BY

I would like to express my sincere thanks and gratitude to my thesis advisor Hans G. Othmer for his constant encouragement, infinite patience and valuable guidance through these years of my graduate study. I would like to thank Hye-Won Kang for her helpful discussion on stochastic background and analysis. I would like to thank Christina Brakken-Thal and Mike O’Connor for the discussion on the single cell model. I would like to thank Jifeng Hu for the discussion on Fourier transformation. I would like to thank all other members of Hans Othmer’s group for their comments and support, especially my officemates Chuan Xue, Qixuan Wang and Hua Nie. I would like to thank my parents for their unconditional trust, love and freedom. I also want to thank my husband Yu Gu for his faithful love and encouragement. I want to thank my sister Yakun Zheng for her support and understanding in the past 20 years. i Dedication Dedicated to my parents. i