Quasi-Steady-State Solutions of Some Population Models

Abstract

We consider a class of first-order differential equations generalizing the logistic equation of population growth, together with a two-point boundary condition of the form y(0)=η(y(1)) (where y(t) is the size of the population at time t). Thus the population, defined for t∈[0,1], resets itself at the end of the unit time interval to its initial value. If y satisfies the boundary condition and we define Y(t+n)=y(t) for t∈[0,1) and n=0,1,…, then Y is a 1-periodic solution of the differential equation (extended to t∈[0,∞ by periodicity) for t≠1,2,… and Y has a jump of magnitude η(y(1))−y(0) at t=1,2,…. This quasi-steady-state solution corresponds to a population growing or declining on n−1<t<n (n=1,2,…) and decreasing or increasing impulsively at t=1,2,…. Y plays a role for the jump condition y(n+)=η(y(n−)) analogous to that played by constant solutions to the differential equation with zero jump condition (i.e., y(n+)=y(n−)). We show, under hypotheses motivated by biological considerations, that a strictly positive solution exists, is unique, and is monotone and continuous in its dependence on η.