Mathematical analysis of species coexistence and segregating pattern formation

2018 Fiscal Year Final Research Report

• Project/Area Number: 16H07254
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Single-year Grants
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Meiji University
• Principal Investigator: Contento Lorenzo 明治大学, 研究・知財戦略機構, 研究推進員 (50782562)
• Research Collaborator: Mimura Masayasu ; Hilhorst Danielle
• Project Period (FY): 2016-08-26 – 2018-03-31
• Keywords: mathematical modelling / competition-diffusion / pattern formation / travelling wave / competitive exclusion / species coexistence / singular limit

Outline of Final Research Achievements

The ecological invasion problem in which a weaker exotic species invades an ecosystem inhabited by two strongly competing native species can be modelled by a three-species competition-diffusion system. We have proved rigorously that when the invader is very strong it will always be able to replace the native species, while it will never survive in the new environment if it is sufficiently weak. In the intermediate cases, coexistence occurs in complex spatio-temporal patterns, such as regular or breathing spirals, periodic multi-core spiral patterns or chaotic spiral turbulence. The origin of and transition between such patterns lies in the interaction of two planarly stable fronts. By studying the bifurcation structure of their one-dimensional equivalent (travelling waves), we can also understand the mechanisms governing the two dimensional case.

Free Research Field

現象数理学

Academic Significance and Societal Importance of the Research Achievements

We have shown how complex spatio-temporal patterns can arise from the interaction of two planarly stable fronts, without the need for instability as in other reaction-diffusion models. We have given rigorous results on the system's limit behaviour even if no vector comparison principle holds.