In this paper, we will study a class of discrete Leslie-Gower prey-predator models, which is a discretization of the continuous model proposed by Leslie and Gower in 1960. First, we find all fixed points, use hyperbol...In this paper, we will study a class of discrete Leslie-Gower prey-predator models, which is a discretization of the continuous model proposed by Leslie and Gower in 1960. First, we find all fixed points, use hyperbolic and non-hyperbolic conditions to give the types of fixed points, and then analyze the bifurcation properties of non-hyperbolic fixed points. The generating conditions of Flip bifurcation and Neimark-Sacker bifurcation at fixed points are studied. Finally, numerical simulations of Flip bifurcation and Neimark-Sacker bifurcation are given.展开更多
Using finite differences and entropy inequalities, the global existence of weak solutions to a multidimensional parabolic strongly coupled prey-predator model is obtained. The nonnegativity of the solutions is also sh...Using finite differences and entropy inequalities, the global existence of weak solutions to a multidimensional parabolic strongly coupled prey-predator model is obtained. The nonnegativity of the solutions is also shown.展开更多
In this manuscript, we have studied a fractional-order tri-trophic model with the help of Caputo operator. The total population is divided into three parts, namely prey, intermediate predator and top predator. In addi...In this manuscript, we have studied a fractional-order tri-trophic model with the help of Caputo operator. The total population is divided into three parts, namely prey, intermediate predator and top predator. In addition, the predator fear impact on prey population is suggested in this paper. Existence and uniqueness along with non-negativity and boundedness of the model system have been investigated. We have studied the local stability at all equilibrium points. Also, we have discussed global stability and Hopf bifurcation of our suggested model at interior equilibrium point. The Adam-Bashforth-Moulton approach is utilized to approximate the solution to the proposed model. With the help of MATLAB, we were able to conduct graphical demonstrations and numerical simulations.展开更多
In this paper, we prove the existence of forced waves for Leslie-Gower prey-predator model with nonlocal effects under shifting environment. By constructing a pair of upper and lower solutions with the method of monot...In this paper, we prove the existence of forced waves for Leslie-Gower prey-predator model with nonlocal effects under shifting environment. By constructing a pair of upper and lower solutions with the method of monotone iteration, we can obtain the existence of forced waves for any positive constant shifting speed. Finally, we show the asymptotical behavior of traveling wave fronts in two tails.展开更多
In this article,we investigate a fractional-order singular Leslie-Gower prey-predator bioeconomic model,which describes the interaction bet ween populations of prey and predator,and takes into account the economic int...In this article,we investigate a fractional-order singular Leslie-Gower prey-predator bioeconomic model,which describes the interaction bet ween populations of prey and predator,and takes into account the economic interest.We firstly obtain the solvability condition and the st ability of the model sys tem,and discuss the singularity induced bifurcation phenomenon.Next,we introduce a st ate feedback controller to elimina te the singularity induced bifurcation phenomenon,and discuss the optimal control problems.Finally,numerical solutions and their simulations are considered in order to illustrate the theoretical results and reveal the more complex dynamical behavior.展开更多
In this paper, the problem of chaos, stability and estimation of unknown parameters of the stochastic lattice gas for prey-predator model with pair-approximation is studied. The result shows that this dynamical system...In this paper, the problem of chaos, stability and estimation of unknown parameters of the stochastic lattice gas for prey-predator model with pair-approximation is studied. The result shows that this dynamical system exhibits an oscillatory behavior of the population densities of prey and predator. Using Liapunov stability technique, the estimators of the unknown probabilities are derived, and also the updating rules for stability around its steady states are derived. Furthermore the feedback control law has been as non-linear functions of the population densities. Numerical simulation study is presented graphically.展开更多
This paper investigates a strongly coupled reaction-diffusion model with Holling-II reaction function in a bounded domain with homogeneous Neumann boundary condition. The sufficient condition for the existence and non...This paper investigates a strongly coupled reaction-diffusion model with Holling-II reaction function in a bounded domain with homogeneous Neumann boundary condition. The sufficient condition for the existence and non-existence of the non-constant positive solutions are obtained. Moreover, we prove that the nonlinear diffusion terms can create non-constant positive equilibrium solutions when the corresponding model without nonlinear diffusion term fails.展开更多
Due to the random search of species and from the economic point of view,combined harvesting is more suitable than selective harvesting.Thus,we have developed and analyzed a prey-predator model with the combined effect...Due to the random search of species and from the economic point of view,combined harvesting is more suitable than selective harvesting.Thus,we have developed and analyzed a prey-predator model with the combined effect of nonlinear harvesting in this research paper.Nonlinear harvesting possesses multiple predator-free and interior equilibrium points in the dynamical system.We have examined the local stability analysis of all the equilibrium points.Besides these various types,rich and complex dynamical behaviors such as backward,saddle-node,Hopf and Bogdanov-Takens(BT)bifurcations,homoclinic loop and limit cycles appear in this model.Furthermore,interesting phenomena like bi-stability and tri-stability occur in our model between the different equilibrium points.Also,we have derived different threshold values of predator harvesting parameters and prey environmental carrying capacity from these bifurcations to obtain the different harvesting strategies for both species.We have observed that the extinction of predator species may not happen due to backward bifurcation,although a stable predator-free equilibrium(PFE)exists.Finally,numerical simulations are discussed using MATLAB to verify all the theoretical results.展开更多
This paper presents the dynamical properties of a discrete-time prey-predator model with refuge in prey under imprecise biological parameters.We consider the refuge concept of prey,which is proportional to the density...This paper presents the dynamical properties of a discrete-time prey-predator model with refuge in prey under imprecise biological parameters.We consider the refuge concept of prey,which is proportional to the density of prey species with interval parameters.The model develops with natural interval parameters since the uncertainties of parameters of any ecological system are a widespread phenomenon in nature.The equilibria of the model are obtained,and the dynamic behaviours of the proposed system are examined.Simulations of the model are performed for different parameters of the model.Numerical simulations show that the proposed discrete model exhibits rich dynamics of a chaotic and complex nature.Our study,through analytical derivation and numerical example,presents the effect of refuge on population dynamics under imprecise biological parameters.展开更多
In this paper,the Caputo fractional derivative is assumed to be the prey-predator model.In order to create Caputo fractional differential equations for the prey-predator model,a discretization process is first used.Th...In this paper,the Caputo fractional derivative is assumed to be the prey-predator model.In order to create Caputo fractional differential equations for the prey-predator model,a discretization process is first used.The fixed points of the model are categorized topologically.We identify requirements for the fixed points of the suggested prey-predator model's local asymptotic stability.We demonstrate analytically that,under specific parametric conditions,a fractional order prey-predator model supports both a Neimark-Sacker(NS)bifurcation and a Flip bifurcation.We present evidence for NS and Flip bifurcations using central manifold and bifurcation theory.The parameter values and the initial conditions have been found to have a profound impact on the dynamical behavior of the fractional order prey-predator model.As the bifurcation parameter is increased,the system displays chaotic behavior.Numerical simulations are shown to demonstrate chaotic behaviors like bifurcations,phase portraits,invariant closed cycles,and attractive chaotic sets in addition to validating analytical conclusions.The suggested prey-predator dynamical system's chaotic behavior will be controlled by the OGY and hybrid control methodology,which will also visualize the chaotic state for various biological parameters.展开更多
In this paper, we study the positive steady states of a prey-predator model with diffusion throughout and a non-monotone conversion rate under the homogeneous Dirichlet boundary condition. We obtain some results of th...In this paper, we study the positive steady states of a prey-predator model with diffusion throughout and a non-monotone conversion rate under the homogeneous Dirichlet boundary condition. We obtain some results of the existence and non-existence of positive steady states. The stability and uniqueness of positive steady states are also discussed.展开更多
We consider a Lotka-Volterra prey-predator model with cross-diffusion and Holling type-II functional response.The main concern is the existence of positive solutions under the combined effect of cross-diffusion and Ho...We consider a Lotka-Volterra prey-predator model with cross-diffusion and Holling type-II functional response.The main concern is the existence of positive solutions under the combined effect of cross-diffusion and Holling type-II functional response.Here,a positive solution corresponds to a coexistence state of the model.Firstly,we study the sufficient conditions to ensure the existence of positive solutions by using degree theory and analyze the coexistence region in parameter plane.In addition,we present the uniqueness of positive solutions in one dimension case.Secondly,we study the stability of the trivial and semi-trivial solutions by analyzing the principal eigenvalue of the corresponding linearized system,and then we characterize the stable/unstable regions of semi-trivial solutions in parameter plane.展开更多
In this paper, we investigate a prey-predator model with diffusion and ratio-dependent functional response subject to the homogeneous Neumann boundary condition. Our main focuses are on the global behavior of the reac...In this paper, we investigate a prey-predator model with diffusion and ratio-dependent functional response subject to the homogeneous Neumann boundary condition. Our main focuses are on the global behavior of the reaction-diffusion system and its corresponding steady-state problem. We first apply various Lyapunov functions to discuss the global stability of the unique positive constant steady-state. Then, for the steady-state system, we establish some a priori upper and lower estimates for positive steady-states, and derive several results for non-existence of positive non-constant steady-states if the diffusion rates are large or small.展开更多
The authors study a diffusive prey-predator model subject to the homogeneous Neumann boundary condition and give some qualitative descriptions of solutions to this reaction-diffusion system and its corresponding stead...The authors study a diffusive prey-predator model subject to the homogeneous Neumann boundary condition and give some qualitative descriptions of solutions to this reaction-diffusion system and its corresponding steady-state problem. The local and global stability of the positive constant steady-state are discussed, and then some results for non- existence of positive non-constant steady-states are derived.展开更多
The stabilization problem of a kind of prey-predator model with Holling fimctional response is investigated. By approximate linearization approach, a feedback control law stabilizing the closed- loop system is obtaine...The stabilization problem of a kind of prey-predator model with Holling fimctional response is investigated. By approximate linearization approach, a feedback control law stabilizing the closed- loop system is obtained. On the other hand, by exact linearization approach, a suitable change of coordinates in the state space and a feedback control law render the complex nonlinear system to be a linear controllable one such that the positive equilibrium point of the closed-loop system is globally asymptotically stable.展开更多
In this paper, a nonlinear predator reproduction and prey competition model with diffusion is discussed. Some existence and non-existence results concerning non-constant positive steady-states are presented using topo...In this paper, a nonlinear predator reproduction and prey competition model with diffusion is discussed. Some existence and non-existence results concerning non-constant positive steady-states are presented using topological degree argument and the energy method, respectively.展开更多
In this paper,we consider a prey-predator fishery model with prey dispersal in a two-patch environment,one is assumed to be a free fishing zone and the other is a reserved zone where fishing and other extractive activ...In this paper,we consider a prey-predator fishery model with prey dispersal in a two-patch environment,one is assumed to be a free fishing zone and the other is a reserved zone where fishing and other extractive activities are prohibited.The existence of possible steady states of the system is discussed.The local and global stability analysis has been carried out.An optimal harvesting policy is given using Pontryagin s maximum principle.展开更多
文摘In this paper, we will study a class of discrete Leslie-Gower prey-predator models, which is a discretization of the continuous model proposed by Leslie and Gower in 1960. First, we find all fixed points, use hyperbolic and non-hyperbolic conditions to give the types of fixed points, and then analyze the bifurcation properties of non-hyperbolic fixed points. The generating conditions of Flip bifurcation and Neimark-Sacker bifurcation at fixed points are studied. Finally, numerical simulations of Flip bifurcation and Neimark-Sacker bifurcation are given.
基金supported by the National Natural Science Foundation of China (Nos. 10701024, 10601011)
文摘Using finite differences and entropy inequalities, the global existence of weak solutions to a multidimensional parabolic strongly coupled prey-predator model is obtained. The nonnegativity of the solutions is also shown.
文摘In this manuscript, we have studied a fractional-order tri-trophic model with the help of Caputo operator. The total population is divided into three parts, namely prey, intermediate predator and top predator. In addition, the predator fear impact on prey population is suggested in this paper. Existence and uniqueness along with non-negativity and boundedness of the model system have been investigated. We have studied the local stability at all equilibrium points. Also, we have discussed global stability and Hopf bifurcation of our suggested model at interior equilibrium point. The Adam-Bashforth-Moulton approach is utilized to approximate the solution to the proposed model. With the help of MATLAB, we were able to conduct graphical demonstrations and numerical simulations.
文摘In this paper, we prove the existence of forced waves for Leslie-Gower prey-predator model with nonlocal effects under shifting environment. By constructing a pair of upper and lower solutions with the method of monotone iteration, we can obtain the existence of forced waves for any positive constant shifting speed. Finally, we show the asymptotical behavior of traveling wave fronts in two tails.
文摘In this article,we investigate a fractional-order singular Leslie-Gower prey-predator bioeconomic model,which describes the interaction bet ween populations of prey and predator,and takes into account the economic interest.We firstly obtain the solvability condition and the st ability of the model sys tem,and discuss the singularity induced bifurcation phenomenon.Next,we introduce a st ate feedback controller to elimina te the singularity induced bifurcation phenomenon,and discuss the optimal control problems.Finally,numerical solutions and their simulations are considered in order to illustrate the theoretical results and reveal the more complex dynamical behavior.
文摘In this paper, the problem of chaos, stability and estimation of unknown parameters of the stochastic lattice gas for prey-predator model with pair-approximation is studied. The result shows that this dynamical system exhibits an oscillatory behavior of the population densities of prey and predator. Using Liapunov stability technique, the estimators of the unknown probabilities are derived, and also the updating rules for stability around its steady states are derived. Furthermore the feedback control law has been as non-linear functions of the population densities. Numerical simulation study is presented graphically.
基金Supported by the National Natural Science Foundation of China (11001160)the Scientific Research Plan Projects of Shaanxi Education Department (09JK480)the President Fund of Xi’an Technological University(XAGDXJJ0830)
文摘This paper investigates a strongly coupled reaction-diffusion model with Holling-II reaction function in a bounded domain with homogeneous Neumann boundary condition. The sufficient condition for the existence and non-existence of the non-constant positive solutions are obtained. Moreover, we prove that the nonlinear diffusion terms can create non-constant positive equilibrium solutions when the corresponding model without nonlinear diffusion term fails.
文摘Due to the random search of species and from the economic point of view,combined harvesting is more suitable than selective harvesting.Thus,we have developed and analyzed a prey-predator model with the combined effect of nonlinear harvesting in this research paper.Nonlinear harvesting possesses multiple predator-free and interior equilibrium points in the dynamical system.We have examined the local stability analysis of all the equilibrium points.Besides these various types,rich and complex dynamical behaviors such as backward,saddle-node,Hopf and Bogdanov-Takens(BT)bifurcations,homoclinic loop and limit cycles appear in this model.Furthermore,interesting phenomena like bi-stability and tri-stability occur in our model between the different equilibrium points.Also,we have derived different threshold values of predator harvesting parameters and prey environmental carrying capacity from these bifurcations to obtain the different harvesting strategies for both species.We have observed that the extinction of predator species may not happen due to backward bifurcation,although a stable predator-free equilibrium(PFE)exists.Finally,numerical simulations are discussed using MATLAB to verify all the theoretical results.
文摘This paper presents the dynamical properties of a discrete-time prey-predator model with refuge in prey under imprecise biological parameters.We consider the refuge concept of prey,which is proportional to the density of prey species with interval parameters.The model develops with natural interval parameters since the uncertainties of parameters of any ecological system are a widespread phenomenon in nature.The equilibria of the model are obtained,and the dynamic behaviours of the proposed system are examined.Simulations of the model are performed for different parameters of the model.Numerical simulations show that the proposed discrete model exhibits rich dynamics of a chaotic and complex nature.Our study,through analytical derivation and numerical example,presents the effect of refuge on population dynamics under imprecise biological parameters.
文摘In this paper,the Caputo fractional derivative is assumed to be the prey-predator model.In order to create Caputo fractional differential equations for the prey-predator model,a discretization process is first used.The fixed points of the model are categorized topologically.We identify requirements for the fixed points of the suggested prey-predator model's local asymptotic stability.We demonstrate analytically that,under specific parametric conditions,a fractional order prey-predator model supports both a Neimark-Sacker(NS)bifurcation and a Flip bifurcation.We present evidence for NS and Flip bifurcations using central manifold and bifurcation theory.The parameter values and the initial conditions have been found to have a profound impact on the dynamical behavior of the fractional order prey-predator model.As the bifurcation parameter is increased,the system displays chaotic behavior.Numerical simulations are shown to demonstrate chaotic behaviors like bifurcations,phase portraits,invariant closed cycles,and attractive chaotic sets in addition to validating analytical conclusions.The suggested prey-predator dynamical system's chaotic behavior will be controlled by the OGY and hybrid control methodology,which will also visualize the chaotic state for various biological parameters.
基金the National Natural Science Foundation of China 10471022the Ministry of Education of China Science and Technology Major Projects Grant 104090the Foundation of Excellent Doctoral Disscrtation of Southeast University YBJJ0405
文摘In this paper, we study the positive steady states of a prey-predator model with diffusion throughout and a non-monotone conversion rate under the homogeneous Dirichlet boundary condition. We obtain some results of the existence and non-existence of positive steady states. The stability and uniqueness of positive steady states are also discussed.
基金supported by National Natural Science Foundation of China(Grant No.11201380)the Fundamental Research Funds for the Central Universities(Grant No.XDJK2012B007)+2 种基金Doctor Fund of Southwest University(Grant No.SWU111021)Educational Fund of Southwest University(Grant No.2010JY053)National Research Foundation of Korea Grant funded by the Korean Government(Ministry of Education,Science and Technology)(Grant No.NRF-2011-357-C00006)
文摘We consider a Lotka-Volterra prey-predator model with cross-diffusion and Holling type-II functional response.The main concern is the existence of positive solutions under the combined effect of cross-diffusion and Holling type-II functional response.Here,a positive solution corresponds to a coexistence state of the model.Firstly,we study the sufficient conditions to ensure the existence of positive solutions by using degree theory and analyze the coexistence region in parameter plane.In addition,we present the uniqueness of positive solutions in one dimension case.Secondly,we study the stability of the trivial and semi-trivial solutions by analyzing the principal eigenvalue of the corresponding linearized system,and then we characterize the stable/unstable regions of semi-trivial solutions in parameter plane.
基金the National Natural Science Foundation of China (Grant Nos. 10801090, 10726016,10771032)the Scientific Innovation Team Project of Hubei Provincial Department of Education (Grant No.T200809)
文摘In this paper, we investigate a prey-predator model with diffusion and ratio-dependent functional response subject to the homogeneous Neumann boundary condition. Our main focuses are on the global behavior of the reaction-diffusion system and its corresponding steady-state problem. We first apply various Lyapunov functions to discuss the global stability of the unique positive constant steady-state. Then, for the steady-state system, we establish some a priori upper and lower estimates for positive steady-states, and derive several results for non-existence of positive non-constant steady-states if the diffusion rates are large or small.
基金Project supported by the National Natural Science Foundation of China (Nos. 10801090, 10726016)
文摘The authors study a diffusive prey-predator model subject to the homogeneous Neumann boundary condition and give some qualitative descriptions of solutions to this reaction-diffusion system and its corresponding steady-state problem. The local and global stability of the positive constant steady-state are discussed, and then some results for non- existence of positive non-constant steady-states are derived.
文摘The stabilization problem of a kind of prey-predator model with Holling fimctional response is investigated. By approximate linearization approach, a feedback control law stabilizing the closed- loop system is obtained. On the other hand, by exact linearization approach, a suitable change of coordinates in the state space and a feedback control law render the complex nonlinear system to be a linear controllable one such that the positive equilibrium point of the closed-loop system is globally asymptotically stable.
基金supported by the National Natural Science Foundation of China(No.10771032)the Natural Science Foundation of Jiangsu province BK2006088the second author was supported by National Natural Science Foundation of China(No.10601011).
文摘In this paper, a nonlinear predator reproduction and prey competition model with diffusion is discussed. Some existence and non-existence results concerning non-constant positive steady-states are presented using topological degree argument and the energy method, respectively.
文摘In this paper,we consider a prey-predator fishery model with prey dispersal in a two-patch environment,one is assumed to be a free fishing zone and the other is a reserved zone where fishing and other extractive activities are prohibited.The existence of possible steady states of the system is discussed.The local and global stability analysis has been carried out.An optimal harvesting policy is given using Pontryagin s maximum principle.
