Spatial patterns created by cross-diffusion for a three-species food chain model (Q2922093)

The authors are concerned with the following food chain model of three species NEWLINENEWLINENEWLINE\[NEWLINE\left.\begin{aligned} \frac{\partial u_1}{\partial t}-d_1\Delta u_1=u_1(a_1-b_{11}u_1-b_{12}u_2),\\ \frac{\partial u_2}{\partial t}-d_2\Delta u_2=u_2(-a_2+b_{21}u_1-b_{22}u_2-b_{23}u_3),\\ \frac{\partial u_3}{\partial t}-d_3\Delta (u_3+d_4u_2u_3)=u_3(-a_3+b_{32}u_2-b_{33}u_3),\\ \frac{\partial u_i}{\partial \nu}=0,\quad i=1,2,3,x\in \partial\Omega. \end{aligned}\right\}x\in \Omega,t>0,NEWLINE\]NEWLINE NEWLINENEWLINENEWLINEHere \(\Omega\subset\mathbb{R}^N\) is a bounded domain with smooth boundary. It is said that the model is with cross-diffusion as \(d_4\not=0\) and without cross-diffusion as \(d_4=0\). Under some conditions on coefficients, there exists a positive stationary uniform equilibrium \(\tilde{\mathbf u}=(\tilde{u}_1,\tilde{u}_2,\tilde{u}_3)\). They showed that \(\tilde{\mathbf u}\) is locally asymptotically stable without cross-diffusion, but unstable with cross-diffusion if the parameters satisfy some restrictions. This result reveal the effect of cross-diffusion. Some numerical simulations are given to illustrate that the spatial patterns are spotted patterns for \(d_4\not=0\). They also showed that nonhomogeneous steady state exists for \(d_4\not=0\), and demonstrated that the cross-diffusion creates nonhomogeneous stationary patterns.