A Periodicity Threshold Theorem for Some Nonlinear Integral Equations

Abstract

We consider a parametrized family of nonlinear integral equations $F_\tau (x) = x$ which arise in the theory of epidemics. We study the equations by means of bifurcation theory and establish a best possible value $\tau _0 $ (a “periodicity threshold”) such that for $\tau > \tau _0 $ the equation has positive, periodic solutions. The value of $\tau _0 $ is determined by the spectral radius $r(L_\tau )$ of $L_\tau $, where $L_\tau $ is an associated family of linear operators, so we also study properties of the map $\tau \to r(L_\tau )$.