Discrete-time Predator Prey Model Research Articles

Dynamics of a discrete-time system with Z-type control

Abstract In community ecology, the stability of a predator–prey system is a considerably desired issue; as a result, population control of a predator–prey system is very important. The dynamics of continuous-time models with Z-type control is studied earlier. But, the effectiveness of the Z-type control mechanism in a discrete-time set-up is lacking. First, we consider a Lotka–Volterra type discrete-time predator–prey model. We observe that without control, the system exhibits rich dynamical behaviors including chaotic oscillations. We apply the Z-control mechanism in both direct and indirect ways to the system and observe that in both cases, controllers have the property to drive the populations of the system to the desired state. We conduct numerical simulation as supporting evidence of our analytical results.

Dynamical analysis and chaos control in discrete-time prey-predator model

In this work, a discretized two-dimensional Leslie-Gower prey-predator model is investigated. The results for the existence and uniqueness and the conditions for the local asymptotic stability of the solutions are determined. It is also exhibited that the discrete system undergoes Neimark-Sacker, flip and fold bifurcation under certain conditions. The discretized system exhibits wide range of complex dynamical behavior viz. periodicity, quasi periodicity and chaos with respect to different parameters. Further, three control methods: state feedback, pole-placement and hybrid control are deployed to control the chaos in the system. Under certain conditions, chaos and bifurcation of the system are stabilized through the control strategies. The extensive numerical simulation is done to demonstrate the analytical findings.

Long-term dynamics of discrete-time predator–prey models: stability of equilibria, cycles and chaos

ABSTRACT In [A.S. Ackleh, M.I. Hossain, A. Veprauskas, and A. Zhang, Persistence and stability analysis of discrete-time predator-prey models: A study of population and evolutionary dynamics, J. Differ. Equ. Appl. 25 (2019), pp. 1568–1603.], we established conditions for the persistence and local asymptotic stability of the interior equilibrium for two discrete-time predator–prey models (one without and with evolution to resist toxicants). In the current paper, we provide a more in-depth analysis of these models, including global stability of equilibria, existence of cycles and chaos. Our main focus is to examine how the speed of evolution ν may impact population dynamics. For both models, we establish conditions under which the interior equilibrium is global asymptotically stable using perturbation analysis together with the construction of Lyapunov functions. For small ν, we show that the global dynamics of the evolutionary system are nothing but a continuous perturbation of the non-evolutionary system. However, when the speed of evolution is increased, we perform numerical studies which demonstrate that evolution may introduce rich dynamics including cyclic and chaotic behaviour that are not observed when evolution is absent.

Stability and bifurcation analysis of a discrete prey–predator model with sigmoid functional response and Allee effect

In this paper, we have considered a discrete-time predator–prey model with sigmoid functional response and Allee effect. We have discussed the conditions of existence of the feasible fixed point. The stability criterion of the fixed point carried out algebraically. Analytically we have shown that the system undergoes flip and Neimark–Sacker bifurcations. We also analyzed Neimark–Sacker bifurcation by center manifold theorem taking discretization factor(generation gap) as a bifurcation parameter. Under a parametric condition, all the bifurcation phenomena and chaotic features have been justified numerically. Finally, we use the hybrid control strategy to control flip and Neimark–Sacker bifurcations.

A discrete-time model with non-monotonic functional response and strong Allee effect in prey

In this paper, we investigate the impact of strong Allee effect on the stability of a discrete-time predator–prey model with a non-monotonic functional response. The dynamics of discrete-time predator–prey models with strong Allee effect is studied earlier. But, the mathematical investigations of predator–prey dynamics in discrete-time set up with Holling type-IV functional response and strong Allee effect in prey are lacking. The proposed model supports the coexistence of two steady states, and the mathematical features of the model are analyzed based on local stability and bifurcation theory. By considering the Allee parameter as the bifurcation parameter, we provide sufficient conditions for the flip and the Neimark–Sacker bifurcations. We observe that Allee parameter plays a significant role in the dynamics of the system.

Bifurcations and hybrid control in a 3×3 discrete-time predator-prey model.

In this paper, we explore the bifurcations and hybrid control in a $3\times3$ discrete-time predator-prey model in the interior of $\mathbb{R}_+^3$. It is proved that $3\times3$ model has four boundary fixed points: $P_{000}(0,0,0)$, $P_{0y0}\left(0,\frac{r-1}{r},0\right)$, $P_{0yz}\left(0,\frac{d}{f},\frac{rf-f-dr}{cf}\right)$, $P_{x0z}\left(\frac{d}{e},0,\frac{a}{b}\right)$, and the unique positive fixed point: $P^+_{xyz}\left(\frac{br(d-f)+f(b+ac)}{ber},\frac{br-b-ac}{br},\frac{a}{b}\right)$ under certain restrictions to the involved parameters. By utilizing method of Linearization, local dynamics along with topological classifications about fixed points have been investigated. Existence of prime period and periodic points of the model are also investigated. Further for $3\times3$ model, we have explored the occurrence of possible bifurcations about each fixed point, that gives more insight about the under consideration model. It is proved that the model cannot undergo any bifurcation about $P_{000}(0,0,0)$ and $P_{x0z}\left(\frac{d}{e},0,\frac{a}{b}\right)$, but the model undergo P-D and N-S bifurcations respectively about $P_{0y0}\left(0,\frac{r-1}{r},0\right)$ and $P_{0yz}\left(0,\frac{d}{f},\frac{rf-f-dr}{cf}\right)$. For the unique positive fixed point: $P^+_{xyz}\left(\frac{br(d-f)+f(b+ac)}{ber},\frac{br-b-ac}{br},\frac{a}{b}\right)$, we have proved the N-S as well as P-D bifurcations by explicit criterion. Further, theoretical results are verified by numerical simulations. We have also presented the bifurcation diagrams and corresponding maximum Lyapunov exponents for the $3\times3$ model. The computation of the maximum Lyapunov exponents ratify the appearance of chaotic behavior in the under consideration model. Finally, the hybrid control strategy is applied to control N-S as well as P-D bifurcations in the discrete-time model.

Discrete-time predator-prey model with flip bifurcation and chaos control.

We explore the local dynamics, flip bifurcation, chaos control and existence of periodic point of the predator-prey model with Allee effect on the prey population in the interior of $\mathbb{R}^*{_+^2}$. Nu-merical simulations not only exhibit our results with the theoretical analysis but also show the complex dynamical behaviors, such as the period-2, 8, 11, 17, 20 and 22 orbits. Further, maximum Lyapunov exponents as well as fractal dimensions are also computed numerically to show the presence of chaotic behavior in the model under consideration.

Stoichiometric knife-edge model on discrete time scale

Ecological stoichiometry is the study of the balance of multiple elements in ecological interactions and processes (Sterner and Elser in Ecological Stoichiometry: The Biology of Elements from Molecules to the Biosphere, 2002). Modeling under this framework enables us to investigate the effect nutrient content on organisms whether the imbalance involves insufficient or excess nutrient content. This phenomenon is called the “stoichiometric knife-edge”. In this paper, a discrete-time predator–prey model that captures this phenomenon is established and qualitatively analyzed. We systematically expound the similarities and differences between our discrete model and the corresponding continuous analog. Theoretical and numerical analyses show that while the discrete and continuous models share many properties, differences also exist. Under certain parameter sets, the models exhibit qualitatively different dynamics. While the continuous model shows limit cycle, Hopf bifurcation, and saddle-node bifurcation, the discrete-time model exhibits richer dynamical behaviors, such as chaos. By comparing the dynamics of the continuous and discrete model, we can conclude that stoichiometric effects of low food quality on predators are robust to the discretization of time. This study can possibly serve as an example for pointing to the importance of time scale in ecological modeling.

Persistence and stability analysis of discrete-time predator–prey models: a study of population and evolutionary dynamics

ABSTRACTProlonged exposure to disturbances, such as toxicants, has the potential to result in evolutionary responses in a population. These evolutionary responses may impact not just that species but also other species with which the population interacts. In this paper, we develop a discrete-time predator–prey model and establish conditions for the existence and stability of equilibria, as well as conditions for the persistence of the prey and predator populations. We then extend this model to an evolutionary model which couples the population dynamics with the dynamics of an evolving phenotypic trait that provides a measure for the level of toxicant resistance developed by the prey. We study the persistence dynamics of this model and establish conditions for when the prey evolves toxicant resistance. We show that the evolution of toxicant resistance may allow both the predator and prey to persist when, without the evolution, both may go extinct.

Stability analysis of a prey refuge predator–prey model with Allee effects

We constructed a discrete-time predator-prey model by adding prey refuge and Allee effects (predator saturation on prey and mate limitation on predator) to an earlier prey-predator model and examined its dynamics. We show the existence of positive fixed points and study the stability properties. The numerical simulations and bifurcation diagrams verify the impact of refuge and the Allee mechanism on the system.

Under the influence of crowding effects: Stability, bifurcation and chaos control for a discrete-time predator–prey model

The interaction between predators and preys exhibits more complicated behavior under the influence of crowding effects. By taking into account the crowding effects, the qualitative behavior of a prey–predator model is investigated. Particularly, we examine the boundedness as well as existence and uniqueness of positive steady-state and stability analysis of the unique positive steady-state. Moreover, it is also proved that the system undergoes Hopf bifurcation and flip bifurcation with the help of bifurcation theory. Moreover, a chaos control technique is proposed for controlling chaos under the influence of bifurcations. Finally, numerical simulations are provided to illustrate the theoretical results. These results of numerical simulations demonstrate chaotic long-term behavior over a broad range of parameters. The presence of chaotic behavior in the model is confirmed by computing maximum Lyapunov exponents.

Flip Bifurcation and Stability in a Discrete-Time Prey-Predator Model with Allee Effect

Bu makalede, Allee etkili kesik zamanlı bir av-avcı modeli ele alındı. Modelin dinamik davranışları incelendi. Modelin her iki türün bir arada olduğu denge noktasının varlığı ve kararlılık şartları elde edildi. Çatallanma teorisi kullanılarak, modelin flip çatallanmaya gittiği gösterildi. Elde edilen teorik sonuçların doğruluğunu göstermek için nümerik gösterimlere yer verildi. 39A33, 37G35, 39A30.

Bifurcations of a two-dimensional discrete-time predator\u2013prey model

We study the local dynamics and bifurcations of a two-dimensional discrete-time predator–prey model in the closed first quadrant mathbb{R}_{+}^{2}. It is proved that the model has two boundary equilibria: O(0,0), A (frac{alpha _{1}-1}{alpha _{1}},0 ) and a unique positive equilibrium B (frac{1}{alpha _{2}},frac{ alpha _{1}alpha _{2}-alpha _{1}-alpha _{2}}{alpha _{2}} ) under some restriction to the parameter. We study the local dynamics along their topological types by imposing the method of linearization. It is proved that a fold bifurcation occurs about the boundary equilibria: O(0,0), A (frac{alpha _{1}-1}{alpha _{1}},0 ) and a period-doubling bifurcation in a small neighborhood of the unique positive equilibrium B (frac{1}{alpha _{2}},frac{alpha _{1} alpha _{2}-alpha _{1}-alpha _{2}}{alpha _{2}} ). It is also proved that the model undergoes a Neimark–Sacker bifurcation in a small neighborhood of the unique positive equilibrium B (frac{1}{ alpha _{2}},frac{alpha _{1}alpha _{2}-alpha _{1}-alpha _{2}}{alpha _{2}} ) and meanwhile a stable invariant closed curve appears. From the viewpoint of biology, the stable closed curve corresponds to the periodic or quasi-periodic oscillations between predator and prey populations. Numerical simulations are presented to verify not only the theoretical results but also to exhibit the complex dynamical behavior such as the period-2, -4, -11, -13, -15 and -22 orbits. Further, we compute the maximum Lyapunov exponents and the fractal dimension numerically to justify the chaotic behaviors of the discrete-time model. Finally, the feedback control method is applied to stabilize chaos existing in the discrete-time model.

Bifurcation, chaos analysis and control in a discrete-time predator\u2013prey system

The dynamical behavior of a discrete-time predator–prey model with modified Leslie–Gower and Holling’s type II schemes is investigated on the basis of the normal form method as well as bifurcation and chaos theory. The existence and stability of fixed points for the model are discussed. It is showed that under certain conditions, the system undergoes a Neimark–Sacker bifurcation when bifurcation parameter passes a critical value, and a closed invariant curve arises from a fixed point. Chaos in the sense of Marotto is also verified by both analytical and numerical methods. Furthermore, to delay or eliminate the bifurcation and chaos phenomena that exist objectively in this system, two control strategies are designed, respectively. Numerical simulations are presented not only to validate analytical results but also to show the complicated dynamical behavior.

Hydra effect and paradox of enrichment in discrete-time predator-prey models

We develop three discrete-time predator-prey models from the Nicholson–Bailey host-parasitoid framework, assuming a type II functional response and logistic prey growth in form of the Beverton–Holt map. Our models show many similarities with the continuous-time Rosenzweig–MacArthur model, not only the same equilibria and sequence of bifurcations, but also phenomena such as the hydra effect and paradox of enrichment. Our three models differ in the order of events, in which the processes of density-dependent prey regulation and predation take place. When their order is reversed, but their relative order remains the same such that only census timing is changed, we observe quantitative differences in population size, but no differences in qualitative behaviour. When a modified order of events induces delayed density dependence, we observe increased stability of population dynamics, which is somewhat contrary to conventional expectation. Overall, our models exhibit behaviour analogous to the Rosenzweig–MacArthur model and highlight the importance of the order of events in discrete-time models.

Chaos control in a discrete-time predator–prey model with weak Allee effect

The stability of the predator–prey model subject to the Allee effect is an interesting topic in recent times. In this paper, we investigate the impact of weak Allee effect on the stability of a discrete-time predator–prey model with Holling type-IV functional response. The mathematical features of the proposed model are analyzed with the help of equilibrium analysis, stability analysis, and bifurcation theory. We provide sufficient conditions for the flip bifurcation by considering Allee parameter as the bifurcation parameter. We observe that the model becomes stable from chaotic dynamics as the Allee parameter increases. Further, we observe bi-stability behavior of the model between only prey existence equilibrium and the coexistence equilibrium. Our analytical findings are illustrated through numerical simulations.

Neimark–Sacker bifurcation and chaos control in discrete-time predator–prey model with parasites

In this paper, we discuss the qualitative behavior of a four-dimensional discrete-time predator–prey model with parasites. We investigate existence and uniqueness of positive steady state and find parametric conditions for local asymptotic stability of positive equilibrium point of given system. It is also proved that the system undergoes Neimark–Sacker bifurcation (NSB) at positive equilibrium point with the help of an explicit criterion for NSB. The system shows chaotic dynamics at increasing values of bifurcation parameter. Chaos control is also discussed through implementation of hybrid control strategy, which is based on feedback control methodology and parameter perturbation. Finally, numerical simulations are conducted to illustrate theoretical results.

Stability, bifurcation analysis and chaos control for a predator-prey system

We study qualitative behavior of a modified prey–predator model by introducing density-dependent per capita growth rates and a Holling type II functional response. Positivity of solutions, boundedness and local asymptotic stability of equilibria were investigated for continuous type of the prey–predator system. In order to discuss the rich dynamics of the proposed model, a piecewise constant argument was implemented to obtain a discrete counterpart of the continuous system. Moreover, in the case of a discrete-time prey–predator model, the boundedness of solutions and local asymptotic stability of equilibria were investigated. With the help of the center manifold theorem and bifurcation theory, we investigated whether a discrete-time model undergoes period-doubling and Neimark–Sacker bifurcation at its positive steady-state. Finally, two novel generalized hybrid feedback control methods are presented for chaos control under the influence of period-doubling and Neimark–Sacker bifurcations. In order to illustrate the effectiveness of the proposed control strategies, numerical simulations are presented.

Qualitative properties and bifurcations of Mistro–Rodrigues–Petrovskii model

In this paper, we consider a discrete-time prey–predator model with a strong Allee effect. At first, applying the center manifold reduction, we discuss the topological structure of orbits near each fixed point, including hyperbolic one and non-hyperbolic one. Then, we apply the center manifold reduction to prove that the model undergoes fold bifurcation and transcritical bifurcation. At last, by the numerical analysis, we illustrate our results and find that the model possesses heteroclinic orbits in the first quadrant. Furthermore, we present a stable 7-periodic orbit and a stable 8-periodic orbit on the invariant cycle produced from the Neimark–Sacker bifurcation.

The snapback repellers for chaos in multi-dimensional maps

The key of Marotto's theorem on chaos for multi-dimensional maps is the existence of snapback repeller. For practical application of the theory, locating a computable repelling neighborhood of the repelling fixed point has thus become the key issue. For some multi-dimensional maps \begin{document}$F$\end{document} , basic information of \begin{document}$F$\end{document} is not sufficient to indicate the existence of snapback repeller for \begin{document}$F$\end{document} . In this investigation, for a repeller \begin{document}$\bar{\bf z}$\end{document} of \begin{document}$F$\end{document} , we start from estimating the repelling neighborhood of \begin{document}$\bar{\bf z}$\end{document} under \begin{document}$F^{k}$\end{document} for some \begin{document}$k ≥ 2$\end{document} , by a theory built on the first or second derivative of \begin{document}$F^k$\end{document} . By employing the Interval Arithmetic computation, we locate a snapback point \begin{document}${\bf z}_0$\end{document} in this repelling neighborhood and examine the nonzero determinant condition for the Jacobian of \begin{document}$F$\end{document} along the orbit through \begin{document}${\bf z}_0$\end{document} . With this new approach, we are able to conclude the existence of snapback repellers under the valid definition, hence chaotic behaviors, in a discrete-time predator-prey model, a population model, and the FitzHugh nerve model.