Detecting minimum energy states and multi-stability in nonlocal advection-diffusion models for interacting species

TL;DR

This paper develops methods to analyze local minimum energy states in multi-species nonlocal advection-diffusion models, revealing rich multi-stability and pattern formation in ecological systems.

Contribution

It introduces a novel approach combining energy functionals and computational algebraic geometry to predict and verify multiple stable spatial patterns in multi-species models.

Findings

Multi-stability with up to four local minima identified.

Spatial patterns include segregation, aggregation, and spike solutions.

Energy functional decreases over time, indicating steady states.

Abstract

Deriving emergent patterns from models of biological processes is a core concern of mathematical biology. In the context of partial differential equations (PDEs), these emergent patterns sometimes appear as local minimisers of a corresponding energy functional. Here we give methods for determining the qualitative structure of local minimum energy states of a broad class of multi-species nonlocal advection-diffusion models, recently proposed for modelling the spatial structure of ecosystems. We show that when each pair of species respond to one another in a symmetric fashion (i.e. via mutual avoidance or mutual attraction, with equal strength), the system admits an energy functional that decreases in time and is bounded below. This suggests that the system will eventually reach a local minimum energy steady state, rather than fluctuating in perpetuity. We leverage this energy functional to develop tools, including a novel application of computational algebraic geometry, for making conjectures about the number and qualitative structure of local minimum energy solutions. These conjectures give a guide as to where to look for numerical steady state solutions, which we verify through numerical analysis. Our technique shows that even with two species, multi-stability with up to four classes of local minimum energy state can emerge. The associated dynamics include spatial sorting via aggregation and repulsion both within and between species. The emerging spatial patterns include a mixture of territory-like segregation as well as narrow spike-type solutions. Overall, our study reveals a general picture of rich multi-stability in systems of moving and interacting species.