Mathematical Model of Predator-Prey Relationship with Human Disturbance

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 Sci-laB in the
simulations as shown in figures 1 to 15.

Keywords: predator-prey model, human disturbance, external periodic force and noise.

TABLE OF CONTENTS

Title Page …………………………………………………………………………………………………. i
Certification ……………………………………………………………………………………………… ii
Dedication ………………………………………………………………………………………………… iii
Acknowledgement …………………………………………………………………………………….. iv
Table of Contents……………………………………………………………………………………….. v
Abstract …………………………………………………………………………………………………… vi
CHAPTER ONE
1.1 Introduction …………………………………………………………………………………………. 1
1.2 Aims of study……………………………………………………………………………………….. 2
1.3 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
4.0 Analysis of Study ……………………………………………………………………………….. 38
4.1 Equilibrium Analysis ………………………………………………………………………… 38
4.2 Stability ………………………………………………………………………………………………. 39
CHAPTER FIVE
5.1 Discussion of Results ……………………………………………………………………………..42
5.6 Physical interpretation/Application of the Study …………………………………………47
5.7 Figures …………………………………………………………………………………………………48
5.2 Summary ………………………………………………………………………….66
5.3 Conclusion………………………………………………………………………..66
5.4 Recommendation …………………………………………………………………67
5.5 Areas of Further Research ……………………………………………………….68
References ……………………………………………………………………………69

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 contestants in an ecological game. This changes the rules for all the other players. If a competitor is taken out, those that are left benefit. Just like when a seed dispersal agent is removed, a plant is not transported. If an enemy is killed, an old victim flourishes. Predators are in a sense arbiter of community structure and local diversity.

In this study, a predator is an animal that hunts, kills and eats other animals for example Lion, Cat, Wolves and other predators. The predator is a carnivorous animal and the prey is a herbivorous animal. An example of simplified predator-prey interaction in our environment is seen in a house where Rats and Cats are living. The population of the Cat and Rat are intertwined in a life and death struggle or fight .It had been predicted by the ecologists that in a sample predator–prey system, that a rise in prey population goes with a move slowly (with a lag) by a rise in the predator population. When the prey population falls, the predator population falls and this allows the prey population to recover and complete one cycle of this interaction.

Predators influence the numbers of prey by removing individuals from the prey population, yet they do not kill off the prey population. This is because under undisturbed conditions, prey population rise steadily thus providing more food for predators. Then the predator population begin to rise, their numbers do not rise immediately since it takes time for the energy from food to be converted into successful reproductive efforts. Because of this time lag, the prey may be well on the road to recovery before the predator population begins to rise. When the predator population finally rises, there is increasing pressure on the prey. Then as the prey begins to be killed off, the predators find themselves with less food and so their own population soon falls off due to starvation or simply a failure to reproduce; this helps the prey to recover. The predator-prey relationships are not often straight forward. Below is the picture of cat pursuing rat.

FIGURE 1

HUMAN DISTURBANCE

The effect of human disturbance on the number of species found in the system is recognized in the intermediate disturbance hypothesis. According to the hypothesis, areas with intermediate levels of disturbance have more species than the areas of lower or higher levels of Figure 1: disturbance. At lower levels, competition is intense and the resulting exclusion yields only a few surviving species. At higher levels, the disturbance itself wipes out all but a few stress-tolerant species. At intermediate levels, not strong enough to kill most species but still strong enough to reduce the competitive impact of dominant species, the number of species is the highest because competitively inferior and superior species as well as stress-intolerant and stress- tolerant species survive. Below is the intermediate disturbance hypothesis showing the number species plotted against the frequency or intensity of disturbance. The diagram is illustrating that greatest number of species found at intermediate frequencies or intensities of disturbance.

Figure 2: An Intermediate Disturbance Hypothesis

OSCILLATION OR PERIODIC FORCE

From the beginning, it will be seen that interaction of the two results in oscillations of constant amplitude which is the time taken for system to ‘‘go round’’ one of the cycles is Number of species

Frequency or intensity of disturbance determined by the prey reproductive rate and the predator death rate. The different cycles representing different amplitudes of oscillation result from the use of different values of an integration constant which depends on the relationship between the rates of increase of the two species. On the common sense grounds, the system of two species would continue to oscillate with constant amplitude. In the more general treatment, what happens in nature is that while the period of oscillation remains constant, the ratio of reproductive rates does not and thus the amplitude of oscillation tends to change progressively so that the system either unwinds, the oscillations becoming greater and greater until one species reaches zero and the system collapses or alternatively, the oscillations tend to die down and the system comes to rest at the singular point in the centre. In the first case, the collapse of the system, if the prey is the species to die out the predator will rapidly follow suit. If the predator is the first to reach zero, the prey population will increase until controlled at a new level by a new density-dependent factor such as food shortage. The implication of this result is that, as soon as the population of one species reaches zero, the whole system collapses. This theory is attributed to the small population involved and the greater liability of the systems becomes extinct as a result. Oscillation will occur depending only on the coefficients of increase of predator and prey and on the initial relative numbers.

The graph that comes from the records of pelts kept by the Hudson’s Bay Company in Canada that is figure (3) will be used to show the oscillating rates of the predator-prey relationship, Wallace [1, 2]. The peaks and crashes in the cat population are definitely dependent on the rat population. The rat population is shown to follow a comparable pattern even in the areas where there are no cats. The rats are responding to cycles in their own part ‘prey’, which themselves seem to reflect climatic variations and changes in insect pest populations. The interrelationships of predator-prey populations are clearly not as simple as it might first be seen.

See the diagram below:

Figure 3: Above is a graph of some analyses of fur data from the records of the Hudson Bay Company in Canada according to Wallace [1,2].

The graph above shows the number of rats and cats living in “NDAAH PACKING SHORE IN OMUANWA” from May 2011 to February 2012. In the graph of Cats and Rats, from May to September, the number of prey (rat) increased. The Cats now had enough to eat, so more of them survived. The growing number of Cats killed more and more Rats. The Rats population decrease. By October, the lack of Rats had greatly affected the Cats. Some Cats starved and others could not raise as many young. Soon the Rats population began to climb again. This cycle for the two species has continued. This shows that the populations of Cats and Rats are related. The Cats population depends on the size of the Rats population and vice versa or predator-prey interactions. The oscillation of numbers of the predatory Cat and its prey (Rat) is almost classic in its characteristics. Prey numbers increase first, followed at once by predator numbers. Then as the predator increase continues, the prey species diminishes in a population crash. The predator’s outcome is relatively similar as its number fall rapidly.

NOISE

Noise is a sound especially when it is unwanted, unpleasant or loud. In this case the noise we need for an effective system is wanted and pleasant. The factor noise seen in this system is exhibited by the predator and prey; when the battle of survival commences. Here, the noise made by the prey is not sustained as to compare with that of the predator. On the part of the prey, it sounds when trying to defend itself and it is caught by its predator. The prey’s noise attracts the predator to itself while the predator’s noise scares them away to their hide-out. Once it is in the domain of predator, killed the noise is terminated because a dead Rat does not make a noise. However, the noise that comes from the predator is from the time of struggling to get hold on the prey to when it is eating the prey; even after that the noise is still sustained and that is the kind of noise that will be evident in our model equation. In this hint, noise in a system increases the dynamics. In this model, the noise in predator’s equation will offset the noise in prey’s equation.

DIFFUSION

Diffusion in this scenario is the process of movement of the predator and prey being spread out and not directed in one place because of the chaos nature of the system. Diffusion is regarded to be the motion of the species in the system. The diagram below depicts that there is no motion at the time zero, which is the point of predation on the prey (Rat). The effect of diffusion in the system is to initiate a travelling wave front which resulted in a smooth travelling wave front solution for the reaction-diffusion equation.

Figure 4

ASSUMPTIONS

In this predator – prey model, we have the basic assumptions for the dynamics of the populations of a predator and its prey species.

Let R(t) be the population density of the Rats (prey) and C(t) be the population density of the Cats (predator) at time t. Thus, we want a mathematical model based on the growth rate for the populations. The models C(r) of functional response are assumed to be continuously differentiable on [0, ∞] and satisfy C(0) = 0, C1(r) ˃ 0 and limı →∞ıııı= ı < ∞
Such models include:

REFERENCES

[1]. Wallace, R. A. (1979): The Ecological and Evolution of Animal Behaviour. 2nd ed. Scott, Foresman, Glenview. New York.

[2]. Wallace, R. A. (1991): Biology the Science of Life. 3rd ed. Harpercollins publishers Inc. New York.

[3]. Mahaffy, J. M. (2000): Lotka-Volterra Models.Williams and Wilkins Co. San Diego State..

[4]. Freedman, H. I. (1980): Deterministic Mathematical Models in Population Ecology. Marcel Dekker. New York.

[5]. Hongler, M. O. and Filliger, R. (2005): An Exactly Soluble Kolmogorov Model for two Interacting Species. Springer, New York.

[6]. Wiens, E. G. (2003):Lotka-Volterra Equation. file://F:/Lotka-Volterra _equation predator prey.htm

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[10]. Sun,G. Q, Jin, Z, Liu, Q.X and Li, B.L. (2010): Rich Dynamics in a Predator-Prey Model with both Noise and Periodic Force. Journal: www.elsevier.com/locate.biosystems Vol. 100, pp 14-22.