In this paper, some feasibly suffcient conditions are obtained for the global asymptotic stability of a positive steady state of a predator-prey system with stage structure for the predator by using the theory of comp...In this paper, some feasibly suffcient conditions are obtained for the global asymptotic stability of a positive steady state of a predator-prey system with stage structure for the predator by using the theory of competitive systems, compound matrices and stability of periodic orbits, and then the work of Wang [4] is improved.展开更多
In this paper, we study a non-autonomous ratio-dependent predator-prey model with exploited term. By means of the coincidence degree theory, we establish a sufficient condition for the existence of at least two positi...In this paper, we study a non-autonomous ratio-dependent predator-prey model with exploited term. By means of the coincidence degree theory, we establish a sufficient condition for the existence of at least two positive periodic solutions of this model.展开更多
In this paper the dynamical behaviors of a predator-prey system with Holling Type-Ⅳ functionalresponse are investigated in detail by using the analyses of qualitative method,bifurcation theory,and numericalsimulation...In this paper the dynamical behaviors of a predator-prey system with Holling Type-Ⅳ functionalresponse are investigated in detail by using the analyses of qualitative method,bifurcation theory,and numericalsimulation.The qualitative analyses and numerical simulation for the model indicate that it has a unique stablelimit cycle.The bifurcation analyses of the system exhibit static and dynamical bifurcations including saddle-node bifurcation,Hopf bifurcation,homoclinic bifurcation and bifurcation of cusp-type with codimension two(ie,the Bogdanov-Takens bifurcation),and we show the existence of codimension three degenerated equilibriumand the existence of homoclinic orbit by using numerical simulation.展开更多
The dynamics of a predator-prey system, where prey population has two stages, an immature stage and a mature stage with harvesting, the growth of predator population is of Lotka-Volterra nature, are modelled by a sys... The dynamics of a predator-prey system, where prey population has two stages, an immature stage and a mature stage with harvesting, the growth of predator population is of Lotka-Volterra nature, are modelled by a system of retarded functional differential equations. We obtain conditions for global asymptotic stability of three nonnegative equilibria and a threshold of harvesting for the mature prey population. The effect of delay on the population at positive equilibrium and the optimal harvesting of the mature prey population are also considered.展开更多
In this paper, a non-autonomous predator-prey model with diffusion andcontinuous time delay is studied, where the prey can diffuse between two pat-ches of a heterogeneous environment with barriers between patches, but...In this paper, a non-autonomous predator-prey model with diffusion andcontinuous time delay is studied, where the prey can diffuse between two pat-ches of a heterogeneous environment with barriers between patches, but for thepredator, the diffusion does not involve a barrier between patches, further itis assumed that all the parameters are time-dependent. It is shown that thesystem can be made persistent under some appropriate conditions. Moreover,sufficient conditions that guarantee the existence of a unique periodic solutionwhich is globally asymptotic stable are derived.展开更多
Based on the facts of releasing natural enemies and spraying pesticides at different time points, we propose a generalized predator-prey model with impulsive interventions. The threshold values for the existence and s...Based on the facts of releasing natural enemies and spraying pesticides at different time points, we propose a generalized predator-prey model with impulsive interventions. The threshold values for the existence and stability of pest eradication periodic solution are provided under the assumptions of releasing natural enemies either more or less frequent than spray. In order to address how the different pulse time points, control tactics affect the pest control （i.e. the threshold value）, the Holling Type II Lotka-Volterra predator- prey system, as an example, with impulsive intervention at different time points axe investigated carefully. The numerical results show how the threshold values are affected by the factors including instantaneous killing rates of pesticides on pests and natural enemies, the release rate of natural enemies and release constant, timing of pesticide application and timing of release period. Furthermore, it is confirmed that the system has the coexistences of pests and natural enemies for a wide range of parameters and with quite different pest amplitudes.展开更多
to biological and chemical control strategy for pest control, a Holling II func- tional response predator-prey system concerning state-dependent impulsive control is investigated. We define the successor functions of ...to biological and chemical control strategy for pest control, a Holling II func- tional response predator-prey system concerning state-dependent impulsive control is investigated. We define the successor functions of semi-continuous dynamic system and give an existence theorem of order 1 periodic solution of such a system. By means of sequence convergence rules and quMitative analysis, we successfully get the conditions of existence and attractiveness of order 1 periodic solution. Our results show that our method used in this paper is more efficient and easier than the existing methods to prove the existence and attractiveness of order 1 periodic solution.展开更多
In this paper, we consider a predator-prey system with stage structure and harvesting (where the predator population has two stages, an immature stage and a mature stage with harvesting, and the growth of the prey pop...In this paper, we consider a predator-prey system with stage structure and harvesting (where the predator population has two stages, an immature stage and a mature stage with harvesting, and the growth of the prey population is of Lotka-Volterra nature). We obtain the conditions of the globally asymptotic stability for three nonne-gative equilibria of this system.展开更多
A discrete predator-prey system with Holling type-IV functional responseobtained by the Euler method is first investigated. The conditions of existence for foldbifurcation, flip bifurcation and Hopf bifurcation are de...A discrete predator-prey system with Holling type-IV functional responseobtained by the Euler method is first investigated. The conditions of existence for foldbifurcation, flip bifurcation and Hopf bifurcation are derived by using the center manifold theoremand bifurcation theory. Furthermore, we give the condition for the occurrence of codimension-twobifurcation called the Bogdanov-Takens bifurcation for fixed points and present approximateexpressions for saddle-node, Hopf and homoclinic bifurcation sets near the Bogdanov-Takensbifurcation point. We also show the existence of degenerated fixed point with codimension three atleast. The numerical simulations, including bifurcation diagrams, phase portraits, and computationof maximum Lyapunov exponents, not only show the consistence with the theoretical analysis but alsoexhibit the rich and complex dynamical behaviors such as the attracting invariant circle,period-doubling bifurcation from period-2,3,4 orbits, interior crisis, intermittency mechanic, andsudden disappearance of chaotic dynamic.展开更多
A stage-structured predator-prey system with impulsive effect and Holling type-II functional response is investigated. By the Floquet theory and small amplitude perturbation skills, it is proved that there exists a gl...A stage-structured predator-prey system with impulsive effect and Holling type-II functional response is investigated. By the Floquet theory and small amplitude perturbation skills, it is proved that there exists a global stable pest-eradication periodic solution when the impulsive period is less than some critical values. Farther, the conditions for the permanence of system are established. Numerical simulations are carried out to illustrate the impulsive effect on the dynamics of the system.展开更多
In this paper, the dynamic properties of a discrete predator-prey model are discussed. The properties of non-hyperbolic fixed points and hyperbolic fixed points of the model are analyzed. First, by using the classic S...In this paper, the dynamic properties of a discrete predator-prey model are discussed. The properties of non-hyperbolic fixed points and hyperbolic fixed points of the model are analyzed. First, by using the classic Shengjin formula, we find the existence conditions for fixed points of the model. Then, by using the qualitative theory of ordinary differential equations and matrix theory we indicate which points are hyperbolic and which are non-hyperbolic and the associated conditions.展开更多
In this paper we analytically and numerically consider the dynamical behavior of a certain predator-prey system with Holling type II functional response, including local and global stability analysis, existence of lim...In this paper we analytically and numerically consider the dynamical behavior of a certain predator-prey system with Holling type II functional response, including local and global stability analysis, existence of limit cycles, transcritical and Hopf bifurcations. Mathematical theory derivation mainly focuses on the existence and stability of equilibrium point as well as threshold conditions for transcritical and Hopf bifurcation, which can in turn provide a theoretical support for numerical simulation. Numerical analysis indicates that theoretical derivation results are correct and feasible. In addition, it is successful to show that the dynamical behavior of this predator-prey system mainly depends on some critical parameters and mathematical relationships. All these results are expected to be meaningful in the study of the dynamic complexity of predatory ecosystem.展开更多
With the help of a continuation theorem based on Gaines and Mawhinscoincidence degree, easily verifiable criteria are established for the global existence of positiveperiodic solutions of the following nonlinear state...With the help of a continuation theorem based on Gaines and Mawhinscoincidence degree, easily verifiable criteria are established for the global existence of positiveperiodic solutions of the following nonlinear state dependent delays predator-prey system{dN_1(t)/dt = N_1(t)[b_1(t) - ∑ from i=1 to n of ai(t)(N_1(t-τ_i(t,N_1(t), N_2(t))))^(α_i) - ∑from j=1 to m of c_j(t)(N_2(t - σ_j(t,N1(t),N_2(t))))^(β_j)] dN_2(t)/dt = N_2(t)[-b_2(t) + ∑ fromi=1 to n of d_i(5)(N_1(t - ρ_i(t,N_1(t),N_2(t))))^(γ_i)], where a_i(t), c_j(t), d_i(t) arecontinuous positive periodic functions with periodic 】 0, b_1(t), b_2(t) are continuous periodicfunctions with periodic ω and ∫_0~ωb_i(t)dt 】 0. τ_i, σ_j, ρ_i (i = 1,2,…,m) are continuousand ω-periodic with respect to their first arguments, respectively. α_i, β_j, γ_i (i = 1,2,…,n,j = 1,2,…,m) are positive constants.展开更多
基金supported by National Natural Science Foundation of China
文摘In this paper, some feasibly suffcient conditions are obtained for the global asymptotic stability of a positive steady state of a predator-prey system with stage structure for the predator by using the theory of competitive systems, compound matrices and stability of periodic orbits, and then the work of Wang [4] is improved.
基金supported by National Natural Science Foundation of China(61425008,61333004,61273054)Top-Notch Young Talents Program of China,and Aeronautical Foundation of China(2013585104)
基金Supported by the National Natural Science Foundation of China （No.19531070）
文摘In this paper, we study a non-autonomous ratio-dependent predator-prey model with exploited term. By means of the coincidence degree theory, we establish a sufficient condition for the existence of at least two positive periodic solutions of this model.
基金Supported by Chinese Academy of Sciences (KZCX2-SW-118)Supported by the NNSF of China (No.10071027No.10231020)
文摘In this paper the dynamical behaviors of a predator-prey system with Holling Type-Ⅳ functionalresponse are investigated in detail by using the analyses of qualitative method,bifurcation theory,and numericalsimulation.The qualitative analyses and numerical simulation for the model indicate that it has a unique stablelimit cycle.The bifurcation analyses of the system exhibit static and dynamical bifurcations including saddle-node bifurcation,Hopf bifurcation,homoclinic bifurcation and bifurcation of cusp-type with codimension two(ie,the Bogdanov-Takens bifurcation),and we show the existence of codimension three degenerated equilibriumand the existence of homoclinic orbit by using numerical simulation.
基金the National Natural Science Foundation of China (No.10171106)the Natural Science Foundation of Henan Province (No.0211010400).
文摘 The dynamics of a predator-prey system, where prey population has two stages, an immature stage and a mature stage with harvesting, the growth of predator population is of Lotka-Volterra nature, are modelled by a system of retarded functional differential equations. We obtain conditions for global asymptotic stability of three nonnegative equilibria and a threshold of harvesting for the mature prey population. The effect of delay on the population at positive equilibrium and the optimal harvesting of the mature prey population are also considered.
基金This work is supported by the Foundation of Ability Person of Fuzhou University under the grant 0030824228 andthe Foundation of Developing Science and Technology of Fuzhou University under the grant 2003-QX-21 and the Foundation of Fujian Education Bur
文摘In this paper, a non-autonomous predator-prey model with diffusion andcontinuous time delay is studied, where the prey can diffuse between two pat-ches of a heterogeneous environment with barriers between patches, but for thepredator, the diffusion does not involve a barrier between patches, further itis assumed that all the parameters are time-dependent. It is shown that thesystem can be made persistent under some appropriate conditions. Moreover,sufficient conditions that guarantee the existence of a unique periodic solutionwhich is globally asymptotic stable are derived.
文摘Based on the facts of releasing natural enemies and spraying pesticides at different time points, we propose a generalized predator-prey model with impulsive interventions. The threshold values for the existence and stability of pest eradication periodic solution are provided under the assumptions of releasing natural enemies either more or less frequent than spray. In order to address how the different pulse time points, control tactics affect the pest control （i.e. the threshold value）, the Holling Type II Lotka-Volterra predator- prey system, as an example, with impulsive intervention at different time points axe investigated carefully. The numerical results show how the threshold values are affected by the factors including instantaneous killing rates of pesticides on pests and natural enemies, the release rate of natural enemies and release constant, timing of pesticide application and timing of release period. Furthermore, it is confirmed that the system has the coexistences of pests and natural enemies for a wide range of parameters and with quite different pest amplitudes.
文摘to biological and chemical control strategy for pest control, a Holling II func- tional response predator-prey system concerning state-dependent impulsive control is investigated. We define the successor functions of semi-continuous dynamic system and give an existence theorem of order 1 periodic solution of such a system. By means of sequence convergence rules and quMitative analysis, we successfully get the conditions of existence and attractiveness of order 1 periodic solution. Our results show that our method used in this paper is more efficient and easier than the existing methods to prove the existence and attractiveness of order 1 periodic solution.
基金This work is supported by the NSF of China and the NSF of Henan Province.
文摘In this paper, we consider a predator-prey system with stage structure and harvesting (where the predator population has two stages, an immature stage and a mature stage with harvesting, and the growth of the prey population is of Lotka-Volterra nature). We obtain the conditions of the globally asymptotic stability for three nonne-gative equilibria of this system.
基金Supported by Chinese Academy Sciences (KZCX2-SW-118)by the National Natural Science Foundation of China (No. 10371037).
文摘A discrete predator-prey system with Holling type-IV functional responseobtained by the Euler method is first investigated. The conditions of existence for foldbifurcation, flip bifurcation and Hopf bifurcation are derived by using the center manifold theoremand bifurcation theory. Furthermore, we give the condition for the occurrence of codimension-twobifurcation called the Bogdanov-Takens bifurcation for fixed points and present approximateexpressions for saddle-node, Hopf and homoclinic bifurcation sets near the Bogdanov-Takensbifurcation point. We also show the existence of degenerated fixed point with codimension three atleast. The numerical simulations, including bifurcation diagrams, phase portraits, and computationof maximum Lyapunov exponents, not only show the consistence with the theoretical analysis but alsoexhibit the rich and complex dynamical behaviors such as the attracting invariant circle,period-doubling bifurcation from period-2,3,4 orbits, interior crisis, intermittency mechanic, andsudden disappearance of chaotic dynamic.
基金Supported by the National Natural Science Foundation of China (Grant Nos.1067120910926064)
文摘A stage-structured predator-prey system with impulsive effect and Holling type-II functional response is investigated. By the Floquet theory and small amplitude perturbation skills, it is proved that there exists a global stable pest-eradication periodic solution when the impulsive period is less than some critical values. Farther, the conditions for the permanence of system are established. Numerical simulations are carried out to illustrate the impulsive effect on the dynamics of the system.
文摘In this paper, the dynamic properties of a discrete predator-prey model are discussed. The properties of non-hyperbolic fixed points and hyperbolic fixed points of the model are analyzed. First, by using the classic Shengjin formula, we find the existence conditions for fixed points of the model. Then, by using the qualitative theory of ordinary differential equations and matrix theory we indicate which points are hyperbolic and which are non-hyperbolic and the associated conditions.
文摘In this paper we analytically and numerically consider the dynamical behavior of a certain predator-prey system with Holling type II functional response, including local and global stability analysis, existence of limit cycles, transcritical and Hopf bifurcations. Mathematical theory derivation mainly focuses on the existence and stability of equilibrium point as well as threshold conditions for transcritical and Hopf bifurcation, which can in turn provide a theoretical support for numerical simulation. Numerical analysis indicates that theoretical derivation results are correct and feasible. In addition, it is successful to show that the dynamical behavior of this predator-prey system mainly depends on some critical parameters and mathematical relationships. All these results are expected to be meaningful in the study of the dynamic complexity of predatory ecosystem.
基金Supported by the National Natural Science Foundation of China (Tian Yuan Foundation) (No.10426010)the Foundation of Science and Technology of Fujian Province for Young Scholars (2004J0002)the Foundation of Fujian Education Bureau (JA04156, JA0301
文摘With the help of a continuation theorem based on Gaines and Mawhinscoincidence degree, easily verifiable criteria are established for the global existence of positiveperiodic solutions of the following nonlinear state dependent delays predator-prey system{dN_1(t)/dt = N_1(t)[b_1(t) - ∑ from i=1 to n of ai(t)(N_1(t-τ_i(t,N_1(t), N_2(t))))^(α_i) - ∑from j=1 to m of c_j(t)(N_2(t - σ_j(t,N1(t),N_2(t))))^(β_j)] dN_2(t)/dt = N_2(t)[-b_2(t) + ∑ fromi=1 to n of d_i(5)(N_1(t - ρ_i(t,N_1(t),N_2(t))))^(γ_i)], where a_i(t), c_j(t), d_i(t) arecontinuous positive periodic functions with periodic 】 0, b_1(t), b_2(t) are continuous periodicfunctions with periodic ω and ∫_0~ωb_i(t)dt 】 0. τ_i, σ_j, ρ_i (i = 1,2,…,m) are continuousand ω-periodic with respect to their first arguments, respectively. α_i, β_j, γ_i (i = 1,2,…,n,j = 1,2,…,m) are positive constants.
