Emma Perracchione

Reliable approximation of separatrix manifolds in competition models with safety niches

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 Wendlands functions as local approximants.

Efficient computation of partition of unity interpolants through a block-based searching technique

In this paper we propose a new efficient interpolation tool, extremely suitable for large scattered data sets. The partition of unity method is used and performed by blending Radial Basis Functions (RBFs) as local approximants and using locally supported weight functions. In particular we present a new space-partitioning data structure based on a partition of the underlying generic domain in blocks. This approach allows us to examine only a reduced number of blocks in the search process of the nearest neighbour points, leading to an optimized searching routine. Complexity analysis and numerical experiments in two- and three-dimensional interpolation support our findings. Some applications to geometric modelling are also considered. Moreover, the associated software package written in Matlab is here discussed and made available to the scientific community.

Robust Approximation Algorithms for the Detection of Attraction Basins in Dynamical Systems

A particular solution of a dynamical system is completely determined by its initial condition. When the omega limit set reduces to a point, the solution settles at steady state. The possible steady states of the system are completely determined by its parameters. However, with the same parameter set, it is possible that several steady states can originate from different initial conditions (multi-stability). In that case the outcome depends on the chosen initial condition. Therefore, it is important to assess the domain of attraction for each possible attractor. The algorithms presented here are general and robust enough so as to solve the problem of reconstructing the basin of attraction of each stable equilibrium point. In order to have a graphical representation of the separatrix manifolds, we focus on systems of two and three ordinary differential equations exhibiting bi- or tri-stability. For this purpose we have implemented several Matlab functions for the approximation of the points lying on the curves or on the surfaces determining the basins of attraction and for the reconstruction of such curves and surfaces. We approximate the latter with the implicit partition of unity method using radial basis functions as local approximants. Numerical results, obtained with a Matlab package made available to the scientific community, support our findings.

Optimal Selection of Local Approximants in RBF-PU Interpolation

The partition of unity (PU) method, performed with local radial basis function (RBF) approximants, has been proved to be an effective tool for solving large scattered data interpolation problems. However, in order to achieve a good accuracy, the question about how many points we have to consider on each local subdomain, i.e. how large can be the local data sets, needs to be answered. Moreover, it is well-known that also the shape parameter affects the accuracy of the local RBF approximants and, as a consequence, of the PU interpolant. Thus here, both the shape parameter used to fit the local problems and the size of the associated linear systems are supposed to vary among the subdomains. They are selected by minimizing an a priori error estimate. As evident from extensive numerical experiments and applications provided in the paper, the proposed method turns out to be extremely accurate also when data with non-homogeneous density are considered.

Partition of unity interpolation using stable kernel-based techniques

Abstract In this paper we propose a new stable and accurate approximation technique which is extremely effective for interpolating large scattered data sets. The Partition of Unity (PU) method is performed considering Radial Basis Functions (RBFs) as local approximants and using locally supported weights. In particular, the approach consists in computing, for each PU subdomain, a stable basis. Such technique, taking advantage of the local scheme, leads to a significant benefit in terms of stability, especially for flat kernels. Furthermore, an optimized searching procedure is applied to build the local stable bases, thus rendering the method more efficient.

Partition of unity interpolation on multivariate convex domains

In this paper, we present an algorithm for multivariate interpolation of scattered data sets lying in convex domains Ω ⊆ ℝN, for any N ≥ 2. To organize the points in a multidimensional space, we build a kd-tree space-partitioning data structure, which is used to efficiently apply a partition of unity interpolant. This global scheme is combined with local radial basis function (RBF) approximants and compactly supported weight functions. A detailed description of the algorithm for convex domains and a complexity analysis of the computational procedures are also considered. Several numerical experiments show the performances of the interpolation algorithm on various sets of Halton data points contained in Ω, where Ω can be any convex domain, like a 2D polygon or a 3D polyhedron. Finally, an application to topographical data contained in a pentagonal domain is presented.

Wild herbivores in forests: four case studies

A three population system with a top predator population, i.e. the herbivores, and two prey populations, grass and trees, is considered to model the interaction of herbivores with natural resources. We apply the model for four natural mountain parks in Northern Italy, three located in the Eastern Alps, two of which in the Dolomites and one in the Julian Alps, and one in the Maritime Alps, Northwest Italy. The simulations, based on actual data gathered from contacts with rangers and parks administrators, field samplings and published material, provide useful information on the behavior of the vegetation-wild herbivores interactions and the possible medium-long term evolution of these ecosystems. At the same time they show that these ecosystems are in a very delicate situation, for which the animal populations could become extinguished in case of adverse environmental conditions. The determination of the so called sensitivity surfaces support our findings and indicate some possible preventive measures to the park admistrators.

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

In this paper we consider the problem of reconstructing separatrices in dynamical systems. 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 \cite{cavoretto11} and the approximation scheme presented in \cite{cavoretto11,C-D-P-V}, 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. The use of the so-called meshfree or meshless methods is used to reconstruct the separatrices.

Positive constrained approximation via RBF-based partition of unity method

In this paper, we discuss the problem of constructing Radial Basis Function (RBF)-based Partition of Unity (PU) interpolants that are positive if data values are positive. More specifically, we compute positive local approximants by adding up several constraints to the interpolation conditions. This approach, considering a global approximation problem and Compactly Supported RBFs (CSRBFs), has been previously proposed in Wu etal. (2010). Here, the use of the PU technique enables us to intervene only locally and as a consequence to reach a better accuracy. This is also due to the fact that we select the optimal number of positive constraints by means of an a priori error estimate and we do not restrict to the use of CSRBFs. Numerical experiments and applications to population dynamics are provided to illustrate the effectiveness of the method in applied sciences.

Competition models with niche for squirrel population dynamics

In this paper we investigate squirrel competition models. More precisely, at first we consider a competition model between red native and grey exotic squirrels, then a model with competition among red native, red indigenous and grey exotic squirrels. We assume that a part of red squirrels can hide in a niche. By adding this hypothesis, we analize if, independently from initial conditions, the grey exotic squirrel population could be prevented from invading the ecosystem and displacing the native populations.