Memory and adaptive behavior in population dynamics: anti-predator behavior as a case study (Q529593)

The aim of the paper is to present an analysis of the asymptotic behavior of a prey-predator system under the superposition that the environment is divided into a number of cells of equal size and in each cell the prey exhibits either a risky or a safe mode of behavior. Some assumptions which characterize the sells are imposed, the simplest of which are the following: \textit{Assumption} 1: At any instant all prey in a given cell are in the same mode of behavior (coloured cell). \textit{Assumption} 2: At any instant prey's mode of behavior can be different in different cells (heterogeneity). \textit{Assumption} 3: Prey in a cell can switch between the safe mode and the risky mode in response to the probability of an individual prey to be attached, that is of the abundance of the predators. \textit{Assumption} 4: There are reactions of prey to other prey behavior (Positive feedback). The final assumption 6, says that prey and predator move freely between and inside the cells according to a diffusive process. Under these assumptions the model takes the form \[ \begin{aligned} \dot{u}&=(b_SP_S(t)+b_R(1-P_S(t))u-(c_SP_S(t)+c_R(1-P_S(t))u^2-(a_SP_S(t)+a_R(1-P_S(t)))uv,\\ \dot{v}&=-dv+e(a_SP_S(t)+a_R(1-P_S(t)))vu, \end{aligned} \] where \(b_S, b_R, c_S, c_R, a_S, a_R\) are constant parameters and \(P_S\) is a probability function given the safe mode when \(P_S(t)=1\) and the risky mode when \(P_S(t)=0.\) These facts are presented in Section 2, where the repelling patches and the flow between the patches is also investigated. The main steady state analysis of the system is presented in Section 3, where it is shown that such a system can have a continuum of equilibrium states. This section closes with several numerical results concerning the global dynamics of some particular cases of the model.