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題名 Wave propagation in predator-prey systems 作者 Fu, Sheng-Chen
符聖珍
Tsai, Je-Chiang貢獻者 應用數學系 日期 2015-11 上傳時間 7-Aug-2017 17:45:21 (UTC+8) 摘要 In this paper, we study a class of predator-prey systems of reaction-diffusion type. Specifically, we are interested in the dynamical behaviour for the solution with the initial distribution where the prey species is at the level of the carrying capacity, and the density of the predator species has compact support, or exponentially small tails near = ± ∞ x . Numerical evidence suggests that this will lead to the formation of a pair of diverging waves propagating outwards from the initial zone. Motivated by this phenomenon, we establish the existence of a family of travelling waves with the minimum speed. Unlike the previous studies, we do not use the shooting argument to show this. Instead, we apply an iteration process based on Berestycki et al 2005 (Math Comput. Modelling 50 1385-93) to construct a set of super/sub-solutions. Since the underlying system does not enjoy the comparison principle, such a set of super/sub-solutions is not based on travelling waves, and in fact the super/sub-solutions depend on each other. With the aid of the set of super/ sub-solutions, we can construct the solution of the truncated problem on the finite interval, which, via the limiting argument, can in turn generate the wave solution. There are several advantages to this approach. First, it can remove the technical assumptions on the diffusivities of the species in the existing literature. Second, this approach is of PDE type, and hence it can shed some light on the spreading phenomenon indicated by numerical simulation. In fact, we can compute the spreading speed of the predator species for a class of biologically acceptable initial distributions. Third, this approach might be applied to the study of waves in non-cooperative systems (i.e. a system without a comparison principle). © 2015 IOP Publishing Ltd & London Mathematical Society Printed in the UK. 關聯 Nonlinearity, 28(12), 4389-4423 資料類型 article DOI http://dx.doi.org/10.1088/0951-7715/28/12/4389 dc.contributor 應用數學系 zh_Tw dc.creator (作者) Fu, Sheng-Chen en_US dc.creator (作者) 符聖珍 zh_TW dc.creator (作者) Tsai, Je-Chiang en_US dc.date (日期) 2015-11 en_US dc.date.accessioned 7-Aug-2017 17:45:21 (UTC+8) - dc.date.available 7-Aug-2017 17:45:21 (UTC+8) - dc.date.issued (上傳時間) 7-Aug-2017 17:45:21 (UTC+8) - dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/111657 - dc.description.abstract (摘要) In this paper, we study a class of predator-prey systems of reaction-diffusion type. Specifically, we are interested in the dynamical behaviour for the solution with the initial distribution where the prey species is at the level of the carrying capacity, and the density of the predator species has compact support, or exponentially small tails near = ± ∞ x . Numerical evidence suggests that this will lead to the formation of a pair of diverging waves propagating outwards from the initial zone. Motivated by this phenomenon, we establish the existence of a family of travelling waves with the minimum speed. Unlike the previous studies, we do not use the shooting argument to show this. Instead, we apply an iteration process based on Berestycki et al 2005 (Math Comput. Modelling 50 1385-93) to construct a set of super/sub-solutions. Since the underlying system does not enjoy the comparison principle, such a set of super/sub-solutions is not based on travelling waves, and in fact the super/sub-solutions depend on each other. With the aid of the set of super/ sub-solutions, we can construct the solution of the truncated problem on the finite interval, which, via the limiting argument, can in turn generate the wave solution. There are several advantages to this approach. First, it can remove the technical assumptions on the diffusivities of the species in the existing literature. Second, this approach is of PDE type, and hence it can shed some light on the spreading phenomenon indicated by numerical simulation. In fact, we can compute the spreading speed of the predator species for a class of biologically acceptable initial distributions. Third, this approach might be applied to the study of waves in non-cooperative systems (i.e. a system without a comparison principle). © 2015 IOP Publishing Ltd & London Mathematical Society Printed in the UK. en_US dc.format.extent 1754677 bytes - dc.format.mimetype application/pdf - dc.relation (關聯) Nonlinearity, 28(12), 4389-4423 en_US dc.title (題名) Wave propagation in predator-prey systems en_US dc.type (資料類型) article - dc.identifier.doi (DOI) 10.1088/0951-7715/28/12/4389 - dc.doi.uri (DOI) http://dx.doi.org/10.1088/0951-7715/28/12/4389 -
