Mathematical Model on Human Population Dynamics Using Delay Differential Equation
Chapter One
OBJECTIVES OF THE WORK
The objectives of this research are to:
- Highlight various characteristics of population growth,
- Use delay population model in describing population growth,
- Proffer solution to the negative consequences of delay population growth and
- Determine the stability in population with respect to changes in age structure of different sex.
CHAPTER TWO
LITERATURE REVIEW
Delay differential equations (DDEs) are widely used in ecology, physiology and many other areas of applied science. Although the form of the DDE model is usually proposed based on scientific understanding of the dynamic system, parameters in the DDE model are often unknown. Thus, it is of great interest to estimate DDE parameters from noisy data. Since the DDE model does not usually have an analytic solution, and the numeric solution requires knowing the history of the dynamic process, the traditional likelihood method cannot be directly applied. Delay differential equations (DDEs) are popular tools for applied scientists to model their dynamic systems. The forms of the DDE models are usually proposed by investigators based on their understanding of the dynamic system. However, the parameters of DDE models are often unknown and often have important scientific interpretations. Therefore it is important to infer their values. It is also a good calibration of the DDE models if the DDE solutions with certain parameter values fit the measurements of the dynamic systems. The objective of this work is to estimate the DDE parameters from noisy data. In 1954 and 1957 Nicholson performed his now classical experiments on laboratory cultures of the sheep blowfly. Since then, attempt to model this experiments have been made by May (1974) and Varley et al (1973) among others. According to Blythe et al (1982), these attempts have produced at best ”generalized insights” and, furthermore no theoretical model has yet yielded a truly satisfactory quantitative fit to the time history of even a single culture, still less has it been possible to formulate a comprehensive framework within which the various subtly different experimental result can systematically be interrelated.
Differential equations have long been used to model various cell populations. In many cases ordinary differential equations (ODEs) are the starting point in the modeling process. When time delays (due to feedback, cells division time lags, etc.) become important, then delay differential equations become a natural tool for modeling in the life sciences. For example, the classic predator-prey ODE model suggested by Lotka and Volterra in the 1920’s. Many consumer species go through two or more life stages as they proceed from birth to death. In order to capture the oscillatory behavior often observed in nature, various models are proposed. They include many difference models and delay differential models. The Hutchinson equationand its variations are among the ones that are most frequently employed in theoretical ecology models. In Equation 1, r is the growth rate, K is the carrying capacity, and τ is a time delay that may have no real biological meaning. Like logistic equations, these models are adhoc and hence can be misleading. Indeed, they produces artificially complex dynamics such as excessive volatility and huge peak-to-valley ratios.
On the other hand, if we assume the adults have a constant birth rate of r, the newborns mature in τ units of time, and the mortality rate is proportional to the adult population density, then the following model will be a reasonable model for the adult population
Therefore, we investigate the growth or decay rate of solutions of ordinary equations with delays which the results can be applied for the case when the characteristic equation of an associated linear equation has complex dominant Eigen value with higher than one multiplicity. Examples are given in describing the asymptotic behavior of solutions in a class of quasi linear equations.
CHAPTER THREE
TERMINOLOGIES AND POPULATION GROWTH MODEL
POPULATION GROWTH
This is the change in population over time, and can be quantified as the change in the number of individuals of any species in a population using “per unit time” for measurement. In biology, population growth is likely to refer to any known organism. In demography, population growth is used to refer specially to the growth of the human population of the world
POPULATION GROWTH RATE (PGR)
This is the rate at which the number of individuals in a population increases in a given time period as a fraction of the initial population. Population growth rate (PGR) ordinarily refers to the change in population over a given time period. It is also expressed as a percentage of the number of individuals in the population at the beginning of that period. The most common way to express population growth is as a percentage.
A positive growth rate indicates that the population is increasing, while a negative growth rate shows that the population is decreasing. A growth rate of zero indicates that there was the same number of people at the two times-net differences between births, and deaths. A growth rate may be zero even when there are significant changes in the birth rates, death rates, immigration rates and age distribution between the two times.
DELAYS IN A POPULATION GROWTH
This is the period in which an organism is not useful within the environment in the ecosystem. It is the time of instability (like age structure) in population growth. This means that stable equilibrium will generally become unstable if a time delay exceeds the dominant time scale of a system.
Time delays are otherwise known as time lags in a population, some examples include: Maturation period e.g. – the time required for larva to become adults, the time between conception and reproductive maturity.
Gestation period e.g. – the time needed by predators to digest prey.
Regeneration period e.g. – the time taken by plants consumed by animals to grow to their earlier levels.
CHAPTER FOUR
POPULATION GROWTH OF MEN USING DELAY DIFFERENTIAL EQUATION MODEL
DELAY DIFFERENTIAL EQUATION FOR JUVENILE
To make ourselves to be more familiar with a delay differential equation (DDE), we have to develop a model for the human being (e.g male) which incorporates age structure into the population while ignoring other details. We approximate the age structure by dividing the
population into “adults” A(t) and juveniles J(t). We choose age twelve as the dividing line for the population since twelve is near the median age that human beings become sexually mature. For this reason, we assume the juveniles do not reproduce. In this example, juveniles are born at a rate proportional to the current adult population and leave the juvenile population by either dying or becoming adults. Here, we neglected immigration and emigration (migration). Hence the delay differential equation for juvenile population growth is:
CHAPTERE FIVE
DISCUSSION OF THE RESULT
In the delay of population growth model, we developed a model of human being which incorporates age structure into the population. The models were subdivided into male delayed differential equation model and female delayed differential equation model.
Ignoring order details using delay differential equation formulation, we are able to incorporate the dependence of birth and death rate as ages without using partial differential equations. For men, the age structure was divided into juvenile and adult whereby the juvenile delayed for twelve years before maturing to adult for reproduction. Therefore it takes juvenile long time before becoming useful to the society.
Also, A(t – 12) , which is the current adult population that is beyond twelve years will become the maturation period of the juvenile.
Adult can also leave its age by dying which can also take time to regenerate in other organisms.
Then, equation (4.1)shows the changes between the becoming and leaving of juvenile
respectively, while equation (4.2) indicates the becoming of adult and leaving the adult as well.
Equation (4.1.8) and (4.2.8) indicate that the juvenile population and adult population respectively increases exponentially as time increases. Meanwhile the system tends towards exponential decay.
CONCLUSION
The use of delay differential equations in the modeling of biological phenomena has become more prevalent in recent years. Analytic results about the behavior of such models is still largely lacking. While numerical simulations provide a basic understanding of these systems, and allow, To be sure, increased computation capacity and speed make the use of such simulations easier. A better analytic understanding of these models, however, would make the use of numeric even more useful, and help in the selection of appropriate models in the first place. Chapter of this project provides the methodology and ordinary differential equation model for human population dynamics. The methods of Chapter 4 provide a straightforward and easily applicable method for analyzing the linear stability of the steady states of such models. The methods for approaching such questions remain quite cumbersome. Ideally, a better understanding of the functional analytic theorems at work here would lead to easier determination of the existence or otherwise of periodic solutions, at least in the case of a system of only two differential equations. For ordinary differential equations, one has theorems such as Poincare-Bendixson which allow one to draw conclusions based solely on global properties (the existence of a trapping region) and linear instability. I hope that continued study of the question of periodicity will lead to steps in the direction of such theorems for delay models. At the very least, a simpler method of determining the ejectivity of a fixed point would be quite welcome. I have spent much time in this thesis attempting to determine the properties of delay differential equations models. I have mentioned that understanding these properties would make it easier to determine the appropriateness of these models for biological phenomena. Much work remains to be done on this question. Although it seems intuitively clear that delays occur in nature, and that they might therefore play a significant role in the dynamics of a given system, the models I have studied are only first approximations. All of the models studied incorporate a discrete delay. In other words, the dynamics depend on the current state of the system and the state of the system exactly _ time units ago. This way of including the delay requires much refinement. Consider the example of human pregnancy. The gestation period is generally stated to be nine months, but this is hardly exact. If such a reproductive delay is significant in the dynamics of some model, then surely the variation about the mean delay time will also be significant. Discrete delays are only an approximation. These systems ought to be studied, since the chance of obtaining concrete results is greater for discrete delays than for their distributed cousins, and knowledge of their behavior provides insight into more complete, distributed models. One suspects that the behavior of the discrete model should correspond to the expected behavior, for example, of a stochastic model, where the length of delay is determined by a probability distribution function. If discrete delay models are to serve as approximations, however, it will be important to determine the extent to which their behavior is an artifact of the essentially discontinuous inclusion of past data. As biologists turn to mathematics to provide a framework for understanding more and more complicated phenomena, it is important to have as many modeling techniques as possible available for use. While the inclusion of delays is but one approach among many, the theory behind it should continue to be developed, with an eye especially toward practical results and the ability to draw applicable conclusions.
RECOMMENDATION
Analysis of time delay in population models usually concentrate upon delaying longer than the dominant time scale and the period of oscillation. Many natural delays can be expected to be short, here the effect of such short delays has been examined in both a single-level model and a Two-level model, and show that short delay due to maturation time can produce a variety of stability changes, including both dramatic decreases and dramatic increases in stability.
I suggest that there is a general rule underlying these examples. When a delay in a population is isolated from the effect of other trophic levels (a rather stringent requirement), then stability is enhanced by short time delay acting through regulatory recruitment. If either the dominant time scale or delay term itself is significantly affected by other trophic levels, then stability is enhanced by nonregulatory recruitment and diminished by regulatory recruitment. Therefore it is found that two processes generally classified as destabilizing, a time lag and nonregulatory recruitment, can interact to increase the stability of population models.
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