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The Stability Analysis on a Predator-Prey Model with Allee Effect

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The Stability Analysis on a Predator-Prey Model with Allee Effect International Conference on Advanced Material Science and Environmental Engineering (AMSEE 2016) The Stability Analysis on a Predator-Prey Model with Allee Effect Dongwei Huang, Dongmei Sun* and Jianguo Tan School of Science, Tianjin Polytechnic University, Tianjin, 300387 *Corresponding author Abstract—This paper mainly focuses on the stability of predator- prey model, in which Allee effect is exerted on preys, people’s Where x represents the population density of the prey at capture of predators and preys in the system, and the time t, y stands for the population density of the predator at demonstration of positive equilibrium point’s existence and stability with the help of Routh-Hurwitz stability discriminance. time t, g() x is the growth rates of the prey, is the coefficient Combining with numerical modeling, the stability of each of the predation, e is conversion rate. All coefficients are population’s evolutionary process is also analyzed through positive constants. choosing Allee effect coefficient and capture coefficient as key parameters, which proves that the two coefficients plays a distinct Also, a literature give the predator-prey model, in which influence on the change of condition of the system. This paper has Allee effect is exerted on preys, the model can look like this particular enlightening significance to the analysis of any condition. 1 1 x( t ) [ r (1 k x ) b ( x a ) ] x xy (2) Keywords-allee effect; predator-prey model; Routh-Hurwitz y() t e xy dy stability; capture; positive equilibrium point I. INTRODUCTION where r is the intrinsic rate of increase of the prey, k represents The interaction between predator and prey is the most the environmental capacity. Allee effects can be divided into fundamental intersperse relationship in ecology, which is also two types: strong Allee effect and weak Allee effect. The considered to be the foundation, on which all biologic chains letter b and a represents Allee effect constant, their value range and the whole ecological system can be established. Lotka- reflects the strength of Allee effects. Voltera model is acknowledged as the most classic predator- Consider applying capture for the predator and prey in the prey model, but it is unstable[1]. Therefore, the derivation of the three model. Let , represent the prey capture coefficient interaction between predator and prey is still an important m p subject of biomathematics [2-4]. There are many researches have and predator capture coefficient respectively, these two values been done till now, getting the model more closer to life [5-8]. are both positive. Then the three model is as follows. Allee Effect can be named as Sparse Effect [9]. Allee once 1 1 pointed out that the so-called Allee Effect referred to the fact x( t ) [ r (1 k x ) b ( x a ) ] x xy mx (3) that the aggregation of population is beneficial to their growth y t)( e xy dy py and existence, but an excessive sparse population will definitely lead to the difficulty of mating. Allee Effect exerts more distinct influence on many endangered species, so it is a In the model(3), let us combined d and p , meanwhile, define hot issue for many[10-11]. Meanwhile, for pursuing economic n p d , then we obtain benefits, people often carry out massive capture of the species resource that is with great economic value in ecological 1 1 system, which directly leads to the extinction of population x( t ) [ r (1 k x ) b ( x a ) ] x xy mx seriously.so there exists great practical significance to (4) researches on capture system [12-14]. y t)( e xy ny II. MODELING III. MODEL ANALYSIS The general predator-prey models can be expressed as A The existence of positive equilibrium point. Considering actual meaning of ecological problem, we We x()() t g x xy will discuss in space R{ x|x 0, y 0}. (1) y() t e xy dy Calculating the model (4), we can get four equilibrium * ** point P1 )0,0( , P4 (,) x y , © 2016. The authors - Published by Atlantis Press 105 P( km ar kr4 k ( b a ( m r )) r ( k ( m r ) ar )2 /(2 r ),0) For the equilibrium points, 2 , , P2 (( km ar kr ) /(2r ),0) P(( km ar kr ) /(2r ),0) P( km ar kr4 k ( b a ( m r )) r ( k ( m r ) ar )2 /(2 r ),0) 3 3 where 4k ( b a ( m r )) r ( k ( m r ) ar )2 .At this time the predator does not exist. In this case, calculating the model (5), From calculating the model (4), we can get the necessary we can get that the necessary and sufficient conditions for the and sufficient conditions for the existence of the positive existence of P2 is equilibrium point ** , 2 2 P3 (,) x y b[ ar k ( r m )] /(4 kr ) , bmin [ ar k ( r m )] /(4 kr ), a ( r m ) and a k()/ m r r .The necessary and sufficient conditions for the existence of P is a( r m ) [ ar k ( r m )]2 /(4 kr ) and b<-) n + ae (bb nr + ekmb ()(/ - ekrb e22 k ) 3 a k()/ m r r .We can get In the model(4), the Jacobi matrix at the any equilibrium Theorem 1 If the predator does not exist and only the prey m r x exists, the equilibrium point is instability. If point is 1 , where , P2 K 2 e x e y n 1 , the equilibrium point of the 2 (kr km ar ) /(2r ) n P3 system is local stability. If b[ ar k ( r m )]2 /(4 kr ) and 1 1 1 2 , the equilibrium point and degenerate to b( x a ) ( 2 k r b ( x a ) ) x y a r k() m r P2 P3 1 a point and are high step singular point of the system. The characteristic equation of the Jacobi matrix K is Proof The Jacobi matrix in system (5) is 2 (trK ) detK 0 m r 3 x Where trK m n r , K 2 1 2 0 n 4 , 2 det K mn nr n1 m 2 r 2 1 2 e xy . Where b( x a )1 ( 2 k1 r b ( x a )2 ) x y , x . Then we get two eigenvalues 3 4 According to the real part of characteristic value of characteristic equation, we can come to the conclusion that the 0.5[( trKK ) (tr )2 4(detK ) ] 1,2 theorem said. If b[ ar k ( r m )]2 /(4 kr ) and a r1 k() m r , the equilibrium point P and P degenerate to the point The stability of positive equilibrium point. 2 3 (0.5r1 k ( r m ) ar ,0) .Substituting it into the Jacobi matrix Considering the local stability of equilibrium point at the K leads to .Therefore it is the high step singular point. system (4).Calculating eigenvalue of the Jacobi matrix for 2 K2 0 each equilibrium point, we can get two eigenvalues, and 1 For the equilibrium points P(,) x** y ,the predator and prey .Then we will use two different methods to analyzes its 4 2 are concomitant. In front of this paper we got that the stability. First of all, using Routh–Hurwitz theorem we can necessary and sufficient conditions for the existence of the judge its stability. positive equilibrium point ** 1 1 2 1 1 For the equilibrium point P1(0,0) , the Jacobi matrix in this P4 (,)(, x y n k nr ()()) r m n a b is r m a1 b 0 system is .At this time , its characteristic 1 1 1 K1 b k r( k n )( n b ) . Then, we have conclusion as 0 n follow equations is 2 (m n a1 b r)( m a1 b r) n 0.We Theorem 2 If the conditions can obtain two eigenvalues, 1 n and 1 1 1 1 2 2 1 2 r m a b .Therefore, when b a() r m , the system is b min k r ( k n )( n b ), rn /( k ) hold, then the local stability. But if b a() r m , the system is instability. positive equilibrium point P4 is existent and local stability. In order to make the research process convenient, the Proof We can get the Jacobi matrix of linear approximate transpose work is as follows: t et , y e1 y , r e1 r , h' () x y m x 1 1 1 system in the system (5): K * , 1 , 1 , 1 .Substituting , , , , , b1 e b m1 e m n1 e n t y r b m n y x n into t , y , r , b , m , n separately lead to 1 1 1 1 1 1 Where h( x ) [ r (1 k1 x ) ( x a )1 b ] x .Then according to the Routh-Hurwitz, we have detK* ( x * , y * ) 2 x * y * 0 , ïìxt()=- [ r (1 k--11 x ) -+ bx ( a ) ] x -b xy - mx ï *** * * 1 * 1 * 2 í trKxy ( , ) hx ( ) ( xhx ) ( ) krxab ( ) . ïyt()=-b xy ny îï (5) 106 In order to make the equilibrium point P4 is local stability, If b 1, m 0 , p 0, and there exists Allee effect but no if and only if trK*** ( x , y ) 0 , it said that b k1 r() x * b 2 . capture, so that the time history graph of preys and predators can be got as follows:(Diagram b) Substituting x* 1 n into it and combining with the The graph shows that when there exists Allee effect but no necessary and sufficient conditions for the existence of the P , capture has been carried out, preys can reach a stable state in we got that the equilibrium point is local stability when P4 no time although it has been affected, while predators can also 1 1 1 2 2 1 b min k r ( k n )( n b ), rn /( k ) .
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