In this paper, a diffusive predator-prey system with general functional responses and prey-tactic sensitivities is studied. Providing such generality, we construct a Lyapunov function and we show that the positive con...In this paper, a diffusive predator-prey system with general functional responses and prey-tactic sensitivities is studied. Providing such generality, we construct a Lyapunov function and we show that the positive constant steady state is locally and globally asymptotically stable. With an eye on the biological interpretations, a numerical simulation is performed to illustrate the feasibility of the analytical findings.展开更多
The stability of the first-order and second-order solution moments for a Harrison-type predator-prey model with parametric Gaussian white noise is analyzed in this paper. The moment equations of the system solution ar...The stability of the first-order and second-order solution moments for a Harrison-type predator-prey model with parametric Gaussian white noise is analyzed in this paper. The moment equations of the system solution are obtained under Ito interpretations. The delay-independent stable condition of the first-order moment is identical to that of the deterministic delayed system, and the delay-independent stable condition of the second-order moment depends on the noise intensities. The corresponding critical time delays are determined once the stabilities of moments lose. Further, when the time delays are greater than the critical time delays, the system solution becomes unstable with the increase of noise intensities. Finally, some numerical simulations are given to verify the theoretical results.展开更多
In this paper, we investigate the solution moment stability for a Harrison-type predator-prey model with parametric dichotomous noises. Using the Shapiro-Loginov formula, the equations for the first-order and second-o...In this paper, we investigate the solution moment stability for a Harrison-type predator-prey model with parametric dichotomous noises. Using the Shapiro-Loginov formula, the equations for the first-order and second-order moments are obtained and the corresponding stable conditions are given. It is found that the solution moment stability depends on the noise intensity and correlation time of noise. The first-order and second-order moments become unstable with the decrease of correlation time. That is, the dichotomous noise can improve the solution moment stability with respect to Gaussian white noise. Finally, some numerical results are presented to verify the theoretical analyses.展开更多
A Holling type III predator-prey model with stage structure for prey is investi-gated. By analyzing the corresponding characteristic equations, the local stability of each of feasible equilibria of the system is discu...A Holling type III predator-prey model with stage structure for prey is investi-gated. By analyzing the corresponding characteristic equations, the local stability of each of feasible equilibria of the system is discussed. By using the uniformly persistence theory, the system is proven to be permanent if the coexistence equilibrium exists. By using Lyapunov functionals and LaSalle’s invariance principle, it is shown that the two boundary equilibria is globally asymptotically stable when the coexistence equilibrium is not feasible. By using compound matrix theory, the sucient conditions are obtained for the global stability of the coexistence equilibrium. At last, numerical simulations are carried out to illustrate the main results.展开更多
This paper deals with the questio n of global stability of the positive locally asymptotically stable equilibrium in a class of predator\|prey system of Gause\|typ e with Holling Ⅲ functional response. The Dulac'...This paper deals with the questio n of global stability of the positive locally asymptotically stable equilibrium in a class of predator\|prey system of Gause\|typ e with Holling Ⅲ functional response. The Dulac's criterion is applied and lia punov functions are constructed to establish the global stability.展开更多
In this paper, the dynamical behaviors of a modified Leslie-Gower predator-prey model incorporating fear effect and prey refuge are investigated. We delve into the construction of the model and its biological signific...In this paper, the dynamical behaviors of a modified Leslie-Gower predator-prey model incorporating fear effect and prey refuge are investigated. We delve into the construction of the model and its biological significance, with preliminary results encompassing positivity, boundedness, and persistence. The stability of the system’s boundary and positive equilibrium points is proven by calculating the real part of the eigenvalues of the Jacobian matrix. At the positive equilibrium point, we demonstrate that the system’s unique positive equilibrium is globally asymptotically stable by using the Dulac criterion. Furthermore, at this equilibrium point, we employ the Implicit Function Theorem to discuss how fear effects and prey refuges influence the population densities of both prey and predators. Finally, numerical simulations are conducted to validate the above-mentioned conclusions and explored the impact of Predator-taxis sensitivity αon dynamics of the system.展开更多
In this paper, a nonautonomous predator-prey system based on a modified version of the Leslie-Gower scheme and Holling-type II scheme with delayed effect is investigated. The general criteria of integrable form on the...In this paper, a nonautonomous predator-prey system based on a modified version of the Leslie-Gower scheme and Holling-type II scheme with delayed effect is investigated. The general criteria of integrable form on the permanence are established. By constructing suitable Lyapunov functionals, a set of easily verifiable sufficient conditions are derived for global stability of any positive solutions to the展开更多
The present paper is concerned with the Holling type IV predator-prey system with diffusion.By analyzing the characteristic equation associated with the positive equilibrium,the conditions for the asymptotic stability...The present paper is concerned with the Holling type IV predator-prey system with diffusion.By analyzing the characteristic equation associated with the positive equilibrium,the conditions for the asymptotic stability of the positive equilibrium is obtained.For the system without delay,it has been shown that the positive equilibrium is stable in certain region of the parameter plane.However,the introducing of the delay can lead to the loss of the stability.We find that in the region where the positive equilibrium is stable for the system without delay,there exists a critical value of the delay and the positive equilibrium is stable when the delay is less than this critical value and becomes unstable when the delay is greater than it.展开更多
In this paper, we considered a homogeneous reaction-diffusion predator-prey system with Holling type II functional response subject to Neumann boundary conditions. Some new sufficient conditions were analytically esta...In this paper, we considered a homogeneous reaction-diffusion predator-prey system with Holling type II functional response subject to Neumann boundary conditions. Some new sufficient conditions were analytically established to ensure that this system has globally asymptotically stable equilibria and Hopf bifurcation surrounding interior equilibrium. In the analysis of Hopf bifurcation, based on the phenomenon of Turing instability and well-done conditions, the system undergoes a Hopf bifurcation and an example incorporating with numerical simulations to support the existence of Hopf bifurcation is presented. We also derived a useful algorithm for determining direction of Hopf bifurcation and stability of bifurcating periodic solutions correspond to j ≠0 and j = 0, respectively. Finally, all these theoretical results are expected to be useful in the future study of dynamical complexity of ecological environment.展开更多
In this paper, a predator-prey model of three species is investigated, the necessary and sucient of the stable equilibrium point for this model is studied. Further, by introduc-ing a delay as a bifurcation parameter, ...In this paper, a predator-prey model of three species is investigated, the necessary and sucient of the stable equilibrium point for this model is studied. Further, by introduc-ing a delay as a bifurcation parameter, it is found that Hopf bifurcation occurs when τ cross some critical values. And, the stability and direction of hopf bifurcation are determined by applying the normal form theory and center manifold theory. numerical simulation results are given to support the theoretical predictions. At last, the periodic solution of this system is computed.展开更多
In this paper, we consider Lotka-Volterra predator-prey model between one and three species. Two cases are distinguished. The first is Lotka-Volterra model of one prey-three predators and the second is Lotka-Volterra ...In this paper, we consider Lotka-Volterra predator-prey model between one and three species. Two cases are distinguished. The first is Lotka-Volterra model of one prey-three predators and the second is Lotka-Volterra model of one predator-three preys. The existence conditions of nonnega-tive equilibrium points are established. The local stability analysis of the system is carried out.展开更多
In this paper, we investigated stability and bifurcation behaviors of a predator-prey model with Michaelis-Menten type prey harvesting. Sufficient conditions for local and global asymptotically stability of the interi...In this paper, we investigated stability and bifurcation behaviors of a predator-prey model with Michaelis-Menten type prey harvesting. Sufficient conditions for local and global asymptotically stability of the interior equilibrium point were established. Some critical threshold conditions for transcritical bifurcation, saddle-node bifurcation and Hopf bifurcation were explored analytically. Furthermore, It should be stressed that the fear factor could not only reduce the predator density, but also affect the prey growth rate. Finally, these theoretical results revealed that nonlinear Michaelis-Menten type prey harvesting has played an important role in the dynamic relationship, which also in turn proved the validity of theoretical derivation.展开更多
In this paper, a stochastic predator-prey model with stage structure for predatorand ratio-dependent functional response is concerned. Sufficient conditions for the globalasymptotic stability of positive equilibrium a...In this paper, a stochastic predator-prey model with stage structure for predatorand ratio-dependent functional response is concerned. Sufficient conditions for the globalasymptotic stability of positive equilibrium are established. Some numerical simulations arecarried out to illustrate the theoretical results.展开更多
This paper deals with the global dynamical behaviors of the positive solutions for a parabolic type ratio-dependent predator-prey system with a crowding term in the prey equation, where it is assumed that the coeffici...This paper deals with the global dynamical behaviors of the positive solutions for a parabolic type ratio-dependent predator-prey system with a crowding term in the prey equation, where it is assumed that the coefficient of the functional response is less than the coefficient of the intrinsic growth rates of the prey species. We demonstrated some special dynamical behaviors of the positive solutions of this system which the persistence of the coexistence of two species can be obtained when the crowding region in the prey equation only is designed suitably. Furthermore, we can obtain that under some conditions, the unique positive steady state solution of the system is globally asymptotically stable.展开更多
One predator two prey system is a research topic which has both the theoretical and practical values. This paper provides a natural condition of the existence of stable positive steady-state solutions for the one pred...One predator two prey system is a research topic which has both the theoretical and practical values. This paper provides a natural condition of the existence of stable positive steady-state solutions for the one predator two prey system. Under this condition we study the existence of the positive steady-state solutions at vicinity of the triple eigenvalue by implicit function theorem, discuss the positive stable solution problem bifurcated from the semi-trivial solutions containing two positive components with the help of bifurcation and perturbation methods.展开更多
Introduction of an invasive prey species into an ecosystem may affect an endemic predator's fitness by altering the prey-predator system. Successful adaptation may allow the endemic predator to eat and control the in...Introduction of an invasive prey species into an ecosystem may affect an endemic predator's fitness by altering the prey-predator system. Successful adaptation may allow the endemic predator to eat and control the invasive species, while unsuccessful adaptation may result in extinction of the predator. We examine the possible effects of the invasive North American bullfrog (Rana [Lithobates] catesbeiana) on the endemic Red-backed rat snake (Oocatochus rufodorsatus) in South Korea. We do so by comparing the morphology and behavior of adult and hatchling snakes from bullfrog-exposed (Taean) and bullfrog-unexposed (Hongcheon) populations. Among the seven morphological characteristics investigated, relative tail length (tail length/snout-vent length) of both adults and hatchlings from Taean was significantly greater than that of adults and hatchlings from Hongcheon. Also, adult snakes from Taean had a signiifcantly shorter latency of ifrst tongue lfick in response to prey compared to adults from Hongcheon. This difference was not observed in hatchlings. In other snake species, a longer relative tail length and shorter latency of ifrst tongue lfick are known to improve foraging efifciency, and these characters may be adaptations ofO. rufodorsatus to prey on bullfrogs. This study provides preliminary evidence that the presence of an invasive prey species may cause morphological and behavioral changes in an endemic predator.展开更多
We formulate and analyze a predator-prey model followed by Leslie-Gower model in which the prey population is infected by disease. We assume that the disease can only spread over prey population. As a result prey popu...We formulate and analyze a predator-prey model followed by Leslie-Gower model in which the prey population is infected by disease. We assume that the disease can only spread over prey population. As a result prey population has been classified into two categories, namely susceptible prey, infected prey where as the predator population remains free from infection. To investigate the behaviour of prey population we incorporate prey refuge in this model. Since the prey refuge decreases the predation rate then it has an important effect in our predator-prey interaction model. We have discussed the existence of various equilibrium points and local stability analysis at those equilibrium points. We investigate the Hopf-bifurcation analysis about the interior equilibrium point by taking the rate of infection parameter and the prey refuge parameter as bifurcation parameters. The numerical analysis is carried out to support the analytical results and to discuss some interesting results that our model exhibits.展开更多
This paper reports the global asymptotic stability of a three-species predator-prey system involving the prey-taxis. With the assumptions, we establish the global asymptotic stability results of its equilibria, respec...This paper reports the global asymptotic stability of a three-species predator-prey system involving the prey-taxis. With the assumptions, we establish the global asymptotic stability results of its equilibria, respectively. Our results illustrate that 1) the global asymptotic stability of the semi-trivial equilibrium does not involve the prey-taxis coefficients χ, ξ;2) the global asymptotic stability of two boundary equilibria relies on a single prey-taxis coefficient χ and ξ, respectively;3) the global asymptotic stability of the unique positive equilibrium depends on two prey-taxis coefficients χ and ξ.展开更多
The current study investigates the predator-prey problem with assumptions that interaction of predation has a little or no effect on prey population growth and the prey’s grow rate is time dependent. The prey is assu...The current study investigates the predator-prey problem with assumptions that interaction of predation has a little or no effect on prey population growth and the prey’s grow rate is time dependent. The prey is assumed to follow the Gompertz growth model and the respective predator growth function is constructed by solving ordinary differential equations. The results show that the predator population model is found to be a function of the well known exponential integral function. The solution is also given in Taylor’s series. Simulation study shows that the predator population size eventually converges either to a finite positive limit or zero or diverges to positive infinity. Under certain conditions, the predator population converges to the asymptotic limit of the prey model. More results are included in the paper.展开更多
Interaction between prey and predator species is a complex and non-linear process. Understanding various phenomena in the dynamics of prey-predator systems is vital to both mathematical ecology and conservation biolog...Interaction between prey and predator species is a complex and non-linear process. Understanding various phenomena in the dynamics of prey-predator systems is vital to both mathematical ecology and conservation biology. Mathematical models on prey-predator systems have been the hot sport providing important information regarding the interactions of prey and predator species in various ecosystems. In this paper, a review of the available mathematical models on prey-predator systems was done. Our aim was to assess their structure, behaviour, available control strategies, population involved and their ability in predicting the future behaviour of the ecosystems. We observed diversities in the reviewed mathematical models, some model incorporated factors such as drought, harvesting and prey refuge as the factors that affect ecosystems, some ignored the contribution of environmental variations while others considered the variable carrying capacity. Most of the models reviewed have not considered the contribution of diseases and seasonal weather variation in the dynamics of prey predator systems. Some of the reviewed models do not match the real situation in most modelled ecosystems. Thus, to avoid unreliable results, this review reveals the need to incorporate seasonal weather variations and diseases in the dynamics of prey predator systems of Serengeti ecosystem.展开更多
文摘In this paper, a diffusive predator-prey system with general functional responses and prey-tactic sensitivities is studied. Providing such generality, we construct a Lyapunov function and we show that the positive constant steady state is locally and globally asymptotically stable. With an eye on the biological interpretations, a numerical simulation is performed to illustrate the feasibility of the analytical findings.
基金supported by the National Natural Science Foundation of China(Grant Nos.11272051 and 11302172)
文摘The stability of the first-order and second-order solution moments for a Harrison-type predator-prey model with parametric Gaussian white noise is analyzed in this paper. The moment equations of the system solution are obtained under Ito interpretations. The delay-independent stable condition of the first-order moment is identical to that of the deterministic delayed system, and the delay-independent stable condition of the second-order moment depends on the noise intensities. The corresponding critical time delays are determined once the stabilities of moments lose. Further, when the time delays are greater than the critical time delays, the system solution becomes unstable with the increase of noise intensities. Finally, some numerical simulations are given to verify the theoretical results.
基金Project supported by the National Natural Science Foundation of China(Grant No.11272051)
文摘In this paper, we investigate the solution moment stability for a Harrison-type predator-prey model with parametric dichotomous noises. Using the Shapiro-Loginov formula, the equations for the first-order and second-order moments are obtained and the corresponding stable conditions are given. It is found that the solution moment stability depends on the noise intensity and correlation time of noise. The first-order and second-order moments become unstable with the decrease of correlation time. That is, the dichotomous noise can improve the solution moment stability with respect to Gaussian white noise. Finally, some numerical results are presented to verify the theoretical analyses.
基金Supported by the NSFC(11371368)Supported by the Basic Courses Department of OEC Foundation(Jcky1302)
文摘A Holling type III predator-prey model with stage structure for prey is investi-gated. By analyzing the corresponding characteristic equations, the local stability of each of feasible equilibria of the system is discussed. By using the uniformly persistence theory, the system is proven to be permanent if the coexistence equilibrium exists. By using Lyapunov functionals and LaSalle’s invariance principle, it is shown that the two boundary equilibria is globally asymptotically stable when the coexistence equilibrium is not feasible. By using compound matrix theory, the sucient conditions are obtained for the global stability of the coexistence equilibrium. At last, numerical simulations are carried out to illustrate the main results.
基金Supported by the National Natural Science Foundation of China(195 310 70 )
文摘This paper deals with the questio n of global stability of the positive locally asymptotically stable equilibrium in a class of predator\|prey system of Gause\|typ e with Holling Ⅲ functional response. The Dulac's criterion is applied and lia punov functions are constructed to establish the global stability.
文摘In this paper, the dynamical behaviors of a modified Leslie-Gower predator-prey model incorporating fear effect and prey refuge are investigated. We delve into the construction of the model and its biological significance, with preliminary results encompassing positivity, boundedness, and persistence. The stability of the system’s boundary and positive equilibrium points is proven by calculating the real part of the eigenvalues of the Jacobian matrix. At the positive equilibrium point, we demonstrate that the system’s unique positive equilibrium is globally asymptotically stable by using the Dulac criterion. Furthermore, at this equilibrium point, we employ the Implicit Function Theorem to discuss how fear effects and prey refuges influence the population densities of both prey and predators. Finally, numerical simulations are conducted to validate the above-mentioned conclusions and explored the impact of Predator-taxis sensitivity αon dynamics of the system.
文摘In this paper, a nonautonomous predator-prey system based on a modified version of the Leslie-Gower scheme and Holling-type II scheme with delayed effect is investigated. The general criteria of integrable form on the permanence are established. By constructing suitable Lyapunov functionals, a set of easily verifiable sufficient conditions are derived for global stability of any positive solutions to the
基金Shanghai Committee of Science and Technology,China (No.11ZR1400200 )Fundamental Research Funds for the Central Universities,China (No.2011D10903)
文摘The present paper is concerned with the Holling type IV predator-prey system with diffusion.By analyzing the characteristic equation associated with the positive equilibrium,the conditions for the asymptotic stability of the positive equilibrium is obtained.For the system without delay,it has been shown that the positive equilibrium is stable in certain region of the parameter plane.However,the introducing of the delay can lead to the loss of the stability.We find that in the region where the positive equilibrium is stable for the system without delay,there exists a critical value of the delay and the positive equilibrium is stable when the delay is less than this critical value and becomes unstable when the delay is greater than it.
文摘In this paper, we considered a homogeneous reaction-diffusion predator-prey system with Holling type II functional response subject to Neumann boundary conditions. Some new sufficient conditions were analytically established to ensure that this system has globally asymptotically stable equilibria and Hopf bifurcation surrounding interior equilibrium. In the analysis of Hopf bifurcation, based on the phenomenon of Turing instability and well-done conditions, the system undergoes a Hopf bifurcation and an example incorporating with numerical simulations to support the existence of Hopf bifurcation is presented. We also derived a useful algorithm for determining direction of Hopf bifurcation and stability of bifurcating periodic solutions correspond to j ≠0 and j = 0, respectively. Finally, all these theoretical results are expected to be useful in the future study of dynamical complexity of ecological environment.
基金Supported by the the NSF of Gansu Province(096RJZE106)
文摘In this paper, a predator-prey model of three species is investigated, the necessary and sucient of the stable equilibrium point for this model is studied. Further, by introduc-ing a delay as a bifurcation parameter, it is found that Hopf bifurcation occurs when τ cross some critical values. And, the stability and direction of hopf bifurcation are determined by applying the normal form theory and center manifold theory. numerical simulation results are given to support the theoretical predictions. At last, the periodic solution of this system is computed.
文摘In this paper, we consider Lotka-Volterra predator-prey model between one and three species. Two cases are distinguished. The first is Lotka-Volterra model of one prey-three predators and the second is Lotka-Volterra model of one predator-three preys. The existence conditions of nonnega-tive equilibrium points are established. The local stability analysis of the system is carried out.
文摘In this paper, we investigated stability and bifurcation behaviors of a predator-prey model with Michaelis-Menten type prey harvesting. Sufficient conditions for local and global asymptotically stability of the interior equilibrium point were established. Some critical threshold conditions for transcritical bifurcation, saddle-node bifurcation and Hopf bifurcation were explored analytically. Furthermore, It should be stressed that the fear factor could not only reduce the predator density, but also affect the prey growth rate. Finally, these theoretical results revealed that nonlinear Michaelis-Menten type prey harvesting has played an important role in the dynamic relationship, which also in turn proved the validity of theoretical derivation.
基金Supported by the National Natural Science Foundation of China(Nos.11371368)The Natural Science Foundation of HeBei(No.A2014506015)
文摘In this paper, a stochastic predator-prey model with stage structure for predatorand ratio-dependent functional response is concerned. Sufficient conditions for the globalasymptotic stability of positive equilibrium are established. Some numerical simulations arecarried out to illustrate the theoretical results.
基金supported by the National Natural Science Foundation of China(11271120,11426099)the Project of Hunan Natural Science Foundation of China(13JJ3085)
文摘This paper deals with the global dynamical behaviors of the positive solutions for a parabolic type ratio-dependent predator-prey system with a crowding term in the prey equation, where it is assumed that the coefficient of the functional response is less than the coefficient of the intrinsic growth rates of the prey species. We demonstrated some special dynamical behaviors of the positive solutions of this system which the persistence of the coexistence of two species can be obtained when the crowding region in the prey equation only is designed suitably. Furthermore, we can obtain that under some conditions, the unique positive steady state solution of the system is globally asymptotically stable.
基金This work is supported by National Science Foundation of China and the Fundes of Institute of Math (opened) Academic Sinica.
文摘One predator two prey system is a research topic which has both the theoretical and practical values. This paper provides a natural condition of the existence of stable positive steady-state solutions for the one predator two prey system. Under this condition we study the existence of the positive steady-state solutions at vicinity of the triple eigenvalue by implicit function theorem, discuss the positive stable solution problem bifurcated from the semi-trivial solutions containing two positive components with the help of bifurcation and perturbation methods.
文摘Introduction of an invasive prey species into an ecosystem may affect an endemic predator's fitness by altering the prey-predator system. Successful adaptation may allow the endemic predator to eat and control the invasive species, while unsuccessful adaptation may result in extinction of the predator. We examine the possible effects of the invasive North American bullfrog (Rana [Lithobates] catesbeiana) on the endemic Red-backed rat snake (Oocatochus rufodorsatus) in South Korea. We do so by comparing the morphology and behavior of adult and hatchling snakes from bullfrog-exposed (Taean) and bullfrog-unexposed (Hongcheon) populations. Among the seven morphological characteristics investigated, relative tail length (tail length/snout-vent length) of both adults and hatchlings from Taean was significantly greater than that of adults and hatchlings from Hongcheon. Also, adult snakes from Taean had a signiifcantly shorter latency of ifrst tongue lfick in response to prey compared to adults from Hongcheon. This difference was not observed in hatchlings. In other snake species, a longer relative tail length and shorter latency of ifrst tongue lfick are known to improve foraging efifciency, and these characters may be adaptations ofO. rufodorsatus to prey on bullfrogs. This study provides preliminary evidence that the presence of an invasive prey species may cause morphological and behavioral changes in an endemic predator.
文摘We formulate and analyze a predator-prey model followed by Leslie-Gower model in which the prey population is infected by disease. We assume that the disease can only spread over prey population. As a result prey population has been classified into two categories, namely susceptible prey, infected prey where as the predator population remains free from infection. To investigate the behaviour of prey population we incorporate prey refuge in this model. Since the prey refuge decreases the predation rate then it has an important effect in our predator-prey interaction model. We have discussed the existence of various equilibrium points and local stability analysis at those equilibrium points. We investigate the Hopf-bifurcation analysis about the interior equilibrium point by taking the rate of infection parameter and the prey refuge parameter as bifurcation parameters. The numerical analysis is carried out to support the analytical results and to discuss some interesting results that our model exhibits.
文摘This paper reports the global asymptotic stability of a three-species predator-prey system involving the prey-taxis. With the assumptions, we establish the global asymptotic stability results of its equilibria, respectively. Our results illustrate that 1) the global asymptotic stability of the semi-trivial equilibrium does not involve the prey-taxis coefficients χ, ξ;2) the global asymptotic stability of two boundary equilibria relies on a single prey-taxis coefficient χ and ξ, respectively;3) the global asymptotic stability of the unique positive equilibrium depends on two prey-taxis coefficients χ and ξ.
文摘The current study investigates the predator-prey problem with assumptions that interaction of predation has a little or no effect on prey population growth and the prey’s grow rate is time dependent. The prey is assumed to follow the Gompertz growth model and the respective predator growth function is constructed by solving ordinary differential equations. The results show that the predator population model is found to be a function of the well known exponential integral function. The solution is also given in Taylor’s series. Simulation study shows that the predator population size eventually converges either to a finite positive limit or zero or diverges to positive infinity. Under certain conditions, the predator population converges to the asymptotic limit of the prey model. More results are included in the paper.
文摘Interaction between prey and predator species is a complex and non-linear process. Understanding various phenomena in the dynamics of prey-predator systems is vital to both mathematical ecology and conservation biology. Mathematical models on prey-predator systems have been the hot sport providing important information regarding the interactions of prey and predator species in various ecosystems. In this paper, a review of the available mathematical models on prey-predator systems was done. Our aim was to assess their structure, behaviour, available control strategies, population involved and their ability in predicting the future behaviour of the ecosystems. We observed diversities in the reviewed mathematical models, some model incorporated factors such as drought, harvesting and prey refuge as the factors that affect ecosystems, some ignored the contribution of environmental variations while others considered the variable carrying capacity. Most of the models reviewed have not considered the contribution of diseases and seasonal weather variation in the dynamics of prey predator systems. Some of the reviewed models do not match the real situation in most modelled ecosystems. Thus, to avoid unreliable results, this review reveals the need to incorporate seasonal weather variations and diseases in the dynamics of prey predator systems of Serengeti ecosystem.