In the running time and in the last few decades, reaction-diffusion model is widely integrated as a major feat in the area of mathematical biology, oceanography, environmental and physical science of mathematical and engineering tasks. Now a day, even the modeling of reaction-diffusion is important in epidemiology, cancer cell growth and in bio medical applications including the development of model for microbial organisms. Most of the research was connected with regular diffusion and in this book we consider the directed dynamics competitive and cooperative model and studied the problem both theoretically and numerically. This book is premeditated to students, teachers, researchers and other professionals who deal with the problems related to mathematical ecology, dynamical system and scientific computing. At a time this book is helpful for the readers to evaluate numerical simulations, advance modeling in biology and other tasks. Most chapters of the book are designed so that readers should not turn out the chapter unnecessarily while ones¿ are studying. I wish to acknowledge the great assistance to all those authors whose texts and articles were consulted.