A mathematical model of the interactions between pollinators and their effects on pollination of almonds

California’s almond industry, valued at $2.3 billion per year, depends on the pollinator services of honey bees, although pollination by other insects, mainly solitary wild bees, is being investigated as an alternative because of recent declines in the number of honey bee colonies. Our objective is to model the movements of honey bees and determine the conditions under which they will forage in less favorable areas of a tree and its surroundings when other pollinators are present. We hypothesize that foraging in less favorable areas leads to increased movement between trees and increased cross pollination between varieties which is required for successful nut production. We use the Shigesada-Kawasaki-Teramoto model (1979) which describes the density of two species in a two-dimensional environment of variable favorableness with respect to intrinsic diffusions and intraand interspecific interactions of species. The model is applied to almond pollination by honey bees and other pollinators with environmental favorableness based on the distribution of flowers in trees. Using the spectral-Galerkin method in a rectangular domain, we numerically approximated the two-dimensional nonlinear parabolic partial differential system arising in the model. When cross-diffusion or interspecific effects of other pollinators was high, honey bees foraged in less favorable areas of the tree. In the model, high cross-diffusion also resulted in increased activity in honey bees which manifested itself in the field in terms of accelerations, decelerations, and changes in direction, indicating rapid redistribution of densities to an equilibrium state. Empirical analysis of the number