MATHEMATICAL MODEL OF PREDATOR-PREY RELATIONSHIP WITH HUMAN DISTURBANCE

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ABSTRACT

The predator-prey model with human disturbance is considered in the model and other factors such as noise, diffusion and external periodic force. The functional response of Holling III is also involved in the study. This predator-prey model involves two species giving us two variables (the predator and prey).  The oscillatory wave in two-dimensional space is shown by the species with time which is obvious when human disturbance and noise are involved.   In this model, the coefficient of diffusion is zero at the point predator is predating on the prey. Also, the effect of the said factor (human disturbance) leads the prey to quick annihilation from the system of interaction at the beginning of the competition and later comes up in its population in an asymptotic and exponential increase respectively. The study when modeled with noise and periodic force showcased a sinusoidal and an exponential increase in the figures below; and without noise and periodic force depicted an asymptotical increase in the shape of the graph figures below.  These results may help us to understand the effects springing up from the true defenselessness to random fluctuations in the real ecosystems. We declared that the human disturbance increases the functional response and the entire processes of motion (diffusion) which showed us that the predator has only one type of food source.  Both the prey and predator will survive the contest. The study has showcased the rate of the predator’s functional response with time, t. We analyzed and discussed the equilibria, stability of the model and solutions of these systems of differential equations. We also used the figures to illustrate the predator-prey interaction in terms of their population which exists in an ecosystem, predator-prey life in an ecological system, a predator predating on its prey and the intensity of human disturbance in the same ecosystem. We performed simulations by illustrating the rate of the predator’s feeding on the prey with time using the Holling-Type III functional response showing the searching time, handling time and total time of the predator in predating on its prey. We used scilab in the simulations as shown in figures 1 to 15.  

TABLE OF CONTENTS

Title Page
Table of Contents
Abstract

CHAPTER ONE
1.1 Introduction
1.2 Aims of study
1.3 Definition of terms in the study

CHAPTER TWO
2.0 Review of Related Literatures

CHAPTER THREE
3.1 The Model

CHAPTER FOUR
4.0    Analysis of Study
4.1      Equilibrium Analysis
4.2 Stability

CHAPTER FIVE
5.1 Discussion of Results
5.6 Physical interpretation/Application of the Study
5.7 Figures
5.2 Summary
5.3 Conclusion
5.4 Recommendation
5.5 Areas of Further Research
References

CHAPTER ONE
                                                               INTRODUCTION
               Predation is the process of removing individuals from a lower trophic level as to prevent monopoly competitive success among the prey.  Predation thus allows increased diversity through what is called ‘‘cropping principle’’.  This effect is demonstrated by removing top predators which results in drastic reductions in prey diversity as successful competitors freed from predation preempt resources. Predation can have a major effect on the size of a population as applied to population that when the death rate exceeds the birth rate in a population, the size of the population usually decreases. If predators are very effective at hunting their prey, the result is often a decrease in the size of the prey population. But a decrease in the prey population in turn affects the predator population. Wolves and Lions preying on ungulates, and Cats preying on Rats have their take limited by the effective defenses of the prey animals such that their predation cannot interrupt rapid population growth of the prey when food and population dynamics produce exponential increase, but relatively high predator densities accentuate population crashes that follow. Predation can be a powerful determinant of community structure.  It has a dynamic influence on the numbers and quality of both predator and prey as it acts as an important agent of natural selection on both groups.    
          However, diversity in ecology is the measure of the number of species coexisting in a community.   An ecosystem is a system of plants, animals and other organisms interacting within themselves and non-living components of their environment; e.g. a lake or forest. There are ‘‘natural’’ and ‘‘managed’’ (that is farms or market gardens) ecosystems.  Today, few ecosystems remain untouched by human activities. Managed ecosystems are essential to our survival by reducing competition through removal of non-useful species (that is weeds). People are able to intensify food and other natural materials production.  These processes more often reduce species diversity but there are instances where human management of ecosystems actually increases species diversity.  No simple relationship exists between the diversity of an ecosystem and ecological processes.  An ecological system is an open system in which the interaction between the component parts is non-linear and the interaction with the environment is noisy. The model will explain the interaction between the species and their natural environment which is the ecological system.                  
Nevertheless, the predator–prey model is the building blocks of the bio and ecosystems as biomasses are grown out of their resource masses. The predator–prey model is a type of mathematical model that involves at least two species (the predator-cat and prey-rat). In the course of the species existence, the species involve compete, develop or evolve and scatter or disperse for the purpose of searching for resources to sustain their living. Based on their specific settings of applications the predator–prey can take the forms of parasite-host, tumor cells (virus)–immune system, resource–consumer, plant–herbivore etc. The predator–prey embark on the business of one specie’s loss is another specie’s gain; interactions may have applications outside the ecosystems.
In the biological point of view, the first rush of ecological theory saw predators as potential controlling agents for populations. Indeed, predators can utterly transform population histories; but the more interesting effects are probably on diversity and structure as predator winnowing of populations alters patterns of competition. It is a truism of history that much of the food of wolves and big cats consists of the old and the sick.

1.2      AIM OF THE STUDY
            Based on the previous works done on investigations, contributions and modifications on predator-prey model, our aim and flair in this model is to find out the effect of human disturbance to the system and proffer solution to or solve the existing equations in two variables and analyze the obtained result. The model will tell us about the effect of human disturbance, periodic force, noise and diffusion.  This will also show that the motion of individual species of the given population is random and isotropic that is no preferred direction.  It will also analyze the state of the system in the presence of human disturbance and the predator’s functional response with the Holling Type-III response.     

1.3      DEFINITION OF TERMS IN THE STUDY
The following terms will be defined in this section:
(i.)                           The predator and Prey
(ii.)                          Human disturbance
(iii.)                        Oscillation or periodic force
(iv.)                        Noise
(v.)                         Diffusion 

THE PREDATOR AND PREY
            A predator is an organism that uses other live organisms as an energy source and in doing so, removes the prey individuals from the population. This definition allows the concept of predation to be extended to include herbivore as well as carnivore. The working ecologists now talk of predation when describing sheep hunting grass, cats hunting rats or squirrels searching for nuts. When predators kill, they remove.....

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