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Cross-diffusion induced Turing instability for a three species food chain model. (English) Zbl 1243.35093
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 $$\cases u_{1t}=\Delta [(k_{11}+k_{12}u_2)u_1]+u_1(d_1-b_{11}u_1-b_{12}u_2)],\\ u_{2t}=\Delta [(k_{21}u_1+k_{22}+k_{23}u_3)u_2]+u_2(d_2+b_{21}u_1-b_{22}u_2-b_{23}u_3)],\\ u_{3t}=\Delta [(k_{32}u_2+k_{33})u_3]+u_3(d_3+b_{32}u_2-b_{33}u_3)] \endcases $$ 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.

MSC:
35K57Reaction-diffusion equations
92D25Population dynamics (general)
35K51Second-order parabolic systems, initial bondary value problems
35K58Semilinear parabolic equations
35B35Stability of solutions of PDE
35Q92PDEs in connection with biology and other natural sciences
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