Predator-prey System Research Articles

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.

Existence, uniqueness and Ulam–Hyers stability result for variable order fractional predator-prey system and it's numerical solution

This study presents an approximate numerical technique for solving time fractional advection-diffusion-reaction predator-prey equations with variable order (VO), where the analyzed fractional derivatives of VO are in the Caputo sense. Results for Ulam–Hyers stability are shown, as well as the existence and uniqueness of solutions. It is suggested to use a numerical approximation based on the shifted second kind of airfoil polynomials to solve the equations under consideration. A fractional derivative operational matrix with VO is derived for shifted airfoil polynomials, which will be used to compute the unknown function. The main equations are transformed into a set of algebraic equations by substituting the aforementioned operational matrix into the equations under consideration and utilizing the properties of the shifted airfoil polynomial along with the collocation points. A numerical solution is obtained by solving the acquired set of algebraic equations. To verify the accuracy and efficiency of the discussed scheme, several illustrative examples have been considered. The results obtained by the proposed method demonstrate the efficiency and superiority of the method compared to other existing methods.

Effect of fear and non-linear predator harvesting on a predator–prey system in presence of environmental variability

In this paper, we have proposed and analyzed a predator–prey system introducing the cost of predation fear into the prey reproduction with Holling type-II functional response in the stochastic environment with the consideration of non-linear harvesting on predators. The system experiences Transcritical, Saddle–node, Hopf, and Bogdanov-Taken (BT) bifurcation with respect to the intrinsic growth rate and competition rate of the prey populations. We have discussed the existence and uniqueness of positive global solution of the stochastic model with the help of Ito’s integral formula and the long-term behavior of the solution is derived here. The existence of stationary distribution and explicit form of the density function is established here when only prey populations survive or both populations. We have shown that due to high fluctuation, the regime changes from one stable state to another state when bistability occurs in the system. The paper ends with some conclusions.

Canard cycle and nonsmooth bifurcation in a piecewise-smooth continuous predator-prey model

This article establishes a bifurcation analysis of a singularly perturbed piecewise-smooth continuous predator–prey system with a sufficiently small parameter. The bifurcation that can generate limit cycles here is our main concern. To achieve this goal, we have developed a lemma that is used to determine the parameter region that can generate limit cycles. Further conclusions indicate that the existence of a 2-shaped critical manifold is required. Based on the Poincaré-Bendixon lemma, Fenichel’s theory and geometric singular perturbation theory, we demonstrate the possibility of generating smooth and nonsmooth bifurcations. In fact, nonsmooth bifurcations only occur in piecewise-smooth systems. More specifically, different types of nonsmooth bifurcations are also presented in this article, including nonsmooth Hopf bifurcation, Hopf-like bifurcation and super-explosion. In addition, this article discusses the existence of crossing limit cycles and explains whether the crossing limit cycle is characterized by canard cycles without head, canard cycles with head or relaxation oscillations. Furthermore, the coexistence of two relaxation oscillations, the coexistence of two canard cycles without head, and the coexistence of one relaxation oscillation and one canard cycle without head are investigated. Moreover, the one-parameter bifurcation diagram is also presented in this paper through numerical simulations.

Complex dynamical properties and chaos control for a discrete modified Leslie-Gower prey-predator system with Holling II functional response

In this study, the semi-discretization technique is employed to establish a discrete representation of a modified Leslie-Gower prey-predator system that includes a Holling II type functional response. The dynamics of this model are then analyzed through the application of center manifold theory and bifurcation theory. We present comprehensive results for the local stability of the fixed points across the entire parameter space. Additionally, we provide sufficient conditions for the occurrence of flip bifurcation and Neimark-Sacker bifurcation. Besides, the system has experienced a flip bifurcation to chaos controlled using the method of chaos control, viz., state feedback method, pole placement technique, and hybrid control strategy. Furthermore, we provide specific conditions to ensure that bifurcation and chaos can be stabilized. Finally, numerical simulations are conducted to validate theoretical analysis and illustrate several new complex dynamical behaviors between two species.

Theoretical and Numerical Bifurcation Analysis of a Discrete Predator–Prey System of Ricker Type with Weak Allee Effect

Theoretical and Numerical Bifurcation Analysis of a Discrete Predator–Prey System of Ricker Type with Weak Allee Effect

Dynamics of a Diffusive Predator–Prey System with Fear Effect in Advective Environments

We explore a diffusive predator–prey system that incorporates the fear effect in advective environments. First, we analyze the eigenvalue problem and the adjoint operator, considering Constant-Flux and Dirichlet boundary conditions, as well as Free-Flow boundary conditions. Next, we use the principle of comparison to prove the non-negativity of the solution. Our investigation focuses on determining the direction and stability of spatial Hopf bifurcation, with the generation delay serving as the bifurcation parameter. Additionally, we examine the influence of both linear and Holling-II functional responses on the dynamics of the model. Through these analyses, we gain better understanding of the intricate relationship among advection, predation, and prey response in this system.

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.

Spatiotemporal flow-induced instability of predator-prey model with Crowley-Martin functional response and prey harvesting.

Ecological systems can generate striking large-scale spatial patterns through local interactions and migration. In the presence of diffusion and advection, this work examines the formation of flow-induced patterns in a predator-prey system with a Crowley-Martin functional response and prey harvesting, where the advection reflects the unidirectional flow of each species migration (or flow). Primarily, the impact of diffusion and advection rates on the stability and the associated Turing and flow-induced patterns are investigated. The theoretical implication of flow-induced instability caused by population migration, mainly the relative migrations between prey and predator, is examined, and it also shows that Turing instability is the particular condition of flow-induced instability. The influence of the relative flow of both species and prey-harvesting effort on the emerging pattern is reported. Advection impacts a wide range of spatiotemporal patterns, including bands, spots, and a mixture of bands and spots in both harvested and unharvested dynamics. We also observe the diagonally bend-type banded patterns and straight-type banded patterns due to positive and negative relative flows, respectively. Here, the increasing relative flow increases the band length. The growing harvesting effort also decreases the band length, producing a thin band and a mixture of spots and bands due to the negative and positive relative flows, respectively. One exciting result observed here is that harvesting effort drives the flow-Turing and flow-Turing-Hopf instability into pure-flow instability.

Spiral organization of quasi-periodic shrimp-shaped domains in a discrete predator-prey system.

In this paper, we report the discovery of some novel dynamical scenarios for quasi-periodic shrimp-shaped structures embedded within chaotic phases in bi-parameter space of a discrete predator-prey system. By constructing high-resolution, two-dimensional stability diagrams based on Lyapunov exponents, we observe the abundance of both periodic and quasi-periodic shrimp-shaped organized domains in a certain parameter space of the system. A comprehensive comparative analysis is conducted to elucidate the similarities and differences between these two types of shrimps. Our analysis reveals that, unlike periodic shrimp, quasi-periodic shrimp induces (i) torus bubbling transition to chaos and (ii) multistability with multi-tori, torus-chaotic, and multi-chaotic coexisting attractors, resulting from the crossing of its two inner antennae. The basin sets of the coexisting attractors are analyzed, and we observe the presence of intriguing basin boundaries. We also verify that, akin to periodic shrimp structures, quasi-periodic shrimps also maintain the three-times self-similarity scaling. Furthermore, we encounter the occurrence of spiral organization for the self-distribution of quasi-periodic shrimps within a large chaotic domain. We believe that these novel findings will significantly enhance our understanding of shrimp-shaped structures and the intricate dynamics exhibited by their distribution in chaotic regimes.

Complex Dynamics of a Discretized Predator–Prey System with Prey Refuge Using a Piecewise Constant Argument Method

The objective of this work is to investigate the complex dynamics of a discrete predator–prey system using the method of piecewise constant argument for discretization. An analysis is conducted to examine the presence and stability of fixed points. Furthermore, the system is shown to undergo period-doubling (PD) and Neimark–Sacker (NS) bifurcations by the use of center manifold and bifurcation theories. The feedback and hybrid control strategies are used to regulate the system’s bifurcating and chaotic behaviors. Both strategies seem to be effective in managing bifurcation and chaos inside the system. Finally, the main results are validated by numerical evidence. Parameters of the system are varied to produce time graphs, phase portraits, bifurcation diagrams, and maximum Lyapunov exponent (MLE) graphs. The discrete model displays rich dynamics, as seen in the numerical simulations and graphs, indicating a complex and chaotic system.

Chaotic Dynamics of the Fractional Order Predator-Prey Model Incorporating Gompertz Growth on Prey with Ivlev Functional Response

This paper examines dynamic behaviours of a two-species discrete fractional order predator-prey system with functional response form of Ivlev along with Gompertz growth of prey population. A discretization scheme is first applied to get Caputo fractional differential system for the prey-predator model. This study identifies certain conditions for the local asymptotic stability at the fixed points of the proposed prey-predator model. The existence and direction of the period-doubling bifurcation, Neimark-Sacker bifurcation, and Control Chaos are examined for the discrete-time domain. As the bifurcation parameter increases, the system displays chaotic behaviour. For various model parameters, bifurcation diagrams, phase portraits, and time graphs are obtained. Theoretical predictions and long-term chaotic behaviour are supported by numerical simulations across a wide variety of parameters. This article aims to offer an OGY and state feedback strategy that can stabilize chaotic orbits at a precarious equilibrium point.

Dynamics of a Predator–Prey System with Impulsive Stocking Prey and Nonlinear Harvesting Predator at Different Moments

In this article, we study a predator–prey system, which includes impulsive stocking prey and a nonlinear harvesting predator at different moments. Firstly, we derive a sufficient condition of the global asymptotical stability of the predator–extinction periodic solution utilizing the comparison theorem of the impulsive differential equations and the Floquet theory. Secondly, the condition, which is to maintain the permanence of the system, is derived. Finally, some numerical simulations are displayed to examine our theoretical results and research the effect of several important parameters for the investigated system, which shows that the period of the impulse control and impulsive perturbations of the stocking prey and nonlinear harvesting predator have a significant impact on the behavioral dynamics of the system. The results of this paper give a reliable tactical basis for actual biological resource management.

A Predator–Prey System with a Modified Leslie–Gower and Prey Stage Structure Scheme in Deterministic and Stochastic Environments

The evolution of the population ecosystem is closely related to resources and the environment. Assuming that the environmental capacity of a predator population is positively correlated with the number of prey, and that the prey population has a sheltered effect, we investigated a predator–prey model with a juvenile–adult two-stage structure. The dynamical behaviour of the model was examined from two distinct environmental perspectives, deterministic and stochastic, respectively. For the deterministic model, the conditions for the existence of equilibrium points were obtained by comprehensive use of analytical and geometric methods, and the local and global asymptotic stability of each equilibrium point was discussed. For the stochastic system, the effect of noise intensity on the long-term dynamic behavior of the population was investigated. By constructing appropriate Lyapunov functions, the criteria that determined the extinction of the system and the ergodic stationary distribution were given. Finally, through concrete examples and numerical simulations, the understanding of the dynamic properties of the model was deepened. The results show that an improvement in the predator living environment would lead to the decrease in the prey population, while more prey shelters could lead to the decline or even extinction of predator populations.

Stability and Hopf Bifurcation of a Delayed Prey-Predator System with Fear, Hunting Cooperative, and Allee Effect

The effect of fear, the hunting cooperative process, and Allee's impact on the behavior of an ecological system are investigated and discussed. The impact of the delay of the prey's response to the predation risk is included. The Leslie-Gower growth is used to describe the growth of the predator population. Firstly, the solutions' existence, positivity, and boundedness within the limits of a suitable region in the parametric space for all time are studied. The stability of all equilibrium points under the surrounding environmental effects is established. Moreover, the occurrence of a Hopf bifurcation is discovered. The stability of the bifurcating periodic dynamic and their dynamical properties are studied. Finally, the obtained theoretical results are confirmed and validated utilizing numerical simulation. It is observed that the system possesses a bi-stable behavior and a Hopf bifurcation.

Turing-Hopf bifurcation analysis and normal form in delayed diffusive predator–prey system with taxis and fear effect

Turing-Hopf bifurcation analysis and normal form in delayed diffusive predator–prey system with taxis and fear effect

Joint Impact of Maturation Delay and Fear Effect on the Population Dynamics of a Predator-Prey System

Joint Impact of Maturation Delay and Fear Effect on the Population Dynamics of a Predator-Prey System

Deciphering two delay dynamics of ecological system with generalist predator incorporating competitive interference

In this manuscript, an attempt has been made to understand the delay induced (gestation and carry-over effect delay) dynamics of an ecological system with generalist predator exerted fear and its carry-over effect with competitive interference. The designed model exhibits finite time blow up depending on large initial data. The stability of both the delayed and non-delayed systems have been analyzed along with Hopf-bifurcation analysis. It has been observed that carry-over and fear effects act in opposite way in context of stability control for non-delayed system. The two delay (carry-over effect and gestation delay) have significant impact on the dynamics. The former exhibits both stabilizing and destabilizing role while the latter has a destabilizing tendency on the system dynamics. The blow up phenomena for predator species have been shown numerically by verifying the analytical conditions. Our study incorporates a diverse array of figures and diagrams to illustrate and support our findings. Through the exploration of non-linear models, our research unveils several intriguing characteristics. These insights can prove invaluable for biologists seeking a more detailed and pragmatic understanding of generalist predator–prey systems. The visual representations provided in our study contribute to a comprehensive analysis, enhancing the accessibility and applicability of the findings for researchers and practitioners in the field.

Dynamic analysis of an infected predator-prey system with prey herd behavior

ABSTRACT To protect themselves from predation, sometimes prey starts roaming in groups and shows herd behavior. This paper addresses the predator-prey system dynamics with prey herd behavior and infection in predators. We explore the conditions under which all equilibria exist and the stability of each equilibrium point of the system. The outcomes of this work suggest that prey herd behavior is beneficial for the prey population when the predators are diseased. Under suitable conditions, the three populations can coexist and maintain the ecological balance of nature.