Global stability and Hopf bifurcation of a diffusive predator–prey model with hyperbolic mortality and prey harvesting

This paper is concerned with a predator&ndashprey model with hyperbolic mortality and prey harvesting. The parameter regions for the stability and instability of the unique positive constant solution of ODE and PDE are derived, respectively, especially the global asymptotical stability of positive constant equilibrium of the diffusive model is obtained by iterative technique. The stability and direction of periodic solutions of ODE and PDE are investigated by center manifold theorem and normal form theory, respectively. Numerical simulations are carried out to depict our theoretical analysis

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

Bifurcation on diffusive Holling–Tanner predator–prey model with stoichiometric density dependence

This paper studies a diffusive Holling–Tanner predator–prey system with stoichiometric density dependence. The local stability of positive equilibrium, the existence of Hopf bifurcation and stability of bifurcating periodic solutions have been obtained in the absence of diffusion. We also study the spatially homogeneous and nonhomogeneous periodic solutions through all parameters of the system, which are spatially homogeneous. In order to verify our theoretical results, some numerical simulations are carried out.&nbsp

Multiple wave solutions in a diffusive predator-prey model with strong Allee effect on prey and ratio-dependent functional response

A thorough analysis is performed in a predator-prey reaction-diffusion model which includes three relevant complex dynamical ingredients: (a) a strong Allee effect; (b) ratio-dependent functional responses; and (c) transport attributes given by a diffusion process. As is well-known in the specialized literature, these aspects capture adverse survival conditions for the prey, predation search features and non-homogeneous spatial dynamical distribution of both populations. We look for traveling-wave solutions and provide rigorous results coming from a standard local analysis, numerical bifurcation analysis, and relevant computations of invariant manifolds to exhibit homoclinic and heteroclinic connections and periodic orbits in the associated dynamical system in $R^4$. In so doing, we present and describe a diverse zoo of traveling wave solutions; and we relate their occurrence to the Allee effect, the spreading rates and propagation speed. In addition, homoclinic chaos is manifested via both saddle-focus and focus-focus bifurcations as well as a Belyakov point. An actual computation of global invariant manifolds near a focus-focus homoclinic bifurcation is also presented to enravel a multiplicity of wave solutions in the model. A deep understanding of such ecological dynamics is therefore highlighted.Comment: 35 pages, 22 figure

Dynamics of a Predator-Prey System with a Mate-Finding Allee Effect

We consider a ratio-dependent predator-prey system with a mate-finding Allee effect on prey. The stability properties of the equilibria and a complete bifurcation analysis, including the existence of a saddle-node, a Hopf bifurcation, and, a Bogdanov-Takens bifurcations, have been proved theoretically and numerically. The blow-up method has been applied to investigate the structure of a neighborhood of the origin. Our mathematical results show the mate-finding Allee effect can reduce the complexity of system behaviors by making the complicated equilibrium less complicated, and it can be a destabilizing force as well, which makes the system has a high possibility of being threatened with extinction in ecology

Discrete Leslie's model with bifurcations and control

We explored a local stability analysis at fixed points, bifurcations, and a control in a discrete Leslie's prey-predator model in the interior of $\mathbb{R}_+^2$. More specially, it is examined that for all parameters, Leslie's model has boundary and interior equilibria, and the local stability is studied by the linear stability theory at equilibrium. Additionally, the model does not undergo a flip bifurcation at the boundary fixed point, though a Neimark-Sacker bifurcation exists at the interior fixed point, and no other bifurcation exists at this point. Furthermore, the Neimark-Sacker bifurcation is controlled by a hybrid control strategy. Finally, numerical simulations that validate the obtained results are given

Dynamics of prey–predator model with strong and weak Allee effect in the prey with gestation delay

This study proposes two prey–predator models with strong and weak Allee effects in prey population with Crowley–Martin functional response. Further, gestation delay of the predator population is introduced in both the models. We discussed the boundedness, local stability and Hopf-bifurcation of both nondelayed and delayed systems. The stability and direction of Hopfbifurcation is also analyzed by using Normal form theory and Center manifold theory. It is shown that species in the model with strong Allee effect become extinct beyond a threshold value of Allee parameter at low density of prey population, whereas species never become extinct in weak Allee effect if they are initially present. It is also shown that gestation delay is unable to avoiding the status of extinction. Lastly, numerical simulation is conducted to verify the theoretical findings.&nbsp