Predator-prey System Research Articles

Ecological and epidemiological ramification of fear: Exploring deterministic and stochastic dynamics in a predator–prey system with predator switching and harvesting

In an ecosystem, harvesting infected prey can assist in managing and containing the spread of the illness within the prey species. On the other hand, the harvesting of predators can be beneficial as it regulates their numbers, preventing them from over-consuming prey and subsequently preserving existence of the prey population. This study introduces a predator–prey model that encompasses prey infection, predator–prey interactions influenced by fear, switching and harvesting. We derive an analytic expression for the basic reproduction number, a critical determinant of disease spread. We investigate the global stability of disease-free and endemic equilibria contingent on the basic reproduction number’s value, highlighting the potential for disease eradication by maintaining it below unity. In-depth analysis of the deterministic model is undertaken, with a focus on Hopf bifurcations that delineate thresholds for disease-free and endemic states. Furthermore, the deterministic model is extended to incorporate environmental stochasticity. We obtain the conditions under which population extinction occurs. Our findings elucidate how the intensity of environmental noise influences population dynamics, providing valuable insights into extinction risks under varying noise levels.

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.

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: I global existence and stability

A diffusive predator-prey system with hunting cooperation in predators and prey-taxis: I global existence and stability

Dynamic Analysis of a Fractional-Modified Leslie-Gower Model with Predator Harvesting

This paper presents an investigation of a modified Leslie-Gower predator-prey model that incorporates fractional discrete-time Michaelis-Menten type prey harvesting. The analysis focuses on the topology of nonnegative interior fixed points, including their existence and stability dynamics. We derive conditions for the occurrence of flip and Neimark-Sacker bifurcations using the center manifold theorem and bifurcation theory. Numerical simulations, conducted with a computer package, are presented to demonstrate the consistency of the theoretical findings. Overall, our study sheds light on the complex dynamics that arise in this model and highlights the importance of considering fractional calculus in predator-prey systems with harvesting.

Modelling the prudent predation in predator–prey interactions

Prudent predators may evolve a strategy of prudent feed at a suitable rate, which is detrimental to their survival, but would not overexploit the prey thus is beneficial to the sustainability of resources. In this paper, by introducing a prey-dependent predation rate function, a two prey and one predator system with prudent predation is established, and the dynamics of the system as well as its subsystems are investigated. The existence and stability of the equilibrium are analyzed and the occurrence of Hopf bifurcation is studied. Numerical simulations are carried out to verify the analytical results and expand the theoretical analyses: (i) In the subsystems, it is possible to have multiple Hopf bifurcation points and prudence acts as a stabilizing factor; (ii) Suitable level of prudence will benefit the predator while sustaining the prey; (iii) Prudent predation can stabilize the system from chaos, which means chaotic dynamics can be controlled by the prudent predation. These results may reveal the important role of predator initiative in predator–prey interactions and enrich the dynamics of predator–prey system.

Dynamics of a Delayed Prey-Predator System With Crowley-Martin Functional Response

ln this paper, an ecological predator-prey model with time delay is proposed and studied analytically and numerically. The stability and bifurcation analysis of the proposed model with a Crowley-Martin response function are covered in this article. In the beginning, equilibrium points of the proposed model are identified. Secondly, the local asymptotic stability of the equilibria and Hopf-bifurcation are discussed by the characteristic equations of the system. Finally, by creating a suitable Lyapunov function and LaSalle’s invariance principle, the equilibria's global asymptotic stability is examined.

Discrete-Time Predator–Prey System Incorporating Fear Effect: Stability, Bifurcation, and Chaos Control

Discrete-Time Predator–Prey System Incorporating Fear Effect: Stability, Bifurcation, and Chaos Control

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.

Bifurcation analysis and exploration of noise-induced transitions of a food chain model with Allee effect

Seeking shelter from threats is a widespread instinct across species, specially employed by prey to avoid direct confrontations with predators. The present investigation centers on a three-species food chain model wherein the basal prey, characterized by logistic growth, seeks refuge to evade the intermediate predator, while the intermediate predator, in turn, seeks refuge to avoid encounters with the top predator. Additionally, our model assumes that the presence of the top predator induces a mate-finding Allee effect among the intermediate predator. We investigate how varying levels of refuge, along with other critical parameters such as the reproduction rate of the basal prey and the natural mortality rate of the top predator, influence system’s dynamics within the biparametric planes. Our model displays multistability and undergoes transcritical, saddle–node, Bogdanov–Takens and cusp bifurcations across different parameters. Moreover, the external environmental noise can induce interesting dynamics in the predator–prey system, resulting in noise-induced frequent transitions between distinct interior attractors or from interior to axial attractors. This phenomenon is particularly notable in scenarios where the deterministic model exhibits tristability. In summary, our findings offer potential new avenues for developing control strategies within the realm of community ecology in constant as well as fluctuating environments.

Inhomogeneous Hopf Bifurcation in a Diffusive Predator–Prey System with Indirect Predator-Taxis

Inhomogeneous Hopf Bifurcation in a Diffusive Predator–Prey System with Indirect Predator-Taxis

Pattern dynamics in a predator–prey model with Smith growth function and prey refuge in predator poisoned environment

In this article, we investigate temporal as well as spatiotemporal dynamics of a predator–prey system where prey grows according to Smith growth function and prey refuges in predator poisoned environment. Self as well as cross diffusion are taken into consideration to describe spatial moment of the species which make the system more realistic. This study aims to investigate the role of prey refuge, toxin-carrying carcass and the impact of cross-diffusion on the spatial distribution of species densities and associated patterns. The existence of equilibria and their stability conditions for non-spatial system are derived. We discuss the occurrence of Hopf bifurcation and detect the stability of limit cycle by computing first Lyapunov number. Temporal dynamics of the system is studied by varying the significant parameters. Study reveals that both prey refuge and fatality rate of predator due to toxin-carrying carcass, initially makes the system stable for low refuge and fatality rate but predator population goes to extinction as those effects increase. Turing instability conditions are listed and different instability regions so formed are discussed. In Turing region, pattern selection with the help of amplitude equation via weakly nonlinear analysis are performed which also numerically supported. Intensity of refuge and fatality rate due to toxin carrying carcass result paradox with real ecological world due to the presence of cross diffusion. Various kind of patterns such as mixture of stripe and spot, spot, irregular chaotic patterns emerge due to Hopf and Turing instability. Particularly, spatiotemporal chaos is a controversial topic because of its significant consequences for population dynamics. The results of this work are expected to contribute to a better understanding the level of prey refuge, use of toxin-carrying carcass considering movement in population and effect of cross-diffusion.

Stability, bifurcation analysis and chaos control in a discrete predator–prey system incorporating prey immigration

Stability, bifurcation analysis and chaos control in a discrete predator–prey system incorporating prey immigration

Some Bifurcations of Codimensions 1 and 2 in a Discrete Predator–Prey Model with Non-Linear Harvesting

This paper delves into the dynamics of a discrete-time predator–prey system. Initially, it presents the existence and stability conditions of the fixed points. Subsequently, by employing the center manifold theorem and bifurcation theory, the conditions for the occurrence of four types of codimension 1 bifurcations (transcritical bifurcation, fold bifurcation, flip bifurcation, and Neimark–Sacker bifurcation) are examined. Then, through several variable substitutions and the introduction of new parameters, the conditions for the existence of codimension 2 bifurcations (fold–flip bifurcation, 1:2 and 1:4 strong resonances) are derived. Finally, some numerical analyses of two-parameter planes are provided. The two-parameter plane plots showcase interesting dynamical behaviors of the discrete system as the integral step size and other parameters vary. These results unveil much richer dynamics of the discrete-time model in comparison to the continuous model.

Predator–prey systems with a variable habitat for predators in advective environments

AbstractCommunity composition in aquatic environments can be shaped by a broad array of factors, encompassing habitat conditions in addition to abiotic conditions and biotic interactions. This paper pertains to reaction–diffusion–advection predator–prey model featuring a variable predator habitat in advective environments, governed by a unidirectional flow. First, we establish the near‐complete global dynamics of the system. In instances where the functional response to predation conforms to Holling‐type I or II, we explore the uniqueness and stability of positive steady‐state solutions via the application of particular auxiliary techniques, the comparison principle for parabolic equations, and perturbation analysis. Furthermore, we obtain the critical positions at the upper and lower boundaries of the predator's habitat, which determine the survival of the prey irrespective of the predator's growth rate. Finally, we show how the location and length of the predator's habitat affect the persistence and extinction of predators and prey in the event of a small population loss rate at the downstream end. From the biological point of view, these results contribute to our deeper understanding of the effects of habitat on aquatic populations and may have applications in aquaculture and the establishment of protection zones for aquatic species.

Minimal wave speed and spreading speed in predator-prey systems with stage structure

This article studies the minimal wave speed of traveling wave solutions as well as spreading speed in predator-prey systems. The model divides the prey into the immaturity and maturity, and the predator feeds on the immature individuals. For the traveling wave solutions connecting the predator-free equilibrium with the coexistence state, we consider two different diffusion modes. For both modes, the minimal wave speed is shown by presenting the existence or nonexistence of nontrivial traveling wave solutions for all wave parameters. When the diffusion is local, the minimal wave speed is the spreading speed of the corresponding initial value problem, in which the initial condition implies that the predator is the invader while the prey is the aborigine in the whole habitat. Under the persistence assumption in the corresponding kinetic systems, our conclusion coincides with the fact that the diffusion of the prey may have less effect on the invasion ability of the predator.

Local and global dynamics of a prey–predator system with fear, Allee effect, and variable attack rate

Fear prompts prey to adopt risk-averse behaviors, such as reduced foraging activity, increased vigilance, and avoidance of areas with high predator presence, which affects its reproduction. In a real scenario, a population requires a minimum density to avoid extinction, known as an Allee threshold. In light of these biological factors, we propose a predator–prey model with (i) a fear effect in a prey population, (ii) an Allee effect in a predator population, and (iii) a non-constant attack rate that modifies the functional response. We ensured the non-negativity and boundedness of the solutions and examined the local and global stability status for each existing steady state solutions. We investigated some deep dynamical properties of the system by varying different parameters, such as cost of fear in prey and strength of the Allee effect in predators and their mortality rate. In codimension one bifurcations, we observed saddle node, Hopf, homoclinic, and coalescence of two limit cycles. Additionally, codimension two bifurcations were observed, including Bautin and Bogdanov Takens bifurcations. To provide a clearer understanding of these bifurcations, we conducted biparametric analysis involving the fear and Allee parameters, as well as the fear parameter and predator mortality rate. Our investigation shows that cost of fear and strength of Allee strongly influences the survival status of the predator. Furthermore, bistability and tristability reveal that the survival and extinction of predator are dependent on the initial population level. Numerical simulations and graphical illustrations are provided to support and validate our theoretical findings.

Dynamical Behaviour of a Predator-Prey System with Holling Type III Functional Response under Harvesting and Self-crowding

Dynamical Behaviour of a Predator-Prey System with Holling Type III Functional Response under Harvesting and Self-crowding

Nonlinear dynamics in a fear‐driven predator–prey system: Bistability, bifurcations, hydra effect, and optimal harvesting

The impact of predator‐driven fear on ecosystems is significant and can encompass both trophic (direct) and nontrophic (indirect) effects. Previous studies have shown that nontrophic fear effects have an important role in predator–prey dynamics. This study investigates the nontrophic fear effect on prey caused by generalist predators and explores optimal harvesting. We assume that the reproduction rate of prey is reduced due to the fear effect, and generalist predator follows Holling type II foraging strategy for predation. Additionally, we assume that predators are commercially valuable and harvested proportionately to their density. We demonstrate the existence of equilibrium points, their local and global stability, and bifurcation analysis. We observe that the model system undergoes a sequence of codimension one and codimension two bifurcations. Our results show that in the absence of predator harvesting, increasing levels of fear destabilize the predator–prey system, and controlled harvesting is beneficial for the coexistence of both populations. Also, the harvesting of predators may produce hydra and multiple hydra effects. We identify different types of bistability phenomena, which emphasize the importance of initial population size. Further, an optimal harvesting policy is also explored by formulating an optimal control problem (OCP). The harvesting cost‐functional is designed by incorporating the bionomic equilibrium state. We use Pontryagin's maximum principle and solve the OCP numerically. It is observed that implementing optimal harvesting not only contributes to the ecological benefits by maintaining a sustainable balance of predator–prey evolution but also plays a significant role in maximizing the economic benefits.

A complete analysis of a slow–fast modified Leslie–Gower predator–prey system

In this paper, we transform a modified Leslie–Gower predator–prey system into the corresponding fast–slow version by assuming that prey reproduces much faster than predator, and then perform a complete analysis about its dynamics. More specifically, we find the necessary and sufficient conditions of the exact number (zero, one or two) of positive equilibria of the slow–fast system and its (or their) location, and then we further fully determine its (or their) dynamics under explicit conditions. Besides, by converting the slow–fast system into its slow–fast normal form, we are able to characterize its rich dynamics completely, including relaxation oscillation, singular Hopf bifurcation, canard explosion, homoclinic orbits, heteroclinic orbits and global attraction of equilibrium. Moreover, the sufficient conditions to guarantee these various rich dynamics are explicitly given, including the explicit conditions to determine whether singular Hopf bifurcation is supercritical or subcritical, which generally cannot be explicitly derived in the existing literatures. Additionally, the cyclicity of diverse canard cycles is found under explicit conditions. Of particular interest is that we show the existence and uniqueness of one canard cycle without head whose cyclicity is at most two under explicit parameters conditions. Our results complement and enrich the previous work (relaxation oscillation, and global attraction of a boundary equilibrium) about this fast–slow system. We also employ numerical simulations to illustrate our theoretical analysis.