Allee effect and bistability in a spatially heterogeneous predator-prey model
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- by Yihong Du and Junping Shi PDF
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Abstract:
A spatially heterogeneous reaction-diffusion system modelling pre-dator-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.References
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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
- MR Author ID: 616436
- ORCID: 0000-0003-2521-9378
- Email: shij@math.wm.edu
- 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. - © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2309198