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Effect of cross-diffusion in the diffusion prey-predator model with a protection zone

The work is supported by the Natural Science Foundation of China (11271236,11401356,11671243,61672021), the Natural Science Basic Research Plan in Shaanxi Province of China (No.2015JQ1023), the Shaanxi New-star Plan of Science and Technology (No.2015KJXX-21).
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  • In this work, we continue the mathematical study started in [K. Oeda, J. Differential Equations 250 (2011) 3988-4009] on the analytic aspects of the diffusion prey-predator system with a protection zone and cross-diffusion. For small birth rates of two species and large cross-diffusion for the prey, the detailed structure of positive solutions is established by the bifurcation theory and the Lyapunov-Schmidt reduction, which is determined by a finite dimensional limiting system. Moreover, we prove that the stability of positive solutions changes only at every turning point by a spectral analysis for the linearized eigenvalue problem of the limiting system and its perturbation.

    Mathematics Subject Classification: Primary:35J65, 35B32;Secondary:92D25.

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