Diffusive Predator Prey Model Research Articles

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.

Dynamic Analysis of a Diffusive Predator–Prey Model with Hunting Cooperation Functional Response and Prey-Taxis

Dynamic Analysis of a Diffusive Predator–Prey Model with Hunting Cooperation Functional Response and Prey-Taxis

Dynamics of a delayed reaction–diffusion predator–prey model with nonlocal competition and double Allee effect in prey

In this paper, we study the effects of the nonlocal competition and double Allee effect in prey on a diffusive predator–prey model. We investigate the local stability of coexistence equilibrium in the predator–prey model by analyzing the eigenvalue spectrum. We study the consequence of double Allee effect on the prey population. Also, we discuss the existence of Hopf bifurcation under different parameters by using the gestation time delay of predators as a bifurcation parameter and analyzing the distribution of eigenvalues. By utilizing the normal form method and center manifold theorem, we have given some conditions that could determine the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions. Through our research, we obtain that the double Allee effect can affect the coexistence of prey and predator and induce periodic oscillations of the densities of prey and predator. In addition, the nonlocal competition can also affect the stability and homogeneity of the solutions, but may receive the consequences of the Allee effect as well as time delay. In the numerical simulation part, we further demonstrate the correctness of this conclusion by comparison.

Stationary distribution of a stochastic reaction–diffusion predator–prey model with additional food and fear effect

Providing additional food supplements to predators is one of the crucial and effective tools for biological conservation and pest management, which has been widely recognized and validated by theoretical and experimental scientists. This paper focuses on the existence and uniqueness of stationary distribution of a stochastic reaction–diffusion predator–prey model that incorporates additional food for predator and fear effect for prey. It reveals that spatial diffusion is beneficial for maintaining the steady state of the model, whereas environment white noise has a negative effect.

Dynamics of a diffusive two predators- one prey system

This paper analyses a diffusive predator-prey model consisting of a single prey species and two predator species with modified Leslie-Gower term Holling type II functional response subject to the homogeneous Neumann boundary condition. Local stability condition is derived by the application of Routh-Hurwitz criterion. Global asymptotic stability of the unique positive steady state is shown by constructing a suitable Lyapunov function when self diffusion is allowed where as non-constant positive steady states can exist due to the presence of cross-diffusion, that means, cross-diffusion can induce stationary pattern. Taking the cross diffusion as a bifurcation parameter, one can show the existence of positive non-constant solutions with the help of bifurcation theory. A brief conclusion completes the paper.

Laplace transform-homotopy perturbation method for fractional time diffusive predator–prey models in ecology

In this paper, we consider some reaction–diffusion systems arising from two-component predator–prey models with various kinetics ranging from prey-dependent, ratio-dependent functional responses and subdiffusion. The goal of the present work is to simulate the time-dependent predator–prey model of Lotka–Volterra. Analytical solutions of this model are performed using the Laplace transform-homotopy perturbation method. The proposed scheme finds the solution without any discretization or restrictive assumptions and avoids the round-off errors. The fact that the proposed technique solves nonlinear problems without using Adomian’s polynomials can be considered as a clear advantage of this algorithm over the decomposition method. We show all the possible equilibria and examine their stability in line with the ecological parameters. Numerical simulations of the diffusive predator–prey model in one-, two- and three-dimensions are given to compare and demonstrate the asymptotic behaviour of the time-dependent reaction–diffusion systems. The results show that proposed technique is a powerful tool to solve systems of nonlinear ordinary and partial differential equations of predator–prey models in ecology.

Pattern dynamics in a reaction–diffusion predator–prey model with Allee effect based on network and non-network environments

In this paper, we establish a predator–prey model with Beddington–Deangelis functional response. For this model, firstly, the existence of the positive equilibrium point and the conditions for the Turing instability are studied. Then the amplitude equation is derived through weakly nonlinear analysis, and the relationship between the selection of pattern and the coefficients of the amplitude equation is obtained. At the same time, through a large number of numerical simulations, we verify the accuracy of the theoretical analysis. Actually, we mainly choose to change the values of parameters r and d2 to study the sensitivity of the pattern to them. When the pattern tends to be stable, there could be the pattern of spots, coexistence of spots and stripes or stripes. Even if they are spot pattern, the spot density will also be different due to the selection of parameters. Finally, we simulate and compare that the network structure(mainly BA and WS) has a certain influence on the time required for pattern stabilization and the distribution of node density. The final results show that the growth rate of prey, diffusion coefficients and network structure all play an important role in the formation of Turing pattern.

Dynamics of a diffusive predator–prey model with nonlocal fear effect

The fear effect on prey populations due to the presence of a predator’s sound or smell can be more impactful than direct predation. In this paper, we formulate a diffusive predator–prey model incorporating the nonlocal fear effect. The existence and boundedness of solutions are showed. We also study the stability of constant steady states by using the characteristic equation and Lyapunov functionals. Steady-state bifurcations are carried out in detail by the Lyapunov-Schmidt method. The analyses demonstrate that the fear effect in the system can change the stability of the constant steady state and there exist spatially nonhomogeneous steady states. Hopf bifurcations are also investigated and the results show that high levels of fear can stabilize the model by excluding the existence of periodic solutions. Finally, numerical simulations are sketched to illustrate our theoretical findings. This work can help us further understand the impact of fear effect on the stability of constant steady states, steady-state bifurcations and Hopf bifurcations of predator–prey models.

Instability and bifurcation analysis of a diffusive predator–prey model with fear effect and delay

In this paper, a delayed diffusive predator–prey model with fear effect under Neumann boundary conditions is considered. For the system without diffusion and delay, the conditions for the existence and local stability of equilibria are obtained by analyzing the eigenvalues. Then, the instability induced by diffusion and delay-diffusion of the positive constant stationary solutions are discussed, respectively. Moreover, the regions of instability and pattern formation can be achieved with respect to diffusion and delay coefficients. Furthermore, the existence and direction of Hopf bifurcation and the properties of the homogeneous/nonhomogeneous bifurcated periodic solutions are driven by using the center manifold theorem and the normal form theory. Finally, some numerical simulations are carried out to verify the theoretical results.

Detection of Hopf bifurcations induced by pregnancy and maturation delays in a spatial predator–prey model via crossing curves method

We investigate the stability switches behaviors due to Hopf bifurcations triggered by maturation and pregnancy delays for a diffusive predator–prey model under the joint impacts of fear and Allee effect of predators. We firstly carry out the linear analysis of the proposed model by applying the recently developed crossing curves method to cope with the corresponding characteristic equation incorporating wave number as well as delay-dependent parameters. We then procure the Hopf bifurcation curves through which delays pass, the stability of the associated constant equilibrium can change accordingly. Thereupon, we deduce the normal form regarding Hopf bifurcation and thus clarify its properties. Finally, we present a numerical example and perform some simulations to verify the accuracy of the obtained results. It is shown that both the maturation and pregnancy periods can respectively induce stability switches.

Dynamic properties of a reaction–diffusion predator–prey model with nonlinear harvesting: A linear and weakly nonlinear analysis

In this work, we have studied the spatiotemporal dynamics of a predator–prey model with selective nonlinear saturated harvesting of both species. The study of the non-diffusive system includes analytical and numerical analysis of local and global bifurcations. Bifurcation analysis of the non-diffusive system unveils that the system may exhibit transcritical, Hopf, saddle–node and homoclinic bifurcation with increased carrying capacity and harvesting efforts. The simulation results further reveal that the paradox of enrichment may disappear in a harvested system for a higher value of carrying capacity. The spatiotemporal study consists of an analysis of different pattern-forming instabilities. A critical ratio of the diffusivities of predator to prey for the occurrence of Turing instability is established. Further, a weakly nonlinear analysis is provided to derive the amplitude equation and predict the form of the pattern. The key highlight of the work is that prey harvesting facilitates spatiotemporal chaos, and predator diffusion controls it. Furthermore, intensive prey harvesting forces spatial segregation, whereas predator harvesting relaxes it. It is also observed that spatial segregation enriches the overall species biomass. A unique observation of this study is the occurrence and nonoccurrence of spatiotemporal chaos in the Hopf-Turing parametric space.

A diffusion-advection predator-prey model with a protection zone

In this paper, we investigate a diffusion-advection predator-prey model imposed the Danckwerts boundary conditions with three different types of protection zone. Some special cases have been extensively studied, such as: diffusion-advection predator-prey model without protection zone [22,29] and diffusive predator-prey model with protection zone without advection in high dimension [5,16]. Some ideas developed in the former works do not appear to work due to the difficulty caused by diffusion, advection and protection zone. By applying the comparison principle for parabolic equations [32] and persistence theory [26,37], we obtain almost complete long-time dynamics, which describes the optimal protection zone. If the predator functional response is Holling-type I, the uniqueness of the positive steady state has been derived, which improves the method proposed by López-Gómez and Pardo [20] and developed by Nie et al. [28]. Finally, new ways of coexistence have been obtained by investigating a special three-species model.