Population models with singular equilibrium

Abstract

A class of models of biological population and communities with a singular equilibrium at the origin is analyzed; it is shown that these models can possess a dynamical regime of deterministic extinction, which is crucially important from the biological standpoint. This regime corresponds to the presence of a family of homoclinics to the origin, so-called elliptic sector. The complete analysis of possible topological structures in a neighborhood of the origin, as well as asymptotics to orbits tending to this point, is given. An algorithmic approach to analyze system behavior with parameter changes is presented. The developed methods and algorithm are applied to existing mathematical models of biological systems. In particular, we analyze a model of anticancer treatment with oncolytic viruses, a parasite-host interaction model, and a model of Chagas' disease.

Fatal disease and demographic Allee effect: population persistence and extinction

“…Using a similar SI model with the strong Allee effect, Thieme et al [16] established by mathematical theorems that the transition from population decline to population collapse is mediated by a Hopf bifurcation and a heteroclinic orbit. The SI models of Hilker et al and Thieme et al are structurally similar to predator-prey models that have an Allee effect in the prey population and a linear functional response (prey eaten per predator per unit time) [2,3,6,10,[12][13][14]17,18]. However, most of these studies on the interplay ofAllee effects and infectious diseases, especially those in the conservation biology literature, seem to be largely concerned with the role of the Allee effect at small population densities [9].…”

Dynamics of a Predator-Prey System with a Mate-Finding Allee Effect

“…According to Lemmas 3 and 4, it then follows from [6] that the following result holds for the origin 0 (0, 0) of system (7), which is equivalent to system (11). Remark 6.…”

“…where is the maximum prey consumption rate and ℎ is the predator handling time, which is proposed by Arditi and Ginzburg [4] and studied by many authors; see, for example, Berezovskaya et al [5,6], Kuang et al [7][8][9][10], and Zhang et al [11,12] in which we know this ratio-dependent functional response provides more reasonable explanations and accurate predictions when community-level situations of food chains and food webs are considered.…”

“…Since the Jacobian matrix cannot be evaluated at 0 (0, 0) (noting that 1 and 2 are not differentiable at (0, 0)), the classical stability anlysis methods are not applied. We apply the algorithm presented in [6] to investigate the structure of a neighborhood of 0 (0, 0) and show that system (7) has a stable 0 (0, 0) for all system parameters. Such result is obtained numerically in Figures 1, 2, 4, and 6(a).…”

Families of traveling impulses and fronts in some models with cross-diffusion

“…9a). The proof of existence of the elliptic sectors can easily be conducted with the methods given in [24]. Asymptotics of homoclinics composing the elliptic sector are u = 0 (trivial) and v = K + u, where K + is the biggest root of the equation…”