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

Abstract

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 intra-and 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 by the GAANN Fellowship, Sloan Fellowship, Graduate College Summer Fellowship, and the Department of Mathematics. Finally, I would like to thank my parents, Sam and Ruth Yong, who have sacrificed many things so that I could get a good education and who have been my biggest supporters throughout my life; my sister, Nani Yong, for always being there; and my wife, Tasha-Leigh Yong, who gives me the strength to get through each day and has been incredibly understanding during the times I had to put my research over her. Thank you and I love you all! v TABLE OF CONTENTS LIST OF TABLES .

Global existence for a strongly coupled reaction diffusion system

“…To approximate µ i,j,k (i = 1, 2), we use the methods described in [40] to express µ 1,j,k as the following ordinary differential equation that can be solved numerically using the Matlab function ode113: A similar method is used to approximate µ 2,j,k .…”