This paper deals with the host-parasitoid model,where the logistic equation governs the host population growth,and a proportion of the host population can find refuge.The equilibrium points'existence,number,and lo...This paper deals with the host-parasitoid model,where the logistic equation governs the host population growth,and a proportion of the host population can find refuge.The equilibrium points'existence,number,and local character are discussed.Taking the parameter regulating the parasitoid's growth as a bifurcation parameter,we prove that Neimark-Sacker and period-doubling bifurcations occur.Despite the complex behavior,it can be proved that the system is permanent,ensuring the long-term survival of both populations.Furthermore,it was observed that the presence of the proportional refuge does not significantly influence the system's behavior compared to the system without aproportional refuge.展开更多
Many discrete systems have more distinctive dynamical behaviors compared to continuous ones,which has led lots of researchers to investigate them.The discrete predatorprey model with two different 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 predatorprey 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.展开更多
This work investigates the bifurcation analysis in a discrete-time Leslie-Gower predatorprey model with constant yield predator harvesting.The stability analysis for the fixed points of the discretized model is shown ...This work investigates the bifurcation analysis in a discrete-time Leslie-Gower predatorprey model with constant yield predator harvesting.The stability analysis for the fixed points of the discretized model is shown briefy.In this study,the model undergoes codimension-1 bifurcation such as fold bifurcation(limit point),flip bifurcation(perioddoubling)and Neimark-Sacker bifurcation at a positive fixed point.Further,the model exhibits codimension-2 bifurcations,including Bogdanov-Takens bifurcation and generalized fip bifurcation at the fixed point.For each bifurcation,by using the critical normal form coefficient method,various critical states are calculated.To validate our analytical findings,the bifurcation curves of fixed points are drawn by using MATCONTM.The system exhibits interesting rich dynamics including limit cycles and chaos.Moreover,it has been shown that the predator harvesting may control the chaos in the system.展开更多
This paper investigates multiple bifurcations analyses and strong resonances of the Bazykin-Berezovskaya predator-prey model in depth using analytical and numerical bifurcation analysis.The stability conditions of fix...This paper investigates multiple bifurcations analyses and strong resonances of the Bazykin-Berezovskaya predator-prey model in depth using analytical and numerical bifurcation analysis.The stability conditions of fixed points,codim-1 and codim-2 bifurcations to include multiple and generic bifurcations are studied.This model exhibits transcritical,fip,Neimark-Sacker,and 1:2,1:3,1:4 strong resonances.The normal form coefficients and their scenarios for each bifurcation are examined by using the normal form theorem and bifurcation theory.For each bifurcation,various types of critical states are calculated,such as potential transformations between the one-parameter bifurcation point and different bifurcation points obtained from the two-parameter bifurcation point.To validate our analytical findings,the bifurcation curves of fixed points are determined by using MatcontM.展开更多
The aim of this paper is to investigate the dynamic behaviors of fractional-order logistic model with Allee effects in Caputo-Fabrizio sense.First of all,we apply the two-step Adams-Bashforth scheme to discretize the ...The aim of this paper is to investigate the dynamic behaviors of fractional-order logistic model with Allee effects in Caputo-Fabrizio sense.First of all,we apply the two-step Adams-Bashforth scheme to discretize the fractional-order logistic differential equation and obtain the two-dimensional discrete system.The parametric conditions for local asymptotic stability of equilibrium points are obtained by Schur-Chon criterion.Moreover,we discuss the existence and direction for Neimark-Sacker bifurcations with the help of center manifold theorem and bifurcation theory.Numerical simulations are provided to illustrate theoretical discussion.It is observed that Allee effect plays an important role in stability analysis.Strong Allee effect in population enhances the stability of the coexisting steady state.In additional,the effect of fractional-order derivative on dynamic behavior of the system is also investigated.展开更多
In this paper,we have derived a discrete evolutionary Beverton-Holt population model.The model is built using evolutionary game theory methodology and takes into consideration the strong Allee effect related to predat...In this paper,we have derived a discrete evolutionary Beverton-Holt population model.The model is built using evolutionary game theory methodology and takes into consideration the strong Allee effect related to predation saturation.We have discussed the existence of the positive fixed point and examined its asymptotic stability.Analytically,we demonstrated that the derived model exhibits Neimark-Sacker bifurcation when the maximal predator intensity is at lower values.All chaotic behaviors are justified numerically.Finally,to avoid these chaotic features and achieve asymptotic stability,we implement two chaos control methods.展开更多
In this study,a conformable fractional order Lotka-Volterra predator-prey model that describes the COVID-19 dynamics is considered.By using a piecewise constant approximation,a discretization method,which transforms t...In this study,a conformable fractional order Lotka-Volterra predator-prey model that describes the COVID-19 dynamics is considered.By using a piecewise constant approximation,a discretization method,which transforms the conformable fractional-order differential equation into a difference equation,is introduced.Algebraic conditions for ensuring the stability of the equilibrium points of the discrete system are determined by using Schur-Cohn criterion.Bifurcation analysis shows that the discrete system exhibits Neimark-Sacker bifurcation around the positive equilibrium point with respect to changing the parameter d and e.Maximum Lyapunov exponents show the complex dynamics of the discrete model.In addition,the COVID-19 mathematical model consisting of healthy and infected populations is also studied on the Erdos Rényi network.If the coupling strength reaches the critical value,then transition from nonchaotic to chaotic state is observed in complex dynamical networks.Finally,it has been observed that the dynamical network tends to exhibit chaotic behavior earlier when the number of nodes and edges increases.All these theoretical results are interpreted biologically and supported by numerical simulations.展开更多
First, a discrete stage-structured and harvested predator-prey model is established, which is based on a predator-prey model with Type III functional response. Then the~ oretical methods are used to investigate existe...First, a discrete stage-structured and harvested predator-prey model is established, which is based on a predator-prey model with Type III functional response. Then the~ oretical methods are used to investigate existence of equilibria and their local proper- ties. Third, it is shown that the system undergoes flip bifurcation and Neimark-Sacker bifurcation in the interior of R~_, by using the normal form of discrete systems, the center manifold theorem and the bifurcation theory, as varying the model parameters in some range. In particular, the direction and the stability of the flip bifurcation and the Neimark -Sacker bifurcation are showed. Finally, numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the com- plex dynamical behaviors, such as cascades of period-doubling bifurcation and chaotic sets. These results reveal far richer dynamics of the discrete model compared with the continuous model. The Lyapunov exponents are numerically computed to confirm fur- ther the complexity of the dynamical behaviors. In addition, we show also the stabilizing effect of the harvesting by using numerical simulations.展开更多
The normal forms of generalized Neimark-Sacker bifurcation are extensively studied using normal form theory of dynamic system. It is well known that if the normal forms of the generalized Neimark-Sacker bifurcation ar...The normal forms of generalized Neimark-Sacker bifurcation are extensively studied using normal form theory of dynamic system. It is well known that if the normal forms of the generalized Neimark-Sacker bifurcation are expressed in polar coordinates, then all odd order terms must, in general, remain in the normal forms. In this paper, five theorems are presented to show that the conventional Neimark-Sacker bifurcation can be further simplified. The simplest normal forms of generalized Neimark-Sacker bifurcation are calculated. Based on the conventional normal form, using appropriate nonlinear transformations, it is found that the generalized Neimark-Sacker bifurcation has at most two nonlinear terms remaining in the amplitude equations of the simplest normal forms up to any order. There are two kinds of simplest normal forms. Their algebraic expression formulas of the simplest normal forms in terms of the coefficients of the generalized Neimark-Sacker bifurcation systems are given.展开更多
This paper deals with a discrete-time predator-prey system which is subject to an Allee effect on prey.We investigate the existence and uniqueness and find parametric conditions for local asymptotic stability of fixed...This paper deals with a discrete-time predator-prey system which is subject to an Allee effect on prey.We investigate the existence and uniqueness and find parametric conditions for local asymptotic stability of fixed points of the discrete dynamic system.Moreover,using bifurcation theory,it is shown that the system undergoes Neimark-Sacker bifurcation in a small neighborhood of the unique positive fixed point and an inv aria nt circle will appear.Then the direction of bifurcation is given.Furthermore,numerical analysis is provided to illustrate the theoretical discussions with the help of Matlab packages.Thus,the main theoretical results are supported with numerical simulations.展开更多
In this paper,we use a semidiscretization method to derive a discrete predator–prey model with Holling type II,whose continuous version is stated in[F.Wu and Y.J.Jiao,Stability and Hopf bifurcation of a predator-prey...In this paper,we use a semidiscretization method to derive a discrete predator–prey model with Holling type II,whose continuous version is stated in[F.Wu and Y.J.Jiao,Stability and Hopf bifurcation of a predator-prey model,Bound.Value Probl.129(2019)1–11].First,the existence and local stability of fixed points of the system are investigated by employing a key lemma.Then we obtain the sufficient conditions for the occurrence of the transcritical bifurcation and Neimark–Sacker bifurcation and the stability of the closed orbits bifurcated by using the Center Manifold theorem and bifurcation theory.Finally,we present numerical simulations to verify corresponding theoretical results and reveal some new dynamics.展开更多
A two-parameter family of discrete models, consisting of two coupled nonlinear difference equations, describing a host-parasite interaction is considered. In particular, we prove that the model has at most one nontriv...A two-parameter family of discrete models, consisting of two coupled nonlinear difference equations, describing a host-parasite interaction is considered. In particular, we prove that the model has at most one nontrivial interior fixed point which is stable for a certain range of parameter values and also undergoes a Neimark-Sacker bifurcation that produces an attracting invariant curve in some areas of the parameter.展开更多
In this paper,a difference-algebraic predator prey model is proposed,and its complex dynamical behaviors are analyzed.The model is a discrete singular system,which is obtained by using Euler scheme to discretize a dif...In this paper,a difference-algebraic predator prey model is proposed,and its complex dynamical behaviors are analyzed.The model is a discrete singular system,which is obtained by using Euler scheme to discretize a differential-algebraic predator-prey model with harvesting that we establish.Firstly,the local stability of the interior equilibrium point of proposed model is investigated on the basis of discrete dynamical system theory.Further,by applying the new normal form of difference-algebraic equations,center manifold theory and bifurcation theory,the Flip bifurcation and Neimark-Sacker bifurcation around the interior equilibrium point are studied,where the step size is treated as the variable bifurcation parameter.Lastly,with the help of Matlab software,some numerical simulations are performed not only to validate our theoretical results,but also to show the abundant dynamical behaviors,such as period-doubling bifurcations,period 2,4,8,and 16 orbits,invariant closed curve,and chaotic sets.In particular,the corresponding maximum Lyapunov exponents are numerically calculated to corroborate the bifurcation and chaotic behaviors.展开更多
We propose and investigate a discrete-time predator-prey system with cooperative hunting in the predator population.The model is constructed from the classical Nicholson-Bailey host-parasitoid system with density depe...We propose and investigate a discrete-time predator-prey system with cooperative hunting in the predator population.The model is constructed from the classical Nicholson-Bailey host-parasitoid system with density dependent growth rate.A sufficient condition based on the model parameters for which both populations can coexist is derived,namely that the predator’s maximal reproductive number exceeds one.We study existence of interior steady states and their stability in certain parameter regimes.It is shown that the system behaves asymptotically similar to the model with no cooperative hunting if the degree of cooperation is small.Large cooperative hunting,however,may promote persistence of the predator for which the predator would otherwise go extinct if there were no cooperation.展开更多
文摘This paper deals with the host-parasitoid model,where the logistic equation governs the host population growth,and a proportion of the host population can find refuge.The equilibrium points'existence,number,and local character are discussed.Taking the parameter regulating the parasitoid's growth as a bifurcation parameter,we prove that Neimark-Sacker and period-doubling bifurcations occur.Despite the complex behavior,it can be proved that the system is permanent,ensuring the long-term survival of both populations.Furthermore,it was observed that the presence of the proportional refuge does not significantly influence the system's behavior compared to the system without aproportional refuge.
基金supported by the National Natural Science Foundation of China(No.12001503)the Project of Beijing Municipal Commission of Education(KM 202110015001)。
文摘Many discrete systems have more distinctive dynamical behaviors compared to continuous ones,which has led lots of researchers to investigate them.The discrete predatorprey 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.
基金supported by Science Engineering Research Board,Government of India (CRG/2021/006380).
文摘This work investigates the bifurcation analysis in a discrete-time Leslie-Gower predatorprey model with constant yield predator harvesting.The stability analysis for the fixed points of the discretized model is shown briefy.In this study,the model undergoes codimension-1 bifurcation such as fold bifurcation(limit point),flip bifurcation(perioddoubling)and Neimark-Sacker bifurcation at a positive fixed point.Further,the model exhibits codimension-2 bifurcations,including Bogdanov-Takens bifurcation and generalized fip bifurcation at the fixed point.For each bifurcation,by using the critical normal form coefficient method,various critical states are calculated.To validate our analytical findings,the bifurcation curves of fixed points are drawn by using MATCONTM.The system exhibits interesting rich dynamics including limit cycles and chaos.Moreover,it has been shown that the predator harvesting may control the chaos in the system.
文摘This paper investigates multiple bifurcations analyses and strong resonances of the Bazykin-Berezovskaya predator-prey model in depth using analytical and numerical bifurcation analysis.The stability conditions of fixed points,codim-1 and codim-2 bifurcations to include multiple and generic bifurcations are studied.This model exhibits transcritical,fip,Neimark-Sacker,and 1:2,1:3,1:4 strong resonances.The normal form coefficients and their scenarios for each bifurcation are examined by using the normal form theorem and bifurcation theory.For each bifurcation,various types of critical states are calculated,such as potential transformations between the one-parameter bifurcation point and different bifurcation points obtained from the two-parameter bifurcation point.To validate our analytical findings,the bifurcation curves of fixed points are determined by using MatcontM.
文摘The aim of this paper is to investigate the dynamic behaviors of fractional-order logistic model with Allee effects in Caputo-Fabrizio sense.First of all,we apply the two-step Adams-Bashforth scheme to discretize the fractional-order logistic differential equation and obtain the two-dimensional discrete system.The parametric conditions for local asymptotic stability of equilibrium points are obtained by Schur-Chon criterion.Moreover,we discuss the existence and direction for Neimark-Sacker bifurcations with the help of center manifold theorem and bifurcation theory.Numerical simulations are provided to illustrate theoretical discussion.It is observed that Allee effect plays an important role in stability analysis.Strong Allee effect in population enhances the stability of the coexisting steady state.In additional,the effect of fractional-order derivative on dynamic behavior of the system is also investigated.
文摘In this paper,we have derived a discrete evolutionary Beverton-Holt population model.The model is built using evolutionary game theory methodology and takes into consideration the strong Allee effect related to predation saturation.We have discussed the existence of the positive fixed point and examined its asymptotic stability.Analytically,we demonstrated that the derived model exhibits Neimark-Sacker bifurcation when the maximal predator intensity is at lower values.All chaotic behaviors are justified numerically.Finally,to avoid these chaotic features and achieve asymptotic stability,we implement two chaos control methods.
文摘In this study,a conformable fractional order Lotka-Volterra predator-prey model that describes the COVID-19 dynamics is considered.By using a piecewise constant approximation,a discretization method,which transforms the conformable fractional-order differential equation into a difference equation,is introduced.Algebraic conditions for ensuring the stability of the equilibrium points of the discrete system are determined by using Schur-Cohn criterion.Bifurcation analysis shows that the discrete system exhibits Neimark-Sacker bifurcation around the positive equilibrium point with respect to changing the parameter d and e.Maximum Lyapunov exponents show the complex dynamics of the discrete model.In addition,the COVID-19 mathematical model consisting of healthy and infected populations is also studied on the Erdos Rényi network.If the coupling strength reaches the critical value,then transition from nonchaotic to chaotic state is observed in complex dynamical networks.Finally,it has been observed that the dynamical network tends to exhibit chaotic behavior earlier when the number of nodes and edges increases.All these theoretical results are interpreted biologically and supported by numerical simulations.
文摘First, a discrete stage-structured and harvested predator-prey model is established, which is based on a predator-prey model with Type III functional response. Then the~ oretical methods are used to investigate existence of equilibria and their local proper- ties. Third, it is shown that the system undergoes flip bifurcation and Neimark-Sacker bifurcation in the interior of R~_, by using the normal form of discrete systems, the center manifold theorem and the bifurcation theory, as varying the model parameters in some range. In particular, the direction and the stability of the flip bifurcation and the Neimark -Sacker bifurcation are showed. Finally, numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the com- plex dynamical behaviors, such as cascades of period-doubling bifurcation and chaotic sets. These results reveal far richer dynamics of the discrete model compared with the continuous model. The Lyapunov exponents are numerically computed to confirm fur- ther the complexity of the dynamical behaviors. In addition, we show also the stabilizing effect of the harvesting by using numerical simulations.
基金Supported by National Natural Science Foundation of China (No10872141)Doctoral Foundation of Ministry of Education of China (No20060056005)Natural Science Foundation of Tianjin University of Science and Technology (No20070210)
文摘The normal forms of generalized Neimark-Sacker bifurcation are extensively studied using normal form theory of dynamic system. It is well known that if the normal forms of the generalized Neimark-Sacker bifurcation are expressed in polar coordinates, then all odd order terms must, in general, remain in the normal forms. In this paper, five theorems are presented to show that the conventional Neimark-Sacker bifurcation can be further simplified. The simplest normal forms of generalized Neimark-Sacker bifurcation are calculated. Based on the conventional normal form, using appropriate nonlinear transformations, it is found that the generalized Neimark-Sacker bifurcation has at most two nonlinear terms remaining in the amplitude equations of the simplest normal forms up to any order. There are two kinds of simplest normal forms. Their algebraic expression formulas of the simplest normal forms in terms of the coefficients of the generalized Neimark-Sacker bifurcation systems are given.
文摘This paper deals with a discrete-time predator-prey system which is subject to an Allee effect on prey.We investigate the existence and uniqueness and find parametric conditions for local asymptotic stability of fixed points of the discrete dynamic system.Moreover,using bifurcation theory,it is shown that the system undergoes Neimark-Sacker bifurcation in a small neighborhood of the unique positive fixed point and an inv aria nt circle will appear.Then the direction of bifurcation is given.Furthermore,numerical analysis is provided to illustrate the theoretical discussions with the help of Matlab packages.Thus,the main theoretical results are supported with numerical simulations.
基金This work is partly supported by the National Natural Science Foundation of China(61473340)the Distinguished Professor Foundation of Qianjiang Scholar in Zhejiang Provincethe National Natural Science Foundation of Zhejiang University of Science and Technology(F701108G14).
文摘In this paper,we use a semidiscretization method to derive a discrete predator–prey model with Holling type II,whose continuous version is stated in[F.Wu and Y.J.Jiao,Stability and Hopf bifurcation of a predator-prey model,Bound.Value Probl.129(2019)1–11].First,the existence and local stability of fixed points of the system are investigated by employing a key lemma.Then we obtain the sufficient conditions for the occurrence of the transcritical bifurcation and Neimark–Sacker bifurcation and the stability of the closed orbits bifurcated by using the Center Manifold theorem and bifurcation theory.Finally,we present numerical simulations to verify corresponding theoretical results and reveal some new dynamics.
文摘A two-parameter family of discrete models, consisting of two coupled nonlinear difference equations, describing a host-parasite interaction is considered. In particular, we prove that the model has at most one nontrivial interior fixed point which is stable for a certain range of parameter values and also undergoes a Neimark-Sacker bifurcation that produces an attracting invariant curve in some areas of the parameter.
基金the National Natural Science Foundation of China(Grant No.11871393)the Key Project of the International Science and Technology Cooperation Program of Shaanxi Research&Development Plan(Grant No.2019KWZ-08)the Science and Technology Project founded by the Education Department of Jiangxi Province(Grant No.GJJ14775).
文摘In this paper,a difference-algebraic predator prey model is proposed,and its complex dynamical behaviors are analyzed.The model is a discrete singular system,which is obtained by using Euler scheme to discretize a differential-algebraic predator-prey model with harvesting that we establish.Firstly,the local stability of the interior equilibrium point of proposed model is investigated on the basis of discrete dynamical system theory.Further,by applying the new normal form of difference-algebraic equations,center manifold theory and bifurcation theory,the Flip bifurcation and Neimark-Sacker bifurcation around the interior equilibrium point are studied,where the step size is treated as the variable bifurcation parameter.Lastly,with the help of Matlab software,some numerical simulations are performed not only to validate our theoretical results,but also to show the abundant dynamical behaviors,such as period-doubling bifurcations,period 2,4,8,and 16 orbits,invariant closed curve,and chaotic sets.In particular,the corresponding maximum Lyapunov exponents are numerically calculated to corroborate the bifurcation and chaotic behaviors.
文摘We propose and investigate a discrete-time predator-prey system with cooperative hunting in the predator population.The model is constructed from the classical Nicholson-Bailey host-parasitoid system with density dependent growth rate.A sufficient condition based on the model parameters for which both populations can coexist is derived,namely that the predator’s maximal reproductive number exceeds one.We study existence of interior steady states and their stability in certain parameter regimes.It is shown that the system behaves asymptotically similar to the model with no cooperative hunting if the degree of cooperation is small.Large cooperative hunting,however,may promote persistence of the predator for which the predator would otherwise go extinct if there were no cooperation.
