Reliable approximation of separatrix manifolds in competition models with safety niches

TL;DR

This paper develops algorithms to accurately detect and refine the separatrix manifolds in population competition models with niches, using the Partition of Unity method with Wendland's functions for both two and three population cases.

Contribution

It introduces a novel algorithmic approach for approximating separatrix manifolds in population models with niches, enhancing detection and reconstruction accuracy.

Findings

Effective detection of separatrix points in two and three population models.

Successful reconstruction of separatrix surfaces using the Partition of Unity method.

Improved understanding of basin boundaries in ecological competition models.

Abstract

In dynamical systems saddle points partition the domain into basins of attractions of the remaining locally stable equilibria. This situation is rather common especially in population dynamics models, like prey-predator or competition systems. Focusing on squirrels population models with niche, in this paper we design algorithms for the detection and the refinement of points lying on the separatrix manifold partitioning the phase space. We consider both the two populations and the three populations cases. To reconstruct the separatrix curve and surface, we apply the Partition of Unity method, which makes use of Wendland's functions as local approximants.