Dynamics of a diffusive predator–prey model with herd behavior

This paper is devoted to considering a diffusive predator–prey model with Leslie–Gower term and herd behavior subject to the homogeneous Neumann boundary conditions. Concretely, by choosing the proper bifurcation parameter, the local stability of constant equilibria of this model without diffusion and the existence of Hopf bifurcation are investigated by analyzing the distribution of the eigenvalues. Furthermore, the explicit formula for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are also derived by applying the normal form theory. Next, we show the stability of positive constant equilibrium, the existence and stability of periodic solutions near positive constant equilibrium for the diffusive model. Finally, some numerical simulations are carried out to support the analytical results

Dynamics of a Leslie-Gower predator-prey system with cross-diffusion

A Leslie–Gower predator–prey system with cross-diffusion subject to Neumann boundary conditions is considered. The global existence and boundedness of solutions are shown. Some sufficient conditions ensuring the existence of nonconstant solutions are obtained by means of the Leray–Schauder degree theory. The local and global stability of the positive constant steady-state solution are investigated via eigenvalue analysis and Lyapunov procedure. Based on center manifold reduction and normal form theory, Hopf bifurcation direction and the stability of bifurcating timeperiodic solutions are investigated and a normal form of Bogdanov–Takens bifurcation is determined as well

Existence and stability of periodic solutions for a delayed prey–predator model with diffusion effects

Existence and stability of spatially periodic solutions for a delay prey-predator diffusion system are concerned in this work. We obtain that the system can generate the spatially nonhomogeneous periodic solutions when the diffusive rates are suitably small. This result demonstrates that the diffusion plays an important role on deriving the complex spatiotemporal dynamics. Meanwhile, the stability of the spatially periodic solutions is also studied. Finally, in order to verify our theoretical results, some numerical simulations are also included

Hopf Bifurcation in a Modified Leslie-Gower Two Preys One Predator Model and Holling Type II Functional Response with Harvesting and Time-Delay

In this paper, a modified Leslie-Gower two preys one predator model and Holling type II functional response with harvesting and time-delay were discussed. Model analysis is carried out by determining fixed points, then analyzing the stability of the fixed points and discussing the existence of the Hopf bifurcation. In some conditions that occur in nature indicate the occurrence of hunting of prey and predator species by humans. Therefore, this model is modified by adding the assumption that prey and predators are being harvested. Another modification given to the model is the use of time delays.The delay time term is for taking into account the case that the members of the predator species need time from birth to predation for being active predators. The first case is a model without time delay, it is obtained that 3 fixed points are unstable and 7 fixed points are stable. One of them is the interior fixed point tested with the Routh-Hurwitz criteria. The second case is a model with a delay time, the critical delay value is obained. Hopf bifurcation occurs when the delay time value is equal to the critical delay value and also fulfills the transversality condition. Observations on the model simulation are carried out by varying the value of the delay time. When the Hopf bifurcation occurs, the graph on the solution plane shows a constant oscillatory movement. If the value of the delay time given is less than the critical value of the delay, the controlled system solution goes to a balanced state. Then when the delay time value is greater than the critical delay value, the system solution continues to fluctuate causing an unstable system condition

Turing instability in a diffusive predator-prey model with multiple Allee effect and herd behavior

Diffusion-driven instability and bifurcation analysis are studied in a predator-prey model with herd behavior and quadratic mortality by incorporating multiple Allee effect into prey species. The existence and stability of the equilibria of the system are studied. And bifurcation behaviors of the system without diffusion are shown. The sufficient and necessary conditions for Turing instability occurring are obtained. And the stability and the direction of Hopf and steady state bifurcations are explored by using the normal form method. Furthermore, some numerical simulations are presented to support our theoretical analysis. We found that too large diffusion rate of prey prevents Turing instability from emerging. Finally, we summarize our findings in the conclusion

Qualitative Analysis of a Modified Leslie-Gower Predator-prey Model with Weak Allee Effect II

The article aims to study a modified Leslie-Gower predator-prey model with Allee effect II, affecting the functional response with the assumption that the extent to which the environment provides protection to both predator and prey is the same. The model has been studied analytically as well as numerically, including stability and bifurcation analysis. Compared with the predator-prey model without Allee effect, it is found that the weak Allee effect II can bring rich and complicated dynamics, such as the model undergoes to a series of bifurcations (Homoclinic, Hopf, Saddle-node and Bogdanov-Takens). The existence of Hopf bifurcation has been shown for models with (with- out) Allee effect and the local existence and stability of the limit cycle emerging through Hopf bifurcation has also been studied. The phase portrait diagrams are sketched to validate analytical and numerical findings

Qualitative Analysis of a Modified Leslie-Gower Predator-prey Model with Weak Allee Effect II

The article aims to study a modified Leslie-Gower predator-prey model with Allee effect II, affecting the functional response with the assumption that the extent to which the environment provides protection to both predator and prey is the same. The model has been studied analytically as well as numerically, including stability and bifurcation analysis. Compared with the predator-prey model without Allee effect, it is found that the weak Allee effect II can bring rich and complicated dynamics, such as the model undergoes to a series of bifurcations (Homoclinic, Hopf, Saddle-node and Bogdanov-Takens). The existence of Hopf bifurcation has been shown for models with (without) Allee effect and the local existence and stability of the limit cycle emerging through Hopf bifurcation has also been studied. The phase portrait diagrams are sketched to validate analytical and numerical findings

The Effect of Delay on a Diffusive Predator-Prey System with Holling Type-Ii Predator Functional Response

A delayed diffusive predator-prey system with Holling type-II predator functional response subject to Neumann boundary conditions is considered here. The stability/instability of nonnegative equilibria and associated Hopf bifurcation are investigated by analyzing the characteristic equations. By the theory of normal form and center manifold, an explicit formula for determining the stability and direction of periodic solution bifurcating from Hopf bifurcation is derived