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Weiming Wang - One of the best experts on this subject based on the ideXlab platform.
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dynamics of a leslie gower predator prey model with additive Allee Effect
Applied Mathematical Modelling, 2015Co-Authors: Weiming Wang, Yongli Cai, Caidi Zhao, Jinfeng WangAbstract:Abstract In this paper, we investigate the complex dynamics of a Leslie–Gower predation model with additive Allee Effect on prey. Without Allee Effect, the model has a unique globally asymptotically stable equilibrium point under some conditions. With Allee Effect on prey, the model has none, one or two positive equilibria. Moreover, we prove the existence of parameter subsets for which the model can have Hopf bifurcation. And we find that when the model exhibits two positive equilibria there is a separatrix curve that separates the behavior of trajectories of the system, implying that the model is highly sensitive to the initial conditions. One of the most interesting findings is that Allee Effect can increase the risk of ecological extinction.
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Dynamics of a Leslie–Gower predator–prey model with additive Allee Effect
Applied Mathematical Modelling, 2015Co-Authors: Yongli Cai, Weiming Wang, Caidi Zhao, Jinfeng WangAbstract:Abstract In this paper, we investigate the complex dynamics of a Leslie–Gower predation model with additive Allee Effect on prey. Without Allee Effect, the model has a unique globally asymptotically stable equilibrium point under some conditions. With Allee Effect on prey, the model has none, one or two positive equilibria. Moreover, we prove the existence of parameter subsets for which the model can have Hopf bifurcation. And we find that when the model exhibits two positive equilibria there is a separatrix curve that separates the behavior of trajectories of the system, implying that the model is highly sensitive to the initial conditions. One of the most interesting findings is that Allee Effect can increase the risk of ecological extinction.
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dynamical complexity induced by Allee Effect in a predator prey model
Nonlinear Analysis-real World Applications, 2014Co-Authors: Weiming Wang, Yanuo Zhu, Yongli Cai, Wenjuan WangAbstract:Abstract In this paper, we investigate the complex dynamics induced by Allee Effect in a predator–prey model. For the non-spatial model, Allee Effect remains the boundedness of positive solutions, and it also induces the model to exhibit one or two positive equilibria. Especially, in the case with strong Allee Effect, the model is bistable. For the spatial model, without Allee Effect, there is the nonexistence of diffusion-driven instability. And in the case with Allee Effect, the positive equilibrium can be unstable under certain conditions. This instability is induced by Allee Effect and diffusion together. Furthermore, via numerical simulations, the model dynamics exhibits both Allee Effect and diffusion controlled pattern formation growth to holes, stripe–hole mixtures, stripes, stripe–spot mixtures, and spots replication. That is to say, the dynamics of the model with Allee Effect is not simple, but rich and complex.
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Dynamical complexity induced by Allee Effect in a predator–prey model
Nonlinear Analysis: Real World Applications, 2014Co-Authors: Weiming Wang, Yanuo Zhu, Yongli Cai, Wenjuan WangAbstract:Abstract In this paper, we investigate the complex dynamics induced by Allee Effect in a predator–prey model. For the non-spatial model, Allee Effect remains the boundedness of positive solutions, and it also induces the model to exhibit one or two positive equilibria. Especially, in the case with strong Allee Effect, the model is bistable. For the spatial model, without Allee Effect, there is the nonexistence of diffusion-driven instability. And in the case with Allee Effect, the positive equilibrium can be unstable under certain conditions. This instability is induced by Allee Effect and diffusion together. Furthermore, via numerical simulations, the model dynamics exhibits both Allee Effect and diffusion controlled pattern formation growth to holes, stripe–hole mixtures, stripes, stripe–spot mixtures, and spots replication. That is to say, the dynamics of the model with Allee Effect is not simple, but rich and complex.
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Allee-Effect-Induced Instability in a Reaction-Diffusion Predator-Prey Model
Abstract and Applied Analysis, 2013Co-Authors: Weiming Wang, Yanuo Zhu, Yongli Cai, Zhengguang GuoAbstract:We investigate the spatiotemporal dynamics induced by Allee Effect in a reaction-diffusion predator-prey model. In the case without Allee Effect, there is nonexistence of diffusion-driven instability for the model. And in the case with Allee Effect, the positive equilibrium may be unstable under certain conditions. This instability is induced by Allee Effect and diffusion together. Furthermore, via numerical simulations, the model dynamics exhibits both Allee Effect and diffusion controlled pattern formation growth to holes, stripes-holes mixture, stripes, stripes-spots mixture, and spots replication, which shows that the dynamics of the model with Allee Effect is not simple, but rich and complex.
Yongli Cai - One of the best experts on this subject based on the ideXlab platform.
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dynamics of a leslie gower predator prey model with additive Allee Effect
Applied Mathematical Modelling, 2015Co-Authors: Weiming Wang, Yongli Cai, Caidi Zhao, Jinfeng WangAbstract:Abstract In this paper, we investigate the complex dynamics of a Leslie–Gower predation model with additive Allee Effect on prey. Without Allee Effect, the model has a unique globally asymptotically stable equilibrium point under some conditions. With Allee Effect on prey, the model has none, one or two positive equilibria. Moreover, we prove the existence of parameter subsets for which the model can have Hopf bifurcation. And we find that when the model exhibits two positive equilibria there is a separatrix curve that separates the behavior of trajectories of the system, implying that the model is highly sensitive to the initial conditions. One of the most interesting findings is that Allee Effect can increase the risk of ecological extinction.
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Dynamics of a Leslie–Gower predator–prey model with additive Allee Effect
Applied Mathematical Modelling, 2015Co-Authors: Yongli Cai, Weiming Wang, Caidi Zhao, Jinfeng WangAbstract:Abstract In this paper, we investigate the complex dynamics of a Leslie–Gower predation model with additive Allee Effect on prey. Without Allee Effect, the model has a unique globally asymptotically stable equilibrium point under some conditions. With Allee Effect on prey, the model has none, one or two positive equilibria. Moreover, we prove the existence of parameter subsets for which the model can have Hopf bifurcation. And we find that when the model exhibits two positive equilibria there is a separatrix curve that separates the behavior of trajectories of the system, implying that the model is highly sensitive to the initial conditions. One of the most interesting findings is that Allee Effect can increase the risk of ecological extinction.
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dynamical complexity induced by Allee Effect in a predator prey model
Nonlinear Analysis-real World Applications, 2014Co-Authors: Weiming Wang, Yanuo Zhu, Yongli Cai, Wenjuan WangAbstract:Abstract In this paper, we investigate the complex dynamics induced by Allee Effect in a predator–prey model. For the non-spatial model, Allee Effect remains the boundedness of positive solutions, and it also induces the model to exhibit one or two positive equilibria. Especially, in the case with strong Allee Effect, the model is bistable. For the spatial model, without Allee Effect, there is the nonexistence of diffusion-driven instability. And in the case with Allee Effect, the positive equilibrium can be unstable under certain conditions. This instability is induced by Allee Effect and diffusion together. Furthermore, via numerical simulations, the model dynamics exhibits both Allee Effect and diffusion controlled pattern formation growth to holes, stripe–hole mixtures, stripes, stripe–spot mixtures, and spots replication. That is to say, the dynamics of the model with Allee Effect is not simple, but rich and complex.
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Dynamical complexity induced by Allee Effect in a predator–prey model
Nonlinear Analysis: Real World Applications, 2014Co-Authors: Weiming Wang, Yanuo Zhu, Yongli Cai, Wenjuan WangAbstract:Abstract In this paper, we investigate the complex dynamics induced by Allee Effect in a predator–prey model. For the non-spatial model, Allee Effect remains the boundedness of positive solutions, and it also induces the model to exhibit one or two positive equilibria. Especially, in the case with strong Allee Effect, the model is bistable. For the spatial model, without Allee Effect, there is the nonexistence of diffusion-driven instability. And in the case with Allee Effect, the positive equilibrium can be unstable under certain conditions. This instability is induced by Allee Effect and diffusion together. Furthermore, via numerical simulations, the model dynamics exhibits both Allee Effect and diffusion controlled pattern formation growth to holes, stripe–hole mixtures, stripes, stripe–spot mixtures, and spots replication. That is to say, the dynamics of the model with Allee Effect is not simple, but rich and complex.
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Allee-Effect-Induced Instability in a Reaction-Diffusion Predator-Prey Model
Abstract and Applied Analysis, 2013Co-Authors: Weiming Wang, Yanuo Zhu, Yongli Cai, Zhengguang GuoAbstract:We investigate the spatiotemporal dynamics induced by Allee Effect in a reaction-diffusion predator-prey model. In the case without Allee Effect, there is nonexistence of diffusion-driven instability for the model. And in the case with Allee Effect, the positive equilibrium may be unstable under certain conditions. This instability is induced by Allee Effect and diffusion together. Furthermore, via numerical simulations, the model dynamics exhibits both Allee Effect and diffusion controlled pattern formation growth to holes, stripes-holes mixture, stripes, stripes-spots mixture, and spots replication, which shows that the dynamics of the model with Allee Effect is not simple, but rich and complex.
Chang-zhong Liu - One of the best experts on this subject based on the ideXlab platform.
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analysis of a discrete time predator prey system with Allee Effect
Ecological Complexity, 2011Co-Authors: Wan-xiong Wang, Yan-bo Zhang, Chang-zhong LiuAbstract:Abstract This paper deals with a discrete-time predator–prey system with Allee Effect. We obtain asymptotically stable conditions of the equilibrium points which are subject to the Allee Effect. The stabilizing Effect of Allee Effect on prey and predator populations is studied, respectively. The Allee Effect which occurs on both of predator and prey populations is also discussed by stability analysis, phase-plane and bifurcation diagram analysis. Our study suggests that Allee Effect has stabilizing Effect on population dynamics.
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Analysis of a discrete-time predator–prey system with Allee Effect
Ecological Complexity, 2011Co-Authors: Wan-xiong Wang, Yan-bo Zhang, Chang-zhong LiuAbstract:Abstract This paper deals with a discrete-time predator–prey system with Allee Effect. We obtain asymptotically stable conditions of the equilibrium points which are subject to the Allee Effect. The stabilizing Effect of Allee Effect on prey and predator populations is studied, respectively. The Allee Effect which occurs on both of predator and prey populations is also discussed by stability analysis, phase-plane and bifurcation diagram analysis. Our study suggests that Allee Effect has stabilizing Effect on population dynamics.
Zhengguang Guo - One of the best experts on this subject based on the ideXlab platform.
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Allee-Effect-Induced Instability in a Reaction-Diffusion Predator-Prey Model
Abstract and Applied Analysis, 2013Co-Authors: Weiming Wang, Yanuo Zhu, Yongli Cai, Zhengguang GuoAbstract:We investigate the spatiotemporal dynamics induced by Allee Effect in a reaction-diffusion predator-prey model. In the case without Allee Effect, there is nonexistence of diffusion-driven instability for the model. And in the case with Allee Effect, the positive equilibrium may be unstable under certain conditions. This instability is induced by Allee Effect and diffusion together. Furthermore, via numerical simulations, the model dynamics exhibits both Allee Effect and diffusion controlled pattern formation growth to holes, stripes-holes mixture, stripes, stripes-spots mixture, and spots replication, which shows that the dynamics of the model with Allee Effect is not simple, but rich and complex.
Andrew Morozov - One of the best experts on this subject based on the ideXlab platform.
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Bifurcation analysis of a ratio-dependent prey–predator model with the Allee Effect
Ecological Complexity, 2012Co-Authors: Malay Banerjee, Andrew MorozovAbstract:Abstract There is a growing body of evidence supporting implementation of ratio-dependent functional response of predators in ecological models. Those models often provide a satisfactory explanation of the observed patterns of dynamics which cannot be done based on the ‘classical’ models using the prey-dependent functional response. Surprisingly enough, all theoretical analysis of ratio-dependant predator–prey interactions has so far been completed only for the simplest case where the prey growth is logistic. In a large number of ecologically relevant situations, however, the growth rate of a population is subject to an Allee Effect and the per capita growth rate increases with population density. Taking into account Allee dynamics for the prey growth in models can alter the previous theoretical findings obtained for the logistic growth paradigm. In this paper, we analyse a ratio-dependent predator–prey system with prey growth subject to an Allee Effect. We both consider the cases of a strong Allee Effect (the population growth rate is negative at low species density) and the case of a weak Allee Effect (the population growth is positive at low population density). For both cases we fulfil a comprehensive bifurcation analysis, constructing the parametric diagrams, and also show possible phase portraits. Then we compare the properties of the ratio-dependant predator–prey model with and without the Allee Effect and show a substantial difference in the dynamical behaviour of those systems. We show that including an Allee Effect in a ratio-dependent predator–prey model removes the possibility of sustainable oscillations of species densities (population cycles). We show that the ratio-dependent predator–prey model with the Allee Effect can solve the paradox of enrichment. However, unlike the same model with logistic growth, incorporating the Allee Effect results in a paradox of biological control.
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Bifurcation analysis of a ratio-dependent prey-predator model with the Allee Effect
2012Co-Authors: Malay Banerjee, Andrew MorozovAbstract:A B S T R A C T There is a growing body of evidence supporting implementation of ratio-dependent functional response of predators in ecological models. Those models often provide a satisfactory explanation of the observed patterns of dynamics which cannot be done based on the ‘classical’ models using the prey-dependent functional response. Surprisingly enough, all theoretical analysis of ratio-dependant predator–prey interactions has so far been completed only for the simplest case where the prey growth is logistic. In a large number of ecologically relevant situations, however, the growth rate of a population is subject to an Allee Effect and the per capita growth rate increases with population density. Taking into account Allee dynamics for the prey growth in models can alter the previous theoretical findings obtained for the logistic growth paradigm. In this paper, we analyse a ratio-dependent predator–prey system with prey growth subject to an Allee Effect. We both consider the cases of a strong Allee Effect (the population growth rate is negative at low species density) and the case of a weak Allee Effect (the population growth is positive at low population density). For both cases we fulfil a comprehensive bifurcation analysis, constructing the parametric diagrams, and also show possible phase portraits. Then we compare the properties of the ratio-dependant predator–prey model with and without the Allee Effect and show a substantial difference in the dynamical behaviour of those systems. We show that including an Allee Effect in a ratio-dependent predator–prey model removes the possibility of sustainable oscillations of species densities (population cycles). We show that the ratio-dependent predator–prey model with the Allee Effect can solve the paradox of enrichment. However, unlike the same model with logistic growth, incorporating the Allee Effect results in a paradox of biological control.
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Original research articleBifurcation analysis of a ratio-dependent prey–predator model with the Allee Effect
Ecological Complexity, 2012Co-Authors: Malay Banerjee, Andrew MorozovAbstract:There is a growing body of evidence supporting implementation of ratio-dependent functional response of predators in ecological models. Those models often provide a satisfactory explanation of the observed patterns of dynamics which cannot be done based on the ‘classical’ models using the prey-dependent functional response. Surprisingly enough, all theoretical analysis of ratio-dependant predator–prey interactions has so far been completed only for the simplest case where the prey growth is logistic. In a large number of ecologically relevant situations, however, the growth rate of a population is subject to an Allee Effect and the per capita growth rate increases with population density. Taking into account Allee dynamics for the prey growth in models can alter the previous theoretical findings obtained for the logistic growth paradigm. In this paper, we analyse a ratio-dependent predator–prey system with prey growth subject to an Allee Effect. We both consider the cases of a strong Allee Effect (the population growth rate is negative at low species density) and the case of a weak Allee Effect (the population growth is positive at low population density). For both cases we fulfil a comprehensive bifurcation analysis, constructing the parametric diagrams, and also show possible phase portraits. Then we compare the properties of the ratio-dependant predator–prey model with and without the Allee Effect and show a substantial difference in the dynamical behaviour of those systems. We show that including an Allee Effect in a ratio-dependent predator–prey model removes the possibility of sustainable oscillations of species densities (population cycles). We show that the ratio-dependent predator–prey model with the Allee Effect can solve the paradox of enrichment. However, unlike the same model with logistic growth, incorporating the Allee Effect results in a paradox of biological control.
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Bifurcations and chaos in a predator-prey system with the Allee Effect.
Proceedings. Biological sciences, 2004Co-Authors: Andrew Morozov, Sergei PetrovskiiAbstract:It is known from many theoretical studies that ecological chaos may have numerous significant impacts on the population and community dynamics. Therefore, identification of the factors potentially enhancing or suppressing chaos is a challenging problem. In this paper, we show that chaos can be enhanced by the Allee Effect. More specifically, we show by means of computer simulations that in a time-continuous predator–prey system with the Allee Effect the temporal population oscillations can become chaotic even when the spatial distribution of the species remains regular. By contrast, in a similar system without the Allee Effect, regular species distribution corresponds to periodic/quasi-periodic oscillations. We investigate the routes to chaos and show that in the spatially regular predator–prey system with the Allee Effect, chaos appears as a result of series of period-doubling bifurcations. We also show that this system exhibits periodlocking behaviour: a small variation of parameters can lead to alternating regular and chaotic dynamics.