Global dynamics of the Lotka–Volterra competition–diffusion system with equal amount of total resources, III

• Published in: Calculus of variations and partial differential equations 2017-10, Vol.56 (5), p.1-26, Article 132
• Main Authors: He, Xiaoqing, Ni, Wei-Ming
• Format: Article
• Language: English
• Subjects: Analysis; Calculus of Variations and Optimal Control; Optimization; Competition; Control; Diffusers; Diffusion effects; Dispersion; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Species diffusion; Systems Theory; Theoretical
• ISSN: 0944-2669; 1432-0835
• DOI: 10.1007/s00526-017-1234-5

In this paper—Part III of this series of three papers, we continue to investigate the joint effects of diffusion and spatial concentration on the global dynamics of the classical Lotka–Volterra competition–diffusion system. To further illustrate the general results obtained in Part I (He and Ni in Commun Pure Appl Math 69:981–1014, 2016 . doi: 10.1002/cpa.21596 ), we have focused on the case when the two competing species have identical competition abilities and the same amount of total resources. In contrast to Part II (He and Ni in Calc Var Partial Differ Equ 2016 . doi: 10.1007/s00526-016-0964-0 ), our results here show that in case both species have spatially heterogeneous distributions of resources, the outcome of the competition is independent of initial values but depends solely on the dispersal rates, which in turn depends on the distribution profiles of the resources—thereby extending the celebrated phenomenon “slower diffuser always prevails!” Furthermore, the species with a “sharper” spatial concentration in its distribution of resources seems to have the edge of competition advantage. Limiting behaviors of the globally asymptotically stable steady states are also obtained under various circumstances in terms of dispersal rates.