Analysis on Bifurcation for a Predator-Prey Model with Beddington-DeAngelis Functional Response and Non-Selective Harves
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Analysis on Bifurcation for a Predator-Prey Model with Beddington-DeAngelis Functional Response and Non-Selective Harvesting Rong Wang1 · Yunfeng Jia1
Received: 14 October 2014 / Accepted: 9 October 2015 © Springer Science+Business Media Dordrecht 2015
Abstract The biomathematical model is an important tool in exploring the dynamic behavior of different species. In this paper, we deal with a predator-prey model with BeddingtonDeAngelis functional response and non-selective harvesting. We discuss the existence and stability of the local bifurcation solution, which emanates from the semi-trivial solution. Moreover, by using the boundedness of positive solutions and the global bifurcation theory, we find that the local bifurcation branch can be extended to the global bifurcation. Keywords Predator-Prey model · Bifurcation · Positive solution · Beddington-DeAngelis functional response · Non-Selective harvesting Mathematics Subject Classification 35K57 · 92D25 · 93C20
1 Introduction With the advanced development in biology, biomathematical models play increasingly important role in explaining some biological effects. Models describing interactions of different species have made significant progress since the pioneering work of Lotka and Volterra [1]. Among numerous mathematical models, predator-prey models have long been and will continue to be one of the dominant themes in both ecology and mathematics. In those related models, functional responses are significant components which describe the relationship among different organisms. Functional response in classical Lotka-Volterra models assumes that populations follow the linear growth, and so the form of this functional The work was supported in part by the National Science Foundation of China (11271236, 11401356), by the Program of New Century Excellent Talents in University of Ministry of Education of China (NCET-12-0894), by the Shaanxi New-star Plan of Science and Technology (2015KJXX-21), and also by the Natural Science Basic Research Plan in Shaanxi Province of China (2015JQ1023).
B Y. Jia
[email protected]
1
School of Mathematics and Information Science, Shaanxi Normal University, Xi’an, Shaanxi 710062, China
R. Wang, Y. Jia
response is very simple, which leads to the specific relationship among populations can not be fully reflected. It is just for this reason, such functional response has certain limitations in characterizing the interrelation for predators and preys. And so it has been challenged in the past few decades. Later, Holling introduced three types functional responses [2–4]. Relatively speaking, these three types functional responses are appropriate in explaining the relationship between predators and preys. Therefore, models with Holling type functional responses improved Lotka-Volterra models. The Holling type I functional response is ax, 0 ≤ x < x0 , ϕ(x) = bx0 , x ≥ x0 , where x is the density or size of the organism. This functional response is appropriate for ax ax 2 the algae, the cell and the low grade organism. 1+bx and 1+bx 2
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