Thesis Abstract

Abstract
The discrete logistic model is a fundamental tool in studying population dynamics and has been widely used in ecological and biological research. This model describes the growth of a population in discrete time steps based on a logistic growth function. However, when populations are structured, with individuals differing in attributes such as age, size, or spatial location, the standard logistic model may not capture the complexity of population dynamics accurately. In this research, we critically analyze the discrete logistic model in the context of structured populations. Structured populations present unique challenges and opportunities for modeling population dynamics. By incorporating individual heterogeneity, spatial structure, or other forms of population structure, researchers can gain insights into how different subpopulations interact and influence overall population growth. The discrete logistic model can be extended to account for structured populations by introducing additional parameters to represent the characteristics of individuals within the population. We conduct a thorough examination of the discrete logistic model with structured populations, focusing on the implications of different types of population structure on model behavior. Age-structured populations, for example, may exhibit different growth patterns compared to unstructured populations due to variations in birth and death rates across age classes. Spatially structured populations may show spatial patterns in population growth and dispersal that are not captured by the standard logistic model. Our analysis also considers the impact of dispersal, migration, and other forms of movement on population dynamics in structured populations. These processes can lead to non-uniform distributions of individuals across space and time, affecting population growth rates and stability. By incorporating dispersal mechanisms into the discrete logistic model, we can better understand how movement influences the persistence and distribution of populations in heterogeneous environments. Overall, our critical analyses shed light on the strengths and limitations of the discrete logistic model when applied to structured populations. By accounting for population structure in ecological and biological studies, researchers can improve the realism and predictive power of population models, leading to more accurate assessments of population dynamics and conservation strategies.

Thesis Overview

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 OBECTIVES

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

Ø Analyse logistic models under different circumstances and values of the rate of population growth.
Ø Solve some problems involving the application of the Discrete logistic model to real life situations and draw conclusions from the solutions of such
Ø Construct models for some structured populations and their behaviours
Ø Solve problems relating the structured populations to real life situations.