# CHAPTER 3

# Ecological Models: Single Species

The fact that ecology is essentially a mathematical subject is becoming ever more widely accepted. Ecologists everywhere are attempting to formulate and solve their problems by mathematical reasoning.

—Evelyn C. Pielou

## I. Introduction

This chapter initiates the study of simple deterministic models for population growth. As Evelyn Pielou notes in her book *An Introduction to Mathematical Ecology*, “The investigation of the growth and decline of population is, historically, the oldest branch of mathematical ecology.” Chapter 3 examines models for the changes in single-species population. The mathematical tools employed are first-order differential equations and first-order difference equations. In Chapter 4, we consider some models for population growth that present important features of interaction between two species occupying the same territory. In particular, we study the oscillation of population sizes of two competing species and the dynamics of predator-prey populations. Here the mathematical tool is an autonomous system of first-order differential equations. Chapter 5 presents some models on the growth of a population of cells making up a tumor. The mathematical analysis is self-contained.

## II. The Pure Birth Process

Imagine a population made up entirely of identical organisms that reproduce at a rate that is the same for every individual and that does not vary with time. If we assume that each individual lives forever, that the organisms ...

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