Global asymptotic stability of a predator-prey model with general functional response and including recruitment and capture in both species

In this paper, we complete the study of the (global) dynamics of predator-prey models introduced in a previous work, which considers the possibility of any (generic) functional response involved, and includes recruitment and capture in both species. More specifically, we provide demonstrations for the global asymptotic stability of all the non-trivial steady states. To do that, we ingeniously divide the invariant set of the system by means of an adequate partition and construct appropriate Lyapunov functions for the subsets of such partition, what leads us to the global stability of each non-trivial equilibria. Thus, the work not only extends the precedent studies but also complete the study of them, providing technical ideas which can be useful in other contexts. Finally, numerical simulations associated to several specific examples are shown to illustrate their coherence with the theoretical findings.