## 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.

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

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## 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

vii

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

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## 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.

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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

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(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

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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:

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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

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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:

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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

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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.

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Figure 4

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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:

(i) ( )

r

p

k

r

r p

kr

p r

C r kr

+

=

÷ø

ö

çè

æ +

=

+

=

1 1

(ii) ( ) 2

2

p r r

C r kr

+ +

=

e

(iii) ( ) 2

2

p r

C r kr

+

=

where k, p and ı are positive constants. k denotes the growth rate of the species and p is the

saturation constant. In population dynamics, a functional response of predator to prey density

refers to the change in the density of prey attached per unit time per predator as the prey density

changes. C(r) denotes the predator response function. Based on the above functional response

assumed, (i) is called the Michaelis-Menten or Holling type-II, (ii) is called the sigmoidal

response function and (iii) is called the Holling type-III function and it satisfies the assumptions

made and will be used for the model. Type III functional response is the type in which the attack

rate accelerates at first and then decelerates towards satiation. Type III functional responses are

typical of generalists natural enemies which readily switch from one food species to another

and/or which concentrate their feeding in areas where certain resources are most abundant

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This rate of change in a population with time is equal to the net increase (births) into the

population minus the net decrease (deaths) of the population.

Applying these assumptions to the population of Rats, the Rats population density

becomes the rate of change of the Rats population with respect to change in time (

dt

dR ). The

primary growth in the Rat population grows in proportion to its own population or size is a1R(t).

Hence, we will assume that the primary loss of Rats is due to predation by the Cats and Human

Beings (Human Disturbance). However, predation is often modeled by assuming random contact

between the species in proportion to their populations with a fixed percentage of those contacts

resulting in death of the prey species. Mathematically, this is given by a negative term − a2

R(t)C(t) and − a3R(t). Here, a3 is the total number of prey, n minus the number predated by

Human Being, m i.e. (n–m). If the Cats population is low such that starvation due to

overcrowding dominates the death rate and no human disturbance, then alternative death terms

would be more appropriate. Combining these terms, we have the growth model for the Rat

population:

a R(t) a R(t)C(t) (n m)R(t)

dt

dR = – – – 1 2

( ) 1 2 a R t a

dt

dR = – ( ) R t C(t) a R(t) 3 – (This is the prey’s equation).

For the predator, we will consider the population dynamics of the Cat as the rate of change

of the Cats population with respect to change in time (

dt

dC ). The primary growth for the Cat

population depends on sufficient food for raising Cats, which means an adequate source of

nutrients from predator or preys. Thus, the growth of Cat population is similar to the death rate

for the Rat population with a different constant of proportionality.

Mathematically, the growth of the Cat population can be expressed as b2 R(t)C(t). The human

predation is negative on the Cat because it reduces the quantity of food meant for the predator

i.e. (n – m)C(t) = b3C(t), where b3 represents the total number of Rats, n minus the number

predated by human beings, m. The loss of Cat is presumed to be a type of reverse growth. That

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is, in the absence of Rats, the Cat population declines in population to their own population

which is expressed by the negative modeling term as − b1 C(t).

If the Cat diet used other animals or crowding factors from other predators and Cats were taken

into consideration or account, then these terms would have to be significantly modified. The

growth model for the Cat population gives:

b C(t) b R(t)C(t) (n m)C(t)

dt

dC = – + – – 1 2

b C(t) b R(t)C

dt

dC

1 2 – + (t) b C(t) 3 – This is the predator’s equation.

The model ignores the role of climate variation and the interactions of other species. Other

significant factors ignored are the ages of the animals and the spatial distribution. The two

differential equations above are intertwined into a system of differential equations with each

growth model depending on the unknown variable (population) of the other.

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