Modified Leslie Gower Predator Prey Model Research Articles

Dinámicas de un modelo de depredación considerando respuesta funcional sigmoidea y alimento alternativo para los depredadores

Interrelationships between two species are a basic theme in Population Dynamics, particularly the interaction between predators and their prey. This importance is due to the fact that it allows a deeper understanding of the behavior of complex food webs. In this paper we extend the analysis of a modified Leslie-Gower predator-prey model by assuming that the functional response is sigmoid or Holling type III and the predator have an alternative food. We show that the system representing the model has up to three positive equilibrium points; we establish conditions to determine the nature of each equilibrium point. In addition, we show the existence of different types of bifurcations, including those of Hopf and the homoclinic. The analytical results are discussed from an ecological perspective.

Optimal Harvesting on a Modified Leslie–Gower Predator–Prey Model Under Fear and Allee Effects on Prey

Optimal Harvesting on a Modified Leslie–Gower Predator–Prey Model Under Fear and Allee Effects on Prey

Canard Cycles and Relaxation Oscillations in a Singularly Perturbed Leslie–Gower Predator–Prey Model with Allee Effect

In this paper, we investigate the dynamics of a modified Leslie–Gower predator–prey model with Allee effect on prey. And the Holling type II functional response is considered in this model. When prey reproduces much faster than predator, by combining the normal form theory of slow–fast systems and the geometric singular perturbation theory, we observe much richer new dynamical phenomena than the existing ones. In the case of strong Allee effect, we prove the existence of canard cycles, homoclinic orbits and heteroclinic orbits. Furthermore, we focus on the case of weak Allee effect. In addition to the similar dynamics of that exhibited by the system with strong Allee effect, we also demonstrate the occurrence of relaxation oscillations created by entry-exit function. Moreover, the existence of canard explosion is further explained analytically and numerically with the help of sophisticated slow–fast techniques.

Michaelis-Menten-Type Prey Harvesting in Discrete Modified Leslie-Gower Predator-Prey Model

A discrete-time Michaelis-Menten-type prey harvesting is discussed in this paper, in the modified Leslie-Gower predator-prey model. Detailed analysis of the topology of nonnegative interior fixed points is given, including their existence and stability dynamics. Also, the conditions for the existence of flip and Neimark-Sacker bifurcations are derived by using the center manifold theorem and bifurcation theory. The numerical simulations are provided, using a computer package, to illustrate the consistency of theoretical results.

Spatial pattern formation and delay induced destabilization in predator–prey model with fear effect

Recent field experiments have shown that predators influence the prey population not only by direct consumption but also by stimulating various defensive strategies. The cost of these defensive strategies can include energetic investment in defensive structures, reduced energy income, lower mating success, and emigration which ultimately reduces the reproduction of prey. To explore the effect of these defensive strategies (anti‐predator behaviors), a modified Leslie–Gower predator–prey model with the cost of fear has been considered. Gestation delay is also incorporated in the system for a more realistic formulation. Boundedness, equilibria and stability analysis are performed for the temporal system. By considering gestation delay as a bifurcation parameter, the existence of Hopf‐bifurcation around the interior equilibrium point is discussed together with the direction, stability, and period of bifurcating solutions arising through Hopf‐bifurcation. The spatial extension of the proposed model incorporating density‐dependent cross‐diffusion is also investigated and the conditions for diffusion‐driven instability are obtained. To illustrate the analytical findings, detailed numerical simulations are performed. Biologically realistic Turing patterns such as spots, spots and stripes mixture, and labyrinthine type patterns are identified. It is found that the fear level has a stabilizing impact on delay induced destabilization and both stabilizing and destabilizing effects on Turing instability.

The Dynamics of Modified Leslie-Gower Predator-Prey Model Under the Influence of Nonlinear Harvesting and Fear Effect

A modified Leslie-Gower predator-prey model with fear effect and nonlinear harvesting is developed and investigated in this study. The predator is supposed to feed on the prey using Holling type-II functional response. The goal is to see how fear of predation and presence of harvesting affect the model's dynamics. The system's positivity and boundlessness are demonstrated. All conceivable equilibria's existence and stability requirements are established. All sorts of local bifurcation occurrence conditions are presented. Extensive numerical simulations of the proposed model are shown in form of Phase portraits and direction fields. That is to guarantee the correctness of the theoretical results of the dynamic behavior of the system and to confirm the existence of various forms of bifurcations. The fear rate is observed to have a stabilizing effect up to a threshold value, after which it leads to prey extinction. The harvesting coefficients, on the other hand, serve as control parameters that, when exceeded, trigger the system to extinction.

Predator-prey interactions under fear effect and multiple foraging strategies

<p style='text-indent:20px;'>We propose and analyze the effects of a generalist predator-driven fear effect on a prey population by considering a modified Leslie-Gower predator-prey model. We assume that the prey population suffers from reduced fecundity due to the fear of predators. We investigate the predator-prey dynamics by incorporating linear, Holling type Ⅱ and Holling type Ⅲ foraging strategies of the generalist predator. As a control strategy, we have considered density-dependent harvesting of the organisms in the system. We show that the systems with linear and Holling type Ⅲ foraging exhibit transcritical bifurcation, whereas the system with Holling type Ⅱ foraging has a much more complex dynamics with transcritical, saddle-node, and Hopf bifurcations. It is observed that the prey population in the system with Holling type Ⅲ foraging of the predator gets severely affected by the predation-driven fear effect in comparison with the same with linear and Holling type Ⅱ foraging rates of the predator. Our model simulation results show that an increase in the harvesting rate of the predator is a viable strategy in recovering the prey population.</p>

Global dynamics of a modified Leslie-Gower predator-prey model with Beddington-DeAngelis functional response and prey-taxis

<abstract><p>In this paper, our purpose is to discuss the global dynamics of a modified Leslie-Gower predator-prey model with Beddington-DeAngelis functional response and prey-taxis under homogeneous Neumann boundary conditions. First, we derive that the global classical solutions of the system are globally bounded by taking advantage of the Morse's iteration of the parabolic equation, which further arrives at the global existence of classical solutions with a uniform-in-time bound. In addition, we establish the global stability of the spatially homogeneous coexistence steady states under certain conditions on parameters by constructing Lyapunov functionals.</p></abstract>

Dynamics Analysis of Modified Leslie-Gower Model with Simplified Holling Type IV Functional Response

In this paper, the modified Leslie-Gower predator-prey model with simplified Holling type IV functional response is discussed. It is assumed that the prey population is a dangerous population. The equilibrium point of the model and the stability of the coexistence equilibrium point are analyzed. The simulation results show that both prey and predator populations will not become extinct as time increases. When the prey population density increases, there is a decrease in the predatory population density because the dangerous prey population has a better ability to defend itself from predators when the number is large enough.

Effect of Fear in Leslie-Gower Predator-Prey Model with Beddington-DeAngelis Functional Response Incorporating Prey Refuge

In the present paper, we study the effect of antipredator behavior due to fear of predation on a modified Leslie-Gower predator-prey model incorporating prey refuge which predation rate of predators follows Beddington-DeAngelis functional response. The biological justification of the model is demonstrated through non-negativity, boundedness, and permanence. Next, we perform the analysis of equilibrium and local stability. We obtain four equilibrium points where two points are locally asymptotically stable and other points are unstable. Besides, we show the effect of the fear in the model and obtain a conclusion that the increased rate of fear can decrease the density of both populations, and prey populations become extinct. Meanwhile, for the case with a constant rate of fear, the prey refuge helpful to the existence of both populations. However, for the case with the fear effect is large, prey refuge cannot cause the extinction of predators. Several numerical simulations are performed to support our analytical results.

A Modified Leslie–Gower Model Incorporating Beddington–DeAngelis Functional Response, Double Allee Effect and Memory Effect

In this paper, a modified Leslie–Gower predator-prey model with Beddington–DeAngelis functional response and double Allee effect in the growth rate of a predator population is proposed. In order to consider memory effect on the proposed model, we employ the Caputo fractional-order derivative. We investigate the dynamic behaviors of the proposed model for both strong and weak Allee effect cases. The existence, uniqueness, non-negativity, and boundedness of the solution are discussed. Then, we determine the existing condition and local stability analysis of all possible equilibrium points. Necessary conditions for the existence of the Hopf bifurcation driven by the order of the fractional derivative are also determined analytically. Furthermore, by choosing a suitable Lyapunov function, we derive the sufficient conditions to ensure the global asymptotic stability for the predator extinction point for the strong Allee effect case as well as for the prey extinction point and the interior point for the weak Allee effect case. Finally, numerical simulations are shown to confirm the theoretical results and can explore more dynamical behaviors of the system, such as the bi-stability and forward bifurcation.

A free boundary problem of some modified Leslie-Gower predator-prey model with nonlocal diffusion term

<p style='text-indent:20px;'>This paper is mainly considered a Leslie-Gower predator-prey model with nonlocal diffusion term and a free boundary condition. The model describes the evolution of the two species when they initially occupy the bounded region <inline-formula><tex-math id="M1">\begin{document}$ [0,h_0] $\end{document}</tex-math></inline-formula>. We first show that the problem has a unique solution defined for all <inline-formula><tex-math id="M2">\begin{document}$ t>0 $\end{document}</tex-math></inline-formula>. Then, we establish the long-time dynamical behavior, including Spreading-vanishing dichotomy and Spreading-vanishing criteria.</p>

一类带有时滞效应的修正的Leslie-Gower捕食者–食饵模型动力学分析

一类带有时滞效应的修正的Leslie-Gower捕食者–食饵模型动力学分析

Impact of the Fear Effect on the Stability and Bifurcation of a Leslie–Gower Predator–Prey Model

In this paper, we investigate analytically and numerically the dynamics of a modified Leslie–Gower predator–prey model which is characterized by the reduction of prey growth rate due to the anti-predator behavior. We prove the existence and local/global stability of equilibria of the model, and verify the existence of Hopf bifurcation. In addition, we focus on the influence of the fear effect on the population dynamics of the model and find that the fear effect can not only reduce the population density of both predator and prey, but also destabilize the coexistence equilibrium, which are beneficial to the occurrence of limit-cycle-induced oscillation, or prevent the occurrence of limit cycle oscillation and increase the stability of the system.

A modified Leslie–Gower predator–prey model with alternative food and selective predation of noninfected prey

Infectious diseases affect prey populations, and predators choosing susceptible prey may lead to the extinction of the prey population. We propose a mathematical model of prey–predator interaction with the prey population divided into two classes: susceptible and infected. The susceptible prey subpopulation becomes infected by direct contact with the infected prey subpopulation, and the predator consumes only susceptible prey. An analysis of the model enabled the establishment of thresholds for the spread of disease in the absence and presence of predators. In the absence of predators, we obtain the classic basic reproduction number ℜ0, and in the presence of predators, we obtain a new threshold , which measures the effect of predators in the disease spread. Furthermore, these thresholds establish conditions for the existence and stability of biologically viable equilibrium points. Using numerical simulations, we determined the different biological characteristics of the model and illustrated the analytical results.

Dynamical study of quadrating harvesting of a predator–prey model with Monod–Haldane functional response

In this study, we have investigated local and global dynamics of a modified Leslie–Gower predator–prey model with Monod–Haldane functional response, where prey is subjected to quadratic harvesting. It is found that the solutions of the proposed system are positive and bounded uniformly. The feasible equilibrium points are also obtained for some suitable and predefined conditions. It is observed that the system exhibits at most three non-zero interior equilibrium points for different choices of parameters under certain conditions. The dynamics of all these feasible equilibrium points have been analysed using Routh–Hurwitz criterion. Local bifurcations analysis such as transcritical and saddle-node bifurcations have been investigated using Sotomayor’s theorem. To demonstrate the analytical results, numerical simulations using some suitable data set are carried out. Optimal harvesting policy has been obtained using Pontryagin’s Maximum Principle to show that the species can be preserved from extinction and a sustainable fishery can be achieved.

Hopf bifurcation analysis in a delayed Leslie–Gower predator–prey model incorporating additional food for predators, refuge and threshold harvesting of preys

In this paper, we formulate and analyze a modified Leslie–Gower predator–prey model. Our model incorporates refuge of preys, additional fixed food for predators, harvesting of preys through a continuous threshold policy and a time delay as to account for predators maturity time. We first carry out a qualitative analysis of the model without time delay, showing existence of extinction, prey-free, predator-free and coexistence equilibria. We further study their stability conditions. Relying only on theoretical results of the model, we construct bifurcation diagrams involving refuge and harvest limit parameters. This led to summarize different scenarios for the model including elimination of one species or competition of both species that are proved possible. Furthermore, considering the time delay as bifurcation parameter, we analyze the stability of the coexistence equilibria and prove the system can undergoes a Hopf bifurcation. The direction of that Hopf bifurcation and the stability of the bifurcated periodic solution are determined by applying the normal form theory and the center manifold theorem. Numerical simulations are presented to illustrate our theoretical results.

Coexistence in seasonally varying predator–prey systems with Allee effect

A general seasonally-varying predator–prey model with Allee effect in the prey growth is investigated. The analysis is performed only on the basis of some properties determining the shape of the prey growth rate and the functional responses. General conditions for coexistence are determined, both in the case of weak and strong Allee effect. Finally, a modified Leslie–Gower predator–prey model with Allee effect is investigated. Numerical results illustrate the qualitative behaviors of the system, in particular the presence of periodic orbits.

A Free Boundary Problem for a Leslie-Gower Predator-Prey Model in Higher Dimensions and Heterogeneous Environment

This paper is mainly concerned with some free boundary problems for a modified Leslie-Gower predator-prey model in higher dimensional and heterogeneous environment. To keep it simple in this article, we assume that the environment and solutions are all radially symmetric. We consider the problem which be used to describe the spreading of an introduced predator species in higher dimensional and heterogeneous environment. We will assume that the prey is initially uniformly well disturbed. The prey undergoes the diffusion and growth in the entire space R^n. The predator is initially introduced in some localized location. We establish that a spreading-vanishing dichotomy is held for this model. We use the comparison principle. we will give the existence, uniqueness and some estimates of the solution to the problem. We study the asymptotic behavior of two species evolving. The free boundary represents the spreading front of the predator species. The boundary condition is described by classic Stefan-like condition. It is proved that the problem addressed is well posed, and that the predator species disperses to all domains in finite time. The long time behaviors of solution and criteria for spreading and vanishing of predator species are also provided. Furthermore, in the case that spreading of predator species happens, we deduce some rough estimates of the spreading speed.

Dynamics of Modified Leslie-Gower Harvested Predator-Prey Model with Beddington-DeAngelis Functional Response

This paper considers a modified Leslie-Gower predator-prey model with Beddington-DeAngelis functional response and nonlinear prey harvesting strategy subject to the zero-flux boundary conditions. To understand the dynamics of the considered system, we derive sufficient conditions for permanence analysis, local stability, global stability and Hopf bifurcation of interior equilibrium point. Further we also investigate the existence and non-existence of non-constant positive steady state solutions.