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This paper concerns a discrete diffusive predator–prey system involving two competing predators and one prey in a shifting habitat induced by the climate change. By assuming that both predators can increase when the prey is at the maximal capacity and the prey can still survive under optimal climatic conditions when these two predators have their maximal densities, we investigate the existence and non-existence for different types of forced traveling waves which describe the conversion from the state of a saturated aboriginal prey with two invading alien predators, an aboriginal co-existent predator–prey with an invading alien predator, and the coexistence of three species to the extinction state, respectively. The existence of supercritical and critical forced waves is showed by applying Schauder's fixed point theorem on various invariant cones via constructing different types of generalized super- and sub-solutions while the non-existence of subcritical forced waves is obtained by contradiction. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the codimension-two bifurcations of a two-dimensional discrete time predator-prey model, which is derived from a classical Lotka-Volterra type predator-prey system by the forward Euler scheme. It is shown that the model undergoes codimension-two bifurcations associated with 1:2, 1:3 and 1:4 resonances. The conditions for strong resonance bifurcations are obtained by using the normal form method and the theory of approximation by a flow. Moreover, the bifurcation curves around 1:2 resonance are obtained and returned to the original parameters. Numerical simulations are provided to illustrate our theoretical results, and the model exhibits complex dynamical behaviors such as quasi-periodic orbits and the chaotic sets. [ABSTRACT FROM AUTHOR]

In this paper, we work on the discrete modified Leslie type predator-prey model with Holling type II functional response. The existence and local stability of the fixed points of this system are studied. According to bifurcation theory and normal forms, we investigate the codimension 1 and 2 bifurcations of positive fixed points, including the fold, 1:1 strong resonance, fold-flip and 1:2 strong resonance bifurcations. In particular, the discussion of discrete codimension 2 bifurcation is rare and difficult. Our work can be seen as an attempt to complement existing research on this topic. In addition, numerical analysis is used to demonstrate the correctness of the theoretical results. Our analysis of this discrete system revealed quite different dynamical behaviors than the continuous one.

A discrete Bazykin predator–prey model with predator intraspecific interactions and ratio-dependent functional response is proposed and investigated. A brief mathematical analysis of the model involves giving fixed points and analyzing local stability. Codimension-one bifurcations such as the transcritical, fold, flip, Neimark-Sacker bifurcations and codimension-two bifurcations, including the fold-flip bifurcation, 1:2, 1:3, and 1:4 strong resonances, are investigated. Accurate theoretical analysis and exquisite numerical simulation are given simultaneously, which support the main content of this manuscript. With the help of several local attraction basins, period, and Lyapunov exponent diagrams, an intriguing set of fascinating dynamics in the two-parameter spaces of integral step size with other parameters is uncovered. The results in this paper reveal that the dynamics of the discrete-time predator–prey system in both single-parameter and two-parameter spaces are inherently rich and complex. [ABSTRACT FROM AUTHOR]

The paper introduces a novel two-dimensional fractional discrete-time predator–prey Leslie–Gower model with an Allee effect on the predator population. The model's nonlinear dynamics are explored using various numerical techniques, including phase portraits, bifurcations and maximum Lyapunov exponent, with consideration given to both commensurate and incommensurate fractional orders. These techniques reveal that the fractional-order predator–prey Leslie–Gower model exhibits intricate and diverse dynamical characteristics, including stable trajectories, periodic motion, and chaotic attractors, which are affected by the variance of the system parameters, the commensurate fractional order, and the incommensurate fractional order. Finally, we employ the 0–1 method, the approximate entropy test and the  C 0  algorithm to measure complexity and confirm chaos in the proposed system. [ABSTRACT FROM AUTHOR]

In this paper, some complicate dynamical behaviors are formulated for a discrete predator–prey model with group defense and nonlinear harvesting in prey. After considering the existence and stability for all of its nonnegative fixed points, our main work is to present those conditions for the occurrences of transcritical bifurcation, saddle-node bifurcation and Neimark–Sacker bifurcation, respectively. Numerical simulations not only verify the theoretical results for saddle-node bifurcation and Neimark–Sacker bifurcation but also display more interesting dynamical properties of the model. [ABSTRACT FROM AUTHOR]

The dynamical behavior of a discrete predator–prey system with a nonmonotonic functional response is investigated in this work. We study the local asymptotic stability of the positive equilibrium of the system by examining the characteristic equation of the linearized system corresponding to the model. By choosing the growth rate as a bifurcation parameter, the existence of Neimark–Sacker and period‐doubling bifurcations at the positive equilibrium is established. Furthermore, the effects of perturbations on the system dynamics are investigated. Finally, examples are presented to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, we study the qualitative structure of a discrete predator-prey model with nonmonotonic functional response near a degenerate fixed point whose eigenvalues are $ \pm1 $. Firstly, the model is transformed into an ordinary differential system by using the normal form theory and the Takens's theorem. Then, the qualitative properties of this ordinary differential system near the degenerate equilibrium are analyzed with the blowing-up method. Finally, according to the conjugacy between the discrete model and the time-one mapping of the vector field, the qualitative structure of this discrete model is obtained. Numerical simulations are also given. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The paper introduces a novel two-dimensional fractional discrete-time predator–prey Leslie–Gower model with an Allee effect on the predator population. The model’s nonlinear dynamics are explored using various numerical techniques, including phase portraits, bifurcations and maximum Lyapunov exponent, with consideration given to both commensurate and incommensurate fractional orders. These techniques reveal that the fractional-order predator–prey Leslie–Gower model exhibits intricate and diverse dynamical characteristics, including stable trajectories, periodic motion, and chaotic attractors, which are affected by the variance of the system parameters, the commensurate fractional order, and the incommensurate fractional order. Finally, we employ the 0–1 method, the approximate entropy test and the C0 algorithm to measure complexity and confirm chaos in the proposed system.

In this paper, a discrete Leslie–Gower model with nonlinear prey harvesting and prey refuge is proposed. We prove the existence and stability of an interior equilibrium of the model. In addition, the existence of flip bifurcation and Neimark–Sacker bifurcation at the interior equilibrium is studied by applying central manifold theory and bifurcation theory. Using Pontryagin's maximum principle, we propose an optimal harvesting problem. [ABSTRACT FROM AUTHOR]

The continuous predator-prey model is one of the main models studied in recent years. The dynamical properties of these models are so complex that it is an urgent topic to be studied. In this paper, we transformed a continuous predator-prey model with modified Leslie-Gower and Hollingtype Ⅲ schemes into a discrete mode by using Euler approximation method. The existence and stability of fixed points for this discrete model were investigated. Flip bifurcation analyses of this discrete model was carried out and corresponding bifurcation conditions were obtained. Provided with these bifurcation conditions, an example was given to carry out numerical simulations, which shows that the discrete model undergoes flip bifurcation around the stable fixed point. In addition, compared with previous studies on the continuous predator-prey model, our discrete model shows more irregular and complex dynamic characteristics. The present research can be regarded as the continuation and development of the former studies.

Abstract In this paper, we consider the complex dynamics of a discrete predator–prey system with a strong Allee effect on the prey and a ratio-dependent functional response, which is the discrete version of the continuous system in (Nonlinear Anal., Real World Appl. 16:235–249, 2014). First, by giving several examples to display the limitations and errors of the local stability of the equilibrium point p obtained in (Nonlinear Anal., Real World Appl. 16:235–249, 2014), we provide an easily verified and complete discrimination criterion for the local stability of this equilibrium. Then we study some properties of its discrete version, especially for the stability and bifurcation for the equilibrium point E1 $E_{1}$, which has not been considered in any literature to the best of our knowledge. By using the center manifold theorem and bifurcation theory, we consider the flip bifurcation of this system at E1 $E_{1}$ and obtain the stability of the closed orbits bifurcated from E1 $E_{1}$. The numerical simulations not only show the correctness of our theoretical analysis, but also we find some new and interesting dynamics of this system.

In this paper, we work on the discrete modified Leslie type predator-prey model with Holling type II functional response. The existence and local stability of the fixed points of this system are studied. According to bifurcation theory and normal forms, we investigate the codimension 1 and 2 bifurcations of positive fixed points, including the fold, 1:1 strong resonance, fold-flip and 1:2 strong resonance bifurcations. In particular, the discussion of discrete codimension 2 bifurcation is rare and difficult. Our work can be seen as an attempt to complement existing research on this topic. In addition, numerical analysis is used to demonstrate the correctness of the theoretical results. Our analysis of this discrete system revealed quite different dynamical behaviors than the continuous one.

The predator–prey/consumer–resource interaction is the most fundamental and important process in population dynamics. Many species, such as monocarpic plants and semelparous animals, have discrete nonoverlapping generations and their births occur in regular breeding seasons. Their interactions are described by difference equations or formulated as discrete-time mappings. In this paper we study bifurcations in a discrete predator–prey model with nonmonotone functional response described by a simplified Holling IV function. It is shown that the model exhibits various bifurcations of codimension 1, including fold bifurcation, transcritical bifurcation, flip bifurcations and Neimark–Sacker bifurcation, as the values of parameters vary. Moreover, we establish the existence of Bogdanov–Takens bifurcation of codimension 2 and calculate the approximate expressions of bifurcation curves. Numerical simulations are also presented to illustrate the theoretical analysis. These results demonstrate that the Bogdanov–Takens bifurcation of codimension 2 at the degenerate singularity persists in all three versions of the predator–prey model with nonmonotone functional response: continuous-time, time-delayed, and discrete-time. [ABSTRACT FROM AUTHOR]

In this paper, the bifurcations and the control of a discrete predator–prey model with strong Allee effect on the prey are investigated. This shows that the model undergoes a supercritical Neimark–Sacker bifurcation. Meanwhile, the explicit approximate expression of the stable closed invariant curve caused by the Neimark–Sacker bifurcation is given. 1:3 strong resonance is investigated through approximation by a flow, and the bifurcation curves around 1:3 resonance are obtained. Moreover, for the sake of regulating the stability of the biological system, we extend the hybrid control strategy to control the Neimark–Sacker bifurcation and 1:3 strong resonance. The theoretical analyses are validated by numerical simulations and are explained from the biological point of view. [ABSTRACT FROM AUTHOR]

In this paper, we investigate dynamical behaviours of a discrete predator–prey model with nonmonotonic functional response. Codimension-2 bifurcations associated with 1:2, 1:3 and 1:4 resonances are analyzed by using bifurcation theory. Codimension-2 bifurcation diagrams, maximum Lyapunov exponents diagrams and phase portraits, which not only illustrate the validity of the theoretical results but also display some interesting complex dynamical behaviours, are obtained by numerical simulations. [ABSTRACT FROM AUTHOR]

We propose a variant of the discrete Lotka–Volterra model for predator–prey interactions. A detailed stability and numerical analysis of the model are presented to explore the long time behaviour as each of the control parameter is varied independently. We show how the condition for survival of the predator depends on the natural death rate of predator and the efficiency of predation. The model is found to support different dynamical regimes asymptotically including predator extinction, stable fixed point and limit cycle attractors for co-existence of predator and prey and more complex dynamics involving chaotic attractors. We are able to locate exactly the domain of chaos in the parameter plane using a dimensional analysis. [ABSTRACT FROM AUTHOR]

In this paper we revisit a discrete predator–prey model with a non-monotonic functional response, originally presented in Hu, Teng, and Zhang (2011) [6] . First, by citing several examples to illustrate the limitations and errors of the local stability of the equilibrium points E 3 and E 4 obtained in this article, we formulate an easily verified and complete discrimination criterion for the local stability of the two equilibria. Here, we present a very useful lemma, which is a corrected version of a known result, and a key tool in studying the local stability and bifurcation of an equilibrium point in a given system. We then study the stability and bifurcation for the equilibrium point E 1 of this system, which has not been considered in any known literature. Unlike known results that present a large number of mathematical formulae that are not easily verified, we formulate easily verified sufficient conditions for flip bifurcation and fold bifurcation, which are explicitly expressed by the coefficient of the system. The center manifold theory and Project Method are the main tools in the analysis of bifurcations. The theoretical results obtained are further illustrated by numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, we consider the dynamics of a discrete predator-prey model which is a result of discretization of the corresponding continuous Lotka-Volterra predator-prey model. Among the topis are equilibria and their stability, existence and stability of a period-2 orbit, as well the chaotic behavior of F. The chaos here is in the sense of topological horseshoe and is obtained for certain range of parameter values by applying a recent result from [5]. Our results are in contrast to the recent ones in [6] which claimed that if the predator-prey interaction is replaced by cooperative or competitive interaction, the discretization preserves the property of convergence to the equilibrium, regardless of the step size. [ABSTRACT FROM AUTHOR]

Abstract: A discrete three trophic level food chain model with ratio-dependent Michaelis–Menten type functional response is investigated. It is shown that under some appropriate conditions the system is permanent. The results indicate that, to make the species coexist in the long run, it is a surefire strategy to keep the death rate of the predator and top predator rather small and the intrinsic growth rate of the prey relatively large. [Copyright &y& Elsevier]

Abstract: Sufficient conditions are established for the permanence in a delayed discrete predator–prey model with Holling type III functional response: Our investigation confirms that when the death rate of the predator is rather small as well as the intrinsic growth rate of the prey is relatively large, the species could coexist in the long run. [Copyright &y& Elsevier]

The continuous predator-prey model is one of the main models studied in recent years. The dynamical properties of these models are so complex that it is an urgent topic to be studied. In this paper, we transformed a continuous predator-prey model with modified Leslie-Gower and Hollingtype III schemes into a discrete mode by using Euler approximation method. The existence and stability of fixed points for this discrete model were investigated. Flip bifurcation analyses of this discrete model was carried out and corresponding bifurcation conditions were obtained. Provided with these bifurcation conditions, an example was given to carry out numerical simulations, which shows that the discrete model undergoes flip bifurcation around the stable fixed point. In addition, compared with previous studies on the continuous predator-prey model, our discrete model shows more irregular and complex dynamic characteristics. The present research can be regarded as the continuation and development of the former studies.