Mathematical analysis of a two species model with a differential nature of interaction (Q1184902)

We model a situation in which the nature of interaction between the species changes when one of the species attains a certain population level. We can view a system where one species \((u)\) helps the other \((x)\) till the latter reaches the population \(\hat x\) (we shall take \(\hat x\) less than the carrying capacity of the environment species \(x\)) and beyond which it is harvested by the first species \(u\). We shall assume that in the absence of either of the species, the other grows logistically. Thus before the second species attains the population level \(\hat x\), the interaction between \(u\) and \(x\) is commensal and thereafter the species \(x\) is harvested by \(u\). Such type of interaction is peculiar due to the dual behaviour of \(u\). In the first section we propose a model while in the second section we search for equilibria. In section 3 we analyse the stability of equilibria if they exist. The section 4 deals with some examples. In the last section we conclude the whole analysis in the form of discussion.