2021/03/22 by M. S. Thorne, Thorne, Michael
Biochemistry, Genetics and Molecular Biology · Environmental Science · Medicine · #93D15 #Classical Analysis and ODEs (math.CA) #Ecosystem dynamics and resilience #Evolution and Genetic Dynamics #FOS: Biological sciences #FOS: Mathematics #Mathematical and Theoretical Epidemiology and Ecology Models #Populations and Evolution (q-bio.PE)
paper · pdf · doi:10.48550/arxiv.2103.12173
openalex publication_date 2021/03/22 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
Ecological systems are studied using many different approaches and mathematical tools. One approach, based on the Jacobian of Lotka-Volterra type models, has been a staple of mathematical ecology for years, leading to many ideas such as on questions of system stability. Instability in such methods is determined by the presence of an eigenvalue of the community matrix lying in the right half plane. The coefficients of the characteristic polynomial derived from community matrices contain information related to the specific matrix elements that play a greater destabilising role. Yet the destabilising circuits, or cycles, constructed by multiplying these elements together, form only a subset of all the feedback loops comprising a given system. This paper looks at the destabilising feedback loops in predator-prey, mutualistic and competitive systems in terms of sets of the matrix elements to explore how sign structure affects how the elements contribute to instability. This leads to quite rich combinatorial structure among the destabilising cycle sets as set size grows within the coefficients of the characteristic polynomial.