Individual-based models solved using fast Fourier transforms

Abstract

We describe and apply an efficient solution method for individual-based models that uses Fourier transforms of convolution integrals. These integrals are commonly employed in quantitative genetic and individual-based structured population models. The convolution integrals define the cumulative intensity of interactions across all phenotypes that interact via reproduction, competition, predation, mutualism, etc. Such equations are efficiently integrated using the Fast Fourier Transform (FFT). As illustrations, we apply the method to a quantitative genetics model for the evolution of character displacement among competitors and to a predator–prey population model. We obtained a 7-fold increase in computation speed using the FFT as compared to direct integration using the trapezoid method. For the population dynamics we examined, the agreement between the FFT method and the trapezoid method is relatively insensitive to the number of intervals into which the functions to be integrated are discretized. However, excluding a portion of the tails of the distributions of the integrand reduces the agreement with the case using a large proportion of the tails. When applied to the character displacement model, eight competing species evolved to be equally spaced in phenotype space. Phenotype variances were not identical: species at the extremes of the linear sequence of phenotypes had a variance 25% larger than species on the interior of the sequence. In the predator–prey model without evolution, prey oscillations were stabilized with large variances in the predator phenotype or when the predator’s and prey’s mean phenotype differed significantly.