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

Stability and Bifurcation Analysis of a Delayed Three Species Food Chain Model with Crowley-Martin Response Function

In this paper we have studied the dynamical behaviors of three species prey-predator system. The interaction between prey and middle-predator is Crowley-Martin type functional response. Positivity and boundedness of the system are discussed. Stability analysis of the equilibrium points is presented. Permanence and Hopf-bifurcation of the system are analyzed under some conditions. The effect of discrete time-delay is studied, where the delay may be regarded as the gestation period of the super-predator. The direction and the stability criteria of the bifurcating periodic solutions are determined with the help of the normal form theory and the center manifold theorem. Extensive numerical simulations are carried out to validate our analytical findings. Implications of our analytical and numerical findings are discussed critically

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

Studying Both Direct and Indirect Effects in Predator-Prey Interaction

Studying and modelling the interaction between predators and prey have been one of the central topics in ecology and evolutionary biology. In this thesis, we study two different aspects of predator-prey interaction: direct effect and indirect effect. Firstly, we study the direct predation between predators and prey in a patchy landscape. Secondly, we study indirect effects between predators and prey. Thirdly, we extend our previous model by incorporating a stage-structure into prey. Finally, we further extend our previous model by incorporating spatial structures into modelling

Stability and Hopf Bifurcation in a Delayed Predator-Prey System with Herd Behavior

A special predator-prey system is investigated in which the prey population exhibits herd behavior in order to provide a self-defense against predators, while the predator is intermediate and its population shows individualistic behavior. Considering the fact that there always exists a time delay in the conversion of the biomass of prey to that of predator in this system, we obtain a delayed predator-prey model with square root functional response and quadratic mortality. For this model, we mainly investigate the stability of positive equilibrium and the existence of Hopf bifurcation by choosing the time delay as a bifurcation parameter

Stability and Bifurcation Analysis of Time Delayed Prey-Predator System with Holling Type-III Response Function

Interaction between prey and predator is a recurring event that occurs continuously and the presence of both can mutually affect each other’s population. This paper discusses the stability and bifurcation analysis of time delayed prey-predator system with Holling type-III response function incorporating a prey refuge. Holling type-III response function is used because the availability of the prey in nature is decreasing. Time delay represents the time for predators to find another prey population when the available population is decreasing. Dynamic analysis is used to study the influence of a given response function. The equilibrium point and stability analysis of the model with and without time delay has been calculated and analyzed. Model analysis under the influence of time delay and coefficient of competition among predators shows an indication of Hopf bifurcation in the neighborhood of the co-existing equilibrium point

Stability and Hopf bifurcation of a ratio-dependent predator-prey model with time delay and stage structure

In this paper, a ratio-dependent predator-prey model described by Holling type II functional response with time delay and stage structure for the prey is investigated. By analyzing the corresponding characteristic equations, the local stability of the coexistence equilibrium of the model is discussed and the existence of Hopf bifurcations at the coexistence equilibrium is established. By using the persistence theory on infinite dimensional systems, it is proven that the system is permanent if the coexistence equilibrium exists. By introducing some new lemmas and the comparison theorem, sufficient conditions are obtained for the global stability of the coexistence equilibrium. Numerical simulations are carried out to illustrate the main results