The controllable transition between Turing and antispiral patterns is studied by using a time-delayed-feedback strategy in a FitzHugh-Nagumo model. We treat the time delay as a perturbation and analyse the effect of t...The controllable transition between Turing and antispiral patterns is studied by using a time-delayed-feedback strategy in a FitzHugh-Nagumo model. We treat the time delay as a perturbation and analyse the effect of the time delay on the Turing and Hopf instabilities near the Turing Hopf codimension-two phase space. Numerical simulations show that the transition between the Turing patterns （hexagon, stripe, and honeycomb）, the dual-mode antispiral, and the antispiral by applying appropriate feedback parameters. The dual-mode antispiral pattern originates from the competition between the Turing and Hopf instabilities. Our results have shown the flexibility of the time delay on controlling the pattern formations near the Turing-Hopf codimension-two phase space.展开更多
This study considers a model which incorporates delays,diffusion and toxicity in a phytoplankton-zooplankton system.Initially,we analyze the global existence,asymptotic behavior and persistence of the solution.We then...This study considers a model which incorporates delays,diffusion and toxicity in a phytoplankton-zooplankton system.Initially,we analyze the global existence,asymptotic behavior and persistence of the solution.We then analyze the equilibria's local stability and investigate the non-delayed system's bifurcation phenomena,including Turing and Hopf bifurcations and their combination.Subsequently,we explore the effects of delays on bifurcation and the global stability of the system using Lyapunov functional,focusing on Hopf and Turing-Hopf bifurcations.Finally,we present numerical simulations to validate the theoretical results and verify the emergence of various spatial patterns in the system.展开更多
In this paper,we consider a Leslie-Gower type reaction-diffusion predator-prey system with an increasing functional response.We mainly study the effect of three different types of diffusion on the stability of this sy...In this paper,we consider a Leslie-Gower type reaction-diffusion predator-prey system with an increasing functional response.We mainly study the effect of three different types of diffusion on the stability of this system.The main results are as follows:(1)in the absence of prey diffusion,diffusion-driven instability can occur;(2)in the absence of predator diffusion,diffusion-driven instability does not occur and the non-constant stationary solution exists and is unstable;(3)in the presence of both prey diffusion and predator diffusion,the system can occur diffusion-driven instability and Turing patterns.At the same time,we also get the existence conditions of the Hopf bifurcation and the Turing-Hopf bifurcation,along with the normal form for the Turing-Hopf bifurcation.In addition,we conduct numerical simulations for all three cases to support the results of our theoretical analysis.展开更多
This is an Open Access article published by World Scientific Publishing Company.It is distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0(CC BY-NC-ND)License,which permits u...This is an Open Access article published by World Scientific Publishing Company.It is distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0(CC BY-NC-ND)License,which permits use,distribution and reproduction,provided that the original work is properly cited,the use is non-commercial and no modifications or adaptations are made.The cognitive abilities of animals,such as memory,have a significant impact on their movement in space.In this paper,we consider a radio-dependent model with memorybased diffusion under the conditions of Neumann boundary.The stability of a positive equilibrium and the existence of the Turing-Hopf bifurcation induced by memory diffusion and memory delay are carried out in details.Notably,our findings indicate that with a relatively short average memory period,the large memory diffusion can stabilize an otherwise unstable equilibrium.In addition,the third-order truncated normal form for the Turing-Hopf bifurcation restricted to the central manifold is derived,which can reveal the generation of some steady-state and time-periodic solutions with spatial heterogeneity.The coefficients within the normal form are systematically determined through matrix operations,and these results can also be applied to other models with memory diffusion.Ultimately,leveraging the theoretical findings,we elucidated the intricate spatial-temporal dynamics and their associated parameter scopes caused by Turing-Hopf bifurcation through numerical simulations.展开更多
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 10975043 and 10947166)the Natural Science Foundation of Hebei Province,China (Grant Nos. A2011201006 and A2010000185)the Science Foundation of Hebei University
文摘The controllable transition between Turing and antispiral patterns is studied by using a time-delayed-feedback strategy in a FitzHugh-Nagumo model. We treat the time delay as a perturbation and analyse the effect of the time delay on the Turing and Hopf instabilities near the Turing Hopf codimension-two phase space. Numerical simulations show that the transition between the Turing patterns （hexagon, stripe, and honeycomb）, the dual-mode antispiral, and the antispiral by applying appropriate feedback parameters. The dual-mode antispiral pattern originates from the competition between the Turing and Hopf instabilities. Our results have shown the flexibility of the time delay on controlling the pattern formations near the Turing-Hopf codimension-two phase space.
基金supported by the National Social Science Fund Youth Project of China[Grant 21CJY040].
文摘This study considers a model which incorporates delays,diffusion and toxicity in a phytoplankton-zooplankton system.Initially,we analyze the global existence,asymptotic behavior and persistence of the solution.We then analyze the equilibria's local stability and investigate the non-delayed system's bifurcation phenomena,including Turing and Hopf bifurcations and their combination.Subsequently,we explore the effects of delays on bifurcation and the global stability of the system using Lyapunov functional,focusing on Hopf and Turing-Hopf bifurcations.Finally,we present numerical simulations to validate the theoretical results and verify the emergence of various spatial patterns in the system.
文摘In this paper,we consider a Leslie-Gower type reaction-diffusion predator-prey system with an increasing functional response.We mainly study the effect of three different types of diffusion on the stability of this system.The main results are as follows:(1)in the absence of prey diffusion,diffusion-driven instability can occur;(2)in the absence of predator diffusion,diffusion-driven instability does not occur and the non-constant stationary solution exists and is unstable;(3)in the presence of both prey diffusion and predator diffusion,the system can occur diffusion-driven instability and Turing patterns.At the same time,we also get the existence conditions of the Hopf bifurcation and the Turing-Hopf bifurcation,along with the normal form for the Turing-Hopf bifurcation.In addition,we conduct numerical simulations for all three cases to support the results of our theoretical analysis.
基金supported by National Natural Science Foundation of China 12101318Natural Science Found of Jiangsu Province BK20200805 and Natural Science Fund Project of Colleges in Jiangsu Province 20KJB110009.
文摘This is an Open Access article published by World Scientific Publishing Company.It is distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0(CC BY-NC-ND)License,which permits use,distribution and reproduction,provided that the original work is properly cited,the use is non-commercial and no modifications or adaptations are made.The cognitive abilities of animals,such as memory,have a significant impact on their movement in space.In this paper,we consider a radio-dependent model with memorybased diffusion under the conditions of Neumann boundary.The stability of a positive equilibrium and the existence of the Turing-Hopf bifurcation induced by memory diffusion and memory delay are carried out in details.Notably,our findings indicate that with a relatively short average memory period,the large memory diffusion can stabilize an otherwise unstable equilibrium.In addition,the third-order truncated normal form for the Turing-Hopf bifurcation restricted to the central manifold is derived,which can reveal the generation of some steady-state and time-periodic solutions with spatial heterogeneity.The coefficients within the normal form are systematically determined through matrix operations,and these results can also be applied to other models with memory diffusion.Ultimately,leveraging the theoretical findings,we elucidated the intricate spatial-temporal dynamics and their associated parameter scopes caused by Turing-Hopf bifurcation through numerical simulations.
