The Hopfbifurcation for the Brusselator ordinary-differential-equation （ODE） model and the corresponding partial-differential-equation （PDE） model are investigated by using the Hopf bifurcation theorem. The stabil...The Hopfbifurcation for the Brusselator ordinary-differential-equation （ODE） model and the corresponding partial-differential-equation （PDE） model are investigated by using the Hopf bifurcation theorem. The stability of the Hopf bifurcation periodic solution is discussed by applying the normal form theory and the center manifold theorem. When parameters satisfy some conditions, the spatial homogenous equilibrium solution and the spatial homogenous periodic solution become unstable. Our results show that if parameters are properly chosen, Hopf bifurcation does not occur for the ODE system, but occurs for the PDE system.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
We are concerned with a reaction-diffusion predator–prey model under homogeneous Neumann boundary condition incorporating prey refuge(proportion of both the species)and harvesting of prey species in this contribution...We are concerned with a reaction-diffusion predator–prey model under homogeneous Neumann boundary condition incorporating prey refuge(proportion of both the species)and harvesting of prey species in this contribution.Criteria for asymptotic stability(local and global)and bifurcation of the subsequent temporal model system are thoroughly analyzed around the unique positive interior equilibrium point.For partial differential equation(PDE),the conditions of diffusion-driven instability and the Turing bifurcation region in two-parameter space are investigated.The results around the unique interior feasible equilibrium point specify that the effect of refuge and harvesting cooperation is an important part of the control of spatial pattern formation of the species.A series of computer simulations reveal that the typical dynamics of population density variation are the formation of isolated groups within the Turing space,that is,spots,stripe-spot mixtures,labyrinthine,holes,stripe-hole mixtures and stripes replication.Finally,we discuss spatiotemporal dynamics of the system for a number of different momentous parameters via numerical simulations.展开更多
In this paper, spatial patterns of a diffusive predator-prey model with sigmoid （Holling type III） ratio-dependent functional response which concerns the influence of logistic population growth in prey and intra-spe...In this paper, spatial patterns of a diffusive predator-prey model with sigmoid （Holling type III） ratio-dependent functional response which concerns the influence of logistic population growth in prey and intra-species competition among predators are investigated. The （local and global） asymptotic stability behavior of the corresponding non- spatial model around the unique positive interior equilibrium point in homogeneous steady state is obtained. In addition, we derive the conditions for Turing instability and the consequent parametric Turing space in spatial domain. The results of spatial pat- tern analysis through numerical simulations are depicted and analyzed. ~rthermore, we perform a series of numerical simulations and find that the proposed model dynamics exhibits complex pattern replication. The feasible results obtained in this paper indicate that the effect of diffusion in Turing instability plays an important role to understand better the pattern formation in ecosystem.展开更多
基金Project supported by the National Natural Science Foundation of China (No.10771032)the Natural Science Foundation of Jiangsu Province (BK2006088)
文摘The Hopfbifurcation for the Brusselator ordinary-differential-equation （ODE） model and the corresponding partial-differential-equation （PDE） model are investigated by using the Hopf bifurcation theorem. The stability of the Hopf bifurcation periodic solution is discussed by applying the normal form theory and the center manifold theorem. When parameters satisfy some conditions, the spatial homogenous equilibrium solution and the spatial homogenous periodic solution become unstable. Our results show that if parameters are properly chosen, Hopf bifurcation does not occur for the ODE system, but occurs for the PDE system.
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the financial support in part from Special Assistance Programme(SAP-III)sponsored by the University Grants Commission(UGC),New Delhi,India(Grant No.F.510/3/DRS-III/2015(SAP-I)).Dr.S.Djilali is partially supported by the DGRSDT of Algeria.
文摘We are concerned with a reaction-diffusion predator–prey model under homogeneous Neumann boundary condition incorporating prey refuge(proportion of both the species)and harvesting of prey species in this contribution.Criteria for asymptotic stability(local and global)and bifurcation of the subsequent temporal model system are thoroughly analyzed around the unique positive interior equilibrium point.For partial differential equation(PDE),the conditions of diffusion-driven instability and the Turing bifurcation region in two-parameter space are investigated.The results around the unique interior feasible equilibrium point specify that the effect of refuge and harvesting cooperation is an important part of the control of spatial pattern formation of the species.A series of computer simulations reveal that the typical dynamics of population density variation are the formation of isolated groups within the Turing space,that is,spots,stripe-spot mixtures,labyrinthine,holes,stripe-hole mixtures and stripes replication.Finally,we discuss spatiotemporal dynamics of the system for a number of different momentous parameters via numerical simulations.
文摘In this paper, spatial patterns of a diffusive predator-prey model with sigmoid （Holling type III） ratio-dependent functional response which concerns the influence of logistic population growth in prey and intra-species competition among predators are investigated. The （local and global） asymptotic stability behavior of the corresponding non- spatial model around the unique positive interior equilibrium point in homogeneous steady state is obtained. In addition, we derive the conditions for Turing instability and the consequent parametric Turing space in spatial domain. The results of spatial pat- tern analysis through numerical simulations are depicted and analyzed. ~rthermore, we perform a series of numerical simulations and find that the proposed model dynamics exhibits complex pattern replication. The feasible results obtained in this paper indicate that the effect of diffusion in Turing instability plays an important role to understand better the pattern formation in ecosystem.
