Mathematics

# Critical Analysis of the Discrete Logistic Model and Structured Populations

## Critical Analysis of the Discrete Logistic Model and Structured Populations

1.0 INTRODUCTION

1.1 Introduction and Motivation

1.2 Aims and objective

1.3 Statement of the problem

1.4 Scope of the study

1.5 Definition of terms

2.0 CHAPTER TWO: LITERATURE REVIEW

2.1 Models and Modelling

2.2 The Logistic model

2.3 The logistic differential equation

2.4 The logistic function

2.5 The logistic graph/map

2.6 Comparison of the discrete and continuous models

CHAPTER THREE

3.0 The discrete logistic models

3.1 Structured Population models

CHAPTER FOUR: PROBLEMS AND SIMULATION

4.0 Data simulation and

4.1 Behaviour of the model

4.2 Graphing the model

## INTRODUCTION

The well-known logistic differential equation was originally proposed by the Belgian mathematician Pierre-François Verhulst (1804–1849) in 1838, to describe the growth of a population under the assumptions that the rate of growth of the population was proportional to the existing population and the number of available resources.

When this scenario is “translated” into mathematics, it results in the differential equation where t denotes time, P0 is the initial population, and r, k are constants associated with the growth rate and the carrying capacity of the population.

Although it can be considered as a simple differential equation, in the sense that it is completely solvable by the use of elementary techniques of the theory of differential equations, it has tremendous and numerous applications in various fields. The first application was already mentioned, and it is connected with population problems, and more generally, problems in ecology. Other applications appear in problems of chemistry, linguistics, medicine (especially in modeling the growth of tumors), pharmacology (especially in the production of antibiotic medicines), epidemiology, atmospheric pollution, flow in a river, and so forth.

Nowadays, the logistic differential equation can be found in many biology textbooks and can be considered as a cornerstone of ecology. However, it has also received much criticism from several ecologists.

However, as it often happens in applications, when modeling a realistic problem, one may decide to describe the problem in terms of differential equations or terms of difference equations. Thus, the initial value problem which describes the population problem studied by Verhulst could be formulated instead as an initial value problem of a difference equation. Also, there is great literature on topics regarding discrete analogs of the differential calculus. In this context, the general difference equation has been known as the discrete logistic equation and it serves as an analog to the initial value problem.

There are several ways to “end up” with (ii) starting (i) from or and some are:

• by iterating the function,F(x) = µX(1 – X), , which gives rise to the difference equation Xn+1 = µXn(1 – Xn)

• by discretizing using a forward difference scheme for the derivative, which gives rise to the differential equation where being the step size of the scheme, or

• by “translating” the population problem studied by Verhulst in terms of differences: if Pn is the population under study at the time, its growth is indicated by. Thus, the following initial value problem appears:

Notice of course that all three equations are special cases of (i)

## AIMS AND OBJECTIVES

This study is being conducted to critically analyze the Discrete Logistic model and structured populations. At the end of the study; we should be able to;

1. Analyze logistic models under different circumstances and values of the rate of population growth.

2. Solve some problems involving the application of the Discrete logistic model to real-life situations and conclude the solutions of such.

3. Construct models for some structured populations and their behaviors

4. Solve problems relating the structured populations to real-life situations.