Reliable approximation of separatrix manifolds in competition models with safety niches

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.

Keywords:

• population theory
• dynamical systems
• competition population models
• competitive exclusion
• scattered data interpolation
• Partition of Unity method
• Wendland's functions

2010 AMS Subject Classifications:

• 65D05
• 65D17
• 92D25

Acknowledgements

The authors gratefully acknowledge the financial support of the Department of Mathematics ‘G. Peano’, University of Torino, project ‘Numerical analysis for life sciences’ (2012).