Persistence and stability for a generalized Leslie-Gower model with stage structure and dispersal (Q1035049)

Summary: A generalized version of the Leslie-Gower predator-prey model that incorporates the prey structure and predator dispersal in two-patch environments is introduced. \[ \begin{aligned} & \dot x_1(t)=\alpha x_2(t)-r_1x_1(t)-\alpha e^{-r_1\tau}x_2(t-\tau),\\ & \dot x_2(t)=\alpha e^{-r_1\tau}x_2(t-\tau)-r_2x_2(t)-r_3x^2_2(t)-\frac{a_1y_1(t)x_2(t)}{x_2(t)+k_1}\,,\\ & \dot y_1(t)=\left(\beta_1 -\frac{a_2y_1(t)}{x_2(t)+k_2}\right) y_1(t)+D_1(y_2(t)-y_1(t)),\\ & \dot y_2(t)=(\beta_2-r_4y_2(t))y_2(t)+D_2(y_1(t)-y_2(t)),\end{aligned}\tag{*} \] where \(x_1(t)\) and \(x_2(t)\) represents the densities of immature and mature individual prey in patch 1 at time \(t\), \(y_i(t)\) represent the densities of immature and mature individual prey in patch 1 at time \(t\), \(y_i(t)\) denote the density of predator species in patch \(i\), \(i=1,2\) at time \(t\), all parameters of (*) are positive constants The focus is on the study of the boundedness of solution, permanence, and extinction of the model. Sufficient conditions for global asymptotic stability of the positive equilibrium are derived by constructing a Lyapunov functional. Numerical simulations are also presented to illustrate our main results.