Dynamics of Leslie–Gower type generalist predator in a tri-trophic food web system

In this paper, the dynamics of a tri-trophic food web system consists of Leslie–Gower type generalist predator has been explored. The system is bounded under certain conditions. The Hopf-bifurcation has been established in the phase planes. The bifurcation diagrams exhibit coexistence of all three species in the form of periodic/chaotic solutions. The “snail-shell” chaotic attractor has very high Lyapunov exponents. The coexistence in the form of stable equilibrium is also possible for lower values of parameters. The two-parameter bifurcation diagrams are drawn for critical parameters.     

Modeling and analysis of a modified Leslie–Gower type three species food chain model with an impulsive control strategy

This paper describes a modified Leslie–Gower type three species food chain model with harvesting. We have incorporated impulsive control strategy to the system. Theories of impulsive differential equations, small amplitude perturbation skills and comparison technique are used to study dynamical behavior of the system. Sufficient conditions are derived to ensure global stability of the lowest-level prey and mid-level predator eradication periodic solution. Sufficient conditions are also derived to examine the permanence of the system. Numerical simulations are carried out to verify the analytical results, and the system is analyzed through graphical illustrations. It is observed that the stability of the system exhibits several states, ranging from stable situation to cyclic oscillatory behavior, under different favorable conditions. These results are useful to study the dynamic complexity of ecological systems. The computation of the largest Lyapunov exponent demonstrates the chaotic dynamic nature of the system. The qualitative nature of strange attractor is examined. It is to be noted that the harvesting effort can cause a stable equilibrium to become unstable and even a switching of stabilities.     

Multiple attractors in a Leslie–Gower competition system with Allee effects

We investigate asymptotic dynamics of the classical Leslie–Gower competition model when both competing populations are subject to Allee effects. The system may possess four interior steady states. It is proved that for certain parameter regimes both competing populations may either go extinct, coexist or one population drives the other population to extinction depending on initial conditions.     

Nonlinear oscillations in the modified Leslie–Gower model

In this paper we study the existence of nonlinear oscillations of a modified Leslie–Gower model around the positive equilibrium point. It is proved that at least one limit cycle can exist bifurcating from it but that this point is never a center, that is, there does not exist an infinite number of nonlinear oscillations around it.     

Dynamics of a Leslie–Gower Holling-type II predator–prey system with Lévy jumps

This paper is concerned with a stochastic predator–prey system with modified Leslie–Gower and Holling-type II schemes with Lévy jumps. First, we prove there is a unique positive solution to the system with a positive initial value. Then we establish the sufficient conditions for stability in mean and extinction of the system. Finally, we introduce some numerical simulations to support the main results. The results shows that the Lévy jumps can change the properties of the population systems significantly.     

Spatiotemporal dynamics of a Leslie–Gower predator–prey model incorporating a prey refuge

In this paper, we investigate the spatiotemporal dynamics of a two-dimensional predator–prey model, which is based on a modified version of the Leslie–Gower scheme incorporating a prey refuge. We establish a Lyapunov function to prove the global stability of the equilibria with diffusion and determine the Turing space in the spatial domain. Furthermore, we perform a series of numerical simulations and find that the model dynamics exhibits complex Turing pattern replication: stripes, cold/hot spots-stripes coexistence and cold/hot spots patterns. The results indicate that the effect of the prey refuge for pattern formation is tremendous. This may enrich the dynamics of the effect of refuge on the predator-prey systems.     

Global stability of a Leslie–Gower predator–prey model with feedback controls

In this work the global stability of a unique interior equilibrium for a Leslie–Gower predator–prey model with feedback controls is investigated. The main result together with its numerical simulations shows that feedback control variables have no influence on the global stability of the Leslie–Gower model, which means that feedback control variables only change the position of the unique interior equilibrium and retain its global stability.     

Dynamical complexities in the Leslie–Gower predator–prey model as consequences of the Allee effect on prey

This work deals with the analysis of a predator–prey model derived from the Leslie–Gower type model, where the most common mathematical form to express the Allee effect in the prey growth function is considered.     

Analysis of a diffusive Leslie–Gower predator–prey model with nonmonotonic functional response

In this paper, a diffusive Leslie–Gower predator–prey system with nonmonotonic functional respond is studied. We obtain the persistence of this model and show the local asymptotic stability of positive constant equilibrium by linearized analysis and the global stability by constructing Liapunov function. Besides, Turing instability of this equilibrium is obtained. The existence and nonexistence of positive nonconstant steady states of this model are established. Furthermore, by numerical simulations we illustrate the patterns of prey and predator.     

Stability of the virus dynamics model with Beddington–DeAngelis functional response and delays

A virus dynamics model with Beddington–DeAngelis functional response and delays is introduced. By analyzing the characteristic equations, the local stability of an infection-free equilibrium and a chronic-infection equilibrium of the model is established. By using suitable Lyapunov functionals and the LaSalle invariance principle, we show that the infection-free equilibrium is globally asymptotically stable if _R0?1$R_0?1$ and the chronic-infection equilibrium is globally asymptotically stable if _R0>1$R_0>1$. Numerical simulations are also given to explain our results.     

A Leslie–Gower predator–prey model with disease in prey incorporating a prey refuge

In this paper, a predator–prey Leslie–Gower model with disease in prey has been developed. The total population has been divided into three classes, namely susceptible prey, infected prey and predator population. We have also incorporated an infected prey refuge in the model. We have studied the positivity and boundedness of the solutions of the system and analyzed the existence of various equilibrium points and stability of the system at those equilibrium points. We have also discussed the influence of the infected prey refuge on each population density. It is observed that a Hopf bifurcation may occur about the interior equilibrium taking refuge parameter as bifurcation parameter. Our analytical findings are illustrated through computer simulation using MATLAB, which show the reliability of our model from the eco-epidemiological point of view.     

Two limit cycles in a Leslie–Gower predator–prey model with additive Allee effect

In this work, a bidimensional continuous-time differential equations system is analyzed which is derived from Leslie type predator–prey schemes by considering a nonmonotonic functional response and Allee effect on population prey. For ecological reason, we describe the bifurcation diagram of limit cycles that appear only at the first quadrant in the system obtained. We also show that under certain conditions over the parameters, the system allows the existence of a stable limit cycle surrounding an unstable limit cycle generated by Hopf bifurcation. Furthermore, we give conditions over the parameters such that the model allows long-term extinction or survival of both populations.     

Competitive exclusion and coexistence in a Leslie–Gower competition model with Allee effects

The classical competitive exclusion principle states that two populations competing for a limited resource cannot coexist, one of the populations will drive the other to extinction. We prove in this work that when one population is subject to Allee effects, then for certain parameter regimes both competing populations may either coexist or one population may drive the other to extinction depending on initial conditions.     

Predator interference in a Leslie–Gower intraguild predation model

The aim of this paper is to analyze general dynamics features of a new Intraguild Predation (IGP) model where the top predator feeds only on the mesopredator and also affects its consumption rate. Important dynamical aspects of the model are described. Specifically, we prove that the trajectories of the associated system are bounded and defined for all positive time; there is a trapping domain; there are open subsets of parameters, such that the system in the first octant has at most five equilibrium solutions and at most three of them are of co-existence. Here we characterize the existence of Hopf bifurcations and we prove that this model exhibits either one, two or three small amplitude periodic solutions which arise from a zero-Hopf bifurcation. We prove the existence of alternative stable states. Finally, some numerical computations have been given in order to support our analytical results. The importance of some parameters of the model is discussed, in particular the role of the interference rate. The numerical exploration suggests that the equilibrium biomass of each of the three species grows as the level of interference grows.     

A modified Leslie–Gower predator–prey model with prey infection

The disease effect on ecological systems is an important issue from mathematical and experimental point of view. In this paper, we formulate and analyze a predator–prey model for the susceptible population, infected population and their predator population with modified Leslie–Gower (or Holling–Tanner) functional response. Mathematical analysis of the model equations with regard to invariance of nonnegativity and boundedness of solutions, local and global stability of the biological feasible equilibria and permanence of the system are presented. When the rate of infection crosses a critical value, we determine that the strictly positive interior equilibrium undergoes Hopf bifurcation. From our numerical simulations, we observe that the predation rate also plays an important role on the dynamic behavior of our system.     

Relaxation oscillations in a slow–fast modified Leslie–Gower model

In this paper, we prove the existence and uniqueness of relaxation oscillation cycle of a slow–fast modified Leslie–Gower model via the entry–exit function and geometric singular perturbation theory. Numerical simulations are also carried out to illustrate our theoretical result.     

Positive solutions of a diffusive Leslie–Gower predator–prey model with Bazykin functional response

In this paper, we consider a diffusive Leslie–Gower predator–prey model with Bazykin functional response and zero Dirichlet boundary condition. We show the existence, multiplicity and uniqueness of positive solutions when parameters are in different regions. Results are proved by using bifurcation theory, fixed point index theory, energy estimate and asymptotical behavior analysis.     

Delay-induced Bogdanov–Takens bifurcation in a Leslie–Gower predator–prey model with nonmonotonic functional response

This work is concerned with the dynamics of a Leslie–Gower predator–prey model with nonmonotonic functional response near the Bogdanov–Takens bifurcation point. By analyzing the characteristic equation associated with the nonhyperbolic equilibrium, the critical value of the delay inducing the Bogdanov–Takens bifurcation is obtained. In this case, the dynamics near this nonhyperbolic equilibrium can be reduced to the study of the dynamics of the corresponding normal form restricted to the associated two-dimensional center manifold. The bifurcation diagram near the Bogdanov–Takens bifurcation point is drawn according to the obtained normal form. We show that the change of delay can result in heteroclinic orbit, homoclinic orbit and unstable limit cycle.