Diffusive Predator Prey Model Research Articles

Bifurcation and stability analysis of a Leslie–Gower diffusion predator–prey model with prey refuge and Beddington–DeAngelis functional response

In this paper, we consider the spatiotemporal dynamics behaviors of a Leslie–Gower diffusion predator–prey system with prey refuge and Beddington–DeAnglis (B‐D) functional response. By using the Poincaré inequality and topological degree theory, we first investigate the Turing instability of the reaction–diffusion system and prove the existence of nonconstant positive steady‐state solutions. Then we discuss the steady‐state bifurcation and the direction to Hopf bifurcation of the PDE model by the local bifurcation theorem and center manifold theory. Finally, some numerical simulations are presented to supplement the analytic results in one dimension which indicates that changes in prey refuge and diffusion coefficient can increase the complexity of the system.

Note on the diffusive prey-predator model with variable coefficients and degenerate diffusion

It is of interest to understand effects of variable coefficients and degenerate diffusion on the longtime behaviors of solutions of reaction diffusion equations. Recently, Yang and Yao (2024) studied a classical prey-predator model and proved that the degradation of the diffusion coefficient of the prey and variable coefficients satisfying the appropriate conditions will not affect dynamical properties. In this note we shall simplify the proof of Yang and Yao (2024) and delete the condition (4).

Effect of discontinuous harvesting on a diffusive predator-prey model

Effect of discontinuous harvesting on a diffusive predator-prey model

Dynamics of a non-local intraspecific competition predator–prey model with memory effect

In the paper, we investigate a diffusive predator–prey model with nonlocal intraspecific prey competition and spatial memory under Neumann boundary conditions. Through stability and bifurcation analysis, we find that the memory-based diffusion coefficient and the spatiotemporal diffusive delay have important effects on the dynamics of the model. By using the spatiotemporal diffusive delay as a bifurcation parameter, the critical values are determined for the stability of the positive constant steady state and the associated Hopf bifurcation. We find that the system may admit no stability switch, one stability switch and multiple stability switches.

Numerical Study of the Reaction Diffusion Prey–Predator Model Having Holling II Increasing Function in the Predator Under Noisy Environment

The current study deals with the stochastic diffusive prey–predator model with Holling II increasing function in the predator. As it is a population dynamics its population can be affected by various factors such as disease in prey–predator species, environmental fluctuation, etc. So, the underlying model is considered under the impact of a temporal multiplicative noise. The numerical solution of the governing model is obtained by finite difference schemes. To check the accuracy of the numerical solution, the stability and consistency of the schemes in the mean square sense are derived. All the schemes are consistent with the model and von Neumann stable. Several simulations are drawn to check the efficacy of the scheme. A test problem is investigated with numerical schemes and for the different values of the parameters, 3D and 2D plots are drawn. A graphical representation of the given system is drawn which shows the behavior of both species and the efficiency of the novel schemes. Hope so, these results will motivate the researcher to consider the stochastic prey–predator model for a better understanding of the ecosystem.

Traveling front in a diffusive predator–prey model with Beddington–DeAngelis functional response

In this paper, we consider a singular diffusive predator–prey model with Beddington–DeAngelis functional response, employing geometric singular perturbation theory and Bendixson's criteria. Our investigation revolves around transforming the reaction–diffusion equation into a multi‐scale four‐dimensional slow–fast system with two different orders of small parameters. Through once singular perturbation analysis, our focus shifts towards exploring the existence of heteroclinic orbits in a three‐dimensional system. We analyze these dynamics through the perspective of the Fisher–KPP equation in two limit cases. In the first case, only the normal to the two‐dimensional slow manifold is unstable. This allows for the deduction of existence of heteroclinic orbits in the three‐dimensional system through investigating the dynamics on the two‐dimensional slow manifold. Consequently, we obtain both monotonic traveling fronts and non‐monotonic fronts with oscillatory tails. In the second case, the normal to the one‐dimensional slow manifold exhibits both stable and unstable directions, then it is impossible to restrict the dynamics of the three‐dimensional system entirely to the slow manifold. Instead, we integrate the slow orbits of the reduced system with the fast orbits of the layer system to construct a singular heteroclinic orbit. According to Fenichel's theorem, we discover the existence of exact heteroclinic orbits of three‐dimensional system and derive the monotonic traveling fronts under weaker parameter conditions. Additionally, we also discuss the nonexistence of traveling fronts. Finally, we demonstrate our theoretical results with numerical simulations.

Dynamics of a size-structured predator–prey model with chemotaxis mechanism

This paper is concerned with a size-structured diffusive predator–prey model with chemotaxis mechanism. The existence, linearized stability and monotonicity with respect to the growth rates of boundary steady-state solutions are analyzed. Moreover, the global stability of trivial steady-state solution under certain conditions is proved by constructing Lyapunov functional. We investigate the local and global bifurcations of positive steady-state solutions that emanate from semi-trivial steady-state solutions using Lyapunov–Schmidt reduction and bifurcation techniques when the fertility intensity of a predator or prey is used as a bifurcation parameter. It is shown that the nonlinear nonlocal chemotaxis term can lead to the emergence of Allee effect.

Hopf bifurcation analysis of a two‐delayed diffusive predator–prey model with spatial memory of prey

In this paper, we consider a diffusive predator–prey model with spatial memory of prey and gestation delay of predator. For the system without delays, we study the stability of the positive equilibrium in the case of diffusion and no diffusion, respectively. For the delayed model without diffusions, the existence of Hopf bifurcation is discussed. Further, we investigate the stability switches of the model with delays and diffusions when two delays change simultaneously by calculating the stability switching curves and obtain the existence of Hopf bifurcation. We also calculate the normal form of Hopf bifurcation to determine the direction of Hopf bifurcation and the stability of bifurcation periodic solutions. Finally, numerical simulations verify the theoretical results.

Stability criterion of a nonautonomous 3-species ratio-dependent diffusive predator-prey model

The global stability of a nonautonomous 3-species ratio-dependent diffusive predator-prey model is studied in this paper. Firstly, some easily verifiable sufficient conditions which guarantee the existence of the strictly positive space homogenous periodic solution (SHPS) of the ratio- dependent predator-prey model (RDPPM) with diffusive and variable coefficient are achieved by using a comparison theorem of differential equation and fixed point theorem. At the same time, some new analysis techniques are developed as a byproduct. Secondly, some sufficient conditions ensuring the globally asymptotically stability of the strictly positive SHPS of the diffusive nonautonomous predator-prey model are given by using the method of upper and lower solutions (UALS) for the parabolic partial differential equations and Lyapunov stability theory. In addition, two numerical examples are given to validate the theoretical results obtained in this paper.

Turing–Hopf bifurcation in a diffusive predator–prey model with schooling behavior and Smith growth

This paper explores the dynamics of a diffusive predator–prey model, considering schooling behavior and Smith growth in prey. Initially, we have formulated the pertinent characteristic equations. Subsequently, We proceed to examine the existence of the Turing bifurcation and Hopf bifurcation, phenomena that describe the emergence of spatial and temporal patterns due to diffusion and oscillations, respectively, and focusing on the parameters of the intrinsic growth rate γ and the diffusion coefficient d2 of the prey. Finally, we conduct numerical simulations to validate our theoretical findings and further illustrate the dynamics of the predator–prey system, considering schooling behavior and Smith growth in prey.

Effect of density-dependent diffusion on a diffusive predator–prey model in spatially heterogeneous environment

The paper presents a class of predator–prey model with density-dependent diffusion in the spatially heterogeneous environment. We first provide the global existence and boundedness of the solution for the model. Then, by taking a variable transformation, the difficulty brought by the cross-diffusion can be overcome, and the existence, stability and local bifurcation of semi-trivial steady-state solutions for the equivalent system are further studied. Finally, the existence of positive solutions of the system is also given by using the Leray–Schauder degree theory and the method of principle eigenvalue, especially for the limit cases when the diffusion coefficient tends to zero or infinite.

Impact of non-smooth threshold control on a reaction–diffusion predator–prey model with time delay

Impact of non-smooth threshold control on a reaction–diffusion predator–prey model with time delay

Bifurcation analysis of a diffusive predator–prey model with fear effect

In this paper, a diffusive predator–prey system with fear factor response subject to Neumann boundary conditions is considered. Bifurcations at the boundary equilibria of the corresponding ODE are confirmed by Sotomayor's theorem. Detailed bifurcation analysis shows that the reaction–diffusion system undergoes Hopf bifurcation and steady‐state bifurcation. The bifurcation direction and the stability of the bifurcating periodic solutions are also given. Finally, numerical simulation results are carried out to verify the theoretical results.

A robust multiplicity result in a generalized diffusive predator-prey model

A robust multiplicity result in a generalized diffusive predator-prey model

Complex dynamic analysis of a reaction–diffusion predator–prey model in the network and non-network environment

This paper considers an improved two-dimensional reaction–diffusion predator–prey model with time delay. Firstly, the distribution of the equilibrium points of the system and the existence conditions are discussed. Secondly, under the assumption of the existence of equilibrium points, the linear approximations of the system in both network and non-network settings are derived. Thirdly, the Turing instability conditions for both non-delay and delay systems are investigated, including cases of diffusion-induced and delay-induced instabilities. Fourthly, amplitude equations are derived based on the non-delay system. Finally, extensive numerical simulations are conducted to validate and illustrate the theoretical results, and to analyze and explain their practical significance. The above results effectively demonstrate that the theoretical findings, simulations, and natural reality are consistent. The numerical modeling conducted on the network structure based on the Monte Carlo method exhibits a more diverse range of dynamic behaviors in the parameter of the Holling III functional response function.

Double Hopf bifurcation induced by spatial memory in a diffusive predator–prey model with Allee effect and maturation delay of predator

In this paper, we delve into double Hopf bifurcation induced by memory-driven directed movement in a spatial predator–prey model with Allee effect and maturation delay of predators. We first adopt a novel technique to handle the associated characteristic equation and thus obtain the crossing curves as well as the double Hopf points. We then calculate explicit formulae of normal form regarding non-resonant double Hopf bifurcation. We thus divide the dynamics of the developed model into several categories near the double Hopf bifurcation points. Our numerical and theoretical results both demonstrate that the model can exhibit various complex phenomena when the parameters are near the double Hopf bifurcation points. For example, the transition from one stable spatially inhomogeneous periodic orbit with mode-5 to another with mode-4 and the coexistence of them can be observed.

Traveling wavefronts in an anomalous diffusion predator–prey model

Abstract In this paper, we study traveling wavefronts in an anomalous diffusion predator–prey model with the modified Leslie–Gower and Holling-type II schemes. We perform a traveling wave analysis to show that the model has heteroclinic trajectories connecting two steady state solutions of the resulting system of fractional partial differential equations and corresponding to traveling wavefronts. This also includes numerical results to show the existence of traveling wavefronts. Furthermore, we obtain the numerical time-dependent solutions in order to show the evolution of wavefronts. We find that wavefronts exist that travel faster in the anomalous subdiffusive regime than in the normal diffusive one. Our results emphasize that the main properties of traveling waves and invasions are altered by anomalous subdiffusion in this model.

Bifurcation analysis of a delayed diffusive predator–prey model with spatial memory and toxins

Bifurcation analysis of a delayed diffusive predator–prey model with spatial memory and toxins

Dynamics for a diffusive prey–predator model with advection and free boundaries

This article deals with a free boundary problem of the Lotka–Volterra type prey–predator model with advection in one space dimension. The model considered here may be applied to describe the expanding of an invasive or new predator species adopting a combination of random movement and advection upward or downward along the resource gradient, with the free boundaries representing expanding fronts of predator species. The main purpose of this article is to understand the influence of the advection environment on the dynamics of the model. We provide sufficient conditions for spreading and vanishing of the predator species, and we find a sharp threshold between the spreading and vanishing concerning this model. Moreover, for the case of successful spreading for the predator, we give estimates of asymptotic spreading speeds and nonlocal long‐time behavior of the prey and the predator.

Existence and bifurcation of positive solutions to a class of predator-prey models with mutual interference among the predators

In this work, we study a class of diffusive predator-prey models with mutual interference among the predators while searching for food. Under Dirichlet boundary condition, by the fixed point index theory, we provide the necessary and sufficient conditions for the existence of coexistence states (i.e., stationary pattern). This is a strong contrast to the corresponding Neumann boundary problem, which exhibits bistability and admits no patterns.