Bifurcation Scenarios of Some Modified Predator Prey Nonlinear Systems

Transcription

1 Bifuration Senarios of Some Modified Predator Prey Nonlinear Systems I. Kusbeyzi*. A. Hainliyan** *Department of Information Systems and Tehnologies, Yeditepe University, 755 Istanbul, TURKEY (Tel: ; ** Department of Information Systems and Tehnologies, Yeditepe University, 755 Istanbul, TURKEY (Tel: ; Abstrat: Bifuration senarios in a series of predator prey models are studied with partiular emphasis on a form inluding a ubi interation that is shown to introdue additional stability in a simple way suggestey Nutku. Eigenvalues of the linearized analysis about equilibrium points and numerial analysis of bifuration using the MatLab pakage MATCONT, inluding the instability introduey this ubi term and the subsequent Hopf bifuration for the proposed model inluding the ubi interation are presented. Lyapunov stability for this ubi model is also studied. The behaviour is omparable to that observed in forms with frational nonmonotoni terms ustomarily used in the literature. Keywords: Asymptoti stability, Dynami models, Dynami stability, Nonlinear systems, Stability Analysis. 1. INTRODUCTION The predator prey problem attempts to model the relationship in the populations of different speies that share the same environment and some of the speies (predators) prey on the others. The prey is assumed to exhibit linear growth given by a positive. Predator speies onsume preys with a nonlinear interation with another set of s that determine the rate of ompetition between predators. The natural death rate of the predator is assumed to be linear and given by a negative. One of the earliest implementations, the Lotka-Volterra model serves as a starting point of more advaned models in the analysis of population namis. Beause of its stability problems, the model and its generalizations serve as a popular objet of stability analysis that has reently gained muh attention. To understand the behavior of a nonlinear system one has to also analyze the existene and stability of equilibrium points whih hange. Changes in the number and stability of equilibrium points as s are varied lead to bifuration. Numerial methods are usually employed to perform this analysis. [Ghosh] A semi-perturbative analyti approah, an alternative to the present stu, is providey the normal form method. [Chen] Well known generalizations of the Lotka-Volterra model inlude the addition of polynomial interations [Cairo], non monotoni response funtions [Broer] [Zhu], time delayed [Xiao] and diffusion effeted time delayed [Yan] non monotoni interations. Bifuration properties of some of these and a generalization of the model with a ubi interation term that is shown to introdue additional stability in a simple way are studied in this work.. THE LOTKA VOLTERRA MODEL The lassi mathematial model of predator prey systems was first developed in 195 by the Amerian biophysiist Alfred Lotka ( ) and Italian mathematiian Vito Volterra ( ) to analyze the yli hanges in the populations of sharks and the fish eaten by sharks in the Adriati Sea. It is given by ax bxy y + y Here, a, b,, d > 0 are s desribed above, x and y are the prey and predator populations. The equilibrium points a are ( x 0, y0 ) (0,0) and (, ). Eigenvalues assoiated with the equilibrium point x, y ) (0,0) is {a,-} and ( 0 0 a eigenvalues assoiated with the equilibrium point (, ) are { i a, i a}. Sine a and are always greater than zero, the origin is a saddle point. The eigenvalues for the a equilibrium point (, ) are purely imaginary, hene all the solutions are periodi whih means the predator and prey populations yle and osillate around this equilibrium. This model is popular as a simple predator-prey system sine it shows yling of predator-prey populations [Pankovi]. However, the model has signifiant problems. Its equilibrium points are not stable; the model does not exhibit strutural

2 stability, the simulated values for the population yle endlessly without settling down. Thus, it fails to desribe realisti predator and prey populations whih typially reah harateristi stea state values. Hene, realisti models should predit a single losed orbit, or perhaps finitely many, but not a ontinuous family of neutrally stable yles. For this reason, the model is generalizey adding additional information.. GENERALIZED QUADRATIC LOTKA VOLTERRA MODEL WITH POLYNOMIAL INTERACTION The simplest polynomial generalization involves the addition of a quadrati term [Broer] and the model beomes a( ( y + y ) + y Two additional equilibrium points are introduey this hange. The equilibrium points are ( a + b) a( + d) ( x 0, y0) (0,0), (1,0), (, ) and (0, 1). Eigenvalues assoiated with the equilibrium point ) (0,0) are {a,-} so that this is a saddle node. Eigenvalues assoiated with the equilibrium point (1,0) are {-a,-+d}. If d> this equilibrium is a saddle node, otherwise it is a stable equilibrium. The eigenvalues assoiated with the ( a + b) a( + d) equilibrium point (, ) form a omplex onjugates pair and the orresponding eigenvetors are omplex onjugates of one another. If a + ab a + ad < 0 then this equilibrium is stable. ( a bd) Eigenvalues assoiated equilibrium (0,-1) are {a+b,}. This equilibrium is unstable. Hene, this generalization an introdue one or two stable equilibrium points in the linearized approximation for appropriate values of the s. Sine the stability of the equilibrium points an hange with the values, bifuration may, in priniple, our.. GENERALIZED QUADRATIC LOTKA-VOLTERRA MODEL WITH NONPOLYNOMIAL INTERACTION As we add more s we add to a nonlinear system, we will get additional bifuration points. Thus, the model beomes loser to the real life problem. For this reason, modified Predator Prey multi systems with non monotoni response funtions are studied. Here is the simplest generalization of the quadrati model. x( a yp( y y + yq( mx P ( 1+ βx + αx, α 0, > 0, 0, β > Q ( P(, > 0 a The equilibrium points are ( x 0, y0) (0,0),(,0),(0, ) and there are further 5 nontrivial equilibrium points. Eigenvalues assoiated with the equilibrium point ) (0,0) are { a, }, the origin is a saddle point sine a and are nonnegative. Eigenvalues assoiated with the a equilibrium point (,0 ) are { a, + }, if a α > this equilibrium is a stable node. a α Eigenvalues assoiated to equilibrium point (0, ) are m {, a + }. This equilibrium is learly unstable. 5. GENERALIZED CUBIC LOTKA-VOLTERRA MODEL WITH POLYNOMIAL INTERACTION Nutku has proposed a generalization where a ubi rather than a quadrati interation is involved. This is an interation that has not been studieefore. It is of interest, sine the non interating parts for both the prey and the predator would undergo pithfork bifuration rather than transritial bifuration along opposite diretions for the predator and prey. a( ( y + y ) + y The equilibrium points are x, y ) (0,0), (1,0), (-1,0), ( 0 0 (0, i),(0,-i) and additionally roots of two ubi equations that produe two real and two omplex onjugate pairs of equations. Eigenvalues assoiated with the equilibrium point ) (0,0) are {a,-} and this is again a saddle point. Eigenvalues assoiated with the equilibrium point (0, -i) are {a+bi,} and it is unstable. The eigenvalues assoiated with the equilibrium point (0,i) are {a-bi,} and it is either a saddle node or unstable. The linearized stability of the eigenvalues assoiated to those equilibrium points that orrespond to the roots of the ubi equations depend on the values. However the equilibrium points (1, 0) and (-1, 0) have the eigenvalues {-a, --d} and {-a, -+d}. One of them is always stable, the other one is stable if d<. We also note that the normal form expansion may or may not have resonant terms, depending on the values of the s. α

3 6. GENERALIZED CUBIC LOTKA-VOLTERRA MODEL WITH NONPOLYNOMIAL INTERACTION The ubi model suggestey Nutku an also be generalized in like fashion. mx P ( 1+ βx + αx x( a x y y ) yp( + yq(, α 0, > 0, 0, β > Q ( P(, > 0 Equilibrium points are a a ( x0, y0 ) (0,0),(-,0),(,0),(0, i ), (0, i ) and there are other 10 nontrivial equilibrium points. Eigenvalues assoiated with the equilibrium point ) (0,0) are { a, }. Hene the origin is again a saddle point. Eigenvalues a assoiated with the equilibrium point (-,0) { a, } and this equilibrium is (1 + ) learly stable. Eigenvalues assoiated with the equilibrium a point (,0) α are are { a, + }. If >, this equilibrium is stable. Eigenvalues assoiated with the equilibrium point (0, i ) i m are {, a + } and this equilibrium is unstable. Eigenvalues assoiated with the equilibrium point (0, i ) i m are {, a }. If i m a < then this equilibrium is a saddle point, otherwise it is unstable. It is lear that the ubi generalization has stable equilibrium points in the linear approximation with or without additional non monotoni terms. 7. NUMERICAL STUDY OF BIFURCATION IN THE CUBIC MODEL We perform the bifuration analysis of the following example of the ubi predator prey model suggestey Nutku. a( ( y + y ) + bxy For simpliity we fix the s as follows: a 1, b,. There are nine equilibrium points for this hoie and we hoose the starting value ( x, (0.7659,0.9) whih is the only equilibrium points of our system that gives rise to Hopf bifuration, sine the linearized eigenvalues are { i, i}. With the help of the Matlab pakage MATCONT [Dhooge], we an start to visualize the bifuration senario of this system by hoosing the s a,b and as free s. We have the following Hopf anranh points shown in the Table below for a as free. Table 1. Positions in Fig.1 x y a Lya. Coef. Eigenvalues P e+000 { 1.619e 009 ±.10e 005} H Fig. 1. Bifuration diagrams when a is hosen as a free Table. Positions in Fig. X y b Eigenvalues P {-, e-016} BP { ,-5.657e-009} LP

4 This approahes to a negative number when time goes to infinity, so we onlude that generalized ubi model is asymptotially stable. 9. INTEGRABILITY OF LOTKA VOLTERRA MODELS Fig.. Bifuration diagrams when b is hosen as a free x y Lyap. Coef. Table. Positions in Fig. Fig.. Bifuration diagrams when is hosen as a free 8. LYAPUNOV FUNCTION STUDY OF BIFURCATION IN THE CUBIC MODEL We define the following salar funtion V : R R R whih is a positive definite funtion satisfying V ( 0,0) 0, V ( x, > 0. Taking time derivate we have V ( x, α x + βy Eigenvalues 1 0 {-,-.6695e-015} BP e {-.897,-.88e-009} P LP {.781e 009 ± } HP {-, e-015} BP x βy {± } {-.897,-.88e-009} dv αx( a( x x ) bx + βy( ( y y ) + α (ax ax bx + β ( y y + y ) NS LP The main advantage of integrability of systems of namial equations is to obtain information about the behaviour of the system in the long term whih is handley the existene of time invariants i.e. onstants of integration. Aording to Nutku [Nutku] the simple Lotka Volterra Model is defined as and these equations give the seperable differential equation whih is integrated to yield x( a b y( + ) + by log( alog( The generalized quadrati model is and has the first integral where - a - d 1 ( a x a b d + bd x + bd, (a x + ab - a + a ab - a a + a d ab, bd a( - a - b + - d) provided 0 x ax bxy y + y a( ( y + y ) + y 1 y y ( 1+ x +, b - bd - + d d + b d x - b d + bd )/ + ab + abd abd x + a An algebrai integral has not been found for the ubi Lotka Volterra Model. Three dimensional predator-prey Lotka-Volterra system whih is defined as )

5 ax bxy xz + exy + fyz dz gz hxz iyz for a, b,, d, e, f, g, h, i > 0, where a, b, d and e are as in the general Lotka Volterra equations and the other s show the interations of the prey and the predator with the third speies population, an be desribed as Hamiltonian systems. The trajetories in xy-plane an be obtained with the help of the seperable equation that has solution x( a by z) y( d + ex + fz) d log( + ex + fz log( alog( + by + z log( Pankovi, V., Banja, D., Glavatovi, R., Predojevi, M., (006). A Simple Solution of the Lotka Volterra Equations. Xiao, D. (001). Multiple Bifurations in a Delayed Predator Prey System with Nonmonotoni Funtional Response. Journal of Differential Equations, 176, Yan, X.P., Li, W.T. (006) Hopf Bifuration and Global Periodi Solutions in a Delayed Predator Prey System. Applied Mathematis and Computation Zhu,H., S. A. Campbell and G. S. K. Wolkowiz (00). Bifuration Analysis of A Predator-Prey System With Nonmonotoni Funtional Response. SIAM J. APPL. MATH., 6, is also seperable and the form of the solutions an be dz found in like fashion. 10. CONCLUSIONS In this work, we have onfirmed that quadratially self oupled two dimensional Lotka Volterra systems do not possess inherent stability and require the addition of non polynomial, non monotoni terms to indue the stability and more realisti bifuration shemes. As suggestey Nutku, a simpler way of induing non monotoniity and stability is hanging the self oupling from quadrati to ubi. This introdues stability at least in the linearized approximation. The normal form method or numerial simulation oule invoked to stu the behavior of the system near these equilibrium points. The authors thank Professor Yavuz Nutku for suggesting this problem. REFERENCES Broer, H.W., Naudot, V., Roussarie, R., Saleh, K. (005). Bifurations of a Predator Prey Model With Nonmonotoni Response Funtion. C. R. Aad. Si. Paris, Ser. I 1, Cairo, L and Feix, M. R. (199) On the Hamiltonian struture of D ODE possessing an invariant. J. Phys. A Math. Gen., 5, L187-LI9. Chen, J. (005). Bifurations, Normal Forms and their Appliations, PHD Thesis. Dhooge,A., W. Govaerts, Yu.A. Kuznetsov, W. Mestrom, A.M. Riet and B. Sautois (006) MATCONT and CL MATCONT: Continuation toolboxes in matlab, Ghent and Utreht Universities Preprint. Ghosh, D. and A. R. Chowdhury (007). On the Bifuration Pattern and Normal Form in a Modified Predator - Prey Nonlinear System. Journal of Computational and Nonlinear Dynamis,, Nutku, Y. (1990). Hamiltonian struture of the Lotka- Volterra equations. Phys. Lett A, 15, 7-8.