Document Type


Date of Degree

Spring 2012

Degree Name

PhD (Doctor of Philosophy)

Degree In

Applied Mathematical and Computational Sciences

First Advisor

Li, Yi

Second Advisor

Hendrix, Stephen D

First Committee Member

Ayati, Bruce P

Second Committee Member

Mitchell, Colleen C

Third Committee Member

Strohmer, Gerhard


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. High cross-diffusion also resulted in increased activity in honey bees in terms of accelerations, decelerations, and changes in direction, indicating rapid redistribution of densities to an equilibrium state. Empirical analysis of the number of honey bees and other visitors in two-minute intervals to almond trees shows a negative relationship, indicating cross-diffusion effects in nature with the potential to increase movement to a different tree with a more favorable environment, potentially increasing nut production.


cross- and self-diffusion, Galerkin method, insect pollination, intraspecific and interspecific interactions, pollination services


x, 69 pages


Copyright 2012 Kamuela Yong