In this paper, we propose a discrete ratio-dependent predator-prey system. The stability of the fixed points of this model is studied. At the same time, it is shown that the discrete model undergoes fold bifurcation a...In this paper, we propose a discrete ratio-dependent predator-prey system. The stability of the fixed points of this model is studied. At the same time, it is shown that the discrete model undergoes fold bifurcation and flip bifurcation by using bifurcation theory and the method of approximation by a flow. Numerical simulations are presented not only to demonstrate the consistence with our theoretical analyses, but also to exhibit the complex dynamical behaviors, such as the cascade of period-doubling bifurcation in period-2 and the chaotic sets. The Maximum Lyapunov exponents are numerically computed to confirm further the complexity of the dynamical behaviors. These results show that the direct discrete method has more rich dynamic behaviors than the discrete model obtained by Euler method.展开更多
This paper considers a class of ratio-dependent Holling-Taner model with infinite delay and prey harvest, which is of periodic coefficients. By means of the coincidence degree theory, a set of sufficient conditions fo...This paper considers a class of ratio-dependent Holling-Taner model with infinite delay and prey harvest, which is of periodic coefficients. By means of the coincidence degree theory, a set of sufficient conditions for the existence of at least two positive periodic solutions of this model is established.展开更多
This article addresses a stochastic ratio-dependent predator-prey system with Leslie-Gower and Holling type II schemes. Firstly, the existence of the global positive solution is shown by the comparison theorem of stoc...This article addresses a stochastic ratio-dependent predator-prey system with Leslie-Gower and Holling type II schemes. Firstly, the existence of the global positive solution is shown by the comparison theorem of stochastic differential equations. Secondly, in the case of persistence, we prove that there exists a ergodic stationary distribution. Finally, numerical simulations for a hypothetical set of parameter values are presented to illustrate the analytical findings.展开更多
We study a non-autonomous ratio-dependent predator-prey model with exploited terms. This model is of periodic coefficients, which incorporates the periodicity of the varying environment. By means of the coincidence de...We study a non-autonomous ratio-dependent predator-prey model with exploited terms. This model is of periodic coefficients, which incorporates the periodicity of the varying environment. By means of the coincidence degree theory, we establish sufficient conditions for the existence of at least four positive periodic solutions of this model.展开更多
This article is concerned with the existence of traveling wave solutions for a discrete diffusive ratio-dependent predator-prey model. By applying Schauder’s fixed point theorem with the help of suitable upper and lo...This article is concerned with the existence of traveling wave solutions for a discrete diffusive ratio-dependent predator-prey model. By applying Schauder’s fixed point theorem with the help of suitable upper and lower solutions, we prove that there exists a positive constant c* such that when c > c* , the discrete diffusive predator-prey system admits an invasion traveling wave. The existence of an invasion traveling wave with c = c* is also established by a limiting argument and a delicate analysis of the boundary conditions.Finally, by the asymptotic spreading theory and the comparison principle, the non-existence of invasion traveling waves with speed c < c* is also proved.展开更多
In this paper, we investigate the dynamics of a stochastic predator-prey model with ratio-dependent functional response and disease in the prey. Firstly, we prove the existence and uniqueness of the positive solution ...In this paper, we investigate the dynamics of a stochastic predator-prey model with ratio-dependent functional response and disease in the prey. Firstly, we prove the existence and uniqueness of the positive solution for the stochastic model by using conventional methods. Then we obtain the threshold <img alt="" src="Edit_0a62b9be-7934-457b-aca3-af3420f5b5ee.png" /> for the infected prey population, that is, the disease will tend to extinction if <img alt="" src="Edit_e6cd63f6-de07-42be-a22a-8750d6c8aac9.png" />< 1, and it will exist in the long time if <img alt="" src="Edit_5964fdd8-a9fe-4dc2-b897-f4206f046f65.png" />> 1. Finally, the sufficient condition on the existence of a unique ergodic stationary distribution is obtained, which indicates that all the populations are permanent in the time mean sense. Numerical simulations are conducted to verify our analysis results.展开更多
A ratio dependent predator-prey system with Holling type Ⅲ functional response is considered. A sufficient condition of the global asymptotic stability for the positive equilibrium and existence of the limit cycle ar...A ratio dependent predator-prey system with Holling type Ⅲ functional response is considered. A sufficient condition of the global asymptotic stability for the positive equilibrium and existence of the limit cycle are given by studying locally asymp- totic stability of the positive equilibrium. The condition under which positive equilibrium is not a hyperbolic equilibrium is investigated using Hopf bifurcation.展开更多
This paper presents a theoretical analysis of evolutionary process that involves organisms distribution and their interaction of spatially distributed population with diffusion in a Holling-III ratio-dependent predato...This paper presents a theoretical analysis of evolutionary process that involves organisms distribution and their interaction of spatially distributed population with diffusion in a Holling-III ratio-dependent predator-prey model, the sufficient conditions for diffusion-driven instability with Neumann boundary conditions are obtained. Furthermore, it presents novel numerical evidence of time evolution of patterns controlled by diffusion in the model, and finds that the model dynamics exhibits complex pattern replication, and the pattern formation depends on the choice of the initial conditions. The ideas in this paper may provide a better understanding of the pattern formation in ecosystems.展开更多
Influences of prey refuge on the dynamics of a predator-prey model with ratio-dependent functional response are investigated. The local and global stability of positive equilibrium of the system are considered. Theore...Influences of prey refuge on the dynamics of a predator-prey model with ratio-dependent functional response are investigated. The local and global stability of positive equilibrium of the system are considered. Theoretical analysis indicates that constant refuge leads to the system undergo supercritical Hopf bifurcation twice with the birth rate of prey species changing continuously.展开更多
In this paper, a SEIR model with ratio-dependent transmission rate in the form ?is studied and the basic reproduction number which determines the disease’s extinction or continued existence is obtained. By constructi...In this paper, a SEIR model with ratio-dependent transmission rate in the form ?is studied and the basic reproduction number which determines the disease’s extinction or continued existence is obtained. By constructing the proper Lyapunov function, we prove that if R0 ≤ 1, the disease-free equilibrium point of the model is globally asymptotically stable and the disease always dies out;if R0 > 1, the endemic equilibrium point is globally asymptotically stable and the disease persists.展开更多
In this paper, the dynamics of a stochastic ratio-dependent predator-prey system with markovian switching and Lévy noise is studied. Firstly, we show the existence condition of global positive solution under the ...In this paper, the dynamics of a stochastic ratio-dependent predator-prey system with markovian switching and Lévy noise is studied. Firstly, we show the existence condition of global positive solution under the given positive initial value. Secondly, sufficient conditions for system extinction and persistence are obtained through some assumptions. Then, the sufficient conditions of stochastically persistence are obtained by combining stochastic analysis technique and M-matrix analysis. In addition, under appropriate conditions, we demonstrate the existence of a unique stationary distribution for a system without Lévy jumps. Finally, the empirical and Mlistein methods are used to verify the theoretical results through numerical simulation.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, 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
In this paper, we consider a three-species ratio-dependent predator-prey model governed by difference equations with periodic coefficients. By using the method of coincidence degree, we discuss the existence of positi...In this paper, we consider a three-species ratio-dependent predator-prey model governed by difference equations with periodic coefficients. By using the method of coincidence degree, we discuss the existence of positive periodic solutions of this system, a set of easily verifiable sufficient conditions are derived.展开更多
A nonautonomous ratio-dependent Leslie system incorporating a prey refuge is studied in this paper. By applying the comparison theorem of diferential equations and constructing a suitable Lyapunov function, a set of s...A nonautonomous ratio-dependent Leslie system incorporating a prey refuge is studied in this paper. By applying the comparison theorem of diferential equations and constructing a suitable Lyapunov function, a set of sufcient conditions which guarantee the persistent property and global attractivity of the system is obtained. Also, by applying the comparison theorem of diferential equations and Fluctuation Lemma, a set of sufcient conditions which ensure the extinction of the prey species and the global attractivity of predator species is obtained. This result shows that for the Lotka-Volterra type predator-prey system, when the value of prey refuge increases, predator species will be driven to extinction due to the lack of food. Our study shows that the alternative food resource predator species is always permanent, which means that prey refuge has no infuence on the permanence of predator species. However, refuge plays an important role in the persistent property of the prey species: large enough prey refuge could keep the persistent property of the prey species, while small enough refuge could lead to the extinction of prey species. Numerical simulations show the feasibility of the main results.展开更多
In this paper, we study a non-autonomous ratio-dependent predator-prey model with predator's harvest. By means of the coincidence degree theory, we establish sufficient conditions for the existence of at least two po...In this paper, we study a non-autonomous ratio-dependent predator-prey model with predator's harvest. By means of the coincidence degree theory, we establish sufficient conditions for the existence of at least two positive periodic solutions of this model.展开更多
In this paper, a nonautonomous ratio-dependent multispecies competitionpredator system with Holling's Ⅲ type functional response is studied, where all parameters are time dependent. It is proved that the system is u...In this paper, a nonautonomous ratio-dependent multispecies competitionpredator system with Holling's Ⅲ type functional response is studied, where all parameters are time dependent. It is proved that the system is uniformly persistent under suitable condition. Furthermore, the sufficient conditions are established for global attractivity of a periodicsolution of the system. response展开更多
A three-species ratio-dependent predator-prey discrete model is studied.As a result,sufficient conditions which guarantee the permanence of the model are obtained. In addition,by constructing a suitable Lyapunov funct...A three-species ratio-dependent predator-prey discrete model is studied.As a result,sufficient conditions which guarantee the permanence of the model are obtained. In addition,by constructing a suitable Lyapunov function,we derive some sufficient conditions,which ensure that the positive solution of the model is stable and attracts all positive solutions.To illustrate the feasibility of the main results,we introduce an example with corresponding numeric simulations.展开更多
This paper is concerned with a ratio-dependent predator-prey system with diffusion and cross- diffusion in a bounded domain with no flux boundary condition. We show that under certain hypotheses, the cross-diffusion c...This paper is concerned with a ratio-dependent predator-prey system with diffusion and cross- diffusion in a bounded domain with no flux boundary condition. We show that under certain hypotheses, the cross-diffusion can create non-constant positive steady states even though the corresponding model without cross-diffusion fails.展开更多
The main purpose of this article is considering the persistence non-autonomous Lotka-Volterra system with predator-prey ratio-dependence and density dependence. We get the sufficient conditions of persistence of syste...The main purpose of this article is considering the persistence non-autonomous Lotka-Volterra system with predator-prey ratio-dependence and density dependence. We get the sufficient conditions of persistence of system, further have the necessary conditions, also the uniform persistence condition, which can be easily checked for the model is obtained.展开更多
文摘In this paper, we propose a discrete ratio-dependent predator-prey system. The stability of the fixed points of this model is studied. At the same time, it is shown that the discrete model undergoes fold bifurcation and flip bifurcation by using bifurcation theory and the method of approximation by a flow. Numerical simulations are presented not only to demonstrate the consistence with our theoretical analyses, but also to exhibit the complex dynamical behaviors, such as the cascade of period-doubling bifurcation in period-2 and the chaotic sets. The Maximum Lyapunov exponents are numerically computed to confirm further the complexity of the dynamical behaviors. These results show that the direct discrete method has more rich dynamic behaviors than the discrete model obtained by Euler method.
文摘This paper considers a class of ratio-dependent Holling-Taner model with infinite delay and prey harvest, which is of periodic coefficients. By means of the coincidence degree theory, a set of sufficient conditions for the existence of at least two positive periodic solutions of this model is established.
基金supported by NSFC of China Grant(11371085)the Fundamental Research Funds for the Central Universities(15CX08011A)
文摘This article addresses a stochastic ratio-dependent predator-prey system with Leslie-Gower and Holling type II schemes. Firstly, the existence of the global positive solution is shown by the comparison theorem of stochastic differential equations. Secondly, in the case of persistence, we prove that there exists a ergodic stationary distribution. Finally, numerical simulations for a hypothetical set of parameter values are presented to illustrate the analytical findings.
基金Supported by the China Postdoctoral Science Foundation (20060400267)
文摘We study a non-autonomous ratio-dependent predator-prey model with exploited terms. This model is of periodic coefficients, which incorporates the periodicity of the varying environment. By means of the coincidence degree theory, we establish sufficient conditions for the existence of at least four positive periodic solutions of this model.
基金supported by NSF of China(11861056)Gansu Provincial Natural Science Foundation(18JR3RA093).
文摘This article is concerned with the existence of traveling wave solutions for a discrete diffusive ratio-dependent predator-prey model. By applying Schauder’s fixed point theorem with the help of suitable upper and lower solutions, we prove that there exists a positive constant c* such that when c > c* , the discrete diffusive predator-prey system admits an invasion traveling wave. The existence of an invasion traveling wave with c = c* is also established by a limiting argument and a delicate analysis of the boundary conditions.Finally, by the asymptotic spreading theory and the comparison principle, the non-existence of invasion traveling waves with speed c < c* is also proved.
文摘In this paper, we investigate the dynamics of a stochastic predator-prey model with ratio-dependent functional response and disease in the prey. Firstly, we prove the existence and uniqueness of the positive solution for the stochastic model by using conventional methods. Then we obtain the threshold <img alt="" src="Edit_0a62b9be-7934-457b-aca3-af3420f5b5ee.png" /> for the infected prey population, that is, the disease will tend to extinction if <img alt="" src="Edit_e6cd63f6-de07-42be-a22a-8750d6c8aac9.png" />< 1, and it will exist in the long time if <img alt="" src="Edit_5964fdd8-a9fe-4dc2-b897-f4206f046f65.png" />> 1. Finally, the sufficient condition on the existence of a unique ergodic stationary distribution is obtained, which indicates that all the populations are permanent in the time mean sense. Numerical simulations are conducted to verify our analysis results.
文摘A ratio dependent predator-prey system with Holling type Ⅲ functional response is considered. A sufficient condition of the global asymptotic stability for the positive equilibrium and existence of the limit cycle are given by studying locally asymp- totic stability of the positive equilibrium. The condition under which positive equilibrium is not a hyperbolic equilibrium is investigated using Hopf bifurcation.
基金supported by the Natural Science Foundation of Zhejiang Province of China (Grant No.Y7080041)
文摘This paper presents a theoretical analysis of evolutionary process that involves organisms distribution and their interaction of spatially distributed population with diffusion in a Holling-III ratio-dependent predator-prey model, the sufficient conditions for diffusion-driven instability with Neumann boundary conditions are obtained. Furthermore, it presents novel numerical evidence of time evolution of patterns controlled by diffusion in the model, and finds that the model dynamics exhibits complex pattern replication, and the pattern formation depends on the choice of the initial conditions. The ideas in this paper may provide a better understanding of the pattern formation in ecosystems.
基金Supported by the NNSF of China(11126284)Supported by the NSF of Department of Education of Henan Province(12A110012)Supported by the Young Scientific Research Foundation of Henan Normal University(1001)
文摘Influences of prey refuge on the dynamics of a predator-prey model with ratio-dependent functional response are investigated. The local and global stability of positive equilibrium of the system are considered. Theoretical analysis indicates that constant refuge leads to the system undergo supercritical Hopf bifurcation twice with the birth rate of prey species changing continuously.
文摘In this paper, a SEIR model with ratio-dependent transmission rate in the form ?is studied and the basic reproduction number which determines the disease’s extinction or continued existence is obtained. By constructing the proper Lyapunov function, we prove that if R0 ≤ 1, the disease-free equilibrium point of the model is globally asymptotically stable and the disease always dies out;if R0 > 1, the endemic equilibrium point is globally asymptotically stable and the disease persists.
文摘In this paper, the dynamics of a stochastic ratio-dependent predator-prey system with markovian switching and Lévy noise is studied. Firstly, we show the existence condition of global positive solution under the given positive initial value. Secondly, sufficient conditions for system extinction and persistence are obtained through some assumptions. Then, the sufficient conditions of stochastically persistence are obtained by combining stochastic analysis technique and M-matrix analysis. In addition, under appropriate conditions, we demonstrate the existence of a unique stationary distribution for a system without Lévy jumps. Finally, the empirical and Mlistein methods are used to verify the theoretical results through numerical simulation.
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Supported by National Natural Sciences Foundation of P.R.China (No.10171010 and 10201005)the Key Project on Science and Technology of the Education Ministry of P.R.China(No.01061)
文摘In this paper, we consider a three-species ratio-dependent predator-prey model governed by difference equations with periodic coefficients. By using the method of coincidence degree, we discuss the existence of positive periodic solutions of this system, a set of easily verifiable sufficient conditions are derived.
文摘A nonautonomous ratio-dependent Leslie system incorporating a prey refuge is studied in this paper. By applying the comparison theorem of diferential equations and constructing a suitable Lyapunov function, a set of sufcient conditions which guarantee the persistent property and global attractivity of the system is obtained. Also, by applying the comparison theorem of diferential equations and Fluctuation Lemma, a set of sufcient conditions which ensure the extinction of the prey species and the global attractivity of predator species is obtained. This result shows that for the Lotka-Volterra type predator-prey system, when the value of prey refuge increases, predator species will be driven to extinction due to the lack of food. Our study shows that the alternative food resource predator species is always permanent, which means that prey refuge has no infuence on the permanence of predator species. However, refuge plays an important role in the persistent property of the prey species: large enough prey refuge could keep the persistent property of the prey species, while small enough refuge could lead to the extinction of prey species. Numerical simulations show the feasibility of the main results.
基金the Scientific Research Foundation of the Doctor Department of Hubei University of Technology
文摘In this paper, we study a non-autonomous ratio-dependent predator-prey model with predator's harvest. By means of the coincidence degree theory, we establish sufficient conditions for the existence of at least two positive periodic solutions of this model.
基金This work is supported by the Foundation of Science and Technology of Fujian Province for Young Scholars (2004J0002) the Foundation of Fujian Education Bureau (JA04156).
文摘In this paper, a nonautonomous ratio-dependent multispecies competitionpredator system with Holling's Ⅲ type functional response is studied, where all parameters are time dependent. It is proved that the system is uniformly persistent under suitable condition. Furthermore, the sufficient conditions are established for global attractivity of a periodicsolution of the system. response
基金Supported by the Foundation of Fujian Education Bureau (JA04156).
文摘A three-species ratio-dependent predator-prey discrete model is studied.As a result,sufficient conditions which guarantee the permanence of the model are obtained. In addition,by constructing a suitable Lyapunov function,we derive some sufficient conditions,which ensure that the positive solution of the model is stable and attracts all positive solutions.To illustrate the feasibility of the main results,we introduce an example with corresponding numeric simulations.
基金Supported in part by the National Natural Science Foundation of China under Grant No.11601542 and 11626238
文摘This paper is concerned with a ratio-dependent predator-prey system with diffusion and cross- diffusion in a bounded domain with no flux boundary condition. We show that under certain hypotheses, the cross-diffusion can create non-constant positive steady states even though the corresponding model without cross-diffusion fails.
文摘The main purpose of this article is considering the persistence non-autonomous Lotka-Volterra system with predator-prey ratio-dependence and density dependence. We get the sufficient conditions of persistence of system, further have the necessary conditions, also the uniform persistence condition, which can be easily checked for the model is obtained.