Wave propagation in predator-prey systems

Sheng Chen Fu, Je-Chiang Tsai

研究成果: 雜誌貢獻期刊論文同行評審

16 引文 斯高帕斯(Scopus)

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).

原文英語
文章編號4389
頁(從 - 到)4389-4423
頁數35
期刊Nonlinearity
28
發行號12
DOIs
出版狀態已發佈 - 2015 十一月 13

ASJC Scopus subject areas

  • 統計與非線性物理學
  • 數學物理學
  • 物理與天文學 (全部)
  • 應用數學

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