Reaction-diffusion Predator-prey System Research Articles

A novel binary data classification algorithm based on the modified reaction-diffusion predator-prey system with Holling-II function.

In this study, we propose a modified reaction-diffusion prey-predator model with a Holling-II function for binary data classification. In the model, we use u and v to represent the densities of prey and predators, respectively. We modify the original equation by substituting the term v with f-v to obtain a stable and clear nonlinear decision surface. By employing a finite difference method for numerical solution of the original model, we conduct various experiments in two-dimensional and three-dimensional spaces to validate the feasibility of the classifier. Additionally, with consideration for wide real applications, we perform classification experiments on electroencephalogram signals, demonstrating the effectiveness and robustness of the classifier in binary data classification.

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.

Global bifurcation in a diffusive Beddington-DeAngelis predator–prey model with population flux by attractive transition

The study involves examining the global bifurcation structure associated with the nonconstant steady states of a reaction-diffusion predator-prey system where both the species interact in accordance with the Beddington DeAngelis response and the movement flux of the predator incorporates attractive transition. We consider the magnitude of population flux by attractive transition as the bifurcation parameter and employ the Crandall-Rabinowitz bifurcation theorem to study the global bifurcation structure associated with the problem. We have also derived some a priori estimates associated with the problem and carried out numerical simulations to support our theoretical results. This work can be regarded as the first step towards inclusion of population flux by attractive transition in scenarios where interactions are governed by complex functional responses.

Parameter identification of a reaction-diffusion predator-prey system based on optimal control theory

This paper takes a reaction-diffusion predator-prey system with ratio-dependent Holling III functional response function and Leslie-Gower term into consideration. First of all, the system model is proposed on the basis of basic biological assumptions and previous work, and the existence conditions of the equilibrium point of the system are discussed. Secondly, under the assumption of the existence of equilibrium point, the Turing instability necessary conditions induced by diffusion are investigated. Thirdly, optimal control theory is derived, and the adjoint system and the first-order optimization condition are established. Fourthly, parameter identification based on optimal control is studied, and the technique is extended to network structure. Finally, extensive numerical simulations, including Turing pattern, Normalized Population High Distribution Area (NPHDA) diagram and parameter identification, are carried out to illustrate and validate the analytical results. For the system dynamics phenomena, the results from different perspectives effectively demonstrate that the theoretical findings, numerical simulations and natural reality are identical. In terms of the parameter identification of continuous model and network model, the efficiency and accuracy of various algorithms are fully tested.

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.

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.

Dynamics of a delayed reaction-diffusion predator-prey model with the effect of the toxins.

In this study, we investigate a delayed reaction-diffusion predator-prey system with the effect of toxins. We first investigate whether the internal equilibrium exists. We then provide certain requirements for the presence of Turing and Hopf bifurcations by examining the corresponding characteristic equation. We also study Turing-Hopf and Hopf bifurcations brought on by delays. Finally, numerical simulations that exemplify our theoretical findings are provided. The quantitatively obtained properties are in good agreement with the findings that the theory had predicted. The effects of toxins on the system are substantial, according to theoretical and numerical calculations.

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 for a reaction–diffusion predator–prey model with Holling-II functional response and prey–taxis

Abstract The aim of this study is to establish a precise illustration of the structure of the nonconstant steady state for a Holling-II predator–prey reaction–diffusion system, with prey-taxis. We treat the nonlinear prey-taxis as a bifurcation parameter to discuss the global bifurcation of the system. Furthermore, the prey-taxis term, under a rather natural condition, offers nonconstant steady states. In the proof, a priori estimates and the regularity of the steady states will play an important role. Numerical simulation is provided to support our main theoretical results. Finally, conclusions are drawn to summarise the main analytical results.

Consequences of refuge and diffusion in a spatiotemporal predator–prey model

In this investigation, we offer and examine a predator–prey interacting model with prey refuge in proportion to both the species and Beddington–DeAngelis functional response. We first prove the well-posedness of the temporal and spatiotemporal models which are restricted in a positive invariant region. Then for the temporal model, we analyse its temporal dynamics including uniform boundedness, permanence, stability of all feasible non-negative equilibria and show that refugia can induce periodic oscillation via Hopf bifurcation around the unique positive equilibrium; for the spatiotemporal model, we not only investigate its permanence, stability of non-negative constant steady states and Turing instability but also study the existence and non-existence of non-constant positive steady states by Leray–Schauder degree theory. The key observation is that the coefficient of refuge cooperates a significant part in modifying the dynamics of the current system and mediates the population permanence, stability of coexisting equilibrium and even the Turing instability parameter space. Finally, general numerical simulation consequences are given to illustrate the validity of the theoretical results. Through numerical simulations, one observes that the model dynamics shows prey refugia and self-diffusion control spatiotemporal pattern growth to spots, stripe–spot mixtures and stripes reproduction. The outcomes assign that the dynamics of the model with prey refuge is not simple, but rich and complex. Additionally, numerical simulations show that the other model parameters have an important effect on species’ spatially inhomogeneous distribution, which results in the formation of spots pattern, mixture of spots and stripes pattern, mixture of spots, stripes and rings pattern and anti-spot pattern. This may improve the model dynamics of the prey refuge on the reaction–diffusion predator–prey system.

Pattern dynamics of a reaction-diffusion predator-prey system with both refuge and harvesting

We are concerned with a reaction-diffusion predator–prey model under homogeneous Neumann boundary condition incorporating prey refuge (proportion of both the species) and harvesting of prey species in this contribution. Criteria for asymptotic stability (local and global) and bifurcation of the subsequent temporal model system are thoroughly analyzed around the unique positive interior equilibrium point. For partial differential equation (PDE), the conditions of diffusion-driven instability and the Turing bifurcation region in two-parameter space are investigated. The results around the unique interior feasible equilibrium point specify that the effect of refuge and harvesting cooperation is an important part of the control of spatial pattern formation of the species. A series of computer simulations reveal that the typical dynamics of population density variation are the formation of isolated groups within the Turing space, that is, spots, stripe-spot mixtures, labyrinthine, holes, stripe-hole mixtures and stripes replication. Finally, we discuss spatiotemporal dynamics of the system for a number of different momentous parameters via numerical simulations.

Dynamical Analysis of a Beddington–DeAngelis Interacting Species System with Prey Harvesting

A reaction–diffusion interacting species system with Beddington–DeAngelis functional response that has been proposed in the environment of mathematical ecology, which provides the rise to spatial pattern formation, is investigated and associated with the models of deterministic dynamics. The dynamical behaviour of a generalist predator–prey system with linear harvesting of each species and predator-dependent functional response is fully analyzed. Conditions of stability behaviour of the interior equilibrium point are established properly. Furthermore, we have recognized that the unique positive equilibrium point of the system is globally stable via appropriate Lyapunov function structure, which signifies that appropriate harvesting has no impact on the persistence property of the harvesting system. Also, we establish the conditions for the existence of bifurcation phenomena including a saddle-node bifurcation and a Hopf bifurcation. Subsequently, complete analysis regarding the impact of harvesting is carried out, and an interesting decision is that under some appropriate constraints, harvesting has immense impact on the final size of the interacting species. In addition, in accordance with Turing’s ideas on morphogenesis , our analysis shows that harvesting effort in a reaction–diffusion predator–prey system plays a vital function for geological conservation of interacting species. Finally, we discuss sufficient conditions for the existence of bionomic equilibrium point and the optimal harvesting policy attained by using the Pontryagin maximal principle.

Pattern formation in diffusive predator-prey systems with predator-taxis and prey-taxis

A reaction-diffusion predator-prey system with prey-taxis and predator-taxis describes the spatial interaction and random movement of predator and prey species, as well as the spatial movement of predators pursuing prey and prey evading predators. The spatial pattern formation induced by the prey-taxis and predator-taxis is characterized by the Turing type linear instability of homogeneous state and bifurcation theory. It is shown that both attractive prey-taxis and repulsive predator-taxis compress the spatial patterns, while repulsive prey-taxis and attractive predator-taxis help to generate spatial patterns. Our results are applied to the Holling-Tanner predator-prey model to demonstrate the pattern formation mechanism.

Global Asymptotic Stability and Hopf Bifurcation in a Homogeneous Diffusive Predator-Prey System with Holling Type II Functional Response

In this paper, we considered a homogeneous reaction-diffusion predator-prey system with Holling type II functional response subject to Neumann boundary conditions. Some new sufficient conditions were analytically established to ensure that this system has globally asymptotically stable equilibria and Hopf bifurcation surrounding interior equilibrium. In the analysis of Hopf bifurcation, based on the phenomenon of Turing instability and well-done conditions, the system undergoes a Hopf bifurcation and an example incorporating with numerical simulations to support the existence of Hopf bifurcation is presented. We also derived a useful algorithm for determining direction of Hopf bifurcation and stability of bifurcating periodic solutions correspond to j ≠ 0 and j = 0, respectively. Finally, all these theoretical results are expected to be useful in the future study of dynamical complexity of ecological environment.

SPATIAL PATTERN FORMATIONS IN DIFFUSIVE PREDATOR-PREY SYSTEMS WITH NON-HOMOGENEOUS DIRICHLET BOUNDARY CONDITIONS

A reaction-diffusion predator-prey system with non-homogeneous Dirichlet boundary conditions describes the persistence of predator and prey species on the boundary. Compared with homogeneous Neumann boundary conditions, the former conditions may prompt or prevent the spatial patterns produced through diffusion-induced instability. The spatial pattern formation induced by non-homogeneous Dirichlet boundary conditions is characterized by the Turing type linear instability of homogeneous state and bifurcation theory. Furthermore, transient spatiotemporal behaviors are observed through numerical simulations.

The Influence of Dirichlet Boundary Conditions on the Dynamics for a Diffusive Predator–Prey System

A reaction–diffusion predator–prey system with homogeneous Dirichlet boundary conditions describes the lethal risk of predator and prey species on the boundary. The spatial pattern formations with the homogeneous Dirichlet boundary conditions are characterized by the Turing type linear instability of homogeneous state and bifurcation theory. Compared with homogeneous Neumann boundary conditions, we see that the homogeneous Dirichlet boundary conditions may depress the spatial patterns produced through the diffusion-induced instability. In addition, the existence of semi-trivial steady states and the global stability of the trivial steady state are characterized by the comparison technique.

Mathematical analysis and numerical simulation of a fractional reaction-diffusion system with Holling-type III functional response

In recent years, many investigators have questioned the use of convectional diffusion equation to model many physical or real life situations. As a result, fractional space derivatives have been proposed to model anomalous diffusion or related processes, where a particle plume spreads at inconsistent rate with the classical Brownian motion model. By replacing the second derivative in the classical diffusion model with fractional derivative, results to enhance a process known as superdiffusion. A high-dimensional predator-prey reaction-diffusion system with Holling-type III functional response, where the usual second-order derivatives give place to a fractional derivative of order α with 1 < α ≤ 2. Analysis of the main equation guides in the correct choice of parameter values. We established the condition for local and global stabilities. We also show that the system undergoes a Hopf bifurcation subject to a small perturbation of the steady-state solution. The complexity of fractional derivative at some instances of order α for the superdiffusive scenario is demonstrated with some numerical experiments in one, two and three dimensions. The effectiveness of the numerical method is demonstrated through numerical simulations to confirm the theoretical results.