A Global Attractor in Some Discrete Contest Competition Models with Delay under the Effect of Periodic Stocking

Abstract

We consider discrete models of the form $x_{n+1}=x_{n}f(x_{n-1})+h_n$, where $h_n$ is a nonnegative p-periodic sequence representing stocking in the population, and investigate their dynamics. Under certain conditions on the recruitment function $f(x)$, we give a compact invariant region and use Brouwer fixed point theorem to prove the existence of a p-periodic solution. Also, we prove the global attractivity of the p-periodic solution when $p=2$. In particular, this study gives theoretical results attesting to the belief that stocking (whether it is constant or periodic) preserves the global attractivity of the periodic solution in contest competition models with short delay. Finally, as an illustrative example, we discuss Pielou's model with periodic stocking.