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.sh y 1.. where y t. is the size of the population at time t.. Thus the population, defined for tgw0, 1x, resets itself at the end of the unit time interval to its initial value. If y satisfies the boundary condition and we define Y tqn.sy t.for tgw0, 1. and ns0, 1, . . . , then Y is a 1-periodic solution of the differential equation extended to tgw0, ` by periodicity. for t/1, 2, . . . and Y has a jump of magnitude h y 1..yy 0. at ts1, 2, . . . . This quasi-steady-state solution corresponds to a population growing or declining on ny1-t-n ns1, 2, . . ..and decreasing or increasing impulsively at ts1, 2, . . . . Y plays a role for the jump condition y nq.sh y ny..analogous to that played by constant solutions to the differential equation with zero jump condition i.e., y nq.sy ny... 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 h.