Discrete Predator-prey Model Research Articles

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.

Complex Dynamics of a Discrete Prey–Predator Model Exposing to Harvesting and Allee Effect on the Prey Species with Chaos Control

Complex Dynamics of a Discrete Prey–Predator Model Exposing to Harvesting and Allee Effect on the Prey Species with Chaos Control

Bifurcation and Stability Analysis of a Discrete Predator–Prey Model with Alternative Prey

Bifurcation and Stability Analysis of a Discrete Predator–Prey Model with Alternative Prey

A discrete-time dynamical model of prey and stage-structured predator with juvenile hunting incorporating negative effects of prey refuge

This paper examines a discrete predator–prey model that incorporates prey refuge and its detrimental impact on the growth of the prey population. Age structure is taken into account for predator species. Furthermore, juvenile hunting along with its negative effects in the form of prey counterattack is considered. This paper provides a comprehensive analysis of the existence and stability conditions pertaining to all possible fixed points of this system. The impact of the parameters reflecting prey growth and prey refuge is thoroughly discussed. This paper delves into the occurrence of the Neimark–Sacker bifurcation and period-doubling bifurcation, specifically in relation to the parameters associated with prey growth rate and prey refuge. Numerous numerical simulations are presented in order to validate the theoretical findings.

Analysis of the Dynamical Properties of Discrete Predator-Prey Systems with Fear Effects and Refuges

This paper examines the dynamic behavior of a particular category of discrete predator-prey system that feature both fear effect and refuge, using both analytical and numerical methods. The critical coefficients and properties of bifurcating periodic solutions for Flip and Hopf bifurcations are computed using the center manifold theorem and bifurcation theory. Additionally, numerical simulations are employed to illustrate the bifurcation phenomenon and chaos characteristics. The results demonstrate that period-doubling and Hopf bifurcations are two typical routes to generate chaos, as evidenced by the calculation of the maximum Lyapunov exponents near the critical bifurcation points. Finally, a feedback control method is suggested, utilizing feedback of system states and perturbation of feedback parameters, to efficiently manage the bifurcations and chaotic attractors of the discrete predator-prey model.

Transcritical bifurcation and Neimark-Sacker bifurcation of a discrete predator-prey model with herd behaviour and square root functional response

Transcritical bifurcation and Neimark-Sacker bifurcation of a discrete predator-prey model with herd behaviour and square root functional response

Bifurcation Patterns in a Discrete Predator–Prey Model Incorporating Ratio-Dependent Functional Response and Prey Harvesting

Bifurcation Patterns in a Discrete Predator–Prey Model Incorporating Ratio-Dependent Functional Response and Prey Harvesting

Qualitative Properties of Equilibrium Point in a Discrete Predator-Prey Model

Qualitative Properties of Equilibrium Point in a Discrete Predator-Prey Model

Optimal control problem of a discrete spatiotemporal prey–predator three-species fishery model

In this work, we discuss a spatiotemporal discrete prey–predator model. It consists of three compartments: prey, predator, and super-predator. The proposed model describes the interaction between prey, predator, and super-predator in a region with a discrete displacement. We also provide research on appropriate regional control strategies. The controls are applied to the predator and the super-predator, respectively; they represent catching these in measured quantities in a space and a time chosen. The aim is to increase the number of prey and reduce the number of predators, restore the food chain system, and ensure its sustainability. Finally, we provide graphical visuals and numerical simulations to support our analytical findings.

Bifurcation and hybrid control in a discrete predator–prey model with Holling type-IV functional response

Bifurcation and hybrid control in a discrete predator–prey model with Holling type-IV functional response

Habitat complexity of a discrete predator-prey model with Hassell-Varley type functional response

Prey-predator models with a refuge effect are particularly significant in ecology. The two common conceptions of refuge in the literature are continual refuge and refuge pro-portionate to prey. Academics are already paying attention to new types of refuge concepts. Prey-predator interaction has become a prominent issue in recent biomathe-matical studies due to its environmental influence. In this paper, the habitat complexity of a predator-prey model with Hassell-Varley type functional response is considered. For this, we focused our study on the question of existence and uniqueness in Sec. 2. And Sec. 3 is devoted to show a generalized stability. Note that this representation also al-lows us to generalize the results obtained recently in the literature. In Sec. 4, we have studied the numerical algorithm for the suggested problem. The paper is ended by two examples illustrating our results.

BIFURCATION ANALYSIS AND CHAOS CONTROL OF DISCRETE PREDATOR–PREY MODEL WITH ADDITIVE ALLEE EFFECT

This paper presents a study of a discrete prey–predator model with additive Allee effect. The model is discretized using the forward Euler method, and the system is analyzed using bifurcation theory and chaos control. The equilibrium point of the discrete system is obtained through equilibrium analysis, and the stability of the equilibrium point is determined by analyzing the parameter conditions. The study also establishes the existence and direction of Neimark–Sacker bifurcation at the positive equilibrium point. The paper proposes two control strategies to manage chaotic behavior and Neimark–Sacker bifurcation: exponential control and hybrid feedback control. These control methods are demonstrated to be effective in controlling the chaotic behavior and bifurcation of the system through numerical simulation. Overall, the results of the study provide important insights into the dynamics of the discrete prey–predator model with additive Allee effect, as well as effective methods for controlling chaos and bifurcation in the system.

Dynamical analysis of a three-species discrete biological system with scavenger

The main focus of this paper is to investigate the dynamical properties of a discrete prey–predator model with scavenger in the interior of R+3. We have explored the local dynamics at equilibria, the existence of bifurcation sets under certain conditions, bifurcation analysis, and chaos control of a three-species discrete prey–predator system with scavenger. By the bifurcation theory, we explored the existence of bifurcations and gave a detailed bifurcation analysis at equilibria accordingly. We also studied the chaos control by feedback control strategy for under-studied discrete system. Some numerical simulations are also presented which verified the theoretical results. The phase portraits, Maximum Lyapunov exponents, time series graphs, and bifurcation diagrams of numerical simulation for discrete system are plotted by using the software Matlab/Mathematica. Finally, theoretical results are interpreted biologically.

Dynamical analysis of a two-dimensional discrete predator–prey model

In this paper, we explore the existence of periodic solutions, local dynamics at equilibrium solutions, chaos and bifurcations of a discrete prey–predator model with Michaelis–Menten type functional response. More specifically, we explore local dynamical characteristics at equilibrium solutions, and existence of periodic solutions for the under consideration model. It is also studied the existence of bifurcations at equilibrium solutions, and investigated that at semitrivial and trivial equilibrium solutions model does not undergo flip bifurcation, but at positive equilibrium solution it undergoes flip and Neimark–Sacker bifurcations when parameters goes through the certain curves. It is also investigated that fold bifurcation does not exist at positive equilibrium, and we have studied these bifurcations by center manifold theorem and bifurcation theory. We also studied chaos by feedback control method. The theoretical results are confirmed numerically. Furthermore, we use 0–1 chaos test in order to quantify whether chaos exists in the model or not.

Analysis of a Discrete Predator-Prey Model for Controlling The Extinction of The Pesut Mahakam (Orcaella brevirostris) with Toxic Effect at The Mahakam River

The Pesut Mahakam (Orcaella brevirostris) is an East Kalimantan animal in critical condition and on the verge of extinction. One of the causes of the extinction of the Mahakam dolphins is the toxic effects of environmental pollution. In this study, an analysis of the discrete predator-prey model using Euler's forward difference between the Pesut Mahakam (Orcaella brevirostris) and its prey was investigated. Dynamic analysis includes determining the equilibrium point, stability analysis of the equilibrium point, and numerical simulation. The value of the Mahakam dolphin growth rate parameter (predator) is obtained by adjusting the curve based on data in the field. Numerical simulations were carried out to describe the results of the analysis. Changes in the value of the toxic effect rate parameter affects the number of Mahakam dolphins and their prey.

On the qualitative study of a discrete fractional order prey–predator model with the effects of harvesting on predator population

This research investigates the discrete prey–predator model by including harvesting on the predator population, in the sense of Caputo fractional derivative. We define the topological categories of the model fixed points. We demonstrate mathematically that, under certain parametric conditions, a fractional order prey-predator model undergoes both a Neimark–Sacker (NS) and a Period-doubling (PD) bifurcations. Using the central manifold and bifurcation theory, we present proof for NS and PD bifurcations. It has been discovered that the fractional order prey-predator model’s dynamical behavior is significantly influenced by the parameter values and the initial conditions. Two chaos control techniques have been used to eliminate the chaos in the model. In order to support our theoretical and analytical results and to illustrate complex and chaotic behavior, numerical simulations have been shown.

Bifurcation and global stability of a discrete prey–predator model with saturated prey refuge

This article aims to describe the qualitative behavior of a discrete‐time prey–predator model including prey refuge. It is assumed that the fraction of prey in refuge is a monotonically increasing but self‐limiting function of prey population. The fixed points of the system and their local stability are investigated. The criteria for Neimark–Sacker bifurcation and period‐doubling bifurcation are presented. The sufficient conditions for global asymptotic stability of the interior fixed point are derived. The chaotic behavior of the system is stabilized by applying a hybrid control strategy. We conclude with a numerical simulation that strengthens our theoretical discussion and provides a way to observe the complex behavior of the system.

Dynamic complexity of a discrete predator-prey model with prey refuge and herd behavior

This paper examines a discrete predator-prey model's complex dynamics. Using bifurcation and center manifold theory, we study period-doubling and Neimark-Sacker bifurcations at positive fixed points and their direction. Numerical simulations confirm the theoretical conclusions that the model's dynamics rely on Euler method step size. The model's behavior is also affected by the prey population's conservation rate. The model suggests that excessive conservation may reduce predator populations, causing food shortages. Thus, predator-prey dynamics management must account for prey conservation rate

Strong Resonance Bifurcations and State Feedback Control in a Discrete Prey-Predator Model with Harvesting Effect

Strong Resonance Bifurcations and State Feedback Control in a Discrete Prey-Predator Model with Harvesting Effect

Bifurcations, chaos analysis and control in a discrete predator–prey model with mixed functional responses

Many discrete systems have more distinctive dynamical behaviors compared to continuous ones, which has led lots of researchers to investigate them. The discrete predator–prey model with two different functional responses (Holling type I and II functional responses) is discussed in this paper, which depicts a complex population relationship. The local dynamical behaviors of the interior fixed point of this system are studied. The detailed analysis reveals this system undergoes flip bifurcation and Neimark–Sacker bifurcation. Especially, we prove the existence of Marotto’s chaos by analytical method. In addition, the hybrid control method is applied to control the chaos of this system. Numerical simulations are presented to support our research and demonstrate new dynamical behaviors, such as period-10, 19, 29, 39, 48 orbits and chaos in the sense of Li–Yorke.