  In dynamical systems saddle points partition the domain into basins of
attractions of the remaining locally stable equilibria. This problem is rather
common especially in population dynamics models. Precisely, a particular
solution of a dynamical system is completely determined by its initial
condition and by the parameters involved in the model. Furthermore, when the
omega limit set reduces to a point, the trajectory of the solution evolves
towards the steady state. But, in case of multi-stability it is possible that
several steady states originate from the same parameter set. Thus, in these
cases the importance of accurately reconstruct the attraction basins follows.
In this paper we focus on dynamical systems of ordinary differential equations
presenting three stable equilibia and we design algorithms for the detection of
the points lying on the manifolds determining the basins of attraction and for
the reconstruction of such manifolds. The latter are reconstructed by means of
the implicit partition of unity method which makes use of radial basis
functions (RBFs) as local approximants. Extensive numerical test, carried out
with a Matlab package made available to the scientific community, support our
findings.
