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Existence and instability of some nontrivial steady states for the SKT competition model with large cross diffusion

  • * Corresponding author: Yaping Wu

    * Corresponding author: Yaping Wu
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  • This paper is concerned with the existence and stability of nontrivial positive steady states of Shigesada-Kawasaki-Teramoto competition model with cross diffusion under zero Neumann boundary condition. By applying the special perturbation argument based on the Lyapunov-Schmidt reduction method, we obtain the existence and the detailed asymptotic behavior of two branches of nontrivial large positive steady states for the specific shadow system when the random diffusion rate of one species is near some critical value. Further by applying the detailed spectral analysis with the special perturbation argument, we prove the spectral instability of the two local branches of nontrivial positive steady states for the limiting system. Finally, we prove the existence and instability of the two branches of nontrivial positive steady states for the original SKT cross-diffusion system when both the cross diffusion rate and random diffusion rate of one species are large enough, while the random diffusion rate of another species is near some critical value.

    Mathematics Subject Classification: Primary: 35B35, 35B40, 35K20; Secondary: 35K57, 35P05.

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  • Figure 1.  (a): $ B<C $ i.e. strong competition; (b): $ B>C $ i.e. weak competition

    Figure 2.  (a): spiky steady state near positive constant steady states $ (u^*, v^*) $ for small $ d_2 $, large enough $ \rho_{12} $ and $ \rho_{12}/d_1 $, (b): large spiky steady state for small $ d_2 $, large enough $ \rho_{12} $ and $ \rho_{12}/d_1 $, (c): positive steady state with singular bifurcation structure when $ d_2 $ is near $ a_2/\pi^2 $, $ \rho_{12} $ and $ \rho_{12}/d_1 $ are large enough

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