**CHAPTER ONE**

**1.0 INTRODUCTION**

Population dynamics is the branch of life sciences that studies the size and age composition of populations as dynamical systems, and the biological and environmental processes driving them (such as birth and death rates, and by immigration and emigration). Example scenarios are ageing populations, population growth, or population decline.

**1 Brief History**

Population dynamics has traditionally been the dominant branch of mathematical biology, which has a history of more than 210 years, although more recently the scope of mathematical biology has greatly expanded. The first principle of population dynamics is widely regarded as the exponential law of Malthus, as modeled by the Malthusian growth model. The early period was dominated by demographic studies such as the work of Benjamin Gompertz and Pierre François Verhulst in the early 19th century, who refined and adjusted the Malthusian demographic model.

A more general model formulation was proposed by F.J. Richards in 1959, further expanded by Simon Hopkins, in which the models of Gompertz, Verhulst and also Ludwig von Bertalanffy are covered as special cases of the general formulation. The Lotka–Volterra predator-prey equations are another famous example, as well as the alternative Arditi–Ginzburg equations. The computer game SimCity and the MMORPG Ultima Online, among others, tried to simulate some of these population dynamics.

**Intrinsic Rate of Increase**

The rate at which a population increases in size if there are no density-dependent forces regulating the population is known as the intrinsic rate of increase. It is

The rate of increase of the population, N is the population size, and r is the intrinsic rate of increase. Thus r is the maximum theoretical rate of increase of a population per individual – that is, the maximum population growth rate. The concept is commonly used in insect population biology to determine how environmental factors affect the rate at which pest population’s increase. See also exponential population growth and logistic population growth. (Jahn Gary et al., 2005)

**Exponential Population Growth**

Exponential growth describes unregulated reproduction. It is very unusual to see this in nature. In the last 100 years, human population growth has appeared to be exponential. In the long run, however, it is not likely. Paul Ehrlich and Thomas Malthus believed that human population growth would lead to overpopulation and starvation due to scarcity of resources. They believed that human population would grow at rate in which they exceed the ability at which humans can find food. In the future, humans would be unable to feed large populations. The biological assumptions of exponential growth is that the per capita growth rate is constant. Growth is not limited by resource scarcity or predation. (Clark, Colin, 1990)

**1.2 GENERAL OBJECTIVES**

The general objectives of this study are to show the application of mathematical model on problems of population dynamics. It will assist in understanding the impact of immigrants on the transmission dynamics of the Nation.

**1.2.1 SPECIFIC OBJECTIVES**

The specific objectives of this study are:

- To determine the population growth rate of Ogbomoso area using mathematical system of equation
- To determine the behavior of each embedded parameter of the mathematical problems on population dynamics,
- Mathematical numerical systemin understanding the uncontrollable increase in population problems.

**1.3 IMPORTANCE OF THE STUDY**

In the past 30 years, population dynamics has been complemented by evolutionary game theory, developed first by John Maynard Smith. Under these dynamics, evolutionary biology concepts may take a deterministic mathematical form. Population dynamics overlap with another active area of research in mathematical biology: mathematical epidemiology, the study of infectious disease affecting populations. Various models of viral spread have been proposed and analyzed, and provide important results that may be applied to health policy decisions.