Quasi-steady-state solutions of some population models

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) = eta(y(1)) (where y(t) is the size of the population at time t). Thus the population, defined for t is an element of [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 is an element of [0, 1) and n = 0, 1,..., then Y is a 1-periodic solution of the differential equation (extended to t is an element of [0, infinity by periodicity) for t not equal 1, 2,... and Y has a jump of magnitude eta(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 +)= eta(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 eta. (C) 1999 Academic Press.