 3.4. As before, when the i n i t i a l conditions tend to the singular point of interest, the coefficients a., tend to the values obtained for the linear solutions. This completes the equivalent linearization solution of O'Brien's model. 3.9 Simulation Study of O'Brien's Model We now wish to evaluate the equivalent linearization solution of O'Brien's model. The equivalent linearization solutions were computed for the same parameters and i n i t i a l conditions as the linear solutions (see section 2.13). The corresponding amplitude and frequency error l o c i are shown in figures 3.13 to 3.15. We note as before, we have an improvement in the size of the amplitude error l o c i , but again at the expense of the frequency error l o c i . Thus, as with the other solutions, we use the equivalent linearization solution amplitude and the linear solution frequency. Again, we have an improvement over the linear solutions. This completes our study and evaluation of the method of equivalent linearization of nonlinear differential equations. 103. CONSTANT R E L A T I V E AHPL1TUOE ERROR L O C I GRAPH ERROR « l O . C O O t S I N G U L A R I T Y AT I X , V I - 1 0 . 5 0 0 0 0 E . 0 1 , O . S O O O O E ' O l l R- 0 . 2 0 0 0 0 E . 0 1 A - 0 . 1 0 0 0 0 E . 0 2 S - 0 . 2 0 0 0 G E . 0 I J - 0 . 5 0 0 0 0 E . 0 1 J W i > A L E N J . i I N E A R _ I J E O . _ S O L U T | O N . . p F _ 0 , O R I E N , 5 . H 3 0 E L j r > I T H .L INEAR. FREQUENCY. _ 3.13 O'Brien's Model Equivalent L i n e a r i z a t i o n S o l u t i o n Amplitude E r r o r L o c i 104. -i...o-_..,.o..;l.*.o.. ! l ^ ^ : . : i ^ : i ^ ! i ; ? ? : i 7 : : ! l ^ ^ 0 0 ! o 6 . 0 0 0 j 7 ^ 3 1 3 1 0.0 I.. 0.0 "j.OO*L ?iPQO ,3.000 . 4 . 0 0 0 5.coo... 1 o.o 9^000 LQ.Q0J3 CONST ANT R E L A T I V E FREOUENCY ERROR L O C ! GRAPH ERROR X 1.0001 S I N G U L A R I T Y AT I X . Y I - 1 O . S O O O j E . O I . 0 . 6 0 0 0 0 E . 0 U » . 0 . 2 0 0 0 0 E . 0 1 A . 0 . 1 0 0 0 0 E . 0 2 S- 0 . 2 0 0 C 0 E . 0 1 J » 0 . 3 0 0 0 0 6 * 0 1 J O U l V l t t N i k l N E ARJ.ieo_.SOLUI. IpN., Of_P.V|R I EN'S .MODEL „-..!< ITM_ L INEAR J R J O J E N C Y . _ F i g . 3.14 O'Brien's Model Equivalent L i n e a r i z a t i o n S o l u t i o n Frequency E r r o r L o c i tt.o i.ooo_ i.OOO.' 3 . 0 0 0 _ . . . 4 . 0 0 0 3 . 0 0 0 . 6 . 0 0 0 ( . 0 0 0 s , 0 0 0 9 . 0 0 0 l _0»Q3O_ R E L A T I V E AHPL1TUDE ERROR L O C I -GRAPH EARO» « 1 0 . 0 0 0 1 S I N G U L A R I T Y AT I A . Y I - 1 O . 5 0 O 0 J E . O 1 , 0 . 6 0 0 0 0 E . 0 1 I R- 0 . 2 0 0 0 0 E . C 1 A * O . I O O O O E . O I S " 0 . 1 0 0 0 0 E . 0 0 J - 0 . 5 0 0 0 0 4 . 0 1 _e0Ul3 , A L E N T . L I N E A R U E 0 . S O L U T I O N .0F..O18RI E N ' S . " O O E L . - . N l T N L I NEAR. .FREQUENCY 3.15 O'Brien's Model Equivalent L i n e a r i z a t i o n Solution Amplitude Error L o c i 106. 3.10 Summary In t h i s chapter we have developed and a p p l i e d the method of equivalent l i n e a r i z a t i o n to the four e c o l o g i c a l models t h a t were introduced i n chapter two. This method of s o l u t i o n y i e l d s an improvement i n the accuracy of the amplitudes of the s o l u t i o n s over the c l a s s i c a l l i n e a r s o l u t i o n . However, the frequency of the equivalent l i n e a r i z e d s o l u t i o n i s found to be l e s s accurate than that of the l i n e a r s o l u t i o n f o r given i n i t i a l c o n d i t i o n s . This apparent t r a d e - o f f between amplitude and frequency e r r o r was r e s o l v e d by combining the amplitude of the equivalent l i n e a r i z a t i o n s o l u t i o n w i t h the frequency of the l i n e a r s o l u t i o n to obtain the best features of both. The t r a d e - o f f between amplitude and frequency i s not a general occurance, but one that depends on the form of the i n d i v i d u a l model. I f the model produces a l a r g e nonsymmetrical o s c i l l a t i o n close to the time o r i g i n , the minimum mean square e r r o r c r i t e r i o n chooses the a_^ _. so that the approximate s o l u t i o n t r a c k s the a c t u a l s o l u t i o n very w e l l near the time o r i g i n . I f the model then continues on to produce symmetrical o s c i l l a t i o n s , (as the models under c o n s i d e r a t i o n do) the o r i g i n a l estimate of the a^ _. i s no longer accurate. As the d e r i v a t i v e s i n the nonsymmetrical o s c i l l a t i o n s are u s u a l l y l a r g e r than those of the symmetrical o s c i l l a t i o n s , we f i n d that t h i s gives an estimate of the d e r i v a t i v e s t h a t i s higher than i t a c t u a l l y i s , which i n t u r n r a i s e s the frequency estimate. This e f f e c t w i l l not n e c e s s a r i l y occur i n a l l n o n l i n e a r d i f f e r e n t i a l equation models that employ t h i s method of s o l u t i o n . I t should be noted that f o r the eq u i v a l e n t l i n e a r i z a t i o n s o l u t i o n of H o l l i n g ' s model and O'Brien's model, i f the i n i t i a l c o n d i t i o n s are l a r g e enough, the denominators of the a.. may become imaginary, i n which case the 107. s o l u t i o n ceases to be meaningful. For i n i t i a l conditions near the singular point of i n t e r e s t , the equivalent l i n e a r i z a t i o n solutions reduce to the l i n e l i n e a r s olutions. With the method of equivalent l i n e a r i z a t i o n , we have a complete l i t e r a l parameter s o l u t i o n , which i n general, w i l l depend i n a nonlinear manner on the i n i t i a l conditions. This s o l u t i o n y i e l d s a r e f i n e d estimate of the a ^ which may now depend on the i n i t i a l conditions, an a f f e c t not found i n l i n e a r i z e d s o l u t i o n s . From the equations for the a^ , one can e a s i l y determine the influence- of i n i t i a l conditions and v a r i a t i o n s of parameters on the various components of the so l u t i o n s . For some parameters combinations, a given a^ _. w i l l be dominated by only one or two terms and may dominate a component of the s o l u t i o n , i n d i c a t i n g which parameters are of major influence on the s o l u t i o n . These formulae save on the trouble of extensive computer s e n s i t i v i t y studies. Refinements to the method of equivalent l i n e a r i z a t i o n w i l l be discussed i n the next chapter. 108. CHAPTER FOUR REFINEMENT OF THE METHOD OF EQUIVALENT LINEARIZATION .4.1 .Introduction Having completed the a p p l i c a t i o n and evaluation of the method of equivalent l i n e a r i z a t i o n , we turn our attention to other techniques of obtaining approximate solutions of nonlinear d i f f e r e n t i a l equations that y i e l d l i t e r a l parameter solutions that would be acceptable over a wider range of i n i t i a l conditions than the previously developed techniques. The f i r s t approach taken towards t h i s end was the e l l i p t i c functions method. The VGW model was used as a prototype for development of the technique. The r e s u l t s may be b r i e f l y summarized by s t a t i n g that they were, at best, only as good as the previously developed r e s u l t s . A b r i e f summary of the methods used, along with the complications and l i m i t a t i o n s encountered are given i n Appendix A for the i n t e r e s t e d reader. The next approach taken to obtain improved solutions i s a completely novel method and i s described i n d e t a i l i n the next sections. 4.2 The Additive Correction Factor While evaluating the methods of l i n e a r i z a t i o n and equivalent l i n e a r i z a t i o n , an i n t e r e s t i n g phenomenon was observed, but at the time, was ignored. For large i n i t i a l conditions, the numerical (true) s o l u t i o n t y p i c a l l y appeared i n the phase plane to be a s p i r a l whose centre of o s c i l l a t i o n was slowly s h i f t i n g down towards the singular point (x . ,y . ). 7 6 e> v \ sing ^ sing I t was conjectured that a transient term i n the form of an exponentially damped ramp could be added to the e x i s t i n g solutions to reproduce t h i s e f f e c t and hopefully reduce the amplitude errors of the equivalent l i n e a r i z a t i o n 109. s o l u t i o n s . Numerical studies were done to evaluate the general v a l i d i t y of this idea. Exponentially damped ramps were extracted from the x(t) and y(t) numerical solutions and then added to the approximate a n a l y t i c a l s o l u t i o n s . This transient was obtained by numerically f i t t i n g (using a least squares method) two pairs of decaying exponentials to the numerical s o l u t i o n maxima and minima, which represent the upper and lower envelopes of each s o l u t i o n . The difference of these two envelope exponentials gives the desired transient. The preliminary studies showed that t h i s was indeed an improve-ment, and should be investigated further. 4.3 A n a l y t i c a l Determination of the Additive Correction Factor Having found an idea that exhibits a p o t e n t i a l improvement for our e x i s t i n g solutions, we now wish to determine a n a l y t i c a l l y the four parameters of the additive corrections f a c t o r s , ° t x = tA e x c x cr t y = tA e v c y y The i n i t i a l goal was to be able to compute the exact values of the s o l u t i o n peaks, which would then y i e l d the correction factors i n a s i m i l a r manner to the preliminary numerical studies. A number of ideas f o r computing the peaks and the correction factors were i n i t i a l l y t r i e d , but these y i e l d e d spurious r e s u l t s , i n terms of accuracy. They are b r i e f l y o utlined i n Appendix B for the i n t e r e s t e d reader. As a means of approximating the extrema of the s o l u t i o n s , we used a three term McLaurin expansion: x = x + F(x , y ) t + F'(x , y ) t 2 / 2 o o o o Jo y = y Q + G ( X q , T o ) t + G ' ( x Q , y Q ) t 2 / 2 110. I t was found that this method yie l d e d good r e s u l t s i f the peak of the solut i o n was within % cycle of the time o r i g i n . Unfortunately, t h i s condition was not met by a l l combinations of i n i t i a l conditions. Also, t h i s method would permit computation of only the f i r s t peak of the x & y solutions, whereas i t was desired to obtain several successive peaks. These problems were circumvented by using the general Taylor series expansion to obtain: x(t) = E x 1 ( x ^ ,y. ) ( t - t ) ± / i ! i = l a a CO . . y(t) = Z y 1 ^ ,y ) ( t - t . ) X / i ! i = l h h b where the higher order derivatives are e a s i l y obtained from the o r i g i n a l d i f f e r e n t i a l equations. There i s a problem i n that the terms i n the serie s are functions of x & y, which are known p r e c i s e l y only at the time o r i g i n . This problem may be resolved by using the equivalent l i n e a r i z a t i o n solutions as an approximation i n evaluating the derivatives i n the Taylor s e r i e s . We choose the points t & t^, about which we develop a Taylor s e r i e s , as follows. From Chapters Three and Four, we know that the phase and frequency of the l i n e a r s o l u t i o n i s accurate, and may use i t to p r e d i c t the l o c a t i o n of the peaks of the true s o l u t i o n . To accomplish t h i s , we f i r s t rewrite the a n a l y t i c a l solutions as follows: x = r e cos ( t o t + 6 ) x x' y = r e C T tsin(cot + 8 ) y y  where r = / ( x -x . ) 2 + ((x -x.. ) (a„ -o)-a (y ,-y . ) ) 2 x / v o sing o s i n g 7 11 ' 12 Jo -'sing 2 r y " / ( y o - y s i n g ) 2 + ( ( y o - y s i n g ) ( a 2 2 ^ ) + a 2 1 ( x o - x s i n g ) ) 2 2 to 111. A V -1 r ^ y o - y s i n g ) a 1 2 - ( x o - X s l n g ) ( air 0 ) , 6 x = t a n [ ~ ( x - x . )_ ~ ] O' sing -, . •' (y -y . )oi 6 = tan [ 7 — TT „\ • / \ — — J y (y -y • ) ( a o o - c 0 + (x -x . ) a 0 . J J J o ; s m g 22 o sing 21 We then locate the extremum points by r e q u i r i n g that x ( t i ) = 0 and y( t ) = 0, from which we obtain: fcx = L - t a n _ 1 ^ - °xi V - ' ^ t t - " 1 0 + 6'y-l These formulae give the l o c a t i o n of the f i r s t peaks of the solutions, and successive ones are located every — time units l a t e r i n time. We f i r s t evaluate our Taylor ser i e s at the points t =t /2 & t, =t 12, which 7 r a x b y ' locates the serie s at least h a l f way between peaks, and then every — l a t e r i n time. The reason that we locate our serie s away from the actual peaks i s that the approximate s o l u t i o n i s accurate near the singular point and permits accurate p r e d i c t i o n of the derivatives i n the Taylor s e r i e s . The peaks of the s o l u t i o n are then given by x ,=x (t /2) & y =y(t /2). p G c l K . X p _ . c L i C y The next question that arises concerns how many terms of the Taylor series are a c t u a l l y needed. Experiments done on the computer showed that terms of order three and higher could be neglected i n most cases. For the cases where the terms of order three and higher are s i g n i f i c a n t , i t was 3 found that by adding on a term of the form c ( t - t ) the accuracy of the X Si Taylor ser i e s was improved. The c o e f f i c i e n t c_^  was chosen so that the Taylor series peak value occurred at t . This a d d i t i o n a l term approximates the ef f e c t s of a l l the neglected terms of the s e r i e s . In many cases, t h i s a d d i t i o n a l term i s not neccessary, and when we compute i t , i t turns out to be zero or n e g l i g i b l e . We obtain c from: X 112. f r c x = " ( F ( x t ,y t ) + F ' ( x t ; , y t ) ( V t a ) ) / 3 ( V t a ) 2 a a - a- a We may compute the term c^ i n a s i m i l a r manner. We now have a means of computing the peak values of the s o l u t i o n s . For large i n i t i a l conditions, when the approximate solutions become inaccurate, so does the Taylor s e r i e s . We may now proceed to determine the actual correction f a c t o r s . We could use the Taylor s e r i e s to generate numerous points of the s o l u t i o n and then proceed i n the manner used i n the numerical study. I f t h i s were done, the formulae f o r the correction factors would become cumbersome, and would not y i e l d any p r a c t i c a l i n s i g h t i n t o the system behaviour. Instead a d i f f e r e n t approach w i l l be pursued. As the major source of error i n the e x i s t i n g solutions usually occurs i n the f i r s t period of the s o l u t i o n , we propose a c o r r e c t i o n factor that w i l l correct only the f i r s t peak i n each s o l u t i o n , which w i l l reduce the o v e r a l l error somewhat. The additive correction factors w i l l take the form: n - n t / t x(t) = x T . + x x = A t e x v / e q l i n c e x , n where . n - n t / t y(t) = y . . + y y = A t e y • ' - ' e q l i n c c y These correction factors are n'th order exponentially damped ramps whose peak values occur at t & t , the times of the f i r s t peaks. The correction x y factors become more l i k e narrow pulses as n becomes l a r g e r . The maximum values of the correction factors are given by A t n e n and A t n e n , which x x y y we would l i k e to be equal to the difference i n peak values between the actual and approximate s o l u t i o n s . The peak values of the approximate solutions are given by: at . J x , = r e x cos (tot . + 8 ) peak x x x y . = r e a t y sinfwt + 8 ) ypeak y y y 113. Thus we obtain: A = (x - x )(e/t ) n x peak Taylor peak Analytic ^ A y ^ypeak Taylor- ^peak Analytic)(e/ty) n We-now wish to choose n so that the correction factor w i l l affect only the f i r s t peak of the solution. When we take n = 11, the correction factor w i l l be reduced to a maximum of 35% of i t s peak value by the time the solution has progressed another quarter period forward in time. When the next peak of the solution occurs, the correction factor w i l l be completely negligible. We may also apply this correction factor to the overdamped solutions First, we must modify the-method whereby we compute the peak times t x & t . We rewrite the overdamped solutions as follows: x(t) = x . + A, e^l* 1 + A i i i eCT2fc sing lx 2x y ( t ) • y S i n g + v v + V s v where = « * 0 - * s i n g > ( a ^ ) ( % - y s l n g ) ) / ( W A 0 = ((x -x . ) (a - a i 1 ) + a 1 0 ( y -y . ))/(G,-°o) 2x o sing 1 11 12 •'o •'sing 7 1 2 A, = ((y -y . ) (a00-,o )+a 0 1 (x -x . ) ) / ( a.-o_) ly o sxng 22 2 21 o sing 1 2 A 0 = ((y -y . ) ( s . - a 0 0 ) - a 0 1 (x -x . ) ) / ( a , ) 2y J o ; s i n g 1 22 21 o sing 1 2 These solutions exhibit a peak point located at T -A. a_ r _ f „ . 2x fcx " ( a L - a 2 ) l n ( A ^ 7 ) 1 - A2 a2 114. If t i s to be greater than zero, we must have (cr - a) > 0 and with similar restrictions on t . Should there be no peak in positive time, we may estimate a point at which we may apply the correction factors. This estimate i s given by: This estimate locates a point on the solution that i s about half way between the i n i t i a l values and the steady state value. For the overdamped case, we don't need such sharp pulses for correction factors, and thus we may reduce n down to n=4. We then proceed with the correction factors as previously outlined for the underdamped case. Thus we have developed a correction factor using a modified Taylor series and an exponentially damped ramp that w i l l correct the f i r s t peak of the existing equivalent linearization solutions and reduce the overall solution error. We now proceed to apply this method to the four ecological models discussed in chapter two, and evauate i t s usefulness. 4.4 Additive Correction Factorfor the VGW Model The f i r s t model we wish to apply the additive correction factor to is the VGW model discussed in section 2.2. For computing the derivatives in the modified Taylor series, we already have the f i r s t derivatives, which are the differential equations. The second derivatives may be computed as t ( I A1* \ I +*A2_<_I> follows: 115. The p a r t i a l derivatives are given i n the A matrices (see chapter two). We now compute the additive correction factors using the method outlined i n section 4.3. To evaluate the additive correction f a c t o r s , we compute the corrected solutions for the same parameter choice and ranges of i n i t i a l ? conditions as f o r the l i n e a r solutions (see section 2.4). The same samples of time plots and phase plane pl o t s are given i n figures 4.1 to 4.3. The corresponding amplitude error l o c i are given i n figure 4.4. We observe that we:have an increase i n the s i z e of region enclosed by the e r r o r l o c i (compared to that of the uncorrected equivalent l i n e a r i z a t i o n s o l u t i o n s ) , i n d i c a t i n g that t h i s method i s d e f i n i t e l y an improvement. As the c o r r e c t i o n factor i s designed to correct only one peak of the s o l u t i o n , the frequency of the s o l u t i o n w i l l not be affected, and hence the corresponding frequency error l o c i w i l l be i d e n t i c a l to those of the uncorrected s o l u t i o n s , and need not be shown here. In the computer algorithm, provisions were made so that i f error improved by the correction factor was l e s s than 10% of the equivalent l i n e a r i z a t i o n s o l u t i o n a correction factor would be applied to the second peak instead of the f i r s t . This was to accomodate the cases where the s o l u t i o n ..started o f f very close (in time) to the f i r s t peak (and incurred l i t t l e error=at that peak)but had a s i g n i f i c a n t error at the second peak. It should be noted that when one peak of the s o l u t i o n i s corrected, the next largest e r r o r (usually on the next peak) becomes the dominating and l i m i t i n g e r r o r i n the s o l u t i o n . I f i t was so desired, additive correction factors could be applied to more than one peak, but t h i s increases the complexity of the complete s o l u t i o n . 116. . „ 1 0 . 0 0 0 CCO 3 0 . 0 0 0 t O ' , ooo 7 3 . 0 0 0 6 0 , 0 0 0 9 0 . 0 0 0 . ^ . 1 0 0 , 0 0 0 ._ . . . I 1 0 3 . 0 0 0 0 . 0 I.. 0 . 0 _ 1 0 , 0 0 0 20 .COO. , M i l l . A N A L Y T I C A L S l ' L U T I u h . 30 .000_ . . . . A O . O t O . M M H I f A I . S l i l l l T I L N . . | | | | | ) . . . . I 1 I I....I 0 . 0 5 0 . 0 0 0 _ 6 0 . 0 3 0 7 U . 0 0 0 8 0 . 0 0 0 _ 9 0 . 0 0 0 1 0 0 . 0 3 0 , A I T I . . . . . » M I I H 1 S C J K H U N TO. B J T H SOLUT IONS A O " 0 . 8 5 0 0 0 E . 0 2 Y O - O.eOOOUfc .02 S I N G U L A R I T Y AT < A , Y I - I 0 . 5 0 U 0 0 E . 0 2 , 0 . A 5 C 0 J E . 0 2 I •TilESi£NrYTAlCr"orjT5oit."02_N^ i , . 3 6 5 2 8 E . 0 1 1 AMPL I TUDE I k R u R I E R I L . - O . 2 5 8 5 2 c . 0 2 « E A B S — 0 . 4 2 2 7 7 E . 0 1 X P E A K N - 0 . 2 6 : 9 5 K C 2 XP t AXA• Q . I 5 1 0 1 f « U 2 X I A . Q . A 1 7 8 9 C . 3 1 T » . 0 . 5 5 1 1 A E - 0 2 N X - U . . X M A X - 2 6 _ V F E A X N - 0 . 1 7 7 3 5 E * 0 2 YPEAXA- 0 . 1 S 3 C 7 E . C 2 YCA- 0.4E3B t^.2« T Y " 0 , . 9 6 0'»E - 0 2 N Y - 1 1 N Y M A X - 2 9 A L P H A . O . i O O O d r . o i - 8 E T A - ' " o - ' l J O C C t » 0 ~ 2 GAHA- O.SOO0GE . L 3 O E L T A . 0 . 1 0 C 0 0 E . 0 2 L A H B O - . O . l O O O O e . O l . . « « < A Q O I T I V E C C S E C T I O M - EQUIVALENT L INEA» I-AT lull S t j L U T l U N OF L 0 T K A - V J L T E 1 » S Y S T E M .1 TH P I U V LAPP.Y INO C A P A C I T Y » > » 4.1 VGW Model; Corrected Equivalent Linearized.'.Solutipn ;for ( x 0,y ) = (85, 117. xm _o.a _0.005._ ..I. . . . _4.0.iQ_. ..I 0 . 0 1 5 _0.020.. ..I.... _0.023 , Q.030 ....I....I — I — I . . . . 0 3 9 . . . . . 0.C4Q 0.045 C.050 . . I . . . . I.... I . . . . I...*l 100.000 H i l l • m.»Tlct| SOLUT ION , 0 . 0 . 6 5 0 0 0 E . C 2 Y 0 - 0 . 6 0 0 0 0 E . 0 2 S I N C U L A P I T Y AT I X . Y 1 - I 0 . 5 0 0 0 0 E . 0 2 • 0 . < . 5 iC0 f c . 021 ' . . . „ . - R i ^ N C T r . K ^ f 5 5 . 4 1 . 0 2 - n U B . 0 . 7 2 T . . E . O Z E K K O K . u . 3 . 5 2 . E . 0 1 . « P L , T _ . • » . « . 1 > » 4 . E . 0 2 X t A A B S . - 0 . . 2 2 T , E . O > « , •= . «• . . 0 . 2 . 2 ^ . . - . 2 » P F » « - 0 . n « n i F . 0 2 X C A - O . A l T b V E . 3 1 T x - 0 . 5 5 »3>fc-0<. I.X-|| . . X H A X . 2 6 , A L P H A " 0 . 5 0 0 0 0 E . 0 3 o t T A - 0 . 1 O 0 0 O E . 0 2 G A M - 0 . 5 G C 0 C E . 0 3 D E L T A - O . 1 O O 0 0 2 . O 2 L A H b l M " O . l O O G O E . C l - < « < < - A O O . T i v r c o r x E C T . O ^ E O u f v A L ^ ^ S O L U T I O N U P X o T K . - V O L T E K F A S Y S T E H - T H P K . Y C.P.KY1NG C A P A C I T Y » » > Fig. 4.2 VGW Model Corrected Equivalent Linearization Solution, for (x Q,y o)-(85,60) 118. V 1 T I . 0 . 0 0 5 ..I _ J U Q » 0 . _ I....I . . 3 . 0 1 5 . . ..I . 1 . 0 2 0 , ..I J . 0 2 5 _ . . 0 . 0 3 0 - t 1 1 . 0 .0 ' . . . I . . 0 . 0 5 0 1 0 0 . 0 0 0 1 11 \ ll \ • f // \ // V - , n - n n o 5 0 . 0 0 0 -i l if/ V If Vl , i.1 i ! V 7 — n 1 ~ " / / \ \ - 4 0 . 0 3 0 4 0 . 0 0 0 -u J t t IL ¥c • // W * X ..' - L ft- W : : " ^ 3 0 . 0 0 0 U M W J \*4 - 3 0 . 0 0 0 ; W / \ / : : AY/ tiOM**il>rr —' 2 0 . 0 0 0 : V ArifiPirictL . . . I O . d 0 0 • —— -'-*-— - in.noo O.O I....I. a.o . . i — i . _ 0 . 0 0 5 ..I I. _ 0 . 0 1 0 „ . . . ..I — I....I....I. 0 . 0 1 5 . 0 . 0 2 0 . ..I....I. 0 . 0 2 5 ..I....I. . 0 . 0 3 0 ..I....I. . . J . 0 3 5 .... ..I....I ••••!• 0 . 0 4 0 0 . 0 * 5 T IME S E C . ..I 0 . 0 . . 0 . 310 m n - A M I TT IC AL SOLUT ION NUHFB1CAL SLJLUT10N . . . . . • P O I N T S C J M M J K T i l BOTH S - l L U T l O N S A O - C . O 5 C O 0 E . C 2 T O - O . o C O O O E . 0 2 S I N C U L A P 1 T Y AT U , Y I - I 0 . 5 C 0 0 0 C 0 2 • 3 . A 5 J 0 0 E . C 2 1 ^ . - f r ^ f N l y r i i ^ ^ i , ^ ^ O V 3 . 5 2 B E . 0 1 . A N P L l T u D : ERROR. " E Y P E L - O . 2 5 . 5 2 E . 0 2 . E Y . » s V - o . » l B « E . O l T P E A . N . O. I 7 7 3 5 E . 0 2 Y P t A n A . 0 . 1 5 3 0 7 1 . 0 2 T C A . 3 . A d 5 6 5 E » 2 B T T . O . I ^ O O V i - 0 2 MT-11 i y f N A K ' 2 . 9 A L P H A . O . 5 C 0 0 C K 0 3 OfcTA- 0 . 1 0 0 0 0 E . C 2 G A h A - O . 5 C 0 0 0 E . 0 3 D E L T A - 0 . 1 0 C 0 0 E . 0 2 L A H b O A - 0 . 1 0 O C 3 E . 0 1 "<<«rTroiTiv^l .TKeTlm S O L U T I O N O f • i o T r . A - V O l t t R k * S Y S T E M . I T H , « « » C A R R Y I N G - C A P A C I T Y > » » F i g . 4.3 VGW Model Corrected Equivalent L i n e a r i z a t i o n S o lution.for ( X 0»Y 0) =(85,60) 119. • 0.0 10 .TOO 20.000 30.000 40.000 5 0 . 3 0 0 60.000 _ 70.000 _ _ 8 0 . 0 0 3 _ J»Q. 000__I 00. QOO i ^ o w i T . ^ ; r r . T . ^ 100.000 ; 9 0 . 0 0 0 - 6 5 4 7 7 7 6 J 9 0 . 0 0 0 6 ! * J 3 3 3 5 5 6 7 8 1 6 0 . 0 0 0 - 6 4 3 3 _ 2 _ 2 4 4 4 3 6 7 8 0 . 0 0 0 4 3 3 S 2 2 \ j 3 3 * 4 5 6 7 0 7 0 0 0 - . r J- 7 - X _ ^ V - X i \i 3 3 * -~~76T6oo 8 ' -/ \ \ 6 0 . 0 0 0 - 6 0 . 0 0 0 \ ^ 1 \ I - S O . 0 0 0 9 0 . 0 0 0 7 X 2 4 0 . 0 0 0 3 - - 4 0 . 0 0 0 =• — 3 0 . 0 0 0 \ - 3 0 . 0 0 0 1 " - t i 1 1 * 1 1 1 — - 2 0 . 0 0 0 2 0 . 0 0 0 - 3 \ 2 1 / 4 : V 2 0 % — V - 7 3 \ j 2 2 2 2 2 2 2 1 0 . 0 0 0 -~ io robo 0 . 0 I 0 . 0 — X l O ) . ' CONSTANT fcELATlVE A H P U T J 3 E E •* «GR. LOC I . \ . ; ! l jk A PH ERROR X 10.OOOC . A L P H A " 0 . 3 0 0 0 O E » 0 3 S E T A " 0 . t O O O O E * 0 2 GA HA- O . 5 0 0 O 0 E » O 3 CELT A * 0 . I j lOOOE *0 2 L **< B D A " 0 . I OOOOE*01 : , « « < A D D I T I V E C O R R E C T I O N - E G J I V & L E N T L I N E A R I Z A T I O N S O L U T t U N UF L O I K A - V O L U R R A S Y S T E M * I T H PREY C A R R Y I N G C A P A C I T Y » » > F i g . 4i'4 VGW Model Corrected Equivalent, L i n e a r i z a t i o n Solution Amplitude Err o r L o c i 120. The corrected solutions were also computed for the overdamped case, and the same samples of time plots and phase plane curves are shown in figures 4.5 to 4.10. The corresponding amplitude error l o c i are shown in figure 4.11. Here also, we have an improvement in the size of the amplitude errorcloci. We now proceed onwards to study the next ecological :model. 4.5 Additive Correction Factor for Holling's Model The second model that we wish to apply an additive correction f factor to i s Holling's model. The correction factors may be computed in the same manner as outlined for the VGW model. To evaluate the additive correction factor, the corrected solutions were computed for the same parameter choice and ranges of i n i t i a l conditions as for the linear solutions (see section 2.7). The same samples of time plots and phase plane curves are given in figures 4.12 to 4.17. The corresponding amplitude error l o c i for the corrected solutions are given in figure 4.18 and 4.19. In both cases, we note that we have an improvement in the size of the amplitude error loci;(compared to the uncorrected equivalent linearized solution). Having found the additive correction factor to be useful improvement for the solutions of Holling's model, we how turn to study the third ecological model. 4.6 Additive Correction Factor for Rosenzweig's Model The next model we wish to apply the additive correction factor to is Rosenzweig's model. The correction factor may be computed in the same manner as outlined for the VGW model. To evaluate the additive correction factor, the corrected solutions were computed for the same parameter choice and ranges of i n i t i a l conditions 1 121. Y l t ) o .o i.poo z.oon lo.oooi. 0 . 0 I.. 0.0 _ 4 . 0 0 0 5 . 0 0 0 6 . 0 0 0 '.000... £ . 0 0 0 „ . 9 . 0 0 0 1 0 . 0 0 0 T....I....I....I 10.000 y.ooo T __*.oqq 3.0,00 *.oop 5 .ooo_ ..I....I....I I- — < • 1 o.o _ 7 . 0 0 0 8 . 0 0 0 9 . 0 0 0 1 0 . 0 0 0 "' X I I I t i l l * » A N A L Y T I C A L S n t U T K I N N ^ ^ I C A L SOLUTION - • POINTS COMMON TO BOTH S O L U T I O N S XO- 0 . 5 5 0 0 0 E . 0 1 Y O . 0 . 1 0 0 0 0 E . 0 1 S I N G U L A R I T Y I T I X . Y I ' I 0 . 5 0 0 1 0 E . 0 1 . 0 . 5 0 0 0 0 E . 0 1 I FILEOUENCYI A N . O T O S U H — a ; * " E R R O R " 07b ~ « A.1PL I T U 1 E E R R O R ! E R E L " 0 . H 9 0 U . 0 M E M S - 0 . 1 W 6 8 E . 0 T " X P N l » . 3 . 5 8 6 6 6 E . n i »°F.A<N- 0 . 5 7 3 1 S F . O I X P E A M - 0 . 5 7 6 5 8 1 . 0 1 X C . . - 0 . 6 1 Q 9 T E . 0 5 TXNUM. 0 . ? A O O O F - O I T X - 0. 1 3 0 0 3 F - 0 I N X . 6 N X N . X-38, Y P N U N - O . I M 8 8 E . 0 1 Y P E A X N . 0 . 1 9 M A E . 0 1 Y P E A R A . 0 . J 6 3 7 9 E . 0 1 7 C 4 . - 0 . 3 6 O 0 2 E . 0 7 I Y N U N - 0 . 3 4 0 0 0 E - 0 1 T Y - 0 . 5 6 W 9 E - 0 I N Y - » — E P H T r T O J W ' E V o r F f E t Y ^ « « < A O O I T I V F : 0 F « F C T I n N - E QUI V At CNT L I N E A R I Z A T I O N SOLUTlO-4 u f L 0 1KA - YOL TE» RA SYSTEM MIT1 P « E T C A R R Y I N G O P A C I T Y >>>>> F i g . 4.5 VGW Model Corrected Equivalent L i n e a r i z a t i o n Solution for ~{^,y ) = Or.5,1) 122. 3.0 1,030 OjSiQ __O.SSQ .0.129 0.15.0. 0.1 B.Q 0.21O_ ~10.300i....I....I....I....I....I....I....II....I....I .... I.-..I. ...I .-• • _ 0 . 2 4 0 _ — 0 . 2 1 0 0 . 3 0 0 ..I I I. . . . I . . . .I 1 0 . 0 0 0 0.0 I.. 0.0 * 0 . 3 3 0 Q . 0 6 0 0 . 0<!0 0 . 1 2 0 0 . 1 5 0 0 . 1 » 0 0 . 2 1 0 _ ..I....I I I....I 0.0 _ 0 . 2 * 0 0 . 2 1 0 0 - 3 0 0 T I N E ' S E C . A l . t t • A N A L Y T I C A L SOLUT ION N. IHERICAL S U L U I 1 0 PO INTS COHHON TO BOTH S O L U T I O N S _ K 0 - 0 . 3 3 0 0 0 E . 0 1 Y O - O . l O O O O t . O l S I N G U L A R I T Y A l I I . I I ' I O . S O O U O t . O l . O . 5 0 3 0 0 E . O I I " E R R O R . OVO «-5l»riTU0r't»»0«r't»«CT«~07»J0d»l»M RTSOSfffYT A N " 0 . 0 NOH- 0 . 0 S t . Q I X P E A X A " 0 . 5 T o 5 S E . p l X C A . - 3 . 6 I 6 9 I E . 0 R TANUH. 0 . 2 A 0 0 O E - 0 1 T x . 0 . 1 3 0 0 3 E - O 1 N X . * W H A » . J j _ APNUH. 0.5B666E.0I XPEAKN. 0.5TJ3 ALPHA. O.2I500E.O3 BETA. 0.10000E.02 GA HA. O.SO'lOOL.OJ DELTA. 0.130006.02 L AH BOA. O.A50O0E.O2 -«<-?XTlWliive-co«iiecTlriN-.-4aJi»«ie«t LINEARIZATION SOLUTION UF'LUIXA-V3LTERRA SYSTEH «ITH PREY CARRYING CAPACITY » » > F i g . 4.6 VGW Model Corrected Equivalent Linearized Solution f o r ( X q , y o ) = (3,i5,1) 123. f / V (T 1 ' 0 . 0 0..T30 0 . 0 6 0 0 . 0 9 0 0 . 1 2 0 0 . 1 5 0 O . l f l O 0 . 2 1 0 Q..240 0.270 0 . 3 0 0 10.000 1 0 . 0 0 0 1 9 . 0 0 0 - f 9 . 0 0 0 i . o o o - 8 . 0 0 0 7 . 0 0 0 - 7 . 0 0 0 6 . 0 0 0 - 6 . 0 0 0 5 . 0 0 0 5 . 0 0 0 -4 . 0 0 0 -4 . OOO 3 . 0 0 0 -- 3 . 0 0 0 * f '-„,,, , r f jt* : . . •——— 1 7 . 0 0 0 • v» if* 1 . 000 1 . 0 0 0 i o . o 0 . 0 0 . 0 3 0 0 . 0 6 0 0 . 0 9 0 0 . 1 2 0 . 0 . 1 3 0 _ 0 . 1 6 0 0.215 ° J \nz~Z C ~ t S S t f - H N M V T 1 C A L SOLUT ION * • * • * - M J H F P 1 C A L S O L U T I O N M f « i • PO 1 * T S._eO_mON TO BOTH S Q L J T . O N S . X O - O . J 5 O O 0 E . O 1 V O - 0 . 1 0 0 0 0 E . 0 1 S I N O I K B 1 I V « t I X , Y I - ( 0 . 5 0 0 0 0 E . O I . 0 . 5 0 0 0 0 E . O I I — F i f l o j E l i c V i AN. o.b~ SuVoTe" ER>o«i T i T o — " « rtPuruoE HMOM* «v»f >>'b.4>tot£»ott •tviSH'b;i*'T<Je»ot Y > N O » . l . l t U S E . O l Y P F A K N - Q . W M i . 0 1 Y P E A K A . 0 .2<. ? 7 1 E » 0 1 Y C A — O . 3 6 0 0 * 5 . 0 7 T Y N U X - 0 . 3-.00QE-O1 I Y - 0 . 5 6 « 9 F , - 0 | N Y - » NY. A L P H A - 0 . 2 7 5 0 1 E . 0 3 B E T A . O . 1 0 0 0 O E . O 2 G A « A - 0 . 5 O 0 0 0 E . O 2 O E L I A - G . I O 0 O 0 E . 0 2 L A N B O A - 0 . 4 3 0 0 0 E . 0 2 ^«r " A Y o m v f C O T R E C U O N ^ ^ bF " T L O t * » - W L T E I < « » ' S Y S t E « I I I IH P B E Y C M R Y I N C C i P X C I I Y ' Fig. 4.7 VGW Model Corrected Equivalent Linearization Solution-for (x ,yo)--(3. 5,1) L 124. 0 ' o " 1.103 ?.000 3.0O0 J , 0 O O . . . „ _ . 3 . M . O . . l _ 6 . 0 0 O . 7.000. 8.000 ... 9.000 . 1 0 . 0 0 0 o i l . . . I . . . . I . . . . I I I I 1 I .... I. ...I I ....I 1 ....)....(....I I 0 . 0 I.. 0 . 0 • ANALYTICAL SOLUTION _Xi^pr) 7 . 0 0 0 3.000... I.... I....I 1 o.o 4 . 0 0 0 5 . 0 0 0 6 . 0 0 0 7 . 0 3 0 8 . 0 0 0 S . 0 0 0 _ 1O.0OJ1 NU1C W IC A L SOLUTION * POINTS C04HQN TP 7 N T H 4011ITIDNS i. 0 . 6 1 0 0 O E . 0 I T O - 0 . 3 5 0 0 3 C . 0 1 S I N G U L A R I T Y AT I X . V L l O . 5 0 O 0 0 E . O I , 0 . 5 0 0 0 0 E . 0 1 I AMPL I TUDE ERROR ! E R E L " " 0 . i 6 6 8 9 E O i l ' I A B > O^loiVoE.OO XO' -WI6ji.MCly« »v''KS SUM-~"6VO ERROR- O.O X P N U H . O . S 3 2 9 B E . O I X P C A X N - 0 . 5 5 4 U 5 E . 0 1 >PEAKA. 0 . 5 1 0 8 5 E . O I X C A - 0 . 5 5 4 I 0 E . 0 8 T . N U N - O . I 2 0 0 0 E - 0 1 T x . 0 . 13333E . -OJ N X - 4 M " A X - » V P N U N * 1 . 4 1 6 0 3 E . 0 1 Y P E A K N - O ^ Z B E . O l Y P E A K A - 0 . 4 3 1 T 2 E . O I T C A - - 0 . 4 8 2 0 7 E . 0 6 . T Y N U 4 - 0 . 5 7 O O 0 E - 0 1 T Y - 0 . 6 0 0 0 3 E - 0 1 N Y - 4 N Y N A X-45 ALPHA. t.WVi^riin7-a^m^r&*^&*^i*i oUfWo.iJoooeoi LAHBOA- O.45OOOE.O2 « < « A O D I T I V E C O R R E C T I O N - EOU IV AL ENT I INE A<t I ZAT ION S O L U T I O N OF l O I K A - W L I F R R A . S Y S T E N . » I TH..PREY_CAR.«V I N G . i A P AC j JJf->>>>2 -Fig. 4.8 VGW Model Corrected Equivalent Linearized Solution for (x ,yo)=(6,3;5) 125. 1 3 . 0 0 0 1 . . . . I I • * * • ! * SOLOT/Dtf tl%%% - ANALVTICAL S G L U T 1 0 N • » H H F R 1 C A L S O U ' " " " ' " " ' C m " 0 H 1 0 B ° ' H M M * . 0 . 1 T 5 . 0 1 . 0 . « T 4 . O . i O O O O E . 0 2 0 » « . C O O O O f O * O H . . . « > . « . . 0 . £ « . l " « . 0 » , 0 . * » 0 0 t . « . _ • - ^ ^ . W K r ™ c ^ RREVCA.R„NO CAR.C, » » > ^ F i g . 4.9 VGW Model Corrected Equivalent L i n e a r i z a t i o n Solution for (x o >y o)-(6,3.5) 126. 0 . 0 0.330 0.060 0.0<*0 0 . 1 2 0 0.1*0 0,1R0 0 . 7 1 0 . ..0.?-.0 0 , ? T 0 . 0 . 3 0 0 , . —r~. 1 . • T i . i 1-——-.- "*i 1 . . 1" i i i i • i i t n nnn I • 9 . 0 0 0 - 9 . 0 0 0 : y M 8 . 3 0 0 - : 8 . 0 0 0 -• 7 . 0 0 0 - T . 0 0 0 6 . 0 0 0 - - 6 . 0 0 0 5 . 0 O 0 - „ A H A L V r f C i i L ***** I 8 . 0 0 0 / J O L y T ^ ^ x ^ 4 . 0 0 0 -4 . 0 0 0 9 . 0 0 0 3 .000 : • '- 2 . 0 0 0 1 . 0 0 0 1 . 0 0 0 0 . 0 0 | i 0 . 0 3 0 0 : 0 6 0 . 0 , 0 90 0 . 1 2 0 0 . 1 5 0 0 . 1 9 0 .0.210 0 . 2 4 0 0_..2_?0_ .1 0 . 0 11111 . A N A L Y T I C A L SOLUT ION • NUHER IC AL S O L U T I O N ' » ' • • PO INTS COMMON 1 0 BOTH S O L U T I O N S _ X 3 . 0 . 6 0 0 0 0 E . 0 1 T O - 0 . 3 5 3 0 3 E . 0 1 S I NIHIL A R IT T AT I X . » l - l 0 . 5 0 0 0 0 C 0 1 . O . 5 0 0 0 O E . O 1 I " E R R O R - 0 . " 6 - ' l AHPLITU. IE" ERROR I E V R E L - ~ 6 . 2 6 6 8 9 E . b l i E Y A 8 S - O . l O l i O E . O O F R E Q U E N C Y * ^ . 0.0 N U M . 0 . 0 Y P N U H . 0 . 4 1 6 0 1 E . 0 1 Y P F A K N - 0 . 4 2 2 2 3 E . Q 1 YPEA<A. 0 . 4 3 3 7 2 E . O I Y C 4 . - 0 . 4 a i 0 7 E . 0 6 TYNUM- 0 . S 7 0 0 0 E - 0 1 TY. 0.600O0E-O1 NY- 4 N Y H A X - 4 3 — A L P H A * 0 . 2 7 S 0 o r . O 3 BETA. 0 . 1 3 0 0 0 E . O 2 04HA. 0 . 5 0 0 O 0 C . 0 2 D E L T A - 0 . 1 0 0 0 0 E . 0 2 LAHBOA. 0 . 4 5 0 0 0 E . 0 2 ^ i<<<t A3"DTT~1 v£"*~t DR R E"C"T I ON -~~6Q~UTV"AL'EN T"~LI NE AR IZ AT I QN"" SOLUT ION" Of LOT AA—V3LT ERAA SYSTEM . ITH PXE Y " C A M V INO C A P A C I TY » > > > Fig. .4.10 VGW Model Corrected Equivalent Linearization Solution for (X q,y Q)=(6,3.5) 127. Y (0) ' p . O I.ton 7 . 0 3 0 3...330. 4 . 0 0 0 5 .000 6.OOO. 7 .000 . . . , — 8 . OOO 9 . .0Q0—_ 10.000 10.0001 10.000 9 . 0 0 0 - 2 1 • ' ' 1 1 1 t » 1 \ - * 1 1 9 . 0 0 0 • B.OOO- ! / 8 . 0 0 0 T.OOO- 1 T . 000 4 6 . 0 0 0 - * l o % - 6.000 5 . 0 0 0 5 . 000 8 X 1 r\ . • \ - ""'V^dao " •V.000 \ • 3 . 000 - 3 . 0 0 0 : \ V —— ^ \ T r-t 1 1 1 l l l l 2.000 - \ 3 2 2 1 2 2 2 2 2 2 - 2 .000 1,000 7 6 5 7 • 7 7 7 7 8 1 . 0 0 0 0 .0 0 . 0 1.000 2.000 3 .000 4 . 0 0 0 5.000 6 . 000 7 . 0 0 0 8 . 000 __?..00P_ I 1 0 . 0 1.000 MOI CONSTANT R E L A T I V E A M P L I T U D E ERROR L O C I GRAPH ERROR X l O . O O O t S I N G U L A R I T Y AT ( X . V ) » ( 0 . 3 0 0 0 0 E * 0 1 * 0 . 3 O O O 0 E » O l l A L P H A - 6 .>>5O0E*<iJ S E T A * 0 . 1 0 0 0 0 f > 0 2 ( JAMA- 0 . 5 0 0 0 0 t * 0 2 U t L T A " O . I O O O O E * 0 2 L A N 8 0 A - O . O O O Q E * Q 2 : « « < A 3 P I T ^ y E _ C 0 R R g C T I 0 N - E Q U I V A L E N T L IN EAR IZ AT ION SOLUT 10.^ OF L 0 T K A - V 3 L T E R R A S Y S T E N MI TH P R E Y C A R R Y ! NG C A P A C I TY_ _>>>>> F i g . 4.11 VGW Model•Corrected Equivalent L i n e a r i z a t i o n Solution Amplitude Error L o c i 128. 0 , J _ _ 5 , 0 0 0 _ _ 1 0 . 0 0 0 . . . . 1 5 . 0 0 0 2 0 . 0 0 0 .. 2 5 . 0 0 0 .... _ 3 0 . 0 0 0 3 P . 0 0 0 4 0 . 0 0 0 4 5 . 0 0 0 . 3 0 . 5 0 . 0 0 0 1 . . . . I I I I . . . . I . . . .I....I - - - - * - - - - • •o.o Bi*-'-i"':[-B"-i--;ii[-a-ayi'" t l l l t - A N A L Y T I C A L SOLUT ION 1 0 . 0 0 0 1 5 . 0 0 0 20.000... ..25.000... __.3.0.000 . . . . . . N U M E R I C A L S O L U T I O N . . . . . • P O I N T S CuMMnN TO BOTH SOLUT IONS X O - 0 . 1 7 5 0 0 E . L 2 Y 0 - 0 . 4 2 5 0 0 E . 0 2 S I N G U L A R I T Y AT 1 X . Y 1 - I 0 . 2 0 0 O 0 E . O 2 , 0 . 3 3 3 3 1 6 . 0 2 1 ^ T R E O U E N C Y . AN- cVilbcst-oTNUMV""c:674i,2E-01 E R R O R - 0 . 2 0 0 8 1 E . 0 2 1 A M P L I T U U E ERKURV E R E L - O . IMO'C.o'S E A B S - - 0 . 5 5 2 6 1 E . 0 1 ..NUM. 0 . 1 0 0 2 7 F . 0 2 X P F A K N - 0.24;3?E.O, X P F A X A . 0 . 2 4 6 4 7 E . Q 2 «f A-0.10497f-P*. TWI- P . W P P f P l 0 . »T W B U l WB-U •!»"» Y P N U M - 0 . 2 2 6 4 2 E . 0 2 Y P E A K N - 0 . 2 4 S . O E . 0 2 Y P E A X A - 0 . 2 B 5 1 1 E . C 2 Y C A - 0 . . 3 2 5 9 E - 0 3 T Y N J M - ..55000E.01 T Y ^ O . » t B 2 E . 0 1 N Y - U NYM»> •«- 0.20000£Jor"x- 0.4jM«^2"«^6Ti0000t»b2"$. d . ' l O O O O t « O t J - 6.2O300E.02 ««l A O O l T l v r CORRECT ION - E Q U I V A L E N T I I NF AR 1 j AT I ON S O I U T I O N OF H Q U I N O ' S "OPf! H ™ l l l f A R fUCMtrlCT » > » • Fig. 4.12 Holling's Model Corrected Equivalent Linearization Solution (x o,y o) = (17.5, 42.5) 129. I I I I . 0 . 0 5 . 0 0 0 1 0 . 0 0 0 1 5 . 0 0 0 2 0 . 0 0 0 2 5 . 0 0 0 . 3 0 . 0 0 0 3 > . 0 0 0 4 0 . 0 0 0 . . 4 5 . 0 0 0 5 0 . 0 0 0 5 0 . 0 0 0 1 5 0 . 0 0 0 45.00(h 4 5 . 0 0 0 4 0 . 0 0 0 - 4 0 . 0 0 0 • : 3 5 . 0 0 0 - 3 5 . 0 0 0 3 0 . 0 0 0 - 3 0 . 0 0 0 j V-» WO ht£fUCAL j \ SOLUTION : 2 3 . 0 0 0 - 2 5 . 0 0 0 : I \ \ r A < 2 0 . 0 0 0 -Ik \ V V» / }y*^y^ v t « i f > 4 i H t i 1 1 1 1 r 1 2 0 . 0 0 0 \ \ ~ 1 5 . 0 0 0 -:\ | xMLHTrc*l- - 1 5 . 0 0 0 •\ jf SOL-Orf OA/ 1 0 . 0 0 0 - 1 0 . 0 0 0 3 . 0 0 0 3 . 0 0 0 0 . 0 1 0 . 0 olo ' frlcco ' lolloOO...' . 1 5 ^ 0 0 0 2Q^00Q_' 2 5 ^ 0 0 0 3 0 ^ J 3 0 ' B J . O O O 4 0 . 0 0 0 4 5 . 0 0 0 , 3 0 . 0 0 0 , ,. T IME S E C . » » m • A N A L Y T I C A L SOLUT ION » « « » * • N U H E B I C A L S O L U T I O N • PO INTS C H H O N TO BOTH S J L U T I O N S / X 0 » 0 . 1 7 S C 0 E + 0 2 Y O - 0 . * 2 5 0 0 E * 0 2 S I N G U L A R I T Y AT U . Y I - l 0 . 2 0 0 0 Q E * 0 2 , d . 3 3 i 3 3 E + 0 2 J " ~ " T R I Q U E N C Y I AN-'"oVai06sE"-or~NUMV"0.674ft2E-0 ' l E R R Q A " 0 . 2 0 0 8 1 6 * 0 2 * AHPL 1 TUOE E R R U R l ~ E X * i E L « - Q . 1 8 4 0 *EVC2 - 1 E X A b j »>^oT5^26 l E ^ O l , J tPNUH* Q . 3 0 0 Z 7 E « C 2 X P L A K N - 0 . 2 * S 3 2 E * q 2 , XP jEAKA - Q . 2 4 6 - . 7 E * 0 2 X C A — 0 . 3 C ( . 9 T E - 0 f r T X N j M " O.eOg&C. E»01 T X * 0 . 8 7 3 0 9 ^ * 0 1 N X M 1 N X H A X . 1 4 3 ft* 0 . 2 0 0 0 0 E * 0 1 K - 0 . 4 5 C 0 0 E * O 2 A - 0 . 1 0 0 0 0 E + 0 2 S - 0 . 1 0 0 0 0 E * 0 1 J - 0 . 2 0 0 0 0 6 * 0 2 < < < « ADO IT Tit" COR ft EC T I O N - E Q g i VAL€NT~L1 h S Y R W AT IGA~ SOL UTICW' OF ' W l U N G * S W00Ei.~k«VTH T l N f A R f iUo^ ENCV7>>>> 4.13 Holling's Model Corrected Equivalent Linearization Solution for (x Q,y o) = (17.5, 42.5) 130. Y I T I - 0 . 0 5 . 0 0 0 1 0 . 0 0 0 . . . 1 5 . 0 0 0 2 0 . 0 0 0 . 2 5 . 0 0 0 . 3 0 . 0 0 0 3 5 . 0 0 0 4 0 . 0 0 0 . 4 5 . 0 0 0 5 0 . 0 0 0 5U.UOU 5 0 . 0 0 0 : -45.000 45.000 > t * * 40.000 40.000 :Vt AMALfTUAL '• 35.000-•—W f-4-—\\ >VWL*<C . *»» . I 35.000 30.000 30.000 25.000- 75.000 20.000 20.000 ' \ 1J.300- 15.000 10.000- . 10.000 3.000 5.000 0.0 0.0 0 , 0 5 . 0 0 0 10.000,. . . . 13 ,000 2 0 . 0 0 0 „ 2 5 . 0 0 0 __ 3 0 . 0 3 0 3 » . 0 0 C i J ) . 3 0 0 _ 4 5 . 0 0 0 „ 5 0 . 0 0 0 T I H e S E C . ttltt . A N A L Y T I C A L S O L U T H 1 N - " " • N U M E R I C A L S O L U T I O N • P U I N T S COMMON T O B O T H S J L U T I O N S X O * 0 . 1 7 5 0 0 6 * 0 2 VO- 0 . < . 2 5 J 0 t * 0 2 S I N G U L A R I T Y A T ( X , Y 1 - I 0 . 2 0 0 0 0 E * 0 2 , 0 . 3 33 j J E * 0 2 i ' f R E U U l N C V i AN-'oTsVOO^yi^lj^ E R R O R - 0 . 2 0 C t t l E * 0 2 * AMPL I TUDE E k R d R ~ E V R E L « 6 . A S i V i E »6l * EYAfaS" -dTf939fE »6l " \ , T P N U M " O t ' ; 2 6'.2E » 0 2 l YPEAH.N* 0 .24 3 f t Q £ » Q 2 Y P E A K A - 0 . 2 8 5 1 I E » 0 2 Y C A - - Q . 6 3 2 5 9 C - 0 3 T Y N j H - 0 . S 5 C 0 0 Q 0 1 T Y - 0 . 5 8 6 8 2 E + 0 1 N Y - U N Y M A X - 2 2 4 A- 0 . 2 0 0 0 0 E * C 1 K- O . * . 5 O O 0 t » O 2 A- 0 . 1 0 0 0 0 E * 0 2 S - 0 . 1 0 0 0 0 E * 0 1 J * 0 « 2 0 0 0 0 £ * 0 2 «<<<"' AOO TT I~VE —COR RECT VON - E ^ i ^ A L l N f ~ L ~ r N E AR FZAY 1 O N S O L J t lOV OF 'HOLL I N C S N O D E * / * I TH L I N E A R F R E Q U E N C Y " >>>>"> ~ ~ ~ 4.14 Holling's Model Corrected Equivalent L i n e a r i z a t i o n Solution for : (x Q,y o) = (17.5, 42.5) ... 131. O.oJ ».000_ _.0.000 __15.000 20.000 . . 25.000 JO.OJO 9>;000 40.000 .45.000. .50.000 50.0C0I | . . . . I . . . . I I I I.... I I. . . . I AM,iV| £P\CftL 10.000 0.0 I.. 0.0 11111 . A N A L Y T I C A L SOLUT ION 1 0 . 0 0 0 . 1 5 . 0 0 0 _ ... I 1 I.... 3 5 . 0 0 0 _ 4 0 . 0 0 0 X i t I I .... I 1 0.0 4 5 . 0 0 0 5 0 . 0 0 0 NUMER ICAL S O L U T I O N Y A K A . PO INTS CJMHON TO hOTH SOLUT IONS A O - O . 3 0 0 0 0 E . 0 2 Y O - 0 . 3 0 0 0 0 E . 02 S I N G U L A R I T Y AT ( X , Y | . | 0 . 2 0 0 0 0 E . 0 2 . 0 . 3 3 3 3 3 t . 0 2 » " fSFoi jENCYl A N i " 0 T t 9 2 0 6 l ^ l _ N U M " . ~ 6 ^ T 9 6 5 t i E ' - 0 1 E R R O R — 0 . 5 4 t « E » 0 0 « A H P L 1 TUOE ERROR i ~ E R E L . - O . l l 7 I S c * 0 2 X E A B S " 0 . 2 7 5 5 3 E . 0 1 X P N U N . 0 . 1 A 6 4 3 E . 0 2 X P E A X N - Q . 1 4 6 S P F . 0 2 X P F A K A . 0 . 1 4 7 2 8 E . Q 2 X C A — 0 • 2 4 0 3 8 E - 0 5 T X N j N . 0 . 7 5 0 0 0 E - 0 1 T X " 0 , t30m>E .QI NX.-11 NXHAA - 193 Y P N U H - 0 . 3 0 9 7 1 E - 0 2 Y P E A K N - 0 . 3 0 1 T 9 E . 0 2 Y P E A K A - 0 . 3 0 3 0 0 E . 0 2 Y C A - — 0 . 7 1 8 4 9 E - 0 7 T T N u M - 0 . 1 0 5 0 0 E . 0 2 T Y - 0 . 1 0 0 0 5 E . 0 2 . N Y - 1 1 N Y H A X - 1 3 0 ft- 0.20000EV61 A. C . 4 5 0 6 0 E . O 2 A - 0 . 1 0 0 C 3 E . 0 2 S> 0 . 1 0 0 0 3 E . 0 1 J- 0 . 2QOOOE.02 < * < « A O O I T I V F C O R R E C T I O N - EQU IVALENT L INEAR IZAT1 CN S O L U T I O N OF H C l L l N G ' S HOOFL XI TH L I N E A R FREQUENCY » > » Fig. 4.15 Holling's Model Corrected Equivalent Linearization Solution for ( x o , y Q ) = (30,30) 4 5 . 0 0 0 ^ X [~C J 4>5.000 , , * — " » • ' •• 40 .000 'fro.ooo- , - • • — " " 7 3 5 . 0 0 0 -35*000 0.0 I....I. o.o  . . 1 . . . 5.000 H i l l • ANAIYTtf.AL S U L L T 1 0 N ...I....| | « | * • t • I ' , „ M l ) 2 5 . 0 0 0 30.030 35.000 40 .000 45 .000 S£._OJJ0_ T I M E S E C . 10.000 15.000 — • N U M E R I C A L S O L U T I O N . . . . . . p n l N T S C iMMON TO 8 Q T H SOLUT IONS 10- O.30O00E.02 TO- 0 .300006 .02 S I N G U L A R I T Y AT l l . T I ' l 0 . 2 O 0 0 0 E . 0 2 . 0 .33333E.02 I — F i R W C N C T I AN.bTT420SE;drruM.-0-:f96 56E-01 ERROR-0 .S4495E . 0 0 1 A H P L 1 T U D E E R R 0 . T E . R E L - O . 11 T i J i . O J I T«ASS--0. l M O i E . O l « , « • • . o . i 6 6 4 4 r . . n » .PE.KN- o . i 4 6 4 2 r . 0 2 XPEARI- o . . 4 7 2 a E . Q 2 XC A - 0 . 2 4 Q 3 SE-05 «.7«IQ<lf»PI T x - P,»WKf"H M ' - l l H « W M 1 R- 0 .20000E.01 K- 0 .45000E.02 A- 0 .10000E .02 S - 0 .10000E.01 J - 0 .20000E.02 ^ . _ — « < « T S T n v E ~ c Z « i r ^ S ' O L J T I O N Of K I L L I N G ' S N O O E L ' H T M L I N E A R F . £ U U E N C V » » > 4.16 Hplling's Model Corrected Equivalent Linearization Solution for (x o,y o) = (30,30) 133. 0.0 5.0C0 10-000 15.000 _20.CQO 25.000 3 0 . 0 3 0 . . . 3>.0Q0 *0.WOO ... ...45.000 5C.0OO 9 0 . 0 0 0 1 4 5 . 0 0 0 ^ 4 9 . 0 0 0 4 0 . 0 0 0 - 4 0 . 0 0 0 3 5 . 0 0 0 - 3 3 . 0 0 0 // 3 0 . 0 0 0 J 3 0 . 0 0 0 2 5 . 0 0 0 - 2 3 . 0 0 0 2 0 . 0 0 0 -i • 2 0 . 0 0 0 1 5 . 0 0 0 - 1 3 . 0 0 0 1 0 .OOO - 10 .OOO 5 . 0 0 0 5 . 0 0 0 0 . 0 1 0 . 0 0,0 5 . 0 0 0 1 0 . 0 0 0 1 5 . 0 0 0 2 0 . 0 0 0 2 5 . 0 0 0 _ _ 3 0 . 0 0 0 3 » . 0 0 0 4.0.000 4 5 . 0 3 0 5 0 . 3 0 0 " T I M E S E C . %%%%% • A N A L Y T I C A L SOLUT ION • NUMERICAL SOLUTION • POINTS C jMMON TO BOTH S J L U T U N S _ XO- 0 . 3 0 0 0 0 6 * 0 2 TO- O . 3 0 0 0 0 £ * 0 2 S I N G U L A R I T Y AT I X i Y I - l 0 . 2 0 Q 0 0 E * 0 2 t 0 . 3 3 3 3 3 E K 2 I T R E Q U E N C V I AN-"6T7920oT-^r^UM^0.79656E-6l E R R 0 R - - 0 . 3 6 4 9 5 E * 0 0 * A M PLITUOE EJVRORT "E'VREL- 0 . 7 5 8 3 c i E * b i l E Y A b S " 0 . 2 7 5 5 3 E * 0 1 »PNUH» Q . 3 0 9 7 1 £ * 0 2 YPEAKN" 0 . 3 0 1 7 9 E * 0 2 Y P E A K A - 0 . 3 0 3 0 0 6 * 0 2 Y C A — 0 . 7 1 6 4 9 E - 0 7 T Y N j H * 0 . 1 0 5 0 0 6 * 0 2 T Y - 0 . 1 0 0 0 5 E * 0 2 N Y . 1 1 1 V M A X M 3 0 » • 0 . 2 0 0 0 0 E * 0 1 K- 0 . 4 5 0 0 0 E * 0 2 A - 0 . 1 0 0 0 0 E » 0 2 S- 0 . 1 0 0 0 Q E * 0 1 J - 0 . 2 0 0 0 C E + 0 2 ^<<<<VoonTvTYa^ SOLUTION'OF H O L L I E S MOOEL K I T H L I N E A R FREQUENCY >>>>> Fig. 4.17 Holling's Model Corrected Equivalent Linearization Solution, for (x o,y Q) = (30,30) ^ V 101 5 0 . 1 0 3 l J5.20Q lu.QOO 15.000 2.Q.000 . . 2 5 . 0 0 0 — 13.000 3 5 . Q 3 0 .40..000 * 5 . 0 0Q 5 0 . 0 0 0 -,.| I....I I I I I I....I 5 0 . 0 0 0 I 2 3 4 5 6 7 9 9 CONSTANT R E L A T I V E AMPL I TUDE E 'tkUH L O C I C » A P H ERRUK X I G . O O O t i I NUULAR1T V AT I X . ¥ | - ( 0 . 2 0 ( 1 0 0 E » 0 2 , 0 . 3 3 3 3 3 F * 0 2 ! H " 0 . 2 0 0 0 0 6 1 - 0 1 0.4 '>00 ' JC»02 A• 0 . 1 0 0 . ) 0 t * - 0 ^ S « 0 . 1 0 U 0 . 1 1 * 0 1 J " 0 . 2 0 0 Q 0 O 0 2 A U D I T I V E C U R k E C M P ^ ^ . ^ > i J V A L . t N T L I VE A-11A11 G-M SOLUT ION O f . H U L L . N S ' S MODEL. h l T H L l N E A f . . F . R C Q U E N C y . . . » » > F i g . 4.18 Holling's Model Corrected Equivalent L i n e a r i z a t i o n Solution Amplitude E r r o r L o c i 135. »<0I / ' 0.0 5.000 10.000 15.. 000 .20.000 25.000 10.000 35.000 40.000 45.Q0JI J3.0O1 5 0 . 0 0 0 3 0 . 0 0 0 4 5 . 0 0 C - 4 5 . 0 0 0 4 0 . 0 0 0 4 0 . 0 0 0 e 3 5 . 0 0 C - S •> 6 « « 3 3 . 0 0 0 6 4 6 6 6 7 7 a « 3 0 . 0 0 0 • 3 0 . 0 0 0 1 V 2 1 2 2 3 3 4 ^ ^ ™ " * ~ < i w . 2 5 . 0 0 0 - ( 3 2 2 1 2 7 2 3 1 / . J 4 . n n n • I ^\ 2 0 . 0 0 0 - + 2 1 / 1 I I 1 1 . V - 2 0 . 0 0 0 I f \ 1. 10% 1 5 . 0 0 0 - 7 3 / ll - 1 5 . 0 0 0 i te-%^J > ^ / 1 0 . 0 0 0 - - 1 0 . 0 0 0 3 X^S^ 2 3 « \  2 ' ~ \^~T*^"C 2 2 3 ^ * ^ ' 3 . 0 0 C - 5.000 0.6 0 0 5 . 0 0 0 1 0 . 0 0 0 1 5 . 0 0 0 2 0 . 0 0 0 2 5 . 0 0 0 3 0 . 0 0 0 3 5 . 0 0 0 4 0 . 0 0 0 . 4 3 . 0 0 0 5 0 0110 X ( 0 ) / CONSTANT R E L A T I V E A H P L I T U O E ERRCR L O C I GRAPH ERROR X 1 0 . 0 0 0 1 S I N G U L A R I T Y AT I X , V I - I 0 . 2 0 0 0 j E » Q 2 , 0 . 1 2 0 0 0 E * 0 2 > R » 0 . 2 0 0 0 0 E * 0 1 K * O . 2 5 0 0 0 E + 0 2 A - 0 . 1 0 0 0 0 E + 0 2 S - 0 . 1 0 0 0 0 E * 0 1 J - 0 . 2 0 0 0 0 6 * 0 2 « « < A O O I T I V E C C R R E C T I O N - E Q U J V A L E N T L .NEAR U A T I O N SQLU.T |QN„ Q F . H Q L l l ' N G ' . S HOOEt, M l T H L INEAR, FREQUENCY » » > . 4.19 Holling's Model Corrected Equivalent L i n e a r i z a t i o n Solution Amplitude E r r o r L o c i 136. as for the linear solutions '(see section 2.10). The amplitude error l o c i for the corrected underdamped case are given in figure 4.20, while the lo c i for the corrected overdamped case are given in figure 4.21. In both cases we observed that wer have an improvement in the size of the amplitude error l o c i , compared with the uncorrected equivalent linearized solutions. Here also we have found the additive correction factor to be a usefull improvement for the solutions of Rosenzweig's model. We now have one remaining model l e f t to study. 4.7 Additive Correction Factor for O'Brien's Model The f i n a l model that we shall apply the additive correction factor to i s O'Brien's model. The correction factors may be computed lin the same manner as outlined for the VGW model. To evaluate the additive correction factor, the corrected solutions were computed for the same parameter choice and ranges of i n i t i a l conditions as for the linear solutions (see section 2.13). The amplitude error l o c i for the corrected underdamped case are given in figure 4.22, while the l o c i for the corrected overdamped case are shown in figure 4.23. In both cases, we find that we have only a very small improvement in the size of the amplitude error loci over the uncorrected equivalent linearized solutions. It should be noted, however, that the uncorrected solutions are useful over a comparatively much wider range of i n i t i a l conditions than the other models previously dicussed, and hence we might not expect that much of an improvement from the correction factors. This completes our evaluation of the additive correction factor as applied to the various ecological models at hand. 137 o ! o ° ' 5 .OHO 1 0 . 0 0 0 . 1 5 . 0 0 0 2 0 . 0 0 0 2 5 . 3 0 0 3 0 . 0 0 0 3 5 . 0 0 0 4 0 . 0 0 0 4 5 . 0 0 0 _ . 5 0 . 0 0 0 . . . 30 .000» 4 5 . 0 0 0 - 5 4 5 . 0 0 0 9 4 3 3 3 9 ^ 4 4 7 4 2 - 4 0 . 0 0 0 7 2 ^ _ 2 ^ 4 4 5 6 7 6 . -( : 3 5 . 0 0 0 - 7 o * ' • 3 5 . 0 0 0 / \ 2 1 l 1_ 1 — 2 \ 2 3 4 = 3 0 . 0 0 0 - / / 1 A 1 " 3 0 . 0 0 0 /•/ \ ' 3 : _2i.noii • n . . . . . 2 0 . 0 0 0 -- t-j-- , " - / 4 4 5 6 - 2 0 , 0 0 0 • • 1 3 . 0 0 0 - 1 9 . 0 0 0 ' . i n . n o o I / 5 . 0 0 0 I- 9 . 0 0 0 0 . 0 1 0 . 0 CUNSTAN I RE LATIV F AMPL ITUDE ERROR L O C I GRAPH ERROR X 1 0 . 0 0 0 1 S I N G U L A R I T Y AT I X . V I - I 0 . 2 0 0 0 3 E . 0 2 . 0 . 2 3 T O O E . 0 2 I 0 . 2 0 0 0 0 E . 0 1 X- 0 . 4 5 0 0 0 E . 0 2 8 - O . l O O O O E . O l C - 0 . 1 0 O 0 0 E . 0 0 S- 0 . I 0 0 0 0 C . 0 1 J " 0 . 2 0 O 0 0 E . 0 2 « < S < _ 4 J O I T l Y C . . . C O » S . F . C T . 1 0 N _ ^ . E Q U . I V A i r i I . L l N F A R U A T I O N S O L U T I O N OF* ROSENZHE I G * S NOOEL U I T H L I N E A R FREQUENCY — . OJADRAT I C . . E » F 0 « . 7 . I _ > : F i g . 4.20 Rosenzweig's Model Corrected Equivalent L i n e a r i z a t i o n Solution Amplitude Err o r Loci '138. -jo-sbi]0—;— 3:000-r-10;000 f ' 5 ; ) 0 0 r " ; o u o , « ; < > ° o - . JO.O»O 35.000 .10.000 «.ooo..._.5o.ooo —S 9 9 9 9 9 J_ 1 2 2 2 0.0 I....I....!.... 1. U 4 0 . 10.000 n - . i n n i 0 . 000._ . 2 5 . 0 0 0 3 0 . 0 0 0 3 5 . 0 3 0 9 0 . 0 0 0 4 5 . 0 0 0 I Q J l f l O _ XIOI CONSTANT R E L A r I V E A M P L I T U D E ERROR L O C I GRAPH ERROR < 1 0 . 0 0 0 1 S I N G U L A R I T Y A T U . V I ' I 0 . 2 0 0 0 3 E . 0 2 , 0 . 9 2 5 2 1 6 * 0 1 1 R- 0 . 2 0 J O O E . 0 1 A . 0 . 2 5 0 0 J E . J 2 8 - O . 1 0 0 0 0 E . O 1 C . 0 . 1 0 3 O 0 E . 0 O S- 0 . 1 0 0 0 0 E . 0 1 J . 0 . 2 0 0 0 0 E . 0 2 : . _ « « S _ . A 3 0 I t l V e _ _ C 0 * R E C I . I O N _ - . . E 0 U I V A l E M . . l I N E A R I 2 « T I O N SOLUT ION OF. A O S E N 2 * 1 0 - S MOOEL » I I H L I N E A R FREQUENCY. - - . . O U A D R A T I C EON F[ 4.21 Rosenzweig's Model Corrected Equivalent Linearization Solution Amplitude Error Loci •> • . . 139. O.QZ 1.000 2 .000 3.000 A . 0 0 0 _ .5 .000 6.0C0 7.000. . 1 0 . 0 0 0 1 . . . . I . . . . I . . . . I - • • • I - - " I 1 • • * * ' • • fl.000 9.000 iO.OOO V T . I . . . . I . . . . I . . . . I . . . . I 10.000 CONSTANT R E L A T I V E AMPL ITUDE ERRCR L O C I GRAPH ERROR » 1 0 . 0 0 0 . S I N G U L A R I T Y AT ( X . Y l . l 0 . 5 3 0 0 J E . 0 1 . O . t O C O O E . Q l l R. 0 . 2 0 0 0 0 E . 0 1 A' O.10OO0E.O2 S . 0 . 2 0 0 0 0 E . 0 1 J " 0 . 5 0 0 0 0 E . 0 1 _«<«_JIODIT|VE..CO.RRECT|ON - _ E 0 U I »ALENT_L 1 N E A R N E 0 S O L U T I O N Of. O'.BRlEVS N 3 0 E L -.MITH.LINEAR, fPAOUENCTJ>>>>._ F i g . 4.22 O'Brien's Model Corrected Equivalent L i n e a r i z a t i o n Solution Amplitude Er r o r L o c i 140. 0 0 0 7 . 0 0 0 8 . 0 0 0 9 . 0 0 0 _ _ _ 1 0 , 0*«_ ...0._2,...-. 3 . 0 0 0 . „ ^ . 0 0 0 . . , . 0 0 0 6 . 0 0 0 . . ; O O O r f TtHoooi I 1 1 1 1 1 — - 1 1 1 * ' l . - . - i . -2 2 2 2 2 2 Z -I 1 1 1 1 -1 0 % K ~T~ 2 " 2 2. 2 » « » 2 2 2 2 5 5 5 5 9 9 * » » 6 6 6 6 6 6 6 6 6 6 . 7 7 7 7 I _ I I 1 1 1 1 1 1 1 1  - - - - " . 0 0 0 LO.OOQ CONSTANT R E L A T I V E A H P L I T U D E ERROR L O C I GRAPH ERROR « 1 0 . 0 0 0 1 S I N G U L A R I T Y AT I X . V I . I 0 . 5 0 0 0 0 E . O I , 0 . 6 0 0 0 0 E . O U R . 0 . 2 0 0 0 0 E . 0 1 A . O . I 0 0 0 O E . 0 2 S - O . IOOOOE .OO J' 0 . 5 0 0 0 0 E . 0 1 _ « « < _ A 0 0 I T I V E . C n . R . E C T . . O N . ^ . E 0 O I V A L E N , _ L . N E . R . 2 E O S O L U T I O N OP O ' . R . E N - S NOOEL - M l TH . L I N E A R 4.23 O.'Brien's Model Equivalent Linearization Solution Amplitude Error Loci 141. 4.8 Summary In this chapter, we have developed an additive correction factor a refinement to the method of equivalent linearization, and i t has been applied to the four ecological models discussed in chapter three. This refinement extends the range of i n i t i a l conditions over which the solutions developed in chapter three are useful by reducing the error of one peak of the solution, which, in turn reduces the overall solution error. The additive correction factor does not affect the frequency of the solution. The additive correction factor increases our understanding of the models behaviour, and the effects of the various model parameters and i n i t i a l conditions on the peaks of the solution o s c i l l i t i o n s . 142. CHAPTER FIVE CONCLUSIONS In t h i s t h e s i s , we have developed and evaluated several d i f f e r e n t a n a l y t i c a l techniques for obtaining l i t e r a l parameter approximate solutions for pairs of f i r s t order, coupled nonlinear d i f f e r e n t i a l equations that model e c o l o g i c a l systems. These solutions permit i n s i g h t i n t o the system dynamics without r e s o r t i n g to numerical computer studies. In a l l the cases considered, the solutions developed were l o c a l to the s i n g u l a r i t y of i n t e r e s t i n the model under consideration. A l l the solutions obtained were u s e f u l l over a range of i n i t i a l conditions l a r g e r than a small e region about the singular point, contrary to what one would normally expect for d i f f e r e n t i a l equations that are dominated by n o n l i n e a r i t i e s . We began by applying the c l a s s i c a l R i t z method for dealing with conservative systems (see chapter one) to the f i r s t e c o l o g i c a l model ever proposed, the Lotka-Volterra model. The solutions obtained by t h i s method were found i d e n t i c a l to those obtained by l i n e a r i z i n g about the s i n g u l a r i t y . The r e s u l t s obtained were'.encouraging, and i t was observed that the amplitude errors were the l i m i t i n g the range of i n i t i a l conditions over which the solutions were useable. These r e s u l t s motivated the develop-ment of a second method of approximating the amplitudes of the s o l u t i o n s . The second method improved the amplitude errors greatly, so that the l i m i t i n g f a c t o r f or these solutions became the frequency errors;.'incurred f or large i n i t i a l conditions. The prototype study of the conservative Lotka-Volterra system was followed by studies of nonconservative systems that incorporated more b i o l o g i c a l phenomenon than contained i n the Lotka-Volterra model. In chapter two, the models of Volterra-Gause-Witt, H o l l i n g , Rosenzweig, and O'Brien 143. were introduced. As a preliminary means of understanding these models, the c l a s s i c a l p r i n c i p l e of l i n e a r i z a t i o n was applied to each model. Again i t was found that these l i t e r a l parameter l i n e a r solutions were useful over a large range of i n i t i a l conditions; much more than the e region about the s i n g u l a r i t y that one would normally expect. I t was also observed that the amplitude er r o r was the l i m i t i n g factor i n determining the range of i n i t i a l conditions for which these solutions were useable. We then had a p r a c t i c a l means of gaining i n s i g h t into the behaviour of the nonconservative e c o l o g i c a l models, without r e s o r t i n g to computer studies. A f t e r completing our preliminary studies of four nonconservative e c o l o g i c a l models, i t was then desired to obtain l i t e r a l parameter solutions that were u s e f u l l over, .a wider range of i n i t i a l conditions than the previously discussed l i n e a r s o l u t i o n s . In chapter three, the method of equivalent l i n e a r i z a t i o n was developed i n d e t a i l and then applied to the previously discussed models. These solutions exhibited an amplitude that had l e s s e r r o r f or given i n i t i a l conditions than the l i n e a r s o l u t i o n s . However, the frequency of the equivalent l i n e a r i z a t i o n solutions had a higher er r o r f than the l i n e a r solutions f or given i n i t i a l conditions. A compromise was made by combining the amplitude of the equivalent l i n e a r i z a t i o n solutions with the frequency of the l i n e a r solutions to obtain superior solutions that possessed the good features of both techniques of s o l u t i o n . Here again, the amplitude er r o r was the l i m i t i n g factor f or determining the u s e f u l l ranges of i n i t i a l conditions for the s o l u t i o n s . The r e s u l t s obtained for the method of equivalent l i n e a r i z a t i o n of nonlinear d i f f e r e n t i a l equations motivated us to develop a method of more accurately computing the amplitudes of the approximate s o l u t i o n s . In chapter four, an additive correction factor for reducing the error.:of one peak of the 144. equivalent l i n e a r i z a t i o n s o l u t i o n was proposed. This additive correction factor consisted of a modified Taylor series used to compute the actual s o l u t i o n peak value, combined with an n'th order exponentially damped ramp, and was designed to a f f e c t only one peak of the s o l u t i o n . Hence the already acceptable frequency of the s o l u t i o n was not i n t e r f e r e d with. This refinement of the method of equivalent l i n e a r i z a t i o n extended the ranges of i n i t i a l conditions over which the solutions were useable. I t was found that f or Rosenzweig's and Holling's models, both the amplitude and frequency errors were s i g n i f i c a n t i n determining the ranges of i n i t i a l conditions for which the solutions were useable. The value of a l l of these solutions i s seen from the fact that we obtain l i t e r a l parameter solutions, which also permits the user to determine which terms dominate a given component of the s o l u t i o n , such as frequency, time constant or singular point, and hence would be s e n s i t i v e to c e r t a i n parameter changes. The user may also f i n d out which terms are n e g l i g i b l e f or a given parameter combinations. In some of the so l u t i o n s , the i n i t i a l conditions appear i n numerous places i n the solutions and i n a nonlinear manner. The user i s thus spared the tedious task of h e u r i s t i c computer s e n s i t i v i t y studies. As a comparative i l l u s t r a t i o n of the improvements gained by the more complex methods developed i n chapter three and four, the 10% amplitude error l o c i for Rosenzweig's model are superimposed for easy assessment i n fig u r e 5.1. Similar comparisons may be made for the other models studies i n this t h e s i s ; In conclusion, we have s u c c e s s f u l l y developed and evaluated some p r a c t i c a l techniques f o r obtaining l i t e r a l parameter approximate solutions for nonlinear d i f f e r e n t i a l equation models of e c o l o g i c a l systems. These 145. Fig. 5.1 Comparison of the 10% Amplitude Error Loci for Rosenzweig's Model for Three Solution Techniques 146. solutions permit the reader to r a p i d l y become f a m i l i a r with the e f f e c t s of the system parameters and i n i t i a l conditions on the model behaviour without re s o r t i n g to the t r a d i t i o n a l numerical computer studies. 147. APPENDIX A ELLIPTIC FUNCTIONS AS APPROXIMATE. SOLUTIONS In this appendix, we w i l l very br i e f l y outline the copious amount of work that was done using e l l i p t i c functions to obtain l i t e r a l parameter/ approximate solutions of nonlinear differential equations. Solutions of the form x(t) = A(t)Cn(cot + 6 x, k) y(t) = B(t)Sn(cot + 9 , k) are postulated, where Cn & Sn are the e l l i p t i c cosine and sine functions, with k being the e l l i p t i c modulus, which may be a time varying quantity. The amplitude functions A(t) & B(t) and the frequency C J , may be functions of k. In a l l the following work, the e l l i p t i c functions were computed by the highly efficient computer oriented algorithms described by King (30). The Lotka-Volterra model (see section 1.3) was selected for a prototype study. The e l l i p t i c functions methods developed by Barkham (2,3 and 4), Soudack''(53 and 54), Lansdowne (31), and Christopher (10) were studied, and motivated us to attempt to extend them to the systems at hand. For the conservative Lotka-Volterra system, the methods reduced to a form of the Ritz method, except using e l l i p t i c functions instead of trigonometric functions. Fi r s t of a l l , the centre singularity of the Lotka-Volterra model was translated to (0,0) and the differential equations were rewritten as a single second order differential equation to obtain: .. 2 • 2 x + cryx + 6ax - <5xx - x /(x + Y/6) = 0 Upon inserting the approximate solutions in the differential equation, we obtain equations for oo and k by matching like powers of e l l i p t i c functions. These equations, are at best, rather clumsy. The parameters cj> and A (which 148. are constants i n t h i s case) were found by. matching i n i t i a l conditions and f i r s t d e r i v a t i v e s . The i n i t i a l major problem encountered with t h i s model was that the parameter k turned out to be greater than unity f o r a l l but the smallest of i n i t i a l conditions, which i s not permitted by the d e f i n i t i o n of the e l l i p t i c functions. In an attempt to resolve t h i s problem, some v a r i a t i o n s i n expanding the terms i n the d i f f e r e n t i a l equation were made so as to hopefully modify the equations for k and eliminate t h i s problem. Unfortunately, t h i s e f f o r t was to l i t t l e a v a i l . Following these attempts at an e l l i p t i c function.:.solution for the Lotka-Volterra system, a s o l u t i o n of the form ( t) = A+B Cn(cot+j),k) X ^ - C+Cn(cot +cj),k) was postulated. This s o l u t i o n would f a c i l i t a t e the previously discussed matching process to obtain the desired equations f o r the so l u t i o n parameters. This y i e l d e d s i x huge equations i n the s i x so l u t i o n parameters, which had to be solved by the computer. The solutions obtained were of next to no improvement over the R i t z method discussed i n chapter one, and c e r t a i n l y not worth a l l the a d d i t i o n a l e f f o r t required. The at temps to develop a p r a c t i c a l e l l i p t i c function s o l u t i o n for the Lotka-Volterra system were then abandoned. The next idea for developing e l l i p t i c function solutions was to modify the Krylov-Boguluibov techniques for nonconservative systems. The VGW model (see section 2.2) was selected for the prototype study. Solutions of the form x = r ( t ) Cn(9(t), k) y = r ( t ) Sn(9(t), k) were proposed. The system and solutions were put i n polar coordinates to 149. obtain: r ( t ) = A2 + y2 0(t) = tn - 1 [ y / x , k ] • • • . r = (xx - yy)/r f l 2~ and 0 = (xy - yx ) / r / r -k y We then used the averaging techniques discussed by L i n (32) to solve f o r r( t ) and 0 ( t ) . To obtain a f i r s t order s o l u t i o n , we have: •='G(1)<V*o> r - r o o where F ( 1 ) ^ 0AQ) = \^ / 4 K[F(r o,(j) o)Cn((t> o,k)+G(r o,({) o)Sn((j> o >k) ]d<f>0 G 7 ( r ,«(....) = ~ / 'A , x [G(r ,cf> )Cn(<f> ,k)-F(r ,<}> )Sn(<j) ,k) ]d<b o o 4K o r Dn(d> ,k) o o o o' To o Tc o o' In the above i n t e g r a l s , we treat r Q and <j>Q as constants. To si m p l i f y the int e g r a l s further, we approximate G'^ (r ,$ Q) as: G ( 1 ) ( t n ' ( ! ) n ) = 7 ~ ^ = / 4 K [ G ( r ,4 )Cn(<j> ,k)-F(r ,<j> )Sn(<j> ,k)]d<}>-° ° 4Kr Dn o o' ro o o o o o o where Dn = ir/2K;(k) A-k2 We s t i l l have to determine the parameter k. The f i r s t method used to obtain k was to minimize the mean square error of the derivatives with respect to k over one period of the s o l u t i o n , 4K, as follows f k [ f e £ 4 K U F ( . , . ) - df(rCnO)] 2+[G(-,') ~ ^ ( r S n 6 ) ] 2}d6] = 0 The equations obtained were extremely messy and the corresponding r e s u l t s 150. were very poor. The problem of k being greater,, than unity was also encountered again. The equations for k (as determined above) did, however, suggest that 2 k would be of the form k = pr(t), with p being a constant. We then assumed this to be the case, and attempted to find an equation for p. This was done by matching i n i t i a l derivatives. The solutions obtained were as good as those obtained by the method of equivalent linearization. However, for some i n i t i a l conditions, the value-of p computed caused k to become greater than unity. This idea for computing k was abandoned for lack of better ideas. As this method of determining e l l i p t i c functions solutions for nonlinear differential equations was going through i t s f i n a l deathbed convulsions, one f i n a l heuristic attempt to find an equation for k was proposed. It was conjectured that the system at hand was equivalent to a 2 3 second order differential equation of the form x + b x + c o x + p x = 0. This differential equation has an e l l i p t i c function solution with k being 2 2 2 2 given by k = pr /2(OJ + pr ) . This form of k was adopted for use in the solution at hand. Again p was obtained by matching i n i t i a l f i r s t derivatives. Unfortunately, for some i n i t i a l conditions, p was negative, which caused the denominator of k to become small and hence k exceeded unity. For the i n i t i a l conditions that this idea worked, the results obtained were only as good as those obtained for the method of equivalent linearization; At this point, the idea of obtaining e l l i p t i c function solutions for the differential equation ecological models was completely abandoned. The problem to be tackled by future researchers i s that of obtaining an 2 equation for k such that k <_ 1 for a l l i n i t i a l conditions. 151. APPENDIX B PRELIMINARY DEVELOPMENTS OF THE ADDITIVE CORRECTION FACTOR In this appendix, we w i l l very bri e f l y outline the chronological attempts to obtain an additive correction factor for the solutions of the method of equivalent linearization. The f i n a l version of the additive correction factor is presented in chapter four. In a l l work, the VGW model (see section 2.2) was used for the prototype study. The f i r s t method evaluated for obtaining the additive correction factors was a modification of the method of equivalent linearization, so that the solutions of the differential equations would contain the desired transient terms. To do this, we started with the desired solutions x(t) = r eCTtcos(cot + EM + A t e ^ X X ' X y(t) = r^e a tsin(cot + 8^) + A^t eay^~ and used Laplace transforms to work backwards to obtain the differential equations that would generate these solutions. We obtained: , « a t . , . a t A ^ c t x = a..nx - a,„y + A e x + (a -a 1 1)tA e x + a 1 0A te y 11 12J x x 11 x 12 y -' I A & _ t _ L tn \ , • cr t . . 0" .t y = a 0 1x + a„„y + A e y + (a - a 0 0 ) t A e y. - a 0 1A te x 21 22 y y 22 y 21 x We then had to determine the parameters of the differential equations. We followed the method outlined in chapter three, and minimized the mean square derivative error J, over one period of the solution, which gives us eight equations for the eight system parameters, which are: J = \ / 2 T T {[F(x,y) - i ] 2 + [G.(x,y) - y] 2} d6 o 152. | f - = \z I21" [F.(x,y)-x][r 1cose - tA e V ^ d9 = 0 | f - = \ - / ? 7 T [F(x,y)-x] [r_sine - tA e°y t ] d6' = 0 da12 2TT 0 2 y |~- = f02lX [G(x,y)-y][r 1cose - t A ^ x d 6 = 0 IZ^" = h C [ G ( x , y ) - y ] [ r 2 s i n 0 - *A^ey. : t] de =-0 JT± = 27 I f[F (x,y)-oc][e x + ( a ^ t e x ] + [ G ( x , y ) - y ] [ - a 2 1 t e a y t ] } dO = 0 ^ = 27 I U F ( x , y ) - x ] [ a 1 2 t e x ] + [G(x,y)-y][e ay t+(a -a ) t e a y t ] } d6 = 0 y ^ laV = h i V U F(x,y)-x][2 t e a x t + ( 0 x - a 1 1 ) t 2 e V ] x + [ G ( x , y ) - y ] [ - a 2 1 t 2 e a x t ] } d6 = 0 3a- 2TT & {[F(x,y)-x] [ a 1 2 t ey ] + [G(x,y)-y][2te gy t + (a y - a 2 2 ) tVy *] } d0 = 0 We make the assumption i n evaluating the i n t e g r a l s f o r the a „ that the transient terms tA^e'^x*' and tA^e ( 7y t may be neglected i n the computations. Thus, the f i r s t four equations reduce to those obtained for the a^.. i n chapter three. To evaluate the remaining four i n t e g r a l s , i t was assumed that the transients were slowly changing and that we could make the approximation e C x t =1. We then obtained four huge nonlinear algebraic equations i n the four unknowns, A , A , a , a . Numerical solutions of these equations x y x y 153. yielded (for some i n i t i a l conditions)"a > 0 and a > 0, which is not x y acceptable as we know the models under analysis are stable. It was then conjectured that the i n i t i a l assumption that e°x t - 1 a t ' • and e y =  1 was invalid. The last four integrals were recomputed without this assumption to yield even more horrendous equations for the unknowns A x > o^, and o However, upon numerical solution of these equations, we obtained reasonable values for a and Q" , but the values for A and A were x y' x y several orders of magnitude to small. At this juncture, i t was decided to try an alternate method for computing the remaining four parameters. By matching i n i t i a l f i r s t and second derivatives of the solution with the original differential equation, we obtain comparatively simple formuae for the parameters. Unfortunately, we run into the problem of ;a x and a^ being greater than zero for some i n i t i a l conditions. For lack of more useful ideas, the whole concept of the modified equivalent linearization differential equation was scrapped. The concept of the peak correction factor was slowly beginning to materialize at this point in time. The f i r s t idea for computing the peak values of the solutions was based on an idea taken from the techniques used to numerically solve differential equations, and was as follows. We rewrite our differential equation in form of an integral equation as x(t) = f*~ F(x,y)dt + X q . To approximate the integral, we use the previously developed equivalent linearization solutions in the integrand, F(x,y). This method accurately predicts the peak values of the solutions i f the peak point is very close to the time origin. This method was not good enough for a l l values of i n i t i a l conditions. The next idea was to approximate the f i r s t fraction of a period of 2 the solution by the polynomial x ( t ) = x + a t + b t , where a and b are chosen 154. so that the f i r s t derivatives of the two solutions match, and the polynomial attains i t s peak value at the time of the actual peak (which i s predicted from the phase of the l i n e a r s o l u t i o n s ) . This polynomial was then used i n the above i n t e g r a l equation. This idea accurately predicted the peaks for most i n i t i a l conditions, but not a l l possible cases. At this point, i t was decided that more terms i n the polynomial were needed to accurately predict the solutions peaks for a l l i n i t i a l conditions. We then began experimenting with three and four term McLaurin expansions. The reader i s now r e f e r r e d back to chapter four f o r the discussion of t h i s idea and the remainder of the development of the additive correction f a c t o r . 155. REFERENCES 1) Ayala, F.J. & Gilpen, M.E. & Ehrenfeld J.C, 1973 Theor. Pop. Bio l . 4 pp. 331-356. 2) Barkham, P.G.D. & Soudack, A.C, 1969 Int. J. Control 4, pp. 377-392. 3) Barkham, P.G.D. & Soudack, A.C, 1970 Int. J . Control 1, pp. 101-114. 4) Barkham, P.G.D. & Soudack, A.C, 1977 Int. J. Control 3, pp. 341-358. 5) Bogoluibov, N.N & Mitropolsky, Y.A., 1961, "Asymptotic Methods in the Theory of Nonlinear Oscillation ", Hindustan Publishing Corp., Delhi. 6) Brauer, F. & Sanchez, D.A., 1975, Theor. Pop. Biol. 1, pp. 12-30. 7) Brauer, F. & Soudack, A.C, & Jarosch, H.S., 1976, Int. J . 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