Stability in Kolmogorov-type quadratic systems describing interactions among two species. A brief revision (Selecciones Matemáticas, 2021)
Population dynamics is a relevant topic in Biomathematics, being the study of the long-term behavior of interaction models between species, one of its central problems. A large part of these relationships are described by ordinary differential equations (ODE), having as main objectives the study of the stability of their solutions.In this document we mainly describe the dynamic behavior of the Volterra predation model. In addition, we make a review of some derived predation models and a brief review of the dynamical properties of models describing other interactions between species such as: competition, mutualism, amensalism, and commensalism; also described by nonlinear ODE systems of the second order of Kolmogorov-type.For each of these models, the non-existence of limit cycles can be demonstrated and in most of  them, there is a globally stable equilibrium point. In one of them, there are  conditions in the parameters for which the only positive equilibrium point is a center, as in the original Lotka-Volterra model. The methodology used is the usual one for the analysis of models with hyperbolic equilibrium points, but it can guide the analysis of other more complicated models.
Keywords: Predator-prey model; functional response; cycles; separatrix curve; stability; Lyapunov function