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## 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.

CERTIFICATION………………………………………………………………………………………………….. I
DEDICATION……………………………………………………………………………………………………… II
ACKNOWLEDGEMENT………………………………………………………………………………………… III
ABSTRACT………………………………………………………………………………………………………….. IV
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
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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
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## 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
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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
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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.

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