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
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.
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CSN Team.
