Global stability and Hopf bifurcation on a predator-prey system with diffusion and delays
The author considers a predator-prey system with diffusion and delays. There are two patches in the system and the system is described by the following system of delay differential equations \[ \begin{aligned} \dot{x}_1(t) & = x_1(t)\left(r_1-a_1x_1(t)-\frac{c_1y(t-\tau_1)}{1+kx_1(t)}\right)+\delta (x_2(t)-x_1(t)), \\ \dot{x}_2(t) & = x_2(t)\left(r_2-a_2x_2(t-\tau_2)\right)+\delta(x_1(t)-x_2(t)), \\ \dot{y} (t) & = y(t)\left(-d_1+\frac {c_2x_1(t)}{1+kx_1(t)}-d_2y(t)\right). \end{aligned} \] Persistence and Hopf bifurcation are investigated. First, the author discusses the existence of possible equilibria; second, he provides a sufficient condition for the system to be persistent; third, by constructing Lyapunov functions, the author studies the local and global stability of equilibria; finally, supposing \(\tau_1=\tau_2=\tau\) and taking \(\tau\) as a parameter, the author studies the Hopf bifurcation.
scientific article; zbMATH DE number 5541290