Leslie-Gower Predator-prey Model Research Articles

Dynamic analysis of a fast slow modified Leslie–Gower predator–prey model with constant harvest and stochastic factor

In this paper, the dynamic properties of a fast slow modified Leslie–Gower predator–prey model with constant harvest are discussed. The results show that the system appears different Hopf bifurcations and limit cycles as parameters change, and the stability of the internal equilibriums changes accordingly. The system also appears Bogdanov–Takens bifurcation phenomenon. For the fast-slow system of this model, the second order approximate perturbation manifolds, and an approximate manifold passing the fold point are constructed. Then using the criterion of inflection point curve and blow-up method, the existence of canard cycles is analyzed. By comparing with the model with nonlinear harvest, the different effects of constant harvest on the model are highlighted. In addition, the effect of stochastic factors on the fast-slow system is considered.

Dynamical analysis of a modified Leslie–Gower predator–prey model with disease in prey incorporating a prey refuge and treatment

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.

Impact of both-density-dependent fear effect in a Leslie–Gower predator–prey model with Beddington–DeAngelis functional response

Animals have evolved fearful behavior throughout a lengthy evolutionary history, and such action should be the result of interactions between prey and predator. At present, the fear factor considered by many scholars in predator–prey models is reasonable to a certain extent, for example in some unicellular and bacterial species, but for most species, the fear factor should be density-dependent of both species due to sexual reproduction. Therefore, based on the existing works, this paper puts forward some more reasonable conditions that should be satisfied by fear factor, and makes theoretical analysis and numerical simulations for specific type of expression. The study shows that fear plays an important role in enriching model dynamics, and that the model dynamics are sensitive to moderate levels of fear. Lower levels of fear maintain the same dynamics as the model without fear, higher levels of fear can exclude periodic solutions (or limit cycles) to promote model stability, while moderate levels of fear may lead to the emergence of Hopf bifurcations. By selecting a different set of parameters, the Hopf bifurcation in our model can be both supercritical and subcritical, contrasting to the typical predator–prey model in which it frequently appears supercritical.

Canard Cycles and Their Cyclicity of a Fast–Slow Leslie–Gower Predator–Prey Model with Allee Effect

We study canard cycles and their cyclicity of a fast–slow Leslie–Gower predator–prey system with Allee effect. More specifically, we find necessary and sufficient conditions of the exact number (zero, one or two) of positive equilibria of the slow–fast system and its location (or their locations), and then we further completely determine its (or their) dynamics under explicit conditions. Besides, by geometric singular perturbation theory and the slow–fast normal form, we find explicit sufficient conditions to characterize singular Hopf bifurcation and canard explosion of the system. Additionally, the cyclicity of canard cycles is completely solved, and of particular interest is that we show the existence and uniqueness of a canard cycle, whose cyclicity is at most two, under corresponding precise explicit conditions.

Coexistence and bistability of a modified Leslie–Gower predator–prey model with mixed movement for the predator and Holling-type II schemes

This paper centers on the analysis of the dynamics of a modified Leslie–Gower predator–prey model employing Holling-type II schemes, with the prey exhibiting pure random diffusion and the predator undergoing a mixed form of movement. The extinction of species and uniform persistence of this system are explored, and several conditions for the stability, uniqueness and multiplicity of positive steady-state solutions are derived. In contrast to the specialist and generalist predator–prey systems in open advective environments, the dynamics of this system are more intricate. It emerges that multiple positive steady-state solutions and the bistable phenomenon exist for this system when a small advection rate and a moderate predation rate are imposed. Numerical simulations reveal that the increase of diffusion rate for prey disadvantages the survival of itself and has no impact on predator invasion, while the increase of diffusion rate for predators favors the invasion of itself.

Canard Cycles and Homoclinic Orbit of a Leslie–Gower Predator–Prey Model with Allee Effect and Holling Type II Functional Response

Canard Cycles and Homoclinic Orbit of a Leslie–Gower Predator–Prey Model with Allee Effect and Holling Type II Functional Response

Bifurcation analysis in a modified Leslie-Gower predator-prey model with fear effect and multiple delays.

In this paper, we explored a modified Leslie-Gower predator-prey model incorporating a fear effect and multiple delays. We analyzed the existence and local stability of each potential equilibrium. Furthermore, we investigated the presence of periodic solutions via Hopf bifurcation bifurcated from the positive equilibrium with respect to both delays. By utilizing the normal form theory and the center manifold theorem, we investigated the direction and stability of these periodic solutions. Our theoretical findings were validated through numerical simulations, which demonstrated that the fear delay could trigger a stability shift at the positive equilibrium. Additionally, we observed that an increase in fear intensity or the presence of substitute prey reinforces the stability of the positive equilibrium.

DYNAMICS OF DELAYED AUTONOMOUS MODEL WITH CARRY-OVER EFFECTS, INCLUDING PREY HARVESTING

This paper proposes a two-dimensional modified Leslie–Gower predator–prey model with carry-over effects, fear, prey protection and additional food, including prey harvesting. For this, it assumes the carry-over effect with delay incorporated in the birth rate of the prey population and gestation delay in Leslie–Gower term of a predator, and supposes that the predator consumes the prey according to Holling type II functional response. This dynamical system is split into two groups, namely non-delayed and delayed systems, respectively. In the non-delayed system, first, we discuss positivity, boundedness solutions, and the existence of equilibria, especially finding the coexistence of interior equilibrium points under certain conditions. After that, local and global stabilities are systematically analyzed. In addition, we obtain the conditions of Hopf-bifurcation in terms of the growth rate of prey species, prey protection and preference rate of predator for the non-delayed system. Whereas in the delayed system, both the delays have an essential role in governing the dynamical system discussed in three cases, and each case found the Hopf-bifurcation parameters for which the system changes stability behavior into instability. At last, the numerical simulations have been carried out to verify the theoretical findings.

Stable spatially inhomogeneous periodic solutions for a diffusive Leslie–Gower predator–prey model

Stable spatially inhomogeneous periodic solutions for a diffusive Leslie–Gower predator–prey model

Model Predator-Prey Leslie-Gower dengan Fungsi Respon Sokol-Howell dan Perilaku Anti Predator

This study discusses the Leslie-Gower predator-prey model with the Sokol-Howell response function and anti-predator behavior. It is assumed that prey has anti-predator behavior that aims to reduce the risk of predation and not as an attempt by prey to find food. This study aims to formulate a Leslie-Gower predator-prey model with the Sokol-Howell response function and anti-predator behavior, analyze the model's equilibrium point and model interpretation. Stability analysis was carried out using the linearization method. The type of stability is determined based on the characteristic eigenvalues ​​obtained using Routh-Hurwitz criteria. The results of the analysis of the equilibrium point show that prey populations will exist and predators will become extinct if the anti-predator coefficient is greater than the intrinsic growth coefficient of predators, while prey and predator populations will always exist if the intrinsic growth coefficient of predators is greater than the anti-predator coefficient and fulfills other conditions required. Based on the numerical simulations performed, the interpretation is that an enlarged anti-predator coefficient increases the number of prey populations until they approach the carrying capacity, while predator populations decrease significantly and over time experience extinction.

Bifurcation analysis of a discrete Leslie–Gower predator–prey model with slow–fast effect on predator

Understanding and accounting for the slow–fast effect are crucial for accurately modeling and predicting the dynamics of predator–prey models, emphasizing the importance of considering the relative speeds of interacting populations in ecological research. This paper examines a predator–prey interaction to study its complex dynamics due to its slow–fast effect on predator populations. The occurrence and stability of equilibria are analyzed. The stability of positive fixed point is dependent on the slow–fast effect parameter , which must fall within a specific range when the generation gap is larger. The positive fixed point becomes unstable for bigger values of because the growth of predators is faster, resulting in the extinction of all prey. Smaller values of cause the positive fixed point to become unstable since the prey grows more quickly while the predator grows more slowly, ultimately causing the extinction of the predator. Moreover, it is shown that Leslie–Gower model experiences Neimark–Sacker and period‐doubling bifurcations at positive equilibrium point. In order to control bifurcation, hybrid control and feedback control methods are employed. Finally, analytical results are confirmed by numerical examples.

Existence and Uniqueness of a Canard Cycle with Cyclicity at Most Two in a Singularly Perturbed Leslie–Gower Predator–Prey Model with Prey Harvesting

Yao and Huzak [2022] proved that the cyclicity of canard cycles in a singularly perturbed Leslie–Gower predator–prey model with prey harvesting is at most two in a region of parameters. In this paper, we further show that there exists only one canard cycle with cyclicity at most two under explicit parameters conditions.

Bifurcation analysis of autonomous and nonautonomous modified Leslie-Gower models.

In ecological systems, the predator-induced fear dampens the prey's birth rate; yet, it fails to extinguish their population, as they endure and survive even under significant fear-induced costs. In this study, we unveil a modified Leslie-Gower predator-prey model by incorporating the fear of predators, cooperative hunting, and predator-taxis sensitivity. We embark upon an exploration of the positivity and boundedness of solutions, unearthing ecologically viable equilibrium points and their stability conditions governed by the model parameters. Delving deeper, we unravel the scenario of transcritical, saddle-node, Hopf, Bogdanov-Takens, and generalized-Hopf bifurcations within the system's intricate dynamics. Additionally, we observe the bistable nature of the system under some parametric conditions. Further, the nonautonomous extension of our model introduces the intriguing interplay of seasonality in some crucial parameters. We establish a set of sufficient conditions that guarantee the permanence of the seasonally driven system. By conducting a numerical study on the seasonally forced model, we observe a myriad of phenomena manifesting the predator-prey dynamics. Notably, periodic solutions, higher periodic solutions, and bursting patterns emerge, alongside intriguing chaotic dynamics. Specifically, seasonal variations of the predator-taxis sensitivity and hunting cooperation can lead to the extinction of prey species and even the control of chaotic (higher periodic) solutions through the generation of a simple periodic solution. Remarkably, the seasonal forcing has the capacity to govern the chaotic behavior, leading to an exceptionally quasi-periodic arrangement in both prey and predator populations.

KONTROL OPTIMAL MODEL LESLIE-GOWER DENGAN PEMANENAN PADA POPULASI PREDATOR DAN PERLINDUNGAN PADA PREY

This paper studied the Leslie-Gower predator-prey model with the effects of prey refuge and linear harvesting on predator populations. Prey populations grow as a logistic model when there is no interaction with predators. The predator population is assumed to have economic value. Analysis of Bionomic and Harvesting Control have been studied further. The results showed that the protective effect of prey had an effect on harvesting activity in the population so that it affected profits in predatory harvests. Simulation of several calculations is given to purchase the analysis results

The Impact of Allee Effect on a Leslie–Gower Predator–Prey Model with Hunting Cooperation

The Impact of Allee Effect on a Leslie–Gower Predator–Prey Model with Hunting Cooperation

Cross-diffusion mediated Spatiotemporal patterns in a predator–prey system with hunting cooperation and fear effect

Cooperation during hunting is a common predation plan of action among several large predators to increase their biomass raising capturing potential which can also produce fear in the prey and dispersal-influenced pattern creation is also relevant from both a fundamental and an applied standpoint. Considering these, we present a modified Leslie–Gower predator–prey model incorporating fear, intra-specific competition and hunting cooperation of predators following Berec’s encounter-driven functional response in presence of cross-diffusion. In the non-spatial system, biologically feasible equilibria and their stability conditions are derived. Taking the level of fear as bifurcation parameter, occurrence of limit cycle via Hopf bifurcation and the stability of limit cycle computing Lyapunov coefficient are discussed. In diffusive model, spacial focus on Turing instability conditions and to determine the relevant intervals at which various patterns will begin to appear. In the two dimensional spatial domain, under Neumann boundary condition, various Turing patterns such as spot, stripe, mixture of spot and stripe, labyrinth patterns are emerged varying fear, hunting cooperation and cross-diffusion coefficient. Hopf–Turing region is also a centre for several complex patterns. The numerical simulations indicate that the patterns are in same or opposite phase according to the sign of cross-diffusion coefficient. Intra-specific competition among predators tend the system towards stability. Cross-diffusion causes a paradox to population density with raising fear and hunting cooperation.

Global Hopf Bifurcation of a Diffusive Modified Leslie–Gower Predator–Prey Model with Delay and Michaelis–Menten Type Prey Harvesting

Global Hopf Bifurcation of a Diffusive Modified Leslie–Gower Predator–Prey Model with Delay and Michaelis–Menten Type Prey Harvesting

Complex dynamics induced by additive Allee effect in a Leslie-Gower predator-prey model

Complex dynamics induced by additive Allee effect in a Leslie-Gower predator-prey model

Dynamics of a Leslie-Gower predator-prey model with advection and free boundaries

This paper investigates a Leslie-Gower predator-prey model with advection and double free boundaries. Firstly, we establish the global existence and uniqueness of the solution. Subsequently, we analyze the long-time behaviors of the solution and demonstrate a spreading-vanishing dichotomy. If the initial habitat size or the expanding capacity of the boundaries is large, the invasive predator will spread throughout the entire space; otherwise, both the prey and predator will eventually vanish. When spreading occurs, two subcases arise, depending on the interaction coefficient $ c $ and the environmental support $ \delta $. If $ c\delta < 1 $, the prey and predator coexist, which we refer to as the weakly hunting case; conversely, if $ c\delta \geq 1 $, the predator survives and spreads throughout the space, while the prey vanishes, which we term the strongly hunting case. In the event of spreading, we also provide an estimate of the spreading speed. Finally, we introduce a front-fixing implicit-explicit finite difference method for the free boundary problem. Numerical simulations validate our theoretical findings and uncover some intriguing new phenomena, prompting further investigation into this and general free boundary problems.

The Impact of Predator-dependent Prey Refuge on the Dynamics of a Leslie-Gower Predator-prey Model

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.