ordinary differential equations – Intuitive explanation for one of the conditions in Kolmogorov’s Predator-Prey theorem

I’m reading Nowak’s “Evolutionary dynamics”. In page 67 he introduces Kolmogorov’s Prey-predator equation, in the form
$$dot{x}=xFleft(x,yright)$$
$$dot{y}=yGleft(x,yright)$$

where $$F$$ and $$G$$ are $$C^{1}$$ functions. The variable $$x$$ is interpreted as the amount of prey while $$y$$ represents the amount of predators. A theorem by Kolmogorov is then stated – That this system has a stable equilibrium or a limit cycle if 8 conditions for $$F$$ and $$G$$ are met. One of these conditions is

$$xfrac{partial F}{partial x}+yfrac{partial F}{partial y}<0$$

And Nowak’s interpretation of this condition is that “For any given ratio of the two species, the rate of increase of prey is a decreasing function of population size”.

Can someone explain why this is a valid interpretation of the condition?