Analyzing the Effect of Fear, and Refuge on Prey, Harvesting, and Hunting Cooperation Among Predators on Leslie–Gower Predator Prey Model

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

In this paper, we present and assess a predator–prey Leslie–Gower model including disease, refuge and treatment in prey population. There are two groups of prey: those who are susceptible and infected. It is hypothesized that prey population is affected by diseases and refuge, and grows logistically in the absence of predators. Infected prey population receives treatment. The predators’ growth rate is governed by the modified Leslie–Gower dynamics. The dynamical attributes of the resulting system are boundedness, positivity of solutions, extinction criteria, existence and (local and global) stability. Biology uses mathematical analysis to identify the possible attributes of equilibrium points. The focus of this study is to assess how treatment and refuge affect the populations of ill prey, susceptible prey, predators and treated prey. The numerical simulation indicates that the influence of treatment, and refuge change the dynamics of the system (2.1). Extensive numerical simulations were performed to validate our analytical findings by using the Mathematica and MATLAB software.

In this paper, we propose a new Leslie-Gower predator-prey model with predator-dependent prey refuge. Firstly, we obtain the positivity and boundedness of the system solution. Secondly, we prove that the origin is unstable using blow-up method, analyze the existence and local stability of the boundary equilibrium point and positive equilibrium point, and prove that the unique positive equilibrium point of the system is globally asymptotically stable by constructing a suitable Dulac function. Finally, mathematic analysis and numerical simulation show that: (1) when the strength of the predator-dependent prey refuge k = 0 , the dynamics of the predator-prey system without predator-dependent prey refuge are consistent with the results obtained from the traditional Leslie-Gower predator-prey system; (2) when k tends to positive infinity, the predator-dependent refuge lead to prey population densities fall somewhere between without prey refuge and with proportional refuge. However, the predator densities within this new form of the predator-dependent prey refuge is greater than the densities of predators without prey refuge and with proportional refuge; (3) increasing the strength k of the predator-dependent prey refuge can increase the densities of predator and prey populations respectively.

In this paper, we propose and analyze a predator–prey model with a prey refuge and additional food for predators. We study the impact of a prey refuge on the stability dynamics, when a constant proportion or a constant number of prey moves to the refuge area. The system dynamics are studied using both analytical and numerical techniques. We observe that the prey refuge can replace the predator–prey oscillations by a stable equilibrium if the refuge size crosses a threshold value. It is also observed that, if the refuge size is very high, then the extinction of the predator population is certain. Further, we observe that enhancement of additional food for predators prevents the extinction of the predator and also replaces the stable limit cycle with a stable equilibrium. Our results suggest that additional food for the predators enhances the stability and persistence of the system. Extensive numerical experiments are performed to illustrate our analytical findings.

A Leslie-Gower predator prey model with density dependent birth rate on prey species and prey refuge is proposed and studied in this paper. Sufficient condition which ensure the global stable of the positive equilibrium is obtained. Our study indicates density dependent birth rate of prey species has negative effect on the final density of both prey and predator species. Density dependent birth rate may lead to the Allee effect of prey species and enhance the extinction chance of the species. Numeric simulations are carried out to show the feasibility of the main results.

This paper investigates the global dynamics of a reaction–diffusion–advection Leslie–Gower predator–prey model in open advective environments. We find that there exist critical advection rates, intrinsic growth rates, diffusion rates and length of the domain, which classify the global dynamics of the Leslie–Gower predator–prey system into three scenarios: coexistence, persistence of prey only and extinction of both species. The results reveal some significant differences with the classical specialist and generalist predator–prey systems. In particular, it is found that the critical advection rates of prey and predator are independent of each other and the parameters about predation rate have no influence on the dynamics of system. The theoretical results provide some interesting highlights in ecological protection in streams or rivers.

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.

Allee effect acting on the prey species in a Leslie–Gower predation model

This paper deals with the dynamics of a prey-dependent two-species model, associating the Holling type II response function. We incorporate the prey refuge to the system with additional food supplement to self-competitive predator. We have found three ecologically significant equilibrium points as well as discussed their stability, instability conditions. The obtained results suggest that the coexisting equilibrium point can go through a Hopf bifurcation for some suitable value of prey refuge. Numerical simulations are performed to support all the analytical findings and to investigate the effects of additional food and self-interactions among predators for various densities of prey refuge. It can be observed that additional food supports coexisting behavior while intraspecific competition does not enhance coexistence so much but reducing the predator population, supports a stable solution. Although a very strong prey refuge forces the predators to extinct in both cases. Moreover, a comparison of four food web models has been performed to elaborate significantly the effects of prey refuge, additional food, and self-competition, respectively, on the system dynamics. The results yield that the model can fairly illustrate a realistic environmental ecology of two interacting populations and may be useful for species conservation.

In ecological environment, Allee effect is one of the important factors which cause significant changes to the system dynamics. In this paper, using the theory of dynamical systems, we analyze a variation of a standard cannibalistic two-dimensional prey–predator model with Holling type-II functional response in the presence of both weak and strong Allee effects. We have analyzed the impact of strong and weak Allee effects on the dynamics of a cannibalistic system, knowing the dynamics of the cannibalistic model without Allee effect. We have deduced that in the presence of cannibalism, both strong and weak Allee effects generate bistability between equilibrium points. For strong Allee effect, bistability occurs between trivial equilibrium point and predator-free equilibrium point as well as between trivial and coexistence equilibrium points. But for weak Allee effect, bistability occurs only between coexistence equilibrium points. We also pointed out that the cannibalistic system without Allee effect exhibits tristability among the trivial equilibrium point, coexistence equilibrium point having low prey concentration and coexistence equilibrium point having comparatively high prey concentration. But in the presence of strong Allee effect, cannibalistic system experiences tristability among trivial and two other stable coexistence equilibrium points. By a comprehensive bifurcation analysis, we have observed that Allee effect enriches both the local and global dynamics of the system. Here, we have reported all possible codimension-one and codimension-two bifurcations extensively by choosing cannibalism, Allee effect and predator natural death rate as the bifurcation parameters. In the analysis of bifurcations, we have explored the existence of transcritical bifurcation, saddle-node bifurcation, Hopf bifurcation, Bogdanov–Takens bifurcation and Bautin bifurcation. Our analytical findings are validated through exhaustive numerical simulations. Finally, we have reported a comparative study between the impacts of strong and weak Allee effects on the dynamics of the cannibalistic system.

Dynamics of generalist predator in a stochastic environment: Effect of delayed growth and prey refuge

Maturation delay induced stability enhancement and shift of bifurcation thresholds in a predator–prey model with generalist predator

Rich dynamics of a delay-induced stage-structure prey–predator model with cooperative behaviour in both species and the impact of prey refuge

By employing a prey refuge mechanism, more preys can be protected from predation. Prey species are also better protected from predation when they congregate in herds. However, what if the prey refuge and herd behavior mechanisms were combined in a system? To investigate this phenomenon, we consider two different prey-predator systems with prey refuge capacity. The first system is a simple prey-predator with prey refuge, whereas the second system considers prey refuge and prey herd behavior mechanisms. Using these models, we explore how different prey refuge strategies affect species interactions in both systems. To accomplish this, we use theoretical techniques (e.g., computing steady states and performing the stability analysis) and numerical bifurcation analysis to demonstrate various dynamical behaviors of these two prey-predator systems. Once prey refuge is treated as a bifurcation parameter, we observe the occurrence of supercritical Hopf and transcritical bifurcations in both systems. Furthermore, we explore the dynamic effects of prey refuge and predator handling time on species population interactions: our findings reveal that using both prey refuge and herd behavior as escape strategies; it is possible to dilute the predation pressure and ensure species biodiversity.

Predator–prey dynamics in general equilibrium and the role of trade