The paper is concerned with a toy model that generalizes the standard Lotka-Volterra equation for a certain population by introducing a competition between instantaneous and accumulative, history-dependent nonlinear feedback the origin of which could be a contribution from any kind of mismanagement in the past. The results depend on the sign of that additional cumulative loss or gain term of strength $\lambda .$ In case of a positive coupling the system offers a maximum gain achieved after a finite time but the population will die out in the long time limit. In this case the instantaneous loss term of strength u is irrelevant and the model exhibits an exact solution. In the opposite case $\lambda <0\mathrm{}$ the time evolution of the system is terminated in a crash after $t_s$ provided $u=0.\mathrm{}$ This singularity after a finite time can be avoided if $u\ne 0.$ The approach may well be of relevance for the qualitative understanding of more realistic descriptions.

• Received 22 January 2002

DOI:https://doi.org/10.1103/PhysRevE.65.056106