Stationary and non-stationary pattern formation over fragmented habitat

TL;DR

This paper investigates how the shape and size of fragmented habitats influence the formation of stationary and dynamic patterns in prey-predator models, combining analytical conditions with numerical simulations.

Contribution

It introduces a prey-predator model on complex habitats with multiple patches, deriving conditions for pattern existence and analyzing habitat shape effects.

Findings

Habitat shape and size significantly affect pattern formation.

Analytical conditions predict stationary pattern existence.

Numerical simulations reveal transient pattern dynamics.

Abstract

Spatio-temporal pattern formation over the square and rectangular domain has received significant attention from researchers. A wide range of stationary and non-stationary patterns produced by two interacting populations is abundant in the literature. Fragmented habitats are widespread in reality due to the irregularity of the landscape. This work considers a prey-predator model capable of producing a wide range of stationary and time-varying patterns over a complex habitat. The complex habitat is assumed to have consisted of two rectangular patches connected through a corridor. Our main aim is to explain how the shape and size of the fragmented habitat regulate the spatio-temporal pattern formation at the initial time. The analytical conditions are derived to ensure the existence of a stationary pattern and illustrate the role of most unstable eigenmodes to determine the number of patches for the stationary pattern. Exhaustive numerical simulations help to explain the spatial domain's size and shape on the transient patterns and the duration of transient states.