Spatial degeneracy vs functional response

Yu-Xia  Wang; Wan-Tong Li

doi:10.3934/dcdsb.2016074

Article Contents

2016, Volume 21, Issue 8: 2811-2837. Doi: 10.3934/dcdsb.2016074

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Spatial degeneracy vs functional response

Yu-Xia Wang^1, and
Wan-Tong Li^2,

1.
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731
2.
School of Mathematics and Statistics, Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, Gansu 730000

Received: September 30, 2015

Revised: May 31, 2016

Published: September 2016

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Abstract

In this paper, we are concerned with a predator-prey model with Beddington-DeAngelis functional response in heterogeneous environment. By the bifurcation theory and some estimates, the global bifurcation of positive stationary solution is shown. Our result shows that new stationary patterns are produced by the spatial degeneracy and the Beddington-DeAngelis functional response. Essentially different from the known results, the two factors generate two critical values for the prey growth rate $\lambda.$ As $\lambda$ crosses each critical value, the positive stationary solution set undergoes a drastic change. In particular, when $\lambda$ is suitably large, the interaction between the two factors yields nonexistence of positive stationary solutions for any $\mu,$ which is in strong contrast to the existence for suitable ranges of $\mu$ corresponding to the Lotka-Volterra or Holling-II functional response. Moreover, which one of the two factors plays a dominating role in the stationary patterns is shown. In addition, we give the asymptotic behavior of the positive stationary solutions as $\mu\rightarrow\infty$. Finally, both uniqueness and multiplicity of the positive stationary solutions are shown as well as their stability.

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Mathematics Subject Classification: Primary: 35K57; Secondary: 92D25, 92D40.

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