Discrete-time Predator Prey Model Research Articles

Examining the influence of prey dynamics on predator–prey interactions

ABSTRACT We develop variations of a discrete-time predator–prey model to examine how intrinsic properties of the prey population may impact overall system dynamics. We focus on two properties of the prey species, namely developmental stage structure and undercompensatory (contest competition) versus overcompensatory (scramble competition) density dependence. Through analysis of these models, we examine how these different prey features affect system stability. Our results show that when prey growth is overcompensatory, the predator may have a stabilizing effect on the system dynamics with increasing predator density reversing the period doubling route to chaos observed with Ricker-type nonlinearities. Moreover, we find that stage structure in the prey does not have a destabilizing effect on the system dynamics unless the prey projection matrix is imprimitive or close to imprimitive, as may occur for semelparous species. In this case, a sufficiently large predator density may stabilize cycles that are otherwise unstable.

An analysis of the stability and bifurcation of a discrete-time predator-prey model with the slow-fast effect on the predator.

In many environments, predators have significantly longer lives and meet several generations of prey, or the prey population reproduces rapidly. The slow-fast effect can best describe such predator-prey interactions. The slow-fast effect ε can be considered as the ratio between the predator's linear death rate and the prey's linear growth rate. This paper examines a slow-fast, discrete predator-prey interaction with prey refuge and herd behavior to reveal its complex dynamics. Our methodology employs the eigenvalues of the Jacobian matrix to examine the existence and local stability of fixed points in the model. Through the utilization of bifurcation theory and center manifold theory, it is demonstrated that the system undergoes period-doubling bifurcation and Neimark-Sacker bifurcation at the positive fixed point. The hybrid control method is utilized as a means of controlling the chaotic behavior that arises from these bifurcations. Moreover, numerical simulations are performed to demonstrate that they are consistent with analytical conclusions and to display the complexity of the model. At the interior fixed point, it is shown that the model undergoes a Neimark-Sacker bifurcation for larger values of the slow-fast effect parameter by using the slow-fast effect parameter ε as the bifurcation parameter. This is reasonable since a large ε implies an approximate equality in the predator's death rate and the prey's growth rate, automatically leading to the instability of the positive fixed point due to the slow-fast impact on the predator and the presence of prey refuge.

Turing-like patterns induced by the competition between two stable states in a discrete-time predator–prey model

Patterns, typical spatiotemporal ordered structures, are widely present in a variety of systems, which are the results of the emergence of complex systems. The theory of Turing, occupying a long-range inhibitor and a short-range activator, provides an explanation for the diversity of Turing patterns. Turing-like patterns, caused by various non-Turing mechanisms, have also been investigated in various models. Patterns resulting from Turing or non-Turing mechanisms have predominantly focused on continuous-time models, however, for populations with non-overlapping generations, discrete-time models are more realistic than continuous-time ones. Here, we investigated a type of Turing-like patterns in a discrete-time model. The results show that the competition between two stable states can lead to the formation of Turing-like patterns. Specifically, we obtained a type of Turing-like patterns in a discrete-time model, which exist outside the parameter regions of Turing instability. Only by applying strong impulse noise to the homogeneous stable state can the patterns be excited. And the excitation threshold is related to the control parameter. Analysis shows that the Turing-like patterns are the results of competition between two stable states corresponding to adjacent orbits, and the excitation threshold is determined by the relative position between orbits. This findings exemplify the multitude of mechanisms observed in nature that give rise to the emergence of Turing or Turing-like patterns, and shed light on the pattern formation for Turing/Turing-like patterns.

Stability, bifurcation, and control: Modeling interaction of the predator-prey system with Alles effect

In this study, we investigate the resilience and bifurcation properties of a discrete-time predator-prey model incorporating an Allee effect, the transfer of prey to predator energy, and a necessary threshold for prey consumption prior to predator reproduction. The discrete-time formulation involves iteratively adjusting the predator and prey populations at distinct time steps. The model encompasses equations that depict the fluctuations in population sizes, considering processes like breeding, predation, death, and the influence of the Allee effects. The investigation results indicate that the system's boundary equilibrium goes through a period-doubling bifurcation. Moreover, when the system is in close proximity to the exclusive positive equilibrium influenced by the prey-predator population, it exhibits Neimark-Sacker bifurcations. It's worth noting that our observations suggest that the predator population might approach extinction or achieve a stable equilibrium when chaotic dynamics appear in the prey population. The validity of this phenomenon is confirmed using the Lyapunov exponents calculation, which verifies the range of diverse dynamic behaviors. The outcomes and analysis of this research offer intriguing insights in both mathematical and biological realms. Implementing methods for chaos control could be beneficial in stabilizing or mitigating the occurrence of chaotic dynamics. Additionally, the theoretical results are supported by numerical simulations, phase portraits, bifurcation diagrams, graphs depicting chaos control, and maximum Lyapunov exponent graphs.

Bifurcation analysis of a discrete-time prey-predator model

This paper investigates the importance of studying the dynamics of predator-prey systems and the specific significance of Neimark-Sacker and period-doubling bifurcations in discrete-time prey-predator models. By conducting a numerical bifurcation analysis and examining bifurcation diagrams and phase portraits, we present important results that differentiate our study from others in the field. Firstly, our analysis reveals the occurrence of Neimark-Sacker and period-doubling bifurcations in the model under certain parameter values. These bifurcations lead to the emergence of stable limit cycles characterized by complex and unpredictable dynamics. This finding emphasizes the inherent complexity and nonlinearity of predator-prey systems and contributes to a deeper understanding of their dynamics. Additionally, our study highlights the advantages and limitations of employing discrete-time models in population dynamics research. The use of discrete-time models allows for a more tractable analysis while still capturing significant aspects of ecological systems. In conclusion, this study holds importance in shedding light on the dynamics of predator-prey systems and the specific role of Neimark-Sacker and period-doubling bifurcations. Our findings contribute to the understanding of predator-prey systems and offer implications for ecological management strategies.

Bifurcation analysis and chaos control of a discrete-time prey-predator model with Allee effect

In this study, a discrete-time prey-predator model based on the Allee effect is presented. We examine the parametric conditions for the local asymptotic stability of the fixed points of this model. Furthermore, with the use of the center manifold theorem and bifurcation theory, we analyze the existence and directions of period-doubling and Neimark-Sacker bifurcations. The plots of maximum Lyapunov exponents provide indications of complexity and chaotic behavior. The feedback control approach is presented to stabilize the unstable fixed point. Numerical simulations are performed to support the theoretical results.

Effects of prey refuge and predator cooperation on a predator–prey system

ABSTRACT We develop and investigate a discrete-time predator–prey model with cooperative hunting among predators and a spatial prey refuge. The system can exhibit two positive equilibria if the magnitude of cooperation is large and the maximal reproduction number of predators is less than one. In such a scenario, the predator population may exhibit a strong Allee effect, and therefore the predator may survive if its density is above the threshold. When the positive equilibrium is unique, we prove that hunting cooperation can destabilize the equilibrium through a Neimark–Sacker bifurcation. Numerical findings indicate that a high degree of predator hunting cooperation can help rescue the predator population if the proportion of prey refuge is large, while hunting cooperation becomes destabilizing when the proportion of refuge is small. Despite hunting cooperation's destabilizing effect, it can facilitate predator persistence, particularly with a large proportion of prey refuge. Furthermore, there exists a wide parameter space where the predator–prey interaction may exhibit chaotic behaviour.

ALLEE EFFECT IN A RICKER TYPE DISCRETE-TIME PREDATOR–PREY MODEL WITH HOLLING TYPE-II FUNCTIONAL RESPONSE

In recent years, the stability of the predator–prey model subject to the Allee effect has become an interesting issue. This study investigates the effect of Allee effect on the stability of a discrete-time predator–prey model with Holling type-II functional response. Using equilibrium analysis, stability analysis and bifurcation theory, the mathematical characteristics of the proposed model are examined. Model experiences flip bifurcation and Neimark–Sacker bifurcation based on the center manifold theorem and bifurcation theory. Our analytical results are demonstrated by numerical simulations.

Codimension-one and -two bifurcation analysis of a discrete-time prey-predator model

Codimension-one and -two bifurcation analysis of a discrete-time prey-predator model

Neimark–Sacker Bifurcation of a Discrete-Time Predator–Prey Model with Prey Refuge Effect

In this paper, we deduce a predator–prey model with discrete time in the interior of R+2 using a new discrete method to study its local dynamics and Neimark–Sacker bifurcation. Compared with continuous models, discrete ones have many unique properties that help to understand the changing patterns of biological populations from a completely new perspective. The existence and stability of the three equilibria are analyzed, and the formation conditions of Neimark–Sacker bifurcation around the unique positive equilibrium point are established using the center manifold theorem and bifurcation theory. An attracting closed invariant curve appears, which corresponds to the periodic oscillations between predators and prey over a long period of time. Finally, some numerical simulations and their biological meanings are given to reveal the complex dynamical behavior.

Strong resonance bifurcations for a discrete-time prey–predator model

Strong resonance bifurcations for a discrete-time prey–predator model

Rich dynamics of a discrete-time prey-predator model

Rich dynamics of a discrete-time prey-predator model

Allee effect in a Ricker type predator-prey model

The stability of the predator-prey model subject to the Allee effect is an interesting topic in recent times. The impact of a weak Allee effect on the stability of a discrete-time predator-prey model is investigated in this paper. Equilibrium analysis, stability analysis, and bifurcation theory are used to examine the mathematical properties of the proposed model. By using the Allee parameter as the bifurcation parameter, we provide sufficient conditions for the flip bifurcation. Numerical simulations are used to demonstrate our analytical conclusions.

Global stability and bifurcation analysis in a discrete-time two prey-one predator model with help

ABSTRACT This article examines a discrete-time predator-prey model where prey teams cooperate. Positivity and boundedness of the model solution are investigated. Positive fixed points are examined. Using an iteration scheme and the comparison principle of difference equations, we determined the sufficient condition for global stability of the positive fixed point. It is shown that the sufficient criterion for Neimark-Sacker bifurcation and flip bifurcation can be established. It is observed that the system behaves in a chaotic manner when a specific set of system parameters is selected, which are controlled by a hybrid control method. Examples are provided to illustrate our conclusions.

Bifurcations and chaos control in a discrete-time prey–predator model with Holling type-II functional response and prey refuge

In this paper, we study the complex dynamical behavior in a discrete prey–predator model with Holling type II functional response with prey refuge. It is exhibited that the system shows different types of bifurcations viz. flip bifurcation, transcritical bifurcation and Neimark–Sacker bifurcation(NSB) by using center manifold theorem and bifurcation theory. Moreover, the chaos occurred in the system has been controlled by deploying control strategies viz. state feedback, pole placement and hybrid control techniques. The certain conditions have been determined under which chaos and bifurcation can be stabilized. The extensive numerical simulation is performed to demonstrate the analytical findings.

Bifurcation, chaos, multistability, and organized structures in a predator-prey model with vigilance.

There is not a single species that does not strive for survival. Every species has crafted specialized techniques to avoid possible dangers that mostly come from the side of their predators. Survival instincts in nature led prey populations to develop many anti-predator strategies. Vigilance is a well-observed effective antipredator strategy that influences predator-prey dynamics significantly. We consider a simple discrete-time predator-prey model assuming that vigilance affects the predation rate and the growth rate of the prey. We investigate the system dynamics by constructing isoperiodic and Lyapunov exponent diagrams with the simultaneous variation of the prey's growth rate and the strength of vigilance. We observe a series of different types of organized periodic structures with different kinds of period-adding phenomena. The usual period-bubbling phenomenon is shown near a shrimp-shaped periodic structure. We observe the presence of double and triple heterogeneous attractors. We also notice Wada basin boundaries in the system, which is quite rare in ecological systems. The complex dynamics of the system in biparameter space are explored through extensive numerical simulations.

Bifurcations and dynamics of a discrete predator–prey model of ricker type

A discrete-time predator–prey model is investigated in this paper. In considered model, the population is assumed to follow the model suggested by Ricker 1954. Existence and stability of equilibria are studied. Numerical simulations reveal that, depending on the parameters, the system has complicated and rich dynamics and can exhibit complex patterns. Also the bifurcation diagrams are presented.

On the analysis of stability, bifurcation, and chaos control of discrete-time predator-prey model with Allee effect on predator

In this paper, a discrete predator-prey model with Allee effect which is obtained by the forward Euler method has been investigated. The local stability conditions of the model at the fixed point have been discussed. In addition, it is shown that the model undergoes Neimark-Sacker bifurcation by using bifurcation theory. Then, the direction of Neimark-Sacker bifurcation has been given. The OGY method is applied in order to control chaos in considered model due to emergence of Neimark-Sacker bifurcation. Some numerical simulations such as phase portraits and bifurcation figures have been presented to support the theoretical results. Also, the chaotic features are justified numerically by computing Lyapunov exponents. Because of consistency with the biological facts, the parameter values have been taken from literature [Controlling chaos and Neimark-Sacker bifurcation discrete-time predator-prey system, Hacet. J. Math. Stat. 49 (5), 1761-1776, 2020].

Multiple Bifurcations of a Discrete-Time Prey–Predator Model with Mixed Functional Response

In this paper, the one- and two-parameter bifurcations of a discrete-time prey–predator model with a mixed functional response are investigated by computing their critical normal form coefficients. The flip, Neimark–Sacker and strong resonance bifurcations are detected for this model. The critical coefficients identify the scenario associated with each bifurcation. The complex dynamical behavior of the model up to the 16th iteration is investigated.

Bifurcation analysis of a three species discrete-time predator-prey model

We explore local dynamics at fixed points, existence of periodic points and bifurcation analysis of a three species discrete predator-prey model. More precisely, it is explored that model has trivial fixed point, five boundary fixed points and the interior fixed point under definite parametric conditions. We have also constructed the linearized system for the model under consideration. Then we have investigated local stability analysis with topological classifications at each fixed points by linear stability theory. Further existence of periodic points and bifurcation analysis at each fixed points are also demonstrated. The extensive numerical simulations are provided to demonstrate theoretical results.