arXiv subjects

Michael Thorne

Publications and source records attributed to Michael Thorne.

4 recordsLinked to original sources

arXiv2026

Will a Large Complex System be Stable? Revisited

Over fifty years ago, Robert May applied random matrix theory to show that as ecological systems grow in size, stability decreases. What emerged from this and the critique that followed was decades of what has been called the complexity-stability debate. However, decades of critique over the assumptions that Robert May applied in carrying out his analysis have not been enough to fully dispel the strength of his conclusion and close the debate. Drawing on a mathematical approach that had not yet been fully developed in the early 70s, it is possible to revisit the argument without the use of random matrix techniques, and provide more detailed understanding of the mechanisms that play a deciding role in stability of ecological systems, countering the broad conclusion that led to the complexity-stability debate.

q-bio.PE↗

arXiv2024

Diagonally forced systems and the spectral signature of matrix cycles

Given any square matrix, $\mathbf{M}$, whose diagonal elements are negative, and which is multiplied by a variable, $\sigma$, we wish to find the minimal $\sigma$ such that the eigenvalue of $\mathbf{M}_{\sigma}$ is exactly zero. By Gershgorin, we know that $\mathbf{M}_{\sigma}$ can be made stable by making $\sigma$ large enough. We prove a relation which analytically determines when and how we are able to find the value of $\sigma$ such that the maximal eigenvalue is exactly zero. In so doing, we prove the equivalence of the roots of the characteristic polynomial of $\mathbf{M}_{\sigma}$ and the eigenvalues that arise from a scaling operation on $\mathbf{M}$. Further, through the characteristic polynomial, we are able to isolate the dominant feedback cycles comprising the elements of the matrix which, under the action of $\sigma$, ensures the stability of the system. We then explore, using the stabilising and destabilising cycles within the coefficients of the characteristic polynomial, an intrinsic spectral signature associated with any matrix based on the size and sign (or zero) of its respective elements.

math.OC↗

arXiv2021

Tipping Cycles

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.

q-bio.PE↗

arXiv2020

$σ$-Stable Matrices

$σ$-Stable matrices are introduced and it is shown that the real roots of the polynomials comprising the coefficients of the characteristic polynomial indicate the coefficient sign changes. A proof of Obrechkoff is then used to show that the largest real root from the coefficients is the point of stability when the maximal eigenvalue of the $σ$-stable matrix is in $\mathbb{R}$. Some implications of the coefficient behaviour for a scaling relation are then discussed.

math.CA↗