An efficient method to reconstruct invariant manifolds of saddle points
In vector field analysis, saddle points have two different types of invariant manifolds, namely stable ones and unstable ones. The invariant manifolds represent separatrices that partition the domain of trajectories into invariant regions of different dynamics. In this work, we analyze the basins of attraction of two different stable nodes by reconstructing the separatrices of a saddle point. To this purpose we present a computational algorithm that detects the points lying on the manifold, considering the plane generated by the two stable eigenvectors of the saddle point. Finally we reconstruct the surface by using the moving least-squares approximant method.
On basins of attraction for a predator-prey model via meshless approximation
Abstract. In this work an epidemiological predator-prey model is studied. It analyzes the spread of an infectious disease with frequency-dependent and vertical transmission within the predator population. In particular we consider social predators, i.e. they cooperate in groups to hunt. The result is a three-dimensional system in which the predator population is divided into susceptible and infected individuals. Studying the dynamical system and bifurcation diagrams, a scenario was identified in which the model shows multistability but the domain of attraction of one equilibrium point can be so small that it is almost the point itself. From a biological point of view it is important to anal…
Separatrix reconstruction to identify tipping points in an eco-epidemiological model
Many ecological systems exhibit tipping points such that they suddenly shift from one state to another. These shifts can be devastating from an ecological point of view, and additionally have severe implications for the socio-economic system. They can be caused by overcritical perturbations of the state variables such as external shocks, disease emergence, or species removal. It is therefore important to be able to quantify the tipping points. Here we present a study of the tipping points by considering the basins of attraction of the stable equilibrium points. We address the question of finding the tipping points that lie on the separatrix surface, which partitions the space of system traj…
Diseased Social Predators
Social predators benefit from cooperation in the form of increased hunting success, but may be at higher risk of disease infection due to living in groups. Here, we use mathematical modeling to investigate the impact of disease transmission on the population dynamics benefits provided by group hunting. We consider a predator-prey model with foraging facilitation that can induce strong Allee effects in the predators. We extend this model by an infectious disease spreading horizontally and vertically in the predator population. The model is a system of three nonlinear differential equations. We analyze the equilibrium points and their stability as well as one- and two-parameter bifurcations. …