Abstract
The Allee effect is incorporated into a predator-prey model with linear functional response. Compared with the predator-prey which only takes the crowding effect and predator partially dependent on prey into consideration, it is found that the Allee effect of the prey species would increase the extinction risk of both the prey and predator. Moreover, by using a center manifold theorem and bifurcation theory, it is shown that the model with Allee effect undergoes the flip bifurcation and Hopf bifurcation in the interior ofR+2with different Allee effect values. In the two bifurcations, we can come to the conclusion that different Allee effect will have different bifurcation value and the increasing of the Allee effect will increase the value of bifurcation, respectively.
Highlights
The predator-prey interaction is a topic of great interest for many ecologists and mathematics
Many researchers have studied the dynamical behavior of the predator-prey system in ecology and contributed to growth of continuous models for large size population [1,2,3,4,5,6,7,8,9,10,11,12,13]
The author got the following conclusion: the systems undergoes flip bifurcation and Hopf bifurcation at fixed points under specific conditions when δ varies in small neighborhood
Summary
The predator-prey interaction is a topic of great interest for many ecologists and mathematics. Many researchers have studied the dynamical behavior of the predator-prey system in ecology and contributed to growth of continuous models for large size population [1,2,3,4,5,6,7,8,9,10,11,12,13]. D denotes competition among individuals of predator species due to overcrowding; l denotes the half saturation constant; m denotes the conversion rate for predator; and k denotes carrying capacity of the prey in a particular habitat In this prey-predator system, due to the impact of their own breeding process or because of the prey itself, the prey population will have the Allee effect. A brief conclusion is given in the last section
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