Reaction Diffusion Predator Prey Research Articles

A diffusive predator-prey system with hunting cooperation in predators and prey-taxis: II stationary pattern formation

This paper is a continuation of the study conducted by Ko and Ryu (2024) [5], which introduces and analyzes a generalized predator-prey reaction-diffusion system incorporating (repulsive) prey-taxis and a hunting cooperation effect in predators, under homogeneous Neumann boundary conditions. In the study, the existence and uniqueness of global and classical solutions for the time- and space-dependent system are analytically examined. Furthermore, the study examines the local and global stability and convergence rate at the constant predator-extinction and coexistence states. In our paper, we analyze the stationary system corresponding to the system in [5], with a specific focus on examining the existence and nonexistence of positive and nonconstant solutions. The nonexistence occurs when the diffusion rate of prey is sufficiently high. On the other hand, the existence occurs when the prey-tactic rate is sufficiently high, indicating a strong repulsive prey-taxis, and the diffusion rate of prey is sufficiently low. For this investigation, we separately employ the energy method and the Leray-Schauder degree theory.

Canard cycle, relaxation oscillation and cross-diffusion induced pattern formation in a slow–fast ecological system with weak Allee effect

In this paper, we consider a Holling type IV functional response that describes a situation in which the predator’s per capita rate of predation decreases at sufficiently high prey density. Meanwhile it can be transformed into a Holling type II functional response in the limiting sense where predator’s attack rate increases at a decreasing rate with prey density until it becomes satiated. To explore the different dynamics of these two functional responses, we consider the slow–fast dynamic behavior of predator–prey system with Holling type IV and II functional responses, respectively, and make some simple comparisons of the dynamics of the system with these two functional responses. Specifically speaking, in the nonspatial case, firstly, the system with Holling type II functional response does not undergo a higher-codimension Hopf bifurcation. Then, the system with Holling type IV is extensively proved the existence of canard cycles and relaxation oscillations by using a range of analytical methods such as geometric singular perturbation technique, normal form of the slow–fast system, and the way in–way out function. In the spatial case, the temporal systems are extended to reaction–diffusion predator–prey systems. For the reaction–diffusion system with Holling IV, the different types of traveling wave are observed. Moreover, for the reaction–diffusion predator–prey system with Holling II, it is demonstrated that Turing instability occurs, which induces spatial heterogeneity patterns. Finally, the comparisons of the above dynamics of Holling Type IV and II functional responses in the non-spatial and spatial cases, respectively, are presented. From these comparisons, different Holling functional responses may be adopted for species at different stages or states, which is more conducive to maintaining species diversity and coexistence.

Well-Possednes of a predator-prey reaction-diffusion model with functional response dependent on species

Well-Possednes of a predator-prey reaction-diffusion model with functional response dependent on species

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

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.

Traveling waves of modified Leslie–Gower predator–prey systems

This paper focuses on the spreading phenomena within modified Leslie–Gower reaction–diffusion predator–prey systems. Our main objective is to investigate the existence of two distinct types of traveling waves. Specifically, with the aid of the upper and lower solution methods, we establish the existence of traveling waves connecting the prey-present state and the coexistence state or the prey-present state and the prey-free state by constructing different and appropriate Lyapunov functions. Moreover, for traveling wave connecting the prey-present state and the prey-free state, more information about the monotonicity of the wave profile can be obtained by analyzing its asymptotic behavior at negative infinity. Finally, our results are applied to modified Leslie–Gower system with Holling-II type functional response or Lotka–Volterra type functional response, and a novel Lyapunov function is constructed for the latter, which further enhances our results. Meanwhile, some numerical simulations are carried out to support our results.

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.

Analysis of patterns in an additional food-provided predator–prey reaction diffusion model using amplitude equations

Analysis of patterns in an additional food-provided predator–prey reaction diffusion model using amplitude equations

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.

Global bifurcation for a Beddington–DeAngelis and Tanner predator–prey reaction–diffusion system with prey‐taxis

The aim of this paper is to establish a precise illustration for the structure of the nonconstant steady states for a Beddington–DeAngelis and Tanner predator–prey reaction–diffusion system with prey‐taxis. We treat the nonlinear prey‐taxis as a bifurcation parameter to analyze the bifurcation structure of the system. Furthermore, the exported global bifurcation theorem, under a rather natural condition, offers the existence of nonconstant steady states. In the proof, a priori estimates of steady states will play an important role. The local stability analysis with a numerical simulation and bifurcation analysis are given. Finally, some conclusions including biological meanings are performed to summarize our main analytic results and future investigations.

Singular Perturbation Analysis for a Holling–Tanner Model with Additive Allee Effect

In this article, a reaction–diffusion predator–prey equation with additive Allee effect acting on prey is investigated by twice singular perturbation analysis. Our focus is to study the traveling fronts of predator invasion under the assumption that the diffusion ability of the predator is much greater than that of the prey. Our results exhibit two kinds of traveling fronts for a Holling–Tanner system in two different limit cases. And these traveling fronts correspond to the heteroclinic connections between a saddle and either a stable node or a stable focus. In addition, we show that heteroclinic connections are formed in different ways for different limit cases. One is formed on the slow manifold, which has one time scale. While the other is formed by the intersection of the slow manifold and the fast manifold, which has two time scales. Furthermore, the existence of traveling front solutions in different limit cases is demonstrated through theoretical analysis and numerical simulation. The main tools are geometric singular perturbation theory and Bendixson’s criteria.

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.

Turing Patterns in a Predator–Prey Reaction–Diffusion Model with Seasonality and Fear Effect

Turing Patterns in a Predator–Prey Reaction–Diffusion Model with Seasonality and Fear Effect

Global dynamics of a stochastic reaction–diffusion predator–prey system with space-time white noise

This paper proposes a stochastic reaction–diffusion predator–prey system with fear under the interference of space-time white noise by using the related theories of stochastic partial differential equations. The motivation of this paper is twofold: (i) mathematically, to try to investigate the effects of environmental noise and diffusion on population dynamics; (ii) biologically, to study how environmental noise affects the permanence of species. First, we analyse the well-posedness of solutions. Then sufficient conditions for extinction and permanence of prey and predator populations are derived. Moreover, we present the existence and uniqueness of stationary distribution of this system. Finally, based on numerical simulations and theoretical analysis, it is revealed that (i) high-intensity white noise can lead to the extinction of predator populations; (ii) low-intensity white noise may prolong the period of periodic solutions. This study provides new theoretical guidance for exploring ecological issues in the face of environmental disturbance and spatial diffusion.

Dynamic characteristics of a hyperbolic reaction–diffusion predator–prey system with self‐diffusion and nonidentical inertia

Hyperbolic reaction–diffusion (HRD) systems have emerged as a better descriptor of the macroscopic spatial interaction models used for studying pattern formation in chemical and biological systems. In contrast to the parabolic reaction–diffusion (PRD) models, the spatial disturbances in an HRD system travel through space with finite velocity. This paper considers the simplest two‐species HRD model with different inertia based on the telegraph equation. The underlying reaction terms are considered as a predator–prey interaction with type III response function and prey refuge. We prescribe the analytical conditions for the existence of diffusion‐driven instabilities of the considered system. Simulation results further verify the theoretical results. A connection between the refuge parameter and inertial time is discussed in generating different spatiotemporal patterns. Extension of the two‐species PRD system to HRD system with diagonal diffusion matrix causes a diffusion‐driven wave instability, which is never possible for its parabolic counterpart. Furthermore, the patterns under pure wave instability are dependent on the initial population density.

Influence of Allee effect on the spatiotemporal behavior of a diffusive predator–prey model with Crowley–Martin type response function

The present paper is dealt with a predator–prey model in which the growth of the prey population is influenced by the Allee effect while the predator species are contended with the prey population following the Crowley–Martin type response function. The proposed model is comprehensively analyzed in terms of stability and manifestation of bifurcation of the system. The system unveils the bi-stability together with the existence of a separatrix. In view of the eminence of spatial ecology, the dynamical complexity emanating from the induction of the Allee effect in prey species of a Crowley–Martin reaction–diffusion predator–prey model is also investigated profoundly. The results of numerical simulations reveal that the present system dynamics is motivated by both the Allee effect and diffusion-controlled pattern formation growth to hot spots, stripe-hot spot mixtures, stripes, labyrinthine, stripe-cold spot mixtures, and cold spots replication. The theoretical consequences of the spatiotemporal model under study are validated through numerical simulations.

Dynamical response of a reaction–diffusion predator–prey system with cooperative hunting and prey refuge

The present research is concerned with the combined outcome of the cooperative hunting and prey refuge in a spatiotemporal predator–prey model. Firstly, the problem is confirmed to be well-posed and some basic preliminaries are provided within the context of the temporal environment. Subsequently, both the local and the global stability of the temporal system including permanence are thoroughly investigated so as to emerge the fact that the competition between the hunting cooperation factor a and the refuge coefficient r can resolve the dynamics of the system. More precisely, global stability for all of the feasible non-negative equilibria corresponding to the temporal environment and the coexistence equilibrium in the spatiotemporal domain are explored in the event of the hunting cooperation factor a not exceeding the prey refuge coefficient r. However, the moment a exceeds r, where both the Hopf bifurcation and the Turing bifurcation are induced by hunting cooperation. Nevertheless, a distinct Turing instability mechanism is emerged when the prey diffusivity exceeds that of predator but interestingly, the opposite is customarily a reasonable constraint in many predator–prey models. Later on, the diffusion coefficient is chosen as a bifurcation parameter interpreting pattern transition and the amplitude equations close to the onset are thereby derived. The stability analysis is made use of to explain the selection of patterns among hot spot patterns, the mixture of hot spots and stripes patterns and the stripe patterns themselves. Finally, numerical simulations are performed to explore pattern selection influenced by the hunting cooperation factor, the prey refuge coefficient and the diffusivity as well. Some interesting dynamical complexities including the variation of the number of equilibria, the bifurcation scenario, etc, also emerge out from such quantitative simulations.

Analysis of a Delayed Reaction-Diffusion Predator–Prey System with Fear Effect and Anti-Predator Behaviour

This paper is devoted to studying the dynamics of a delayed reaction-diffusion predator–prey system incorporating the effects of fear and anti-predator behaviour. First, based on its mathematical model, the global attractor is analyzed and the local stability of its positive equilibria is derived. Moreover, the Hopf bifurcation induced by the time delay variable is also investigated. Furthermore, the existence and non-existence of non-constant positive solutions are analyzed. Finally, numerical simulations are presented to validate the theoretical analysis.

Global bifurcation and pattern formation for a reaction–diffusion predator–prey model with prey-taxis and double Beddington–DeAngelis functional responses

Of concern in this paper is to propose an accurate description for the global bifurcation structure of the nonconstant steady states for a reaction–diffusion predator–prey model with prey-taxis and double Beddington–DeAngelis functional responses systematically. By treating the coefficient of nonlinear prey-taxis as a bifurcation parameter and utilizing the user-friendly version of Crandall–Rabinowitz bifurcation theory, we study the global bifurcation theory of the system. Meanwhile, the existence of nonconstant steady states will be offered by the exported global bifurcation theorem under a rather natural condition. In the proof, a priori estimate of steady states will play an important role. The local stability analysis with a numerical simulation and bifurcation analysis are given.