Critical Analyses Of The Discrete Logistic Model And Structured Populations

Project and Seminar Material for Statistics

Critical Analyses Of The Discrete Logistic Model And Structured Populations


Chapter One


Introduction

The well-known logistic differential equation was originally proposed by the Belgian mathematician Pierre-François Verhulst (1804–1849) in 1838, in order 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 amount of available resources.

When this scenario is “translated” into mathematics, it results to the differential equation

—- (i)

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 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 modelling 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 by several ecologists.

However, as it often happens in applications, when modelling a realistic problem, one may decide to describe the problem in terms of differential equations or in 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 a great literature on topics regarding discrete analogues of the differential calculus. In this context, the general difference equation

……….. (ii)

has been known as the discrete logistic equation and it serves as an analogue 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 difference 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 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 analyse the Discrete Logistic model and structured populations. At the end of the study; we should be able to

  1. Analyse 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 draw conclusions from the solutions of such
  3. Construct models for some structured populations and their behaviours
  4. Solve problems relating the structured populations to real life situations.

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