This paper considers the Holling-Tanner model for predator-prey with self and cross-diffusion. From the Turing theory, it is believed that there is no Turing pattern formation for the equal self-diffusion coefficients...This paper considers the Holling-Tanner model for predator-prey with self and cross-diffusion. From the Turing theory, it is believed that there is no Turing pattern formation for the equal self-diffusion coefficients. However, combined with cross-diffusion, it shows that the system will exhibit spotted pattern by both mathematical analysis and numerical simulations. Furthermore, asynchrony of the predator and the prey in the space. The obtained results show that cross-diffusion plays an important role on the pattern formation of the predator-prey system.展开更多
Pattern formation of a spatial epidemic model with nonlinear incidence rate hI^2 S/（1 ＋ αI^2） is investigated. Our results show that strange spatial dynamics, i.e., filament-like pattern, can be obtained by both m...Pattern formation of a spatial epidemic model with nonlinear incidence rate hI^2 S/（1 ＋ αI^2） is investigated. Our results show that strange spatial dynamics, i.e., filament-like pattern, can be obtained by both mathematical analysis and numerical simulation, which are different from the previous results in the spatial epidemic model such as stripe-like or spotted or coexistence of both pattern and so on. The obtained results well extend the finding of pattern formation in the epidemic model and may well explain the distribution of the infected of some epidemic.展开更多
Pattern formation of a spatial epidemic model with both self- and cross-diffusion is investigated. From the Turing theory, it is well known that Turing pattern formation cannot occur for the equal self-diffusion coeff...Pattern formation of a spatial epidemic model with both self- and cross-diffusion is investigated. From the Turing theory, it is well known that Turing pattern formation cannot occur for the equal self-diffusion coefficients. However, combined with cross-diffusion, the system will show emergence of isolated groups, i.e., stripe-like or spotted or coexistence of both, which we show by both mathematical analysis and numerical simulations. Our study shows that the interaction of self- and cross-diffusion can be considered as an important mechanism for the appearance of complex spatiotemporal dynamics in epidemic models.展开更多
It has been reported that the minimal spatially extended phytoplankton-zooplankton system exhibits both temporal regular/chaotic behaviour, and spatiotemporal chaos in a patchy environment. As a further investigation ...It has been reported that the minimal spatially extended phytoplankton-zooplankton system exhibits both temporal regular/chaotic behaviour, and spatiotemporal chaos in a patchy environment. As a further investigation by means of computer simulations and theoretical analysis, in this paper we observe that the spiral waves may exist and the spatiotemporal chaos emerge when the parameters are within the mixed Turing-Hopf bifurcation region, which arises from the far-field breakup of the spiral waves over a large range of diffusion coefficients of phytoplankton and zooplankton. Moreover, the spatiotemporal chaos arising from the far-field breakup of spiral waves does not gradually invade the whole space of that region. Our results are confirmed by nonlinear bifurcation of wave trains, We also discuss ecological implications of these spatially structured patterns.展开更多
基金Project supported by the National Natural Science Foundation of China (Grant No 60771026)Program for New Century Excellent Talents in University of China (Grant No NCET050271)the Special Scientific Research Foundation for the Subjects of Doctors in University of China (Grant No 20060110005)
文摘This paper considers the Holling-Tanner model for predator-prey with self and cross-diffusion. From the Turing theory, it is believed that there is no Turing pattern formation for the equal self-diffusion coefficients. However, combined with cross-diffusion, it shows that the system will exhibit spotted pattern by both mathematical analysis and numerical simulations. Furthermore, asynchrony of the predator and the prey in the space. The obtained results show that cross-diffusion plays an important role on the pattern formation of the predator-prey system.
基金Supported by the National Natural Science Foundation of China under Grant No 60771026, the Programme for New Century Excellent Talents in University （NCET050271）, and the Special Scientific Research Foundation for the Subjects of Doctors in University （20060110005）.
文摘Pattern formation of a spatial epidemic model with nonlinear incidence rate hI^2 S/（1 ＋ αI^2） is investigated. Our results show that strange spatial dynamics, i.e., filament-like pattern, can be obtained by both mathematical analysis and numerical simulation, which are different from the previous results in the spatial epidemic model such as stripe-like or spotted or coexistence of both pattern and so on. The obtained results well extend the finding of pattern formation in the epidemic model and may well explain the distribution of the infected of some epidemic.
基金Supported by the National Natural Science Foundation of China under Grant No 60771026, the Programme for New Century Excellent Talents in University （NCET050271）, and the Special Scientific Research Foundation for the Subjects of Doctors in University （20060110005）.
文摘Pattern formation of a spatial epidemic model with both self- and cross-diffusion is investigated. From the Turing theory, it is well known that Turing pattern formation cannot occur for the equal self-diffusion coefficients. However, combined with cross-diffusion, the system will show emergence of isolated groups, i.e., stripe-like or spotted or coexistence of both, which we show by both mathematical analysis and numerical simulations. Our study shows that the interaction of self- and cross-diffusion can be considered as an important mechanism for the appearance of complex spatiotemporal dynamics in epidemic models.
基金supported by the National Natural Science Foundation of China (Grant No 60771026)the Program for New Century Excellent Talents in University (Grant No NCET050271)+2 种基金the Natural Science Foundation of Shan’xi Province, China(Grant No 2006011009)US National Science Foundation Biocomplexity Program (DEB0421530)LTER Program (Grant NoDEB0620482)
文摘It has been reported that the minimal spatially extended phytoplankton-zooplankton system exhibits both temporal regular/chaotic behaviour, and spatiotemporal chaos in a patchy environment. As a further investigation by means of computer simulations and theoretical analysis, in this paper we observe that the spiral waves may exist and the spatiotemporal chaos emerge when the parameters are within the mixed Turing-Hopf bifurcation region, which arises from the far-field breakup of the spiral waves over a large range of diffusion coefficients of phytoplankton and zooplankton. Moreover, the spatiotemporal chaos arising from the far-field breakup of spiral waves does not gradually invade the whole space of that region. Our results are confirmed by nonlinear bifurcation of wave trains, We also discuss ecological implications of these spatially structured patterns.
