Analysis of a Delayed Two-Stage Population Model with Space-Limited Recruitment

In this paper we perform a global qualitative analysis of a delayed two-stage population model, which was proposed by Bence and Nisbet [Ecology, 70 (1989), pp. 1434–1441] to study the dynamic behavior of open systems where older or larger individuals can inhibit the recruitment of juveniles or smaller ones into the population, such as an open marine population with space-limited recruitment. It is known that the time delay between settlement and recruitment into the adult population is a crucial factor needed to produce the observed cyclic fluctuations. In addition to showing that solutions with positive initial conditions are positive and bounded, we establish sufficient conditions for the (i) persistence of the system (the adult population eventually staying above a certain level); (ii) local stability and instability of the positive equilibrium; and (iii) global stability of the positive equilibrium. Biological implications of these conditions are discussed.