Pattern dynamics of a predator-prey system with Ivlev-type functional response (Q6587537)

This paper studies the dynamics and pattern formation of a reaction-diffusion predator-prey model with Ivlev-type functional response and homogeneous Neumann boundary conditions: \N\[\N\left\{ \begin{aligned} &\frac{\partial u}{\partial t} = d_1 \Delta u + u (1-u) - (1-e^{-\gamma u}) v,\quad x\in \Omega, t > 0,\\\N&\frac{\partial v}{\partial t} = d_2 \Delta v - \beta v + \alpha \beta (1 - e^{-\gamma u}) v, \quad x \in \Omega, t > 0,\\\N&\partial_v u = \partial_v v = 0, \quad x \in \partial \Omega, t > 0. \end{aligned} \right. \N\]\NHere, \(\Omega\) is a bounded domain with smooth boundary.\N\NThis study considers the global existence and boundedness of nonnegative solutions and discusses the global stability of nonnegative steady states. Some classical techniques are used in the study, such as comparison principles, the Lyapunov functional method, and energy estimates, as well as the Leray-Schauder degree theory. This study theoretically proves the existence and nonexistence of nonconstant steady states and performs some numerical simulations on a two-dimensional circular domain.\N\NThe results of this study are consistent with the ``activation-inhibition'' mechanism, where the prey is treated as an activator and the predator is treated as an inhibitor.