Abstract
This paper deals with a diffusive predator–prey model with two delays. First, we consider the local bifurcation and global dynamical behavior of the kinetic system, which is a predator–prey model with cooperative hunting and Allee effect. For the model with weak cooperation, we prove the existence of limit cycle, and a loop of heteroclinic orbits connecting two equilibria at a threshold of conversion rate p=p#, by investigating stable and unstable manifolds of saddles. When p>p#, both species go extinct, and when p<p#, there is a separatrix. The species with initial population above the separatrix finally become extinct, and the species with initial population below it can be coexisting, oscillating sustainably, or surviving of the prey only. In the case with strong cooperation, we exhibit the complex dynamics of system, including limit cycle, loop of heteroclinic orbits among three equilibria, and homoclinic cycle with the aid of theoretical analysis or numerical simulation. There may be three stable states coexisting: extinction state, coexistence or sustained oscillation, and the survival of the prey only, and the attraction basin of each state is obtained in the phase plane. Moreover, we find diffusion may induce Turing instability and Turing–Hopf bifurcation, leaving the system with spatially inhomogeneous distribution of the species, coexistence of two different spatial-temporal oscillations. Finally, we consider Hopf and double Hopf bifurcations of the diffusive system induced by two delays: mature delay of the prey and gestation delay of the predator. Normal form analysis indicates that two spatially homogeneous periodic oscillations may coexist by increasing both delays.
Highlights
A predator–prey model can be described by the following equations [1]:u = f (u)u − G(u, v)v, v = dG(u, v)v − mv, (1)where u and v stand for the densities of prey and predator, respectively. f (u) represents the per capita prey growth rate in the absence of predators, and G(u, v) is the functional response characterizing predation. d describes the rate of biomass conversion from predation, and m is the death rate of predator
This paper investigate the dynamics of system (4) with cooperative hunting and Allee effect from the point of view bifurcation analysis, which can provide useful insights into the intersection of the predator and prey, and help us figure out how to prevent such a prey population from extinction
We investigate the dynamics of a diffusive predator–prey model with two delays
Summary
A predator–prey model can be described by the following equations [1]:. where u and v stand for the densities of prey and predator, respectively. f (u) represents the per capita prey growth rate in the absence of predators, and G(u, v) is the functional response characterizing predation. d describes the rate of biomass conversion from predation, and m is the death rate of predator. V = p1[b1 + c1v]uv − m1v, where r1 is the per capita intrinsic growth rate of the prey, K1 is the environmental carrying capacity for the prey, a1 (a1 < K1) is the Allee threshold of the prey population, b1 is the attack rate per predator and prey, c1 measures the degree of cooperation of predator, p1 is the prey conversion to predator, and m1 is the per capita death rate of predator They discussed the existence and stability of the equilibria and the occurrence of Hopf bifurcation. This paper investigate the dynamics of system (4) with cooperative hunting and Allee effect from the point of view bifurcation analysis, which can provide useful insights into the intersection of the predator and prey, and help us figure out how to prevent such a prey population from extinction.
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