Stabilization of Prolific Populations Through Migration and Long-Lived Propagules

Abstract

A spatial analogue of the logistic model is constructed, with sub-populations located on a 10 by 10 grid, with terms representing long lived propagules such as seeds left in the seed bank by plants or eggs left by many insects and other animals, and with terms representing migration between sub-populations. Both long lived propagules and migration proved sufficient, individually, to stabilize the population when ‘r’ is set much greater than 4. The usual sequence of bifurcation to chaos is seen as ‘r’ is increased to 4, but the degree of chaos diminishes greatly as ‘r’ is increased beyond 4, ultimately becoming locked in a period 3 cycle. As would be expected, low migration rates allowed each sub-population to operate as an independent oscillator while high migration rates synchronized the sub-populations. Estimation of the correlation dimension is not feasible for this model since the combination of the memory induced by the persistent propagules with the spatial distribution results in very high dimensional behaviour.