Predator-prey systems with defense switching and density-suppressed dispersal strategy

Abstract

<abstract><p>In this paper, we consider the following predator-prey system with defense switching mechanism and density-suppressed dispersal strategy</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \begin{cases} u_t = \Delta(d_1(w)u)+\frac{\beta_1 uvw}{u+v}-\alpha_1 u, &amp; x\in \Omega, \; \; t&gt;0, \\ v_t = \Delta(d_2(w)v)+\frac{\beta_2 uvw}{u+v}-\alpha_2 v, &amp; x\in \Omega, \; \; t&gt;0, \\ w_t = \Delta w-\frac{\beta_3 uvw}{u+v}+\sigma w\left(1-\frac{w}{K}\right), &amp; x\in \Omega, \; \; t&gt;0, \\ \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = \frac{\partial w}{\partial \nu} = 0, &amp; x\in\partial\Omega, \; \; t&gt;0, \\ (u, v, w)(x, 0) = (u_0, v_0, w_0)(x), &amp; x\in\Omega, \ \end{cases} \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>where $ \Omega\subset{\mathbb{R}}^2 $ is a bounded domain with smooth boundary. Based on the method of energy estimates and Moser iteration, we establish the existence of global classical solutions with uniform-in-time boundedness. We further prove the global stability of co-existence equilibrium by using the Lyapunov functionals and LaSalle's invariant principle. Finally we conduct linear stability analysis and perform numerical simulations to illustrate that the density-suppressed dispersal may trigger the pattern formation.</p></abstract>