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
This paper mainly focus on the research of a predator⁃prey system with Gompertz growth of prey.When the system does not contain diffusion,the stability conditions of positive equilibrium and the occurring condition of...This paper mainly focus on the research of a predator⁃prey system with Gompertz growth of prey.When the system does not contain diffusion,the stability conditions of positive equilibrium and the occurring condition of the Hopf bifurcation are obtained.When the diffusion term of the system appears,the stable conditions of positive equilibrium and the Turing instability condition are also obtained.Turing instability is induced by the diffusion term through theoretical analysis.Thus,the region of parameters in which Turing instability occurs is presented.Then the amplitude equations are derived by the multiple scale method.The results will enrich the pattern dynamics in predator⁃prey systems.展开更多
Self-diffusion prerequisite is obtained as the spreading approach of biological populations.Cooperative hunting is a common behavior in predator populations that promotes predation and the coexistence of the prey-pred...Self-diffusion prerequisite is obtained as the spreading approach of biological populations.Cooperative hunting is a common behavior in predator populations that promotes predation and the coexistence of the prey-predator system.On the other side,the Allee effect among prey may cause the system to become unstable.In this paper,a difusive prey predator system with cooperative hunting and the weak Allee effect in prey populations is discussed.The linear stability and Hopf-bifurcation analysis had been used to examine the system's stability.From the spatial stability of the system,the conditions for Turing instability have been derived.The multiple-scale analysis has been used to derive the amplitude equations of the system.The stability analysis of these amplitude equations leads to the formation of Turing patterns.Finally,numerical simulations are used to analyze spatial patterns forming in 1-D and 2-D.The studies indicate that the model can generate a complex pattern structure and that self-diffusion has a drastic impacton species distribution.展开更多
The bifurcation solution of the nonnegative steady state of a reaction diffusion system was investigated. The combination of the sturm type eigenvalue and the theorem of bifurcation was used to study the local coex...The bifurcation solution of the nonnegative steady state of a reaction diffusion system was investigated. The combination of the sturm type eigenvalue and the theorem of bifurcation was used to study the local coexistence solutions, and obtain the stability of bifurcation solutions. The system model describes predator prey interaction in an unstirred chemostat.展开更多
This article studies a nonlinear fractional order Lotka-Volterra prey-predator type dynamical system.For the proposed study,we consider the model under the conformable fractional order derivative(CFOD).We investigate ...This article studies a nonlinear fractional order Lotka-Volterra prey-predator type dynamical system.For the proposed study,we consider the model under the conformable fractional order derivative(CFOD).We investigate the mentioned dynamical system for the existence and uniqueness of at least one solution.Indeed,Schauder and Banach fixed point theorems are utilized to prove our claim.Further,an algorithm for the approximate analytical solution to the proposed problem has been established.In this regard,the conformable fractional differential transform(CFDT)technique is used to compute the required results in the form of a series.Using Matlab-16,we simulate the series solution to illustrate our results graphically.Finally,a comparison of our solution to that obtained for the Caputo fractional order derivative via the perturbation method is given.展开更多
This paper investigates a nonautonomous Volterra predator Prey system with undercrowding effect. A set of sufficient conditions for the existence and globally asymptotic stability of positive solution, which is easy t...This paper investigates a nonautonomous Volterra predator Prey system with undercrowding effect. A set of sufficient conditions for the existence and globally asymptotic stability of positive solution, which is easy to be verified, is obtained.展开更多
This paper studies the dynamics of the generalist predator–prey systems modeled in[E.Alexandra,F.Lutscher and G.Seo,Bistability and limit cycles in generalist predator–prey dynamics,Ecol.Complex.14(2013)48–55].When...This paper studies the dynamics of the generalist predator–prey systems modeled in[E.Alexandra,F.Lutscher and G.Seo,Bistability and limit cycles in generalist predator–prey dynamics,Ecol.Complex.14(2013)48–55].When prey reproduces much faster than predator,by combining the normal form theory of slow-fast systems,the geometric singular perturbation theory and the results near non-hyperbolic points developed by Krupa and Szmolyan[Relaxation oscillation and canard explosion,J.Differential Equations174(2)(2001)312–368;Extending geometric singular perturbation theory to nonhyperbolic points—fold and canard points in two dimensions,SIAM J.Math.Anal.33(2)(2001)286–314],we provide a detailed mathematical analysis to show the existence of homoclinic orbits,heteroclinic orbits and canard limit cycles and relaxation oscillations bifurcating from the singular homoclinic cycles.Moreover,on global stability of the unique positive equilibrium,we provide some new results.Numerical simulations are also carried out to support the theoretical results.展开更多
In this paper,a difference-algebraic predator prey model is proposed,and its complex dynamical behaviors are analyzed.The model is a discrete singular system,which is obtained by using Euler scheme to discretize a dif...In this paper,a difference-algebraic predator prey model is proposed,and its complex dynamical behaviors are analyzed.The model is a discrete singular system,which is obtained by using Euler scheme to discretize a differential-algebraic predator-prey model with harvesting that we establish.Firstly,the local stability of the interior equilibrium point of proposed model is investigated on the basis of discrete dynamical system theory.Further,by applying the new normal form of difference-algebraic equations,center manifold theory and bifurcation theory,the Flip bifurcation and Neimark-Sacker bifurcation around the interior equilibrium point are studied,where the step size is treated as the variable bifurcation parameter.Lastly,with the help of Matlab software,some numerical simulations are performed not only to validate our theoretical results,but also to show the abundant dynamical behaviors,such as period-doubling bifurcations,period 2,4,8,and 16 orbits,invariant closed curve,and chaotic sets.In particular,the corresponding maximum Lyapunov exponents are numerically calculated to corroborate the bifurcation and chaotic behaviors.展开更多
Dissimilar vulnerabilities of different prey types and preferences of predators are factors likely to contribute to community dynamics.This may happen via differential individual properties of prey animals(e.g.vigilan...Dissimilar vulnerabilities of different prey types and preferences of predators are factors likely to contribute to community dynamics.This may happen via differential individual properties of prey animals(e.g.vigilance,escape)or via habitat effects making hunting by a predator easier and more rewarding in some habitats,or both.Furthermore,community dynamics may be influenced by predator mediated apparent competition,in which an increase in one prey type has negative effects on another prey type indirectly via the shared predator.We summarize the current knowledge from the field in a model predator–prey system consisting of sympatric boreal vole species and their common specialist predator and review field studies using predator manipulation and studies on the responses of individuals in the laboratory and in outdoor enclosures.The vole species studied represent different prey types that are thought to have different vulnerabilities.Our observations on the main resident specialist predator,the least weasel(Mustela nivalis nivalis L.),show that it hunts according to prey availability and suitability of the hunting habitat.Prey voles respond to the presence of the predator behaviorally in various ways to avoid predation.We conclude that even if the least weasel is a specialized predator of small rodents it acts like a generalist predator within the small rodent guild and may facilitate the coexistence of prey species via predator switching.This may lead to interspecific synchrony between prey populations,which has often been observed.We suggest that the processes determining the community impact of predator–prey interactions are driven by the behavioral arms race between the predator and the prey,together with the habitat-dependent density of prey and net gain for the predator.展开更多
A recent global review of birds caught in spider webs reported only three Asian cases. Given this surprisingly low number, I made a concerted effort to obtain additional Asian cases from the literature, the internet, ...A recent global review of birds caught in spider webs reported only three Asian cases. Given this surprisingly low number, I made a concerted effort to obtain additional Asian cases from the literature, the internet, and field workers. I present a total of 56 Asian cases which pertain to 33 bird species. As in the global dataset, mostly small bird species were caught in spider webs, with a mean body mass of 17.5 g and a mean wing chord length of 73.1 mm. Conse?quently, birds with a body mass >30 g were very rarely caught. This Asian review corroborates the global review that smaller birds are more likely to be caught and that Nephila spiders are most likely to be the predators. Continuous monitoring of spider webs is recommended to ascertain the frequency of these events.展开更多
The main purpose of this paper is to explore the influence of predation on the spread of a disease developed in the prey population where we assume that the prey has a social behavior.The memory of the prey and the pr...The main purpose of this paper is to explore the influence of predation on the spread of a disease developed in the prey population where we assume that the prey has a social behavior.The memory of the prey and the predator measured by the time fractional derivative plays a crucial role in modeling the dynamical response in a predator–prey interaction.This memory can be modeled to articulate the involvement of interacting species by the presence of the time fractional derivative in the considered models.For the purpose of studying the complex dynamics generated by the presence of infection and the time-fractional-derivative we split our study into two cases. The first one is devotedto study the effect of a non-selective hunting on the spread of the disease, where the localstability of the equilibria is investigated. Further the backward bifurcation is obtainedconcerning basic reproduction rate of the infection. The second case is for explaining theimpact of selecting the weakest infected prey on the edge of the herd by a predator onthe prevalence of the infection, where the local behavior is scrutinized. Moreover, for thegraphical representation part, a numerical simulation scheme has been achieved usingthe Caputo fractional derivative operator.展开更多
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.
基金National Natural Science Foundation of China(No.11971143)。
文摘This paper mainly focus on the research of a predator⁃prey system with Gompertz growth of prey.When the system does not contain diffusion,the stability conditions of positive equilibrium and the occurring condition of the Hopf bifurcation are obtained.When the diffusion term of the system appears,the stable conditions of positive equilibrium and the Turing instability condition are also obtained.Turing instability is induced by the diffusion term through theoretical analysis.Thus,the region of parameters in which Turing instability occurs is presented.Then the amplitude equations are derived by the multiple scale method.The results will enrich the pattern dynamics in predator⁃prey systems.
文摘Self-diffusion prerequisite is obtained as the spreading approach of biological populations.Cooperative hunting is a common behavior in predator populations that promotes predation and the coexistence of the prey-predator system.On the other side,the Allee effect among prey may cause the system to become unstable.In this paper,a difusive prey predator system with cooperative hunting and the weak Allee effect in prey populations is discussed.The linear stability and Hopf-bifurcation analysis had been used to examine the system's stability.From the spatial stability of the system,the conditions for Turing instability have been derived.The multiple-scale analysis has been used to derive the amplitude equations of the system.The stability analysis of these amplitude equations leads to the formation of Turing patterns.Finally,numerical simulations are used to analyze spatial patterns forming in 1-D and 2-D.The studies indicate that the model can generate a complex pattern structure and that self-diffusion has a drastic impacton species distribution.
文摘The bifurcation solution of the nonnegative steady state of a reaction diffusion system was investigated. The combination of the sturm type eigenvalue and the theorem of bifurcation was used to study the local coexistence solutions, and obtain the stability of bifurcation solutions. The system model describes predator prey interaction in an unstirred chemostat.
文摘This article studies a nonlinear fractional order Lotka-Volterra prey-predator type dynamical system.For the proposed study,we consider the model under the conformable fractional order derivative(CFOD).We investigate the mentioned dynamical system for the existence and uniqueness of at least one solution.Indeed,Schauder and Banach fixed point theorems are utilized to prove our claim.Further,an algorithm for the approximate analytical solution to the proposed problem has been established.In this regard,the conformable fractional differential transform(CFDT)technique is used to compute the required results in the form of a series.Using Matlab-16,we simulate the series solution to illustrate our results graphically.Finally,a comparison of our solution to that obtained for the Caputo fractional order derivative via the perturbation method is given.
文摘This paper investigates a nonautonomous Volterra predator Prey system with undercrowding effect. A set of sufficient conditions for the existence and globally asymptotic stability of positive solution, which is easy to be verified, is obtained.
文摘This paper studies the dynamics of the generalist predator–prey systems modeled in[E.Alexandra,F.Lutscher and G.Seo,Bistability and limit cycles in generalist predator–prey dynamics,Ecol.Complex.14(2013)48–55].When prey reproduces much faster than predator,by combining the normal form theory of slow-fast systems,the geometric singular perturbation theory and the results near non-hyperbolic points developed by Krupa and Szmolyan[Relaxation oscillation and canard explosion,J.Differential Equations174(2)(2001)312–368;Extending geometric singular perturbation theory to nonhyperbolic points—fold and canard points in two dimensions,SIAM J.Math.Anal.33(2)(2001)286–314],we provide a detailed mathematical analysis to show the existence of homoclinic orbits,heteroclinic orbits and canard limit cycles and relaxation oscillations bifurcating from the singular homoclinic cycles.Moreover,on global stability of the unique positive equilibrium,we provide some new results.Numerical simulations are also carried out to support the theoretical results.
基金the National Natural Science Foundation of China(Grant No.11871393)the Key Project of the International Science and Technology Cooperation Program of Shaanxi Research&Development Plan(Grant No.2019KWZ-08)the Science and Technology Project founded by the Education Department of Jiangxi Province(Grant No.GJJ14775).
文摘In this paper,a difference-algebraic predator prey model is proposed,and its complex dynamical behaviors are analyzed.The model is a discrete singular system,which is obtained by using Euler scheme to discretize a differential-algebraic predator-prey model with harvesting that we establish.Firstly,the local stability of the interior equilibrium point of proposed model is investigated on the basis of discrete dynamical system theory.Further,by applying the new normal form of difference-algebraic equations,center manifold theory and bifurcation theory,the Flip bifurcation and Neimark-Sacker bifurcation around the interior equilibrium point are studied,where the step size is treated as the variable bifurcation parameter.Lastly,with the help of Matlab software,some numerical simulations are performed not only to validate our theoretical results,but also to show the abundant dynamical behaviors,such as period-doubling bifurcations,period 2,4,8,and 16 orbits,invariant closed curve,and chaotic sets.In particular,the corresponding maximum Lyapunov exponents are numerically calculated to corroborate the bifurcation and chaotic behaviors.
基金We are grateful to the Academy of Finland(projects 52045,44887 and 208478)for funding our research.Konnevesi Research Station has provided facilities for experimentation and thinking,as has Helsinki.
文摘Dissimilar vulnerabilities of different prey types and preferences of predators are factors likely to contribute to community dynamics.This may happen via differential individual properties of prey animals(e.g.vigilance,escape)or via habitat effects making hunting by a predator easier and more rewarding in some habitats,or both.Furthermore,community dynamics may be influenced by predator mediated apparent competition,in which an increase in one prey type has negative effects on another prey type indirectly via the shared predator.We summarize the current knowledge from the field in a model predator–prey system consisting of sympatric boreal vole species and their common specialist predator and review field studies using predator manipulation and studies on the responses of individuals in the laboratory and in outdoor enclosures.The vole species studied represent different prey types that are thought to have different vulnerabilities.Our observations on the main resident specialist predator,the least weasel(Mustela nivalis nivalis L.),show that it hunts according to prey availability and suitability of the hunting habitat.Prey voles respond to the presence of the predator behaviorally in various ways to avoid predation.We conclude that even if the least weasel is a specialized predator of small rodents it acts like a generalist predator within the small rodent guild and may facilitate the coexistence of prey species via predator switching.This may lead to interspecific synchrony between prey populations,which has often been observed.We suggest that the processes determining the community impact of predator–prey interactions are driven by the behavioral arms race between the predator and the prey,together with the habitat-dependent density of prey and net gain for the predator.
基金I acknowledge financial support from Taipei Medical University through a SEED Grant
文摘A recent global review of birds caught in spider webs reported only three Asian cases. Given this surprisingly low number, I made a concerted effort to obtain additional Asian cases from the literature, the internet, and field workers. I present a total of 56 Asian cases which pertain to 33 bird species. As in the global dataset, mostly small bird species were caught in spider webs, with a mean body mass of 17.5 g and a mean wing chord length of 73.1 mm. Conse?quently, birds with a body mass >30 g were very rarely caught. This Asian review corroborates the global review that smaller birds are more likely to be caught and that Nephila spiders are most likely to be the predators. Continuous monitoring of spider webs is recommended to ascertain the frequency of these events.
文摘The main purpose of this paper is to explore the influence of predation on the spread of a disease developed in the prey population where we assume that the prey has a social behavior.The memory of the prey and the predator measured by the time fractional derivative plays a crucial role in modeling the dynamical response in a predator–prey interaction.This memory can be modeled to articulate the involvement of interacting species by the presence of the time fractional derivative in the considered models.For the purpose of studying the complex dynamics generated by the presence of infection and the time-fractional-derivative we split our study into two cases. The first one is devotedto study the effect of a non-selective hunting on the spread of the disease, where the localstability of the equilibria is investigated. Further the backward bifurcation is obtainedconcerning basic reproduction rate of the infection. The second case is for explaining theimpact of selecting the weakest infected prey on the edge of the herd by a predator onthe prevalence of the infection, where the local behavior is scrutinized. Moreover, for thegraphical representation part, a numerical simulation scheme has been achieved usingthe Caputo fractional derivative operator.