Mathematical Model on Human Population Dynamics Using Delay Differential Equation

Mathematical Model on Human Population Dynamics Using Delay Differential Equation

ABSTRACT

Simple population growth models involving birth rate, death rate, migration, and carrying capacity of the environment were considered. Furthermore, the particular case where there is discrete delay according to the sex involved in the population growth were treated.

The equilibrium and stability analysis of each of the cases were considered also. The stability analysis shows that the discrete delays in the population growth lead to instability in the growth.

TABLE OF CONTENTS

CERTIFICATION………………………………………………………………………………………………….. I
DEDICATION……………………………………………………………………………………………………… II
ACKNOWLEDGEMENT………………………………………………………………………………………… III
ABSTRACT………………………………………………………………………………………………………….. IV
TABLE OF CONTENTS………………………………………………………………………………………….. V
CHAPTER ONE …………………………………………………………………………………………….. 1
1.0 INTRODUCTION …………………………………………………………………………………………… 1
1.1 Objective of the Work ………………………………………………………………………………….. 2
1.2 Significance of the Work ………………………………………………………………………………… 2
1.3 Scope of the Work ………………………………………………………………………………………… 3
CHAPTER TWO ……………………………………………………………………………………………. 4
2.0 Literature Reviews ……………………………………………………………………………………….. 4
CHAPTER THREE ………………………………………………………………………………………….. 8
3.0 Terminologies and Population Growth Model ……………………………………………….. 8
3.1 Population Growth ………………………………………………………………………………………… 8
3. 2 Population Growth Rate (PGR) ……………………………………………………………………… 8
3.3 Delays in a Population Growth ………………………………………………………………………. 9
3.4.0 Determination of Population Growth …………………………………………………………… 9
3.4. 1 Birth rate ……………………………………………………………………………………………… 9
3.4.2 Death rate ……………………………………………………………………………………………… 10
3.4.3 Migration ………………………………………………………………………………………………… 10
3.4.4 Carrying Capacity …………………………………………………………………………………… 10
3.5 Population Growth Model using Birth and Death Rates ……………………………… 11
3.6 Population Growth Model using Birth, Death and Migration ……………………… 13
3.7 Population Growth Model using Birth, Death, Migration and Carrying Capacity. 13
3.8 Basic Concept of Delay Different Equations ………………………………………………….. 15
3. 9 Biological Mechanism Responsible for Time Delay ……………………………………… 16
CHAPTER FOUR ……………………………………………………………………………………………… 17
4.1.0 Population Growth of Men using Delay Differential Equation ………………………… 17
4.1.1 Delay Differential Equation for Juvenile …………………………..………………………… 17
4.1. 2 Delay Differential Equation for Adult ………………………………………………………… 18
4.2.0 Population growth of women using Delay Different Equation …………………… 21
4.2.1 Delay Differential Equation for Juvenile …………………………………………………….. 21
4.2.2 Delay Differential Equation for Child Bearing Age ……………………………………. 21
4.2.3 Delay Differential Equation for Adult ………………………………………………… 22
4. 3.0 Equilibrium analysis ……………………………………………………………………………………… 25
4.4.0 Stability analysis …………………………………………………………………………………………. 27
4.4.1 Stability analysis for Men…………………………………………………………………………….. 27
4.4.2 Stability analysis for Women………………………………………………………………………… 29
CHAPTER FIVE …………………………………………………………………………………………….. 31
5.1.0 Discussion of the Result ……………………………………………………………………………… 31
5.1.1 Conclusion ………………………………………………………………………………………………….. 32
5.1.2 Recommendation ………………………………………………………………………………………… 34
Reference …………………………………………………………………………………………………… 35

CHAPTER ONE

1.0 INTRODUCTION

One of the most generally accepted ideas of population in ecology is that time delays are potent sources of instability in population growth system. If true, this statement has important consequences for our understanding of population dynamics, since time delays are ubiquitous in ecological systems.

All species exhibit a delay due to maturation time. Whenever specie has a recognizable breeding season there can be a lag between an environmental change and the reproductive response of the specie. They also exhibit a delay due to gestation period and regeneration period.

The destabilizing effect of time delays is often expressed by the rule that an otherwise stable equilibrium will generally become unstable if a time delay exceeds the dominant time scale of a system (May 1973a, b; Mayriard Smith 1974).

A dynamic system has two basic time scales namely: the return time, which reflects the rapidity with which the system returns to equilibrium following a small perturbation, and the natural period, which is the period of oscillation exhibited by a perturbed system.

Recently the use of “delay logistic” model has been criticized and alternative models suggested (Cushing 1980, Gurney et al. 1982, Blythe et al. 1982, Nunney 1983). These more realistic models show that time delays do not inevitably turn to instability.

Blithe et al (1982) showed that common competition can make stability resilient to the preservation of long delays due to maturation time, and Nunney (1983) has shown that resource recovery time, which has been cited as potentially important sources of delay (May 1973a, b) need not destabilize a system even when the delay is long.

Similar effect has been observed in the analysis of the predator-prey system which includes maturation time lags. Hastings has shown that if the Lotka-Volterra model is stable by the addition of a type 3 functional response, then the instability can be resistant to very long delays during prey maturation time (Nunney, 1985a).

However the biological mechanism which accounts for time lags is age structure while in physiology, time lags arise from the delay caused by the finite time taken in transmission of message through nerves or hormones. They also arise when populations are distributed over space, resulting in delays because of the finite time of transmission of message from one region to another.

In a more recent paper, Gurney et al (1983) pointed out that the failure of these models lies in their lack of a mathematically rigorous foundation. Nisbet and Gurney remarks that in their (1983) paper with Lawton “that if the

life history of an insect involved developmental stages of arbitrary duration, then the normal integro-differential equation describing a population with over lapping generations reduced to a set of coupled ordinary delay-differential equations, provided only that all individuals in a particular age class have the same birth and death rates”.

1.1 OBJECTIVES OF THE WORK

The objectives of this research are to:

i Highlight various characteristics of population growth,

ii Use delay population model in describing population growth,

iii Proffer solution to the negative consequences of delay population growth and

iv Determine the stability in population with respect to changes in age structure of different sex.

1.2 SIGNIFICANCE OF THE WORK

The significance of this work is based on the population parameter stipulated by the delay models. It determines the effect of delay in biological mechanism of population growth. It also accounts for time lag caused by reproductive response of the species with respect to gestation and regeneration period.

1.3 SCOPE OF THE WORK

This work is centered on Characterization and description of population growth of human beings using logistics and delay-differential equation over a given period of time.