SIAM J. APPL. MATH. Vol. 52, No. 3, pp. 855-869, June 1992 ()1992 Society for Industrial and Applied Mathematics 016 ANALYSIS OF A MODEL REPRESENTING STAGE-STRUCTURED POPULATION GROWTH WITH STATE-DEPENDENT TIME DELAY* WALTER G. AIELLOt, H. I. FREEDMAN$, AND J. WU Abstract. A stage-structured model of population growth is proposed, where the time to ma- turity is itself state dependent. It is shown that under appropriate assumptions, all solutions are positive and bounded. Criteria for the existence of positive equilibria, and further conditions for the uniqueness of the equilibria are given. The stability of the equilibria are also discussed. In addition, an attracting region is determined for solutions, such that this region collapses to the unique positive equilibrium in the state-independent case. Key words, attractor, bounded, equilibrium, positivity, single-species, stability, stage struc- ture, time delay AMS(MOS) subject classifications. 92A17, 34K20 1. Introduction. In [2], a stage-structured model of population growth consist- ing of immature and mature individuals was analyzed, where the stage-structure was modeled by the introduction of a constant time delay. Previously, other models of population growth with time delays were considered in the literature [1], [7], [8], [10], [11], [14], [16], [17], [20]. Age- and stage-structured models of various types (discrete and distributed time delays, Stochastic, etc.) have also been utilized [4], [12], [15], [18], [21]. In [9], it was observed that for Antarctic whale and seal populations, the length of time to maturity is a function of the amount of food (mostly krill) available. Prior to World War II, it was observed that individual seals took five years to mature, small whales took seven to ten years, and large whale species took twelve to fifteen years to reach maturity. Subsequent to the introduction of factory ships after the war, and with it a depletion of the large whale populations, there was an increase in the krill available for the seals and the remaining whales. It was then noted that seals took three to four years to mature and small whales now only took five years. Maturation time for large whales also significantly decreased. Since the amount of food available per biomass for a fixed food supply in a closed environment is a function of the total consumer biomass, we modify the model considered in [2] to include a monotonically decreasing, state-dependent time delay. The existence of a such monotonically increasing time to maturity has been observed in other contexts as well. For example, Andrewartha and Birch [3, p. 370] describe how the duration of larval development of flies is a nonlinear increasing function of larval density. We believe that this is the first time such a population model has *Received by the editors November 12, 1990; accepted for publication (in revised form) May 2, 1991. Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada T6G 2G1. Applied Mathematics Institute, Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada T6G 2G1. This author’s research was partly supported by Natural Sciences and Engineering Research Council of Canada grant NSERC A4823. Department of Mathematics, York University, North York, Ontario, Canada M3J 1P3. This author’s research was carried out while on a G. Gordin Kaplan Memorial Postdoctoral Fellowship. 855 Downloaded 11/13/15 to 130.63.174.132. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php