# American Institute of Mathematical Sciences

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December  2017, 22(10): 3653-3661. doi: 10.3934/dcdsb.2017145

## A PDE model of intraguild predation with cross-diffusion

 1 Institute for Mathematical Sciences, Renmin University of China, Beijing 100872, China 2 Department of Mathematics, University of Miami, Coral Gables, FL 33124, USA 3 Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA

* Corresponding author

Received  November 2016 Revised  January 2017 Published  April 2017

Fund Project: RSC is partially supportted by Beijing Municipal Foreign Expert Bureau, and NSF grants DMS-11-18623 and DMS-15-14752. XC is supported by the Research Funds of Renmin University of China (15XNLF21), and by China Postdoctoral Science Foundation (2016M591319). KYL is partially supported by NSF grant DMS-1411476. TX is supported by NSF of China (No. 11601516,11571364 and 11571363) and research Funds of Renmin University of China (No. 15XNLF10)

This note concerns a quasilinear parabolic system modeling an intraguild predation community in a focal habitat in $\mathbb{R}^n$, $n ≥ 2$. In this system the intraguild prey employs a fitness-based dispersal strategy whereby the intraguild prey moves away from a locale when predation risk is high enough to render the locale undesirable for resource acquisition. The system modifies the model considered in Ryan and Cantrell (2015) by adding an element of mutual interference among predators to the functional response terms in the model, thereby switching from Holling Ⅱ forms to Beddington-DeAngelis forms. We show that the resulting system can be realized as a semi-dynamical system with a global attractor for any $n ≥ 2$. In contrast, the orginal model was restricted to two dimensional spatial habitats. The permanance of the intraguild prey then follows as in Ryan and Cantrell by means of the Acyclicity Theorem of Persistence Theory.

Citation: Robert Stephen Cantrell, Xinru Cao, King-Yeung Lam, Tian Xiang. A PDE model of intraguild predation with cross-diffusion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3653-3661. doi: 10.3934/dcdsb.2017145
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