This paper deals with the stability analysis to a three-species food chain model with crossdiffusion, the results of which show that there is no Turing instability but crossdiffusion makes the model instability possib...This paper deals with the stability analysis to a three-species food chain model with crossdiffusion, the results of which show that there is no Turing instability but crossdiffusion makes the model instability possible. We then show that the spatial patterns are spotted patterns by using numerical simulations. In order to understand why the spatial patterns happen, the existence of the nonhomogeneous steady states is investigated. Finally, using the Leray-Schauder theory, we demonstrate that cross-diffusion creates nonhomogeneous stationary patterns.展开更多
A competitive LotkaVolterra reactiondiffusion system with two delays subject to Neumann boundary conditions is considered. It is well known that the positive con stant steady state of the system is globally asymptotic...A competitive LotkaVolterra reactiondiffusion system with two delays subject to Neumann boundary conditions is considered. It is well known that the positive con stant steady state of the system is globally asymptotically stable if the interspecies competition is weaker than the intraspecies one and is unstable if the interspecies com petition dominates over the intraspecies one. If the latter holds, then we show that Hopf bifurcation can occur as the parameters （delays） in the system cross some critical val ues. In particular, we prove that these Hopf bifurcations are all spatially homogeneous if the diffusive rates are suitably large, which has the same properties as Hopf bifur cation of the corresponding delayed system without diffusion. However, if the diffusive rates are suitably small, then the system generates the spatially nonhomogeneous Hopf bifurcation. Furthermore, we derive conditions for determining the direction of spatially nonhomogeneous Hopf bifurcations and the stability of bifurcating periodic solutions. These results indicate that the diffusion plays an important role for deriving the complex spatiotemporal dynamics.展开更多
文摘This paper deals with the stability analysis to a three-species food chain model with crossdiffusion, the results of which show that there is no Turing instability but crossdiffusion makes the model instability possible. We then show that the spatial patterns are spotted patterns by using numerical simulations. In order to understand why the spatial patterns happen, the existence of the nonhomogeneous steady states is investigated. Finally, using the Leray-Schauder theory, we demonstrate that cross-diffusion creates nonhomogeneous stationary patterns.
文摘A competitive LotkaVolterra reactiondiffusion system with two delays subject to Neumann boundary conditions is considered. It is well known that the positive con stant steady state of the system is globally asymptotically stable if the interspecies competition is weaker than the intraspecies one and is unstable if the interspecies com petition dominates over the intraspecies one. If the latter holds, then we show that Hopf bifurcation can occur as the parameters （delays） in the system cross some critical val ues. In particular, we prove that these Hopf bifurcations are all spatially homogeneous if the diffusive rates are suitably large, which has the same properties as Hopf bifur cation of the corresponding delayed system without diffusion. However, if the diffusive rates are suitably small, then the system generates the spatially nonhomogeneous Hopf bifurcation. Furthermore, we derive conditions for determining the direction of spatially nonhomogeneous Hopf bifurcations and the stability of bifurcating periodic solutions. These results indicate that the diffusion plays an important role for deriving the complex spatiotemporal dynamics.
