Delayed density dependence at the immature stage in insects and the dynamic behavior of nonlinear difference equations

RESUMO

A growing body of evidence indicates that the conditions experienced by immatures in insects, in particular crowding, have a lasting consequence for the population dynamics o adults. In this case, as first demonstrated by Prout (1), the dynamic characteristics of populations sampled at the adult stage may not be derived. We examine the dynamic properties of the model proposed by Prout to take into account the delayed effect of two life-history traits, survival and fecundity, occurring at the immature stage. Two parameters are present in the model> Beta, which describes the rate of change in survival and fecundity with respect to increasing density of immatures, and alpha which combines maximum survival and fecundity. The latter parameter is found to determine the dynamic behavior of Prount'a model, and this model is shown to tbe a reparametrization of the classical discrete logistic equation. In the interval 1 < alpha < e2 there is one fixed point, at alpha = e2 there is period doubling bifuraction, and due to the appearance of period three Prout's model shows chaotic behavior. The theoretical results are briefly discussed in the light of data on the equilibrium dynamics of Drosophila and blowflies.