Mathematical Model of Predator-Prey Relationship with Human Disturbance.
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………….. i
Certification…………. ii
Dedication………….. iii
Acknowledgement…………. iv
Table of Contents…….. v
Abstract……… vi
CHAPTER ONE
- Introduction……………… 1
- Aims of study….. 2
- Definition of terms in the study………… 2
CHAPTER TWO
2.0 Review of Related Literatures…………. 13
CHAPTER THREE
3.1 The Model………….. 24
CHAPTER FOUR
- Analysis of Study…………………….. 38
- Equilibrium Analysis…………….. 38
- Stability……………………………… 39
CHAPTER FIVE
5.1 Discussion of Results… 42
- Physical interpretation/Application of the Study………… 47
- Figures…………….. 48
- Summary…………………… 66
- Conclusion……………………… 66
- Recommendation………………………….. 67
- Areas of Further Research………. 68
References………… 69
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.
REFERENCES
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Wallace, R. A. (1991): Biology the Science of Life. 3rd ed. Harpercollins publishers Inc. New York.
Mahaffy, J. M. (2000): Lotka-Volterra Models.Williams and Wilkins Co. San Diego State..
Freedman, H. I. (1980): Deterministic Mathematical Models in Population Ecology. Marcel Dekker. New York.
Hongler, M. O. and Filliger, R. (2005): An Exactly Soluble Kolmogorov Model for two Interacting Species. Springer, New York.
Wiens, E. G. (2003):Lotka-Volterra Equation. file://F:/Lotka-Volterra _equation predator prey.htm