Limit cycles in a general Kolmogorov model (Q1776916)

The authors study the planar system of differential equations \[ x' = \phi(x)[F(x) - \pi(y)], \quad y' = \rho(y)[-r + \psi(x) + \xi(y)], \] which contains many model predator-prey systems that appear in the literature. Their main result is a list of four conditions that guarantee that any limit cycle appearing in the first quadrant is unique. Rather than applying a transformation of coordinates that produces a Liénard system, they introduce auxiliary functions in order to estimate the integral of the divergence around any such cycle, thus determining its stability, and then produce a contradiction if more than one cycle is assumed to exist.