A Predator-prey System With Individual Behavior

The aim of this work is to construct a prey-predator model with individual behavior, to analyze the asymptotic behavior of the model and to study the effect of individual behavior on the prey-predator system.We consider a system with one kind of prey and one kind of predator. It is assumed that only prey has individual behavior or only predator has individual behavior, and that each individual of predator use hawk and dove tactics when only predator has individual behavior, and that each individual of prey has two individual behavior: altruistic and selfish.According to the study of P. Auger, we consider two time scales: the fast time scale and the slow time scale. A fast time scale at which predator play the hawk-dove game. A slow time scale at which the growth of the total density of the populations is consider. Consequently, according to the equation t =ετ, the aggregated systems are obtained. It is composed of two parts: a fast part and a slow part. We consider type II response function. First, we consider the situation when the predator's gain G and the cost C satisfy the inequality G > C, i.e., all the predator using the hawk tactics is the evolutional stable strategy. Prom the study, we find that the positive equilibrium is globally stable or there is a stable limit cycle when the positive equilibrium is unstable. After that, we consider the situation when the predator's gain G and the cost C satisfy the inequality G < C, i.e., the predator using both the hawk and dove tactics is the evolutional stable strategy. We find that the equilibrium (K, 0) may be stable, i.e., the prey will tend to its carry campacity and the predator will extinctive. When there are two positive equilibria, we study their stabih'ty and the system's Hopf bifurcation. When there are one positive equilibrium, we study the system's Hopf bifurcation and Bogdanov-Takens bifurcation.Furthermore, we find the versal unfolding in terms of the original parameters in the system. In this way, we know the approximate homoclinic bifurcation curve.Next, we obtain a new prey-predator model which prey has two individual behavior: altruistic behavior and selfish behavior. We define a coefficient of interference m≥0 which relates to the individual behavior of the prey. Then m > 0 for the altruistic prey and m = 0 for the selfish prey. The predator's rate of feeding upon the altruistic prey in this model is bx_c/(1+mx_c) , where bx_c measures the predator's rate without interference and 1/(1+mx_c) measures the effect from the behavioral change of the prey. We consider the effect of the enviorment on the prey-predator system. From the study of the system which prey plays only one individual behavior we obtain the global stability of the boundary equilibria of the prey-predator model. We also study the boundedness and the permanence of the new system.