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

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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