Stationary solutions and spatial-temporal dynamics of a shadow system of LV competition models

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• 作者：Yuanyuan Zhang ; Li Xia
• 关键词：steady state ; spatial ; temporal dynamics ; Lotka ; Volterra competition system ; shadow system
• 刊名：Advances in Difference Equations
• 出版年：2017
• 出版时间：December 2017
• 年：2017
• 卷：2017
• 期：1
• 全文大小：2909KB
• 刊物主题：Difference and Functional Equations; Mathematics, general; Analysis; Functional Analysis; Ordinary Differential Equations; Partial Differential Equations;
• 出版者：Springer International Publishing
• ISSN：1687-1847
• 卷排序：2017

文摘

The concern of the paper is nonconstant positive solutions of a class of Lotka-Volterra competition systems over 1D domains. We prove the existence of a positive monotonous solution to the shadow system for each small diffusion rate \(\epsilon >0\). Our theoretical results provide a foundation for further theoretical analysis on the shadow system and give insights on how diffusion and advection rates affect the pattern formation in the advective Lotka-Volterra competition systems. The second part of this paper includes numerical simulations of the nontrivial patterns to the shadow system and its original model. It is demonstrated that nontrivial patterns can develop from small perturbations of the homogeneous solution. Our numerics suggest that this system admits very interesting and complicated spatial-temporal dynamics even over 1D domains.