Allee effect and bistability in a spatially heterogeneous predator-prey model

Authors:
Yihong Du and Junping Shi

Journal:
Trans. Amer. Math. Soc. **359** (2007), 4557-4593

MSC (2000):
Primary 35J55, 92D40; Secondary 35B30, 35B32, 35J65, 92D25

DOI:
https://doi.org/10.1090/S0002-9947-07-04262-6

Published electronically:
April 17, 2007

MathSciNet review:
2309198

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Abstract: A spatially heterogeneous reaction-diffusion system modelling predator-prey interaction is studied, where the interaction is governed by a Holling type II functional response. Existence of multiple positive steady states and global bifurcation branches are examined as well as related dynamical behavior. It is found that while the predator population is not far from a constant level, the prey population could be extinguished, persist or blow up depending on the initial population distributions, the various parameters in the system, and the heterogeneous environment. In particular, our results show that when the prey growth is strong, the spatial heterogeneity of the environment can play a dominant role for the presence of the Allee effect. Our mathematical analysis relies on bifurcation theory, topological methods, various comparison principles and elliptic estimates. We combine these methods with monotonicity arguments to the system through the use of some new auxiliary scalar equations, though the system itself does not keep an order structure as the competition system does. Among other things, this allows us to obtain partial descriptions of the dynamical behavior of the system.

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Additional Information

**Yihong Du**

Affiliation:
School of Mathematics, Statistics and Computer Sciences, University of New England, Armidale, NSW2351, Australia – and – Department of Mathematics, Qufu Normal University, Qufu, Shandong 273165, People’s Republic of China

Email:
ydu@turing.une.edu.au

**Junping Shi**

Affiliation:
Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187 – and – School of Mathematics, Harbin Normal University, Harbin, Heilongjiang 150025, People’s Republic of China

Email:
shij@math.wm.edu

DOI:
https://doi.org/10.1090/S0002-9947-07-04262-6

Keywords:
Reaction-diffusion system,
predator-prey model,
spatial heterogeneity.

Received by editor(s):
April 6, 2005

Received by editor(s) in revised form:
February 10, 2006

Published electronically:
April 17, 2007

Additional Notes:
The first author was partially supported by the Australia Research Council

The second author was partially supported by United States NSF grants DMS-0314736 and EF-0436318, College of William and Mary junior research leave, and a grant from Science Council of Heilongjiang Province, China.

Article copyright:
© Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.