# Spatial memory and taxis-driven pattern formation in model ecosystems

## TL;DR

This paper develops diffusion-taxis models incorporating memory effects to explain how animal populations form spatial patterns, revealing conditions for stationary and oscillatory pattern emergence, including chaos, on short timescales.

## Contribution

It introduces a new class of models that include memory and inter-population movement, providing a comprehensive analysis of pattern formation in multi-species systems.

## Key findings

- For two populations, only stationary patterns form asymptotically.
- For three or more populations, oscillatory and chaotic patterns can emerge.
- The models highlight the importance of movement responses in spatial distribution modeling.

## Abstract

Mathematical models of spatial population dynamics typically focus on the interplay between dispersal events and birth/death processes. However, for many animal communities, significant arrangement in space can occur on shorter timescales, where births and deaths are negligible. This phenomenon is particularly prevalent in populations of larger, vertebrate animals who often reproduce only once per year or less. To understand spatial arrangements of animal communities on such timescales, we use a class of diffusion-taxis equations for modelling inter-population movement responses between $N \geq 2$ populations. These systems of equations incorporate the effect on animal movement of both the current presence of other populations and the memory of past presence encoded either in the environment or in the minds of animals. We give general criteria for the spontaneous formation of both stationary and oscillatory patterns, via linear pattern formation analysis. For $N=2$, we classify completely the pattern formation properties using a combination of linear analysis and non-linear energy functionals. In this case, the only patterns that can occur asymptotically in time are stationary. However, for $N \geq 3$, oscillatory patterns can occur asymptotically, giving rise to a sequence of period-doubling bifurcations leading to patterns with no obvious regularity, a hallmark of chaos. Our study highlights the importance of understanding between-population animal movement for understanding spatial species distributions, something that is typically ignored in species distribution modelling, and so develops a new paradigm for spatial population dynamics.