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Stationary patterns of a prey-predator system with a protection zone and cross-diffusion of fractional type. (English) Zbl 1442.92136

Summary: This paper is concerned with the stationary patterns of a prey-predator model with a protection zone and fractional type cross-diffusion for the prey. It is shown that the fractional type cross-diffusion has negative effects on the survival of the prey when the intrinsic growth rate of the predator is positive. Moreover, our mathematical analysis shows that, compared with the results obtained in [K. Oeda, J. Differ. Equations 250, No. 10, 3988–4009 (2011; Zbl 1210.35266); Adv. Math. Sci. Appl. 22, No. 2, 501–520 (2012; Zbl 1300.92084)], the large cross-diffusion coefficient and large growth rate of the predator species have some essentially different effects on the profiles of the solutions.

MSC:

92D25 Population dynamics (general)
35Q92 PDEs in connection with biology, chemistry and other natural sciences
35B32 Bifurcations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35K57 Reaction-diffusion equations

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References:

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