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.

Graphical Representation of Separatrices of Attraction Basins in Two and Three-Dimensional Dynamical Systems

“…It was already used in previous papers (see [3,6]), but here we considered an extension and a refinement to account for a different model. In particular, the approximation scheme has been improved as far as portability is concerned, in that now it works also for a dynamical system of dimension three with three stable equilibria.…”

“…The problem of the reconstruction of the surface separating two Eq. Stability E 0 unstable E 1 r < f u, q < cu E 2 r < vg, p < av E 3 q < ew, p < bw E 4 q > cu, p > av, r(cuva − pq) > pvg(cu − q) + uf q(va − p) E 5 p > bw, r > f u, q(f uwb − pr) > wpe(f u − r) + rcu(wb − p) E 6 q > we, r > vg, p(gvwe − rq) > bwq(vg − r) + avr(we − q) Table 1: Stability conditions for the equilibria of the system (8).…”

“…In particular, here we aim at partitioning the domain approximating the boundaries of the basins of attraction of different stable equilibria. We start from the 2D case sketched in [3] and the approximation scheme presented in [3,6], and then we extend the reconstruction scheme of separatrices in the cases of three dimensional models with two and three stable equilibria. For this purpose we construct computational algorithms and procedures for the detection and the refinement of points located on the separatrix manifolds that partition the phase space.…”

Ecoepidemiology: a More Comprehensive View of Population Interactions

“…eutrophication, it may further have a destabilizing effect, potentially leading to the ecosystem extinction; finally, in these conditions, predators can survive if their initial state lies within the domain of attraction of a suitable equilibrium, in view of a bistability phenomenon. In this respect, we mention that the accurate calculation of separatrix surfaces has been dealt with in [28]. For the following phytoplankton-zooplankton model with infection in the former, [78], with HTII grazing and standard incidence:…”

“…In (5.12) there are also persistent limit cycles, which are shown to be absent in the corresponding purely demographic case. Bistability is discovered among some equilibria, leading to situations in which the computation of their basins of attraction is relevant for the system outcome in terms of its biological implications, [28].…”