Abstract
In the present article, a new fractional order predator-prey model with group defense is put up. The dynamical properties such as the existence, uniqueness and boundness of solution, the stability of equilibrium point and the existence of Hopf bifurcation of the involved predator-prey model have been discussed. Firstly, we establish the sufficient conditions that guarantee the existence, uniqueness and boundness of solution by applying Lipschitz condition, inequality technique and fractional order differential equation theory. Secondly, we analyze the existence of various equilibrium points by basic mathematical analysis method and obtain some sufficient criteria which guarantee the locally asymptotically stability of various equilibrium points of the involved predator-prey model with the aid of linearization approach. Thirdly, the existence of Hopf bifurcation of the considered predator-prey model is investigated by using the Hopf bifurcation theory of fractional order differential equations. Finally, simulation results are presented to substantiate the theoretical findings.
Highlights
In the present article, a new fractional order predator-prey model with group defense is put up
Γ2u1σ(t)u γ2γ4u1σ(t)u2(t), 2(t where u1 and u2 stand for the densities of prey and predator population, respectively. γ1 represents the logistic growth rate, γ2 stands for the search efficiency of predator for prey, γ3 stands for the mortality rate of predator species, γ4 stands for the biomass conversion coefficient, κ is the carrying capacity of the environment, and σ denotes aggregation efficiency
We mainly focus on the existence, uniqueness and boundness of solution, the stability of equilibrium point and the existence of Hopf bifurcation of fractional order predator-prey model
Summary
A new fractional order predator-prey model with group defense is put up. The dynamical properties such as the existence, uniqueness and boundness of solution, the stability of equilibrium point and the existence of Hopf bifurcation of the involved predator-prey model have been discussed. Considering that the harvesting play an important role in describing the evolution process of a population, Kumar and Kharbanda[11] introduced the Michaelis-Menten type harvesting into predator-prey model (1.1). They established the following predator-prey model with group defense and Michaelis-Menten type harvesting: u1(t) γ2γ4u1σ(t γ2u1σ(t )u2(t),. We think that the investigation on dynamical behavior of fractional-order differential equations have important theoretical significance and broad potential value
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