Cross-diffusion induced Turing instability for a three species food chain model
The author studies a three species food chain model in which the third species preys on the second one and the second species preys on the first one. This is modeled by the following strongly coupled reaction-diffusion system \[ \begin{cases} u_{1t}=\Delta [(k_{11}+k_{12}u_2)u_1]+u_1(d_1-b_{11}u_1-b_{12}u_2)],\\ u_{2t}=\Delta [(k_{21}u_1+k_{22}+k_{23}u_3)u_2]+u_2(d_2+b_{21}u_1-b_{22}u_2-b_{23}u_3)],\\ u_{3t}=\Delta [(k_{32}u_2+k_{33})u_3]+u_3(d_3+b_{32}u_2-b_{33}u_3)] \end{cases} \] on \(\Omega\times (0,\infty)\) under the Neumann boundary condition \(\displaystyle {\partial u_1\over \partial n}={\partial u_2\over \partial n}={\partial u_3\over \partial n}=0\) in \(\partial \Omega\times (0,\infty)\) and the initial condition \(u_i(x,0)=u_{i0}(x)\) in \(\Omega\) for \(i=1,2,3\). Here \(n\) is the unit outward normal to \(\partial \Omega\), \(b_{ij}, d_i, k_{ii}>0\), and \(k_{ij}\geq 0\) for \(i\neq j\). It is shown that the system has a unique positive equilibrium that is globally asymptotically stable for the corresponding ODE system. It remains linearly stable for the reaction-diffusion system without cross-diffusion (\(k_{ij}=0\) for \(i\neq j\)). But with increasing cross-diffusion rates \(k_{21}\) and \(k_{32}\), the equilibrium can become unstable. This shows that the instability is not of the classical Turing type, but solely driven by cross-diffusion.
scientific article; zbMATH DE number 6005711
scientific article