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Open AccessArticle

A Simple Predator-Prey Population Model with Rich Dynamics

by Bing Li 1,2, Shengqiang Liu 1,*, Jing’an Cui 3 and Jia Li 4,*
1
Academy of Fundamental and Interdisciplinary Science, Harbin Institute of Technology, Harbin 150080, China
2
School of Mathematical Science, Harbin Normal University, Harbin 150025, China
3
College of Science, Beijing University of Civil Engineering and Architecture, Beijing 100044, China
4
Department of Mathematical Sciences, The University of Alabama in Huntsville, Huntsville, AL 35899, USA
*
Authors to whom correspondence should be addressed.
Academic Editor: Yang Kuang
Appl. Sci. 2016, 6(5), 151; https://doi.org/10.3390/app6050151
Received: 31 March 2016 / Revised: 7 May 2016 / Accepted: 10 May 2016 / Published: 16 May 2016
(This article belongs to the Special Issue Dynamical Models of Biology and Medicine)
A non-smooth switched harvest on predators is introduced into a simple predator-prey model with logistical growth of the prey and a bilinear functional response. If the density of the predator is below a switched value, the harvesting rate is linear; otherwise, it is constant. The model links the well studied predator-prey model with constant harvesting to that with a proportional harvesting rate. It is shown that when the net reproductive number for the predator is greater than unity, the system is permanent and there may exist multiple positive equilibria due to the effects of the switched harvest, a saddle-node bifurcation, a limit cycle, and the coexistence of a stable equilibrium and a unstable circled inside limit cycle and a stable circled outside limit cycle. When the net reproductive number is less than unity, a backward bifurcation from a positive equilibrium occurs, which implies that the stable predator-extinct equilibrium may coexist with two coexistence equilibria. In this situation, reducing the net reproductive number to less than unity is not enough to enable the predator to go extinct. Numerical simulations are provided to illustrate the theoretical results. It seems that the model possesses new complex dynamics compared to the existing harvesting models. View Full-Text
Keywords: predator-prey model; switched harvest; limit cycle; rich dynamics predator-prey model; switched harvest; limit cycle; rich dynamics
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Li, B.; Liu, S.; Cui, J.; Li, J. A Simple Predator-Prey Population Model with Rich Dynamics. Appl. Sci. 2016, 6, 151.

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