INTRAGUILD PREDATION MODEL WITH DISEASE
JUFIZA BINTI A. WAHAB
UNIVERSITI TEKNOLOGI MALAYSIA
INTRAGUILD PREDATION MODEL WITH DISEASE
JUFIZA BINTI A. WAHAB
A dissertation submitted in partial fulfilment of the
requirements for the award of the degree of
Master of Science (Mathematics)
Faculty of Science
Universiti Teknologi Malaysia
NOVEMBER 2009
iii
Dedicated to my beloved husband,
En. Abdul Hafiz Abdul Raof
my daughter,
Adillya Batrisya Abdul Hafiz
my parent,
En. A. Wahab Mat Tahir and Pn. Hamidah Mat Yatin,
my sister,
Cik Juyana A. Wahab
&
my supervisor,
Dr. Faridah Mustapha
iv
ACKNOWLEDGEMENTS
Alhamdulillah, with His will has allowed me to complete this research.
First and foremost, I would like to thank Dr. Faridah Mustapha, my supervisor, for her
support throughout this project. Her advices on both content and presentation has been
superb, and quite simply, this project would not have been possible without her help.
She also provided invaluable feedback on the strength and weakness of initial drafts.
Thank also goes to Miss Nurul Aini Mohd Fauzi, research assistant of Faculty of
Sciences (UTM) who provided expert assistance in completing this research. She also
provided extremely constructive comments on the draft. I am also indebted to Universiti
Teknologi Mara (UiTM) for funding my M.Sc study. My appreciation also goes to my
parent for their unconditional loving support. A special note of gratitude goes to my
dear husband, Abdul Hafiz Abdul Raof for his support, understanding and
encouragement for all I needed during my graduate study. I would also like to thank my
friends especially Nazihah Ismail, Azwani Alias and Nor Hidayu Nawi who were
always with me during this semester for their kind support and for everything they
taught me. It is impossible to list the many friends and colleagues who over the year
have assisted the development of ideas that have resulted in this research. To each of
these people I express my sincere appreciation.
v
ABSTRACT
Ecosystem stability is an important issue in conservation of biodiversity. The
stability of predator prey systems or competitive systems has been studied extensively.
Although the two fields have been the subject of widespread research recently, no work
has been done to study the effect of a disease on an environment where three species;
predator, prey and resource present. Here we analyze modification of Intraguild
Predation (IGP) model to account for a disease spreading among prey. We chose the
simplest epidemiological model, SI model. Here, we consider the simple mass action
incidence. We analyze the stability equilibrium points by using Ruth-Hurwitz criteria.
Numerical examples will be introduced to show the stability point. The result seems to
indicate that either the disease dies out or both species eventually become infected.
vi
ABSTRAK
Kestabilan ekosistem adalah isu penting dalam pemuliharaan kepelbagaian
biologi. Sistem mangsa pemangsa dan kompetatif telah pun dikaji secara meluas. Pada
masa kini, walaupun kedua-dua bidang tersebut dikaji secara terperinci, masih belum
ada sebarang kajian tentang kesan penyakit dalam persekitaran kehidupan di mana
wujud ketiga-tiga spesis iaitu mangsa, pemangsa dan sumber makanan semulajadi.
Dalam kajian ini, kami menganalisa model pemangsaan ‘intraguild’ mengenai penyakit
yang tersebar dikalangan pemangsa. Kami telah memilih model epidemologi ringkas
iaitu model SI . Dalam kajian ini, kami mengambil kira jisim pergerakan mudah. Kami
menganalisa kestabilan titik keseimbangan menggunakan aplikasi criteria RuthHurwitz. Contoh berangka digunakan untuk menunjukkan titik kestabilan. Dalam
penemuan kajian ini, kite dapat lihat sama ada penyakit lenyap atau pun kedua-dua
sepsis dijangkiti.
vii
TABLE OF CONTENTS
CHAPTER
1
TITLE
PAGE
ACKNOWLEDGEMENT
iv
ABSTRACT
v
ABSTRAK
vi
TABLE OF CONTENTS
vii
LIST OF FIGURES
x
LIST OF SYMBOLS
xii
INTRODUCTION
1.1
Background of Study
1
1.2
Problem Statements
3
1.3
Objective of Study
4
1.4
Scope of Study
4
1.5
Significance of Study
5
viii
2
LITERATURE REVIEWS
2.1
2.2
3
4
5
Inter-Specific Interactions
6
2.1.1 Competition Model
9
2.1.2
Predator-Prey Relations
12
2.1.3
Intraguild Predation (IGP)
16
2.1.4
Other Interactions
21
Simple Model of Epidermics
23
2.2.1
Types of Epidermics Model
24
2.2.2
SIR Model
25
2.2.3
SIS Model
26
METHODOLOGY
3.1
Relating Dynamical Systems to Biological Model
29
3.2
Steady-State and Equilibrium Point
30
3.3
Eigenvalue and Stability
31
3.4
Ruth-Hurwitz Criteria
32
INTRAGUILD PREDATION
4.1
Intraguild Predation Model
37
4.2
Stability Analysis
38
INTRAGUILD PREDATION WITH DISEASE
5.1
Intraguild Predation Model with Disease
44
5.2
Stability Analysis
48
ix
6
CONCLUSION AND DISCUSSION
6.1
Introduction
57
6.2
Conclusions and Discussion
57
6.3
Recommendations
60
REFERENCES
61
x
LIST OF FIGURES
FIGURE NO.
TITLE
2.1
Predator-prey population dynamics
2.2
Changes in the abundance of pelts received by the
Hudson’s Bay Company
2.3
PAGE
13
14
Isoclines of the predator and prey for the system (2.3)
16
2.4
(a) Symmetris IGP without age structure, (b)
Symmetric IGP with age structure, (c) Asymmetric IGP
without age structure and (d) asymmetric IGP with age
structure. a → b means that b preys on a , ↔ means
competition
4.1
The stability region in a ρ and β parameter space
when τ = δ = 0.4 , χ = 1.0 and α = 0.5
5.1
17
43
Stability region in ρ and β parameter space when the
other parameter is fixed
52
xi
5.2
Density of each species versus time when
α = 0.5 , β = 1.5 , χ = 0.6 , ω = 0.6 , τ = 0.4 , δ = 0.4 ,
ε = 0.9 , υ = 0.8 , ρ = 0.4
5.3
53
Density of each species versus time when α = 0.5 ,
β = 1.5 , χ = 0.6 , ω = 0.6 , τ = 0.4 , δ = 0.4 , ε = 0.9 ,
υ = 0.8 , ρ = 0.4
55
xii
LIST OF SYMBOLS
Ni
-
number of individuals in species i
ri
-
intrinsic per capita growth rate per species i
Ki
-
environmental carrying capacity for species i
αij
-
per capita inhibiting effect of species j on the population growth
rate of species i
1
Ki
-
inhibition of species i on its own growth
αij
-
inhibition of species j on the growth of species i
Ei
-
equilibrium point for solution i
N
-
biomass density of prey
NS
-
biomass density of susceptible IG prey
NI
-
biomass density of infected IG prey
P
-
biomass density of predator
Z
-
biomass density of basal resource
S
-
number of individuals not yet infected with the disease
I
-
number of individuals who have been infected with the disease
R
-
individuals who have been infected and then recovered from the
Ki
disease
λi
-
eigenvalue i
a
-
predation rates on the common resource for the IG prey
a'
-
predation rates on the common resource for the IG predator
d
-
IG predation rates on n
m
-
density independent mortality rates for IG prey
m'
-
density independent mortality rates for IG prey for IG predator
b
-
predation rate into offspring for IG prey
b'
-
predation rate into offspring for IG predator by consuming basal
resource
c
-
predation rate into offspring for IG predator
by consuming IG prey
K
-
carrying capacity for the common resource
α
-
non-dimensionalized predation rate into offspring for IG predator
by consuming basal resource
β
-
non-dimensionalized predation rate into offspring for IG predator
by IG prey
δ
-
non-dimensionalized mortality rate for IG predator
ρ
-
non-dimensionalized predation rate into offspring for IG predator
by consuming basal resource
χ
-
non-dimensionalized IG predation rate onto IG prey
τ
-
non-dimensionalized mortality rate for IG prey
ε
-
non-dimensionalized simple mass action among the IG prey
υ
-
non-dimensionalized conversion rate of predation upon infected
IG prey
ω
-
non-dimensionalized predation rate upon infected IG prey
θ
-
non-dimensionalized recovery rate from infective to susceptible
CHAPTER 1
INTRODUCTION
1.1
Background of Study
Venturino (1992) in his study said that the study of interacting species has
already begun in the first part of the century. It has received a renewed interest in the
past fifteen years in the mathematical literature. Major discoveries in biology have
changed the direction of science. From the study of the sexual life of oysters, which was
in some sense boring for the previous generations, biology has become today the Queen
of Science. All hardcore fields, such as physics, mathematics, chemistry, and computer
science are now necessary for the big adventure of unraveling the secrets of life and
conversely, the mathematical sciences are all now enthusiastically inspired by
biological concepts, to the extent that more and more theoreticians are interacting with
biologists. Actually, it is not an understatement to say that biology has a Viagra effect
on the old classical fields. What is today the role of a theoretician among the biologists,
eager to incorporate new concepts? An important part of biology, besides amassing new
experimental information, is the explanation and prediction of new phenomena by
2
applying the quantitative laws of physical chemistry, that is, by quantifying phenomena
in mathematical terms, not by merely fitting curve with Numerical Recipes in Matlab.
Theory is not a painting of the real but it gives the framework for quantitative
computations, analysis and prediction. Data analysis is only small fraction of statistics.
The putting together of the pieces of the puzzle of life begins with the understanding the
life of a protein, a microstructure, a cell, a network, and finally, the life of a living
organism. In order to explain how a pure theoretician, can contribute to the analysis of
biological systems, let us review some selected open questions.
Predation is one of the examples of interaction. Predation occurs when one
animal (the predator) eats another living animal (the prey) to utilize the energy and
nutrients from the body of the prey for growth, maintenance or reproduction. Predation
is often distinguished from herbivory by requiring that the prey be an animal rather than
a plant or other type of organism (bacteria). Population dynamics refers to changes in
the sizes of populations of organisms through time, and predator-prey interactions may
play an important role in explaining the population dynamics of many species. They are
a type of antagonistic interaction, in which the population of one species (predators) has
a negative effect on the population of a second (prey), while the second has a positive
effect on the first. For population dynamics, predator-prey interactions are similar to
other types of antagonistic interactions, such as pathogen-host and herbivore-plant
interactions.
Many insect predators that share the same prey species are also quite likely to
kill and devour each other. This is called Intraguild Predation (IGP), since it is predation
within the guild of predators. IGP is a composition of three species community
consisting of resource, consumer and predator. IGP is a special case of omnivory,
induces two major differences with traditional linear food chain models: the potential
for the occurrence of two alternative stable equilibria at intermediate levels of resource
productivity and the extinction of the consumer at high productivities. At low
3
productivities, the consumer dominates, while at intermediate productivities, the
predator and the consumer can coexist. These theoretical results indicate that the
conditions for stable food chains involving IGP cannot involve strong competition for
the bottommost resources. Predator-prey interactions may have a large impact on the
overall properties of a community. For example, most terrestrial communities are green;
suggesting that predation on herbivores is great enough to stop them from consuming
the majority of plant material. In contrast, the biomass of herbivorous zooplankton in
many aquatic communities is greater than the biomass of the photosynthetic
phytoplankton, suggesting that predation on zooplankton is not enough to keep these
communities green.
1.2
Problem Statements
Interaction between individuals and species in the real world are complex
processes. Every living creature grows, reproduces and eventually dies. In order to
survive, an individual uses its environment for food and protection to its own
advantage. Population sizes of species are affected not only by ecological interactions
such as competition, predation and parasitism, but also by the effects of infectious
diseases. One host species can exclude another by means of a shared infectious disease.
This model suggests that apparent competitive dominance can result if individuals of
one species, as compared to individuals of other species, have a higher growth rate
when uninfected, are less susceptible to becoming infected, or have a higher tolerance
to the disease. The higher tolerance to disease of individuals of one species may result
from their faster recovery, lower death rates or higher reproductive rates. For many
diseases, long time behaviour of disease transmission is related to initial positions. If the
initial value of infective numbers is large, which means we have a large invasion of a
disease, the disease will be persistent. If the initial value of infective numbers is small,
4
which corresponds to a small invasion of a disease, the disease will be extinct. The
study of such population ecology can help us understand the growth, extinctions and
changes in distribution of populations and the underlying processes which determine
these changes.
1.3
Objectives of Study
The objectives of this study are:
1.
To formulate a mathematical model of Intraguild Predation (IGP) population
with infectious diseases.
2.
To find the equilibrium points of the IGP model with disease.
3.
To analyze the stability of the equilibrium points of IGP model with disease.
1.4
Scope of Study
This study will be focused on unstructured IGP populations. For the purpose of
this study, we shall only concentrate on two species population with an infectious
disease for one species at one time. We only consider SI model and only one type of
ways in which individual contract the disease which is mass action incidence.
5
1.5
Significant of Study
The findings from this study will contribute towards an enhanced understanding
of IGP among species and the effect of diseases on the dynamics of the population. The
key result in this model is that the diseases must either die out in both species or remain
endemic in both species.
CHAPTER 2
LITERATURE REVIEWS
2.1
Inter-Specific Interactions
A population is a group of individuals of the same species that have high
probability of interacting with each other. A simple example would be trout in a lake, or
moose on Isle Royale, although in many cases the boundaries delineating a population
are not as clear cut. Population biology is simply the study of biological populations.
Population biology is by its nature a science that focuses on numbers. Thus, we will be
interested in understanding, explaining and predicting changes in size of populations.
The goals of population biology are to understand and predict the dynamics of
populations. Understanding, explaining, and predicting dynamics of biological
population will require models that are expressed in the language of mathematics.
Interaction between individuals and species in the real world are complex
processes. Every living creature grows, reproduces and eventually dies. In order to
survive, an individual uses its environment for food and protection to its own
8
advantage. Venturino (1994) state that biological interactions result from the fact that
organisms in an ecosystem interact with each other, in the natural world, no organism is
an autonomous entity isolated from its surroundings. It is part of its environment, rich in
living and non living elements all of which interact with each other in some fashion. An
organism's interactions with its environment are fundamental to the survival of that
organism and the functioning of the ecosystem as a whole. Sign-mediated interactions
in which molecules serve as signs are the characteristic feature of communicative
interactions.
Interaction between species refers to positive and negative associations between
species that favour or inhibit mutual growth and evolution of populations. It may take
the form of competition, predation, commensalism, amensalism or mutualism.
Mustapha (2001) in her thesis categorized the interactions as follows:
•
Competition (-,-) where both species cause demonstrable reduction in
each other’s survival, growth or fecundity.
•
Predation (+,-) where the first species will gain benefit from this
interaction, whereas the second species will suffer from it.
•
Commensalism (+,0) where one of the species will benefit from the
interaction, without any adverse effect on the other one.
•
Amensalism (-,0) is the reverse of commensalism. In this interaction,
presence of species A will have a negative effect on the species B, while
species B has no effect on species A.
•
Mutualism (+,+) where both species benefit from the interaction.
9
2.1.1
Competition Model
Competition among species usually happens when two or more species live in
proximity and share the same basic requirements. They usually compete for resources,
habitat, or territory. Sometimes only the stronger prevails, driving the weaker
competitor to extinction. One species win because its members are more efficient at
finding or exploiting resources, which leads to an increase in population. Indirectly this
means that a population of competitors finds less of the same resources and cannot grow
at its maximal capacity.
Farkas (2000) stated in his book that the classical competition model is due to
Alfred Lotka and Vito Volterra. The each formulated model independent of each other
around 1925 and 1926. The Lotka-Volterra competition model for two species
competing for a limited resource such as food or habitat is
dN1
dt
⎡ (N + α12 N 2 )⎤
= r1 N1 ⎢1 − 1
⎥
K1
⎣
⎦
dN 2
dt
⎡ ( N + α21 N1 ) ⎤
= r2 N 2 ⎢1 − 2
⎥
K2
⎣
⎦
where Ni is the number of individuals in species i , ri is the intrinsic per capita growth
rate per species i , and Ki is the environmental carrying capacity for species i . The
parameter αij gives the per capita inhibiting effect of species j on the population
growth rate of species i , as compared to the effect of species i on its own population
10
growth rate. One can interpret
αij
Ki
1
as the inhibition of species i on its own growth and
Ki
as the inhibition of species j on the growth of species i .
The competing species model always has three equilibrium points, E 0 = ( 0 ,0 ) ,
E1 = ( K 1 ,0 ) and E 2 = ( 0 , K 2 ) , on the boundary of the positively invariant first
quadrant. These boundary equilibria correspond to either both species being absent, or
one species being absent while the other is at its carrying capacity. Without species 2 (
N 2 = 0 ), species 1 will grow logistically and vice versa. Looking at the nullclines of the
equations, Mustapha (2001), Saenz (2006) explained that there are four possible
outcomes from the model.
1.
If K 1 >
K2
K
and K 2 < 1
α 21
α12
which species 1 inhibits species 2 more than it
inhibits itself and species 2 inhibit itself more that it inhibits species 1
respectively, species 1 wins the competition and all paths with N 1 ( 0 ) > 0
approach the equilibrium E1 = ( K 1 ,0 ) .
2.
If K 1 <
K2
K
and K 2 > 1 which species 1 inhibits itself more than it inhibits
α 21
α12
species 2 and species 2 inhibit species 1 more than it inhibit itself respectively,
species 2 wins the competition and all paths with N 2 ( 0 ) > 0 approach the
boundary equilibrium E 2 = ( 0 , K 2 ) .
3.
If K 1 >
K2
K
and K 2 > 1 which each species inhibits the other more than it
α 21
α12
inhibits itself, the nullclines intersect at an unstable saddle interior equilibrium
E3 = ( N1e , N 2e ) . In this case, there is a separatrix curve through the interior
equilibrium and the origin with solution starting below the saperatrix going to
11
equilibrium E1 = ( K 1 ,0 ) , and solution starting above it going to the boundary
equilibrium E 2 = ( 0 , K 2 ) . Intuitively, whichever species is initially dominant is
the winner of the competition.
4.
If K 1 <
K2
K
and K 2 < 1 which each species inhibits itself more than it inhibits
α 21
α12
the other species, the interior equilibrium is attractive, and all solution starting
with
N1( 0 ) > 0
and
N 2( 0 ) > 0
approach
this
interior
equilibrium
E3 = ( N1e , N 2e ) . In this case the two species coexist and approach coexistence
equilibrium. The interior equilibrium is found as the intersection of the straight
line nullclines N 1 + α12 N 2 = K 1 and N 2 + α 21 N 1 = K 2 . Thus
N1e =
(K1 − α12 K 2 )
(1 − α12α21 )
N 2e =
(K 2 − α21 K1 )
(1 − α12α21 )
where the numerators and denominators are negative in third and positive in
forth case.
If one of the species is more aggressive in competing with the other such as case
1 or 2, then the less aggressive species will be excluded. In case 4, we can see that the
condition for coexistence is
12
α12 <
K1
1
<
K 2 a 21
or
a12 a21 < 1
which mean that the effect of each species on the other is small, indicating that
competition is less intense than self inhibition. In this Lotka-Volterra competition
model, the two species can coexist only if self inhibition is greater than inter-specific
competition.
2.1.2
Predator-prey Relations
Predation occurs when one animal (the predator) eats another living animal (the
prey) to utilize the energy and nutrients from the body of the prey for growth,
maintenance, or reproduction. In the special case in which both predator and prey are
from the same species, predation is called cannibalism. Sometimes the prey is actually
consumed by the predator's offspring. This is particularly prevalent in the insect world.
Insect predators that follow this type of lifestyle are called parasitoids, since the
offspring grow parasitically on the prey provided by their mother.
Predation is often distinguished from herbivory by requiring that the prey be an
animal rather than a plant or other type of organism (bacteria). To distinguish predation
from herbivory, the prey animal must be killed by the predator. Some organisms occupy
a gray area between predator and parasite. Finally, the requirement that both energy and
nutrients be assimilated by the predator excludes carnivorous plants from being
predators, since they assimilate only nutrients from the animals they consume.
13
Population dynamics refers to changes in the sizes of populations of organisms
through time, and predator-prey interactions may play an important role in explaining
the population dynamics of many species. They are a type of antagonistic interaction, in
which the population of one species (predators) has a negative effect on the population
of a second (prey), while the second has a positive effect on the first. For population
dynamics, predator-prey interactions are similar to other types of antagonistic
interactions, such as pathogen-host and herbivore-plant interactions.
Figure 2.1:
Predator-Prey Population Dynamics.
The fact that predator-prey systems have a tendency to oscillate has been
observed for well over a century. The Hudson Bay Company, which traded in animal
furs in Canada, kept records dating back, explained by Farkas (2000). In these records,
oscillations in the populations of lynx and its prey the snowshoe hare are remarkably
regular (see Figure 2.1).
14
According to Venturino (1994), the Lotka–Volterra equations, also known as the
predator–prey equations, are a pair of first-order, non-linear, differential equations
frequently used to describe the dynamics of biological systems in which two species
interact, one a predator and one its prey. They were proposed independently by Alfred
J. Lotka in 1925 and Vito Volterra in 1926.
dN
dt
= aN − bNP
dP
dt
=
(2.1)
cNP − dP
where N and P are the biomass density of prey and predator, respectively, and a and
d are their per-capita rates of change in the absence of each other. Their respective
rates of change due to interaction are b and c . Equations (2.1) assume that in the
absence of the predator, the prêt population will grow exponentially and without the
prey, the predator will become extinct. The interaction will increase the predator
population, and decreases the prey population. These equations have generated a huge
literature, and we shall analyze in detail below.
Figure 2.2: Changes in the abundance of the lynx and the showshoe hare, as
indicated by the number of pelts received by the Hudson’s Bay Company.
15
Volterra assumed that the interaction rate (predation) is proportional to the
product of the densities of the prey and predator. This form of interaction rate is derived
from the law of mass action that states the rate of molecular reactions if two chemicals
species in a dilute gas or solution is proportional to the product of the two
concentrations. Venturino (1994) stated Lotka and Volterra assumed that both species
move randomly and are uniformly distributed over their habitat. This form of
interaction is inaccurate in decreasing the motion of the two species. Most species do
not move randomly but usually move to the save places or to the areas with more
resources.
In the system (2.1) above, the coexistence rest point N * =
d
a
, P* =
is
c
b
neutrally stable (a centre), surrounded by closed trajectories. A closed trajectory in the
N , P planes implies periodic solutions in t for N and P . The isocline for prey is a
horizontal line is a vertical line that goes through
vertical line that goes through
a
and the predator isoclines is a
b
d
as in Figure 2.2. From the isoclines, system (2.3)
c
predicts what is known as paradox of enrichment.
Rosenweig (1971) explained Paradox of enrichment predicts that
•
Equilibrium prey density is dependent of prey growth rate,
•
Sufficient enrichment of the prey population will result in limit cycle
osicillations that grow rapidly in amplitude with further enrichment
(destabilize the equilibrium).
16
P
Predator
a/b
Prey
N
d/c
Figure 2.3:
Isoclines of the predator and prey for the system (2.3).
When we enrich the system, we increase the value of a . As we can see in Figure
2.3, when a increases, the density of the predator increases but not the density of the
prey at the ret point.
2.1.3 Intraguild Predation (IGP)
According to Pimm and Lawton (1991), in food web theory, omnivory is
defined as the act of feeding by one species on resources at different trophic levels. One
of the simplest conceivable examples of omnivory is a constellation of three species: a
predator, a consumer and a resource that is common to both consumer and predator.
This case is also known as “intraguild predation” (Polis and Holt, 1992). By definition,
17
IGP is a combination of exploitative competition and predation interaction. It is
distinguished from competition by the immediate energetic gains for the predator and
differs from classical predation because the predation interaction reduces potential
exploitative competition (Polis, Myers and Holt, 1989).
Predator 1
Predator 2
Adult
predator 1
Adult
predator 2
Juvenile
predator 1
Juvenile
predator 2
Resource
(b)
(a)
IG predator
Adult IG predator
(b)
IG prey
Resource
(c)
Figure 2.4:
Juvenile IG
IG prey
predator
Resource
(d)
(a) symmetric IGP without age structure, (b) Symmetric IGP with age
structure, (c) Asymmetric IGP without age structure and (d) Asymmetric IGP with age
structure. a → b means that b preys on a , ↔ means competition.
This interaction is common among terrestrial arthropods, granivores, terrestrial
carnivores, freshwater and marine invertebrates (Polis and Strong, 1996). Preying on
your potential competitors may be a learned mechanism or it might be simply a by-
18
product of the fact that encounter rates is increased among species that use the same
niche and consume the same resource.
Polis, Myers and Holt (1989) stated that relative body size and degree of trophic
specialization are the two most important factors influencing the frequency and the
direction of IGP. In IGP, Holt and Polis (1997) called the competing species that prey
on its competitor the IG predator, and the competing species that is being preyed on the
IG prey. Most IGP occurs in the system with size-structured populations by generalist
predators that are usually larger than their IG prey. It often occurs among species that
eat the same food resources but differ in body size such that the smaller species or stage
class falls within the normal prey size range of the larger.
In Figure 2.4, we collect different varieties of IGP. Asymmetric IGP is the
situation when only one species practices on its competitor (2.4(c) and (d)), and
symmetric IGP between two species is when the predation is mutual, as in (2.4(a) and
(b)). In symmetric IGP, as in Figure 2.4(c), IG prey has two negative effects from IGP
interactions: the competition for the resource and predation by the IG predator. In order
for both species to coexist, IG prey must be the more efficient competitor (Holt and
Polis, 1997). This is explained in detail in Chapter 4.
From more empirical observations on population dynamics, IGP can lead to a
rich array of possible outcomes such as exclusion, coexistence, priority effects,
alternative stable states and an increase in resource levels (Polis and Holt, 1992). In
asymmetric IGP, as in Figure 2.4(c), when predation is severe enough, it can reduce the
density of the IG prey drastically of even exclude the IG prey completely. When IG
prey is far too superior in competing for the same resource, the opposite situation might
occur: the population of IG predator might decrease.
19
This happened when red shiners were introduced in the Canadian lakes as food
for resident rainbow trout. Even though adult trout (IG predator) prey on shiners,
juveniles trout were outcompeted by shiners (IG prey). Thus, the trout population
decreased, which was opposite to the expected outcome. The same disastrous result
occur red when Mysis, an opossum shrimp, was introduced into several lake in North
America as food for Salmonids (IG predator). Mysis outcompete young salmonids in
exploiting plankton, resulting in the decline of salmonids population, due to the low
growth rate of the young salmonids. This interaction is represented in Figure 2.4(d). if
IG predator is a more efficient competitor for the common resource then the juvenile IG
predator, developmental bottlenecks can occur. This happen when IG prey reduces the
growth rate of the juvenile IG predator, thus producing less predatory reproductive adult
IG predator.
IGP can also cause the shared resource level to increase. This happens when the
IG prey is the superior competitor of the shared resource, or when IG predator prefers
IG prey to the shared resource. Decreasing population of the IG prey due to IGP, causes
the population of the shared resource to increase. This interaction is represented in
Figure 2.4(c), and it can affect the efficacy of biological pest control system. The
density of the biological control target pest species increases when another predator
species is introduced to the system. For example, the fly species Blaesoxipha (IG prey)
preys on the grasshopper Milanoplus sanguinipes (pest). Introducing another predatory
fly species Asilidae (IG predator) reduces the percentages reduction in the grasshopper
(Roseheim et al., 1995). Direct predation on the Blaesoxipha by asilid flies reduces the
Blaesoxipha population and cause an increase in the grasshoppers population.
Symmetric IGP occurs when species 1 and species 2 are mutual predators of one
another. Larger individuals from any species can eat smaller individuals from ather
species. In this case, individual of species 1 starts life as potential prey of species 2 and
finished as the predator of species 2. Symmetric or mutual IGP is common among
20
marine communities (Hadeler and Freedman, 1989), plankton communities, terrestrial
vertebrate carnivores, inserts and amphibians (Polis, Myers and Holt, 1989).
Priority effect can also be found in summetric IGP. In the symmetric IGP, the
established species dominates by preying on the second introduced species. In this case,
high rates of IGP by the adults of the established species on the juveniles of the
introduced species will lower the number if its adults. Although adults of the introduced
species prey on the juveniles of the established species, they are not numerically
dominant enough to reduce the population of the established species through IGP. Preexisting colonies of ants almost always prey on founding queens of any other species of
ants (Polis, Myers and Holt, 1989). Another example is the case of IGP of whelks on
lobster on Marcus Island. The management goal to establish harvestable populations of
lobster on the island by introducing lobster in the island failed (Hadeler and Freedman,
1989). This is due to the IGP by whelks (the established species) on lobster (the
introduced species).
The exact outcome between the two populations in symmetric IGP will depend
on the flow rate to adulthood, the initial densities, the degree of IGP, and the
competition among predator 1 and predator 2. Species 1 can exclude species 2 if its
predation rates are higher than the predation rate of species 2. This will lead to fewer
juveniles of species 2, and hence less numbers of adults of species 2. With less adult of
species 2, juveniles of species 2 will increase which in turn will increase the amount of
adults of species 1. This will further decrease and eventually eliminate species 2.
Despite having a lower predation rate than species 1, coexistence might be
achieved if species 2 is more efficient competitor in exploiting the common resource.
This will lead to less juveniles of species 1, and hence less adults of species 1. Less
adults to species 1 lead to less predation on juveniles of species 2 which lead to
21
coexistence. Even though species 1 decreases more through predation relative to
predation of species 2 on species 1, species 2 can reduce the number of adults of species
1 via competition. This can lead to coexistence, either stable or unstable.
2.1.4
Other interactions
The other category of interactions between species is mutualism. Mutualism is a
biological interaction between two organisms, where each individual derives a fitness
benefit (i.e. increased survivorship). Similar interactions within a species are known as
co-operation. It can be contrasted with interspecific competition, in which each species
experiences reduced fitness, and exploitation, in which one species benefits at the
expense of the other where both species benefit from their interactions. For example,
the relationship between insects and plants. Most insects eat plants and live closely
interconnected with plants. Both insects and plants influence each other in the process
of the diversity (coevolution). For example, the mutualistic interactions between figs
(genus Ficus, Moraceae) and the fig-pollinating wasps (family Agaonidae) are
extremely interesting. They share a highly developed pollination system. Fig flowers
are gathered and hidden in the organ (syconium) like a fruit. For the figs, the pollinating
wasps deliver the pollen through the small hole (ostiole) in the pointed end of the
syconium which is called a tomb flower. In return, the wasps receive some seeds of the
figs as food for their larvae.
In ecology, Commensalism is a class of relationship between two organisms
where one organism benefits but the other is unaffected. An example of commensalism
would be birds following army ant raids on a forest floor. As the army ant colony
22
travels on the forest floor, they stir up various flying insect species. As the insects flee
from the army ants, the birds following the ants catch the fleeing insects. In this way,
the army ants and the birds are in a commensal relationship because the birds benefit
while the army ants are unaffected. The D. folliculorum mites living in your eyelash
follicles share a similar relationship with you. Orchids and mosses are plants that can
have a commensal relationship with trees. The plants grow on the trunks or branches of
trees. They get the light they need as well as nutrients that run down along the tree. As
long as these plants do not grow too heavy, the tree is not affected.
Amensalism between two species involves one impeding or restricting the
success of the other without being affected positively or negatively by the presence of
the other. It is a type of symbiosis. Usually this occurs when one organism exudes a
chemical compound as part of its normal metabolism that is detrimental to another
organism. The bread mold Penicillium is a common example of this; penicillium
secretes penicillin, a chemical that kills bacteria. A second example is the black walnut
tree (Juglans nigra), which secrete juglone, a chemical that harms or kills some species
of neighbouring plants, from its roots. This interaction may still increase the fitness of
the non-harmed organism though, by removing competition and allowing it access to
greater scarce resources. In this sense the impeding organism can be said to be
negatively affected by the other's very existence, making it a +/- interaction. A third
simple example is when sheep or cattle make trails in grass that they trample on, and
without realizing, they are killing the grass.
23
2.2
Simple Models of Epidermics
The outbreak and spread of disease has been questioned and studied for many
years. The ability to make predictions about diseases could enable scientists to evaluate
inoculation or isolation plans and may have a significant effect on the mortality rate of a
particular epidemic. The modeling of infectious diseases is a tool which has been used
to study the mechanisms by which diseases spread, to predict the future course of an
outbreak and to evaluate strategies to control an epidemic (Delay and Gani, 2005).
The first scientist who systematically tried to quantify causes of death was John
Graunt in his book Natural and Political Observations made upon the Bills of Mortality,
in 1662. The bills he studied were listings of numbers and causes of deaths published
weekly. Graunt’s analysis of causes of death is considered the beginning of the “theory
of competing risks” which according to Daley and Gani (2005) is “a theory that is now
well established among modern epidemiologists”.
The earliest account of mathematical modeling of spread of disease was carried
out in 1766 by Daniel Bernoulli. Trained as a physician, Bernoulli created a
mathematical model to defend the practice of inoculating against. The calculations from
this model showed that universal inoculation against smallpox would increase the life
expectancy from 26 years 7 months to 29 years 9 months (Bernoulli and Blower, 2004).
Following Bernoulli, other physicians contributed to modern mathematical
epidemiology. Among the most acclaimed of these were A. G. McKendrick and W. O.
Kermack, whose paper A Contribution to the Mathematical Theory of Epidemics was
24
published in 1927. A simple deterministic (compartmental) model was formulated in
this paper. The model was successful in predicting the behavior of outbreaks very
similar to that observed in many recorded epidemics (Brauer and Castillo-Chavez,
2001).
2.2.1 Types of Epidermics Models
There are two types of epidermics models. First, we call “Stochastic” which
means being or having a random variable. A stochastic model is a tool for estimating
probability distributions of potential outcomes by allowing for random variation in one
or more inputs over time. Stochastic models depend on the chance variations in risk of
exposure, disease and other illness dynamics. They are used when these fluctuations are
important, as in small populations (Trottier and Philippe, 2001).
Second term is what we call “Deterministic”. When dealing with large
populations, as in the case of tuberculosis, deterministic or compartmental mathematical
models are used. In the deterministic model, individuals in the population are assigned
to different subgroups or compartments, each representing a specific stage of the
epidemic. Letters such as M , S , E , I , and R are often used to represent different
stage. The transition rates from one class to another are mathematically expressed as
derivatives, hence the model is formulated using differential equations. While building
such models, it must be assumed that the population size in a compartment is
differentiable with respect to time and that the epidemic process is deterministic. In
other words, the changes in population of a compartment can be calculated using only
the history used to develop the model (Brauer and Castillo-Chavez, 2001).
25
2.2.2 SIR Model
In 1927, W. O. Kermack and A. G. McKendrick created a model in which they
considered a fixed population with only three compartments, susceptible: S which is
used to represent the number of individuals not yet infected with the disease at time t, or
those susceptible to the disease, infected, I denotes the number of individuals who
have been infected with the disease and are capable of spreading the disease to those in
the susceptible category and lastly, removed, R is the compartment used for those
individuals who have been infected and then recovered from the disease. Those in this
category are not able to be infected again or to transmit the infection to others.
The flow of this model may be considered as follows:
S → I → R
Using a fixed population, M = S + I + R , Kermack and McKendrick derived the
following equations:
dS
= − βSI
dT
dI
= βSI − γI
dT
dR
= γI
dT
where β and γ represent the contact rate and recovery rate, respectively. Several
assumptions
were
made
in
the
formulation
of
these
equations:
First, an individual in the population must be considered as having an equal probability
26
as every other individual of contracting the disease with a rate of β , which is
considered the contact or infection rate of the disease. Therefore, an infected individual
makes contact and is able to transmit the disease with βN others per unit time and the
fraction of contacts by an infected with a susceptible is S N . The number of new
infections in unit time per infective then is ( βN )(S N ) , giving the rate of new infections
(or those leaving the susceptible category) as ( βN )(S N )I = βSI (Brauer and CastilloChavez, 2001). For the second and third equations, consider the population leaving the
susceptible class as equal to the number entering the infected class. However, a number
equal to the fraction ( 1γ the mean infective period) of infectives are leaving this class
per unit time to enter the removed class. These processes which occur simultaneously
are referred to as the Law of Mass Action, a widely accepted idea that the rate of
contact between two groups in a population is proportional to the size of each of the
groups concerned (Delay and Gani, 2005). Finally, it is assumed that the rate of
infection and removal is much faster than the time scale of births and deaths and
therefore, these factors are ignored in this model.
2.2.3
SIS Model
The SIS model can be easily derived from the SIR model by simply
considering that the individuals recover with no immunity to the disease, that is,
individuals are immediately susceptible once they have recovered. The flow of this
model may be considered as follows:
S → I
→ S
27
Removing the equation representing the recovered population from the SIR
model and adding those removed from the infected population into the susceptible
population gives the following differential equations:
dS
= − βSI + μ (M − S ) + γI
dT
dI
= βSI − γI − μI
dT
Now, again β and γ represent the contact rate and recovery rate, respectively. The
addition notation μ denotes the mortality rate. If γ = 0 , which means that no recovery
from the infective to susceptible population, the model becomes a simple SI model.
The well-known and most researched interactions are predation and competition.
In IGP, species compete for the same resource and at the same time prey on their
competitor. IGP is different from pure competition, due to the immediate energetic gain
from predation for one participant, and it is different from pure competition, because of
the act of predation reduces potential competitor. Holt and Polis (1992) incorporated
IGP into standard models of competition and predation.
IGP for unstructured environment leads to varieties of outcomes to the
ecosystem. These include extinction of species, coexistence, priority effect and stable
limit cycle. For IGP in age-structured population model, Mustapha (2001) used
difference equations models showed that IGP can lead to a variety of outcomes;
extinction, coexistence, synchronous 2-cycles and chaos. For IGP with size-structured
population model, Mustapha has used integro-partial differential equations. From the
simulations of the model (integrated using escalator boxcar train (EBT) package (De
Roos, 1998)), it is shown that less efficient species can persists by switching to IGP.
28
The effects of disease on competing species have been studied by Hochberg and
Holt (1990), and Venturino (2001). System where one disease-free species competes
with another host which is infected by epidemics is considered by Begon and Bowers
(1995). Anderson and May (1986) considers two competitors, one of which is effected
by a disease, which assumed to annihilate the reproductive rate of the infected
individuals. The possiblity that an infection of superior competitor favours coexistence
with another one, which otherwise be wiped out, is inferred from the study.
Investigations on predator-prey system with disease are presented by Venturino (1994),
Holt and Pickering (1986), and Hadeler and Freedman (1989).
IGP is common among many species, especially the freshwater and marine
invertebrates (Polis and Holt, 1992), so it is of great importance to study the effect of
disease on the dynamics of the IGP populations. Furthermore predator usually preys on
smaller individuals, thus it is more appropriate to include size-structured in the model.
29
CHAPTER 3
METHODOLOGY
3.1
Relating Dynamical Systems to Biological Models
Dynamical systems approaches are essential for understanding ecological food
webs. As has been increasingly recognized, the traditional approach of examining
conditions leading to stable equilibria is not sufficient for understanding what allows
species to coexist in natural systems. There are many open questions concerning the
nonlinear dynamics of interacting species. Another related problem that will require
similar approaches understands the consequences of invading species for the
ecosystems they enter. As the effects of these species cascade through the ecosystem,
the nonlinear effects have the potential of leading to surprising consequences, and
predictions of the effects will again require the tools of dynamical systems.
Unification of genetics and natural selection was achieved, most successfully,
in the thirties, through a dynamical system model. Ever since, the use of Dynamical
System ideas in analyzing ecological and biological models has led to a remarkable
30
improvement of our understanding of their evolution. Besides the now classical
studies of limit cycles in Lotka-Volterra type equations, and the use of homoclinic or
heteroclinic cycles (a notion originated from Celestial Mechanics) in the description of
invasion processes, we could mention more recent applications of results from chaotic
dynamics (strange attractors) in modeling the evolution of animal populations. These
ideas are showing to be equally useful in understanding natural evolution, at the level
of both species and molecules (Shuster, 2001).
3.2
Steady-State and Equilibrium Point
A system in a steady state has numerous properties that are unchanging in time.
The concept of steady state has relevance in many fields, in particular thermodynamics.
Steady state is a more general situation than dynamic equilibrium. If a system is in
steady state, then the recently observed behavior of the system will continue into the
future. In stochastic systems, the probabilities that various different states will be
repeated will remain constant. In many systems, steady state is not achieved until some
time has elapsed after the system is started or initiated. This initial situation is often
identified as a transient state, start-up or warm-up period.
While a dynamic equilibrium occurs when two or more reversible processes
occur at the same rate, and such a system can be said to be in steady state, a system that
is in steady state may not necessarily be in a state of dynamic equilibrium, because
some of the processes involved are not reversible. For example, the flow of fluid
through a tube, or electricity through a network, could be in a steady state because there
is a constant flow of fluid, or electricity. Conversely, a tank which is being drained or
31
filled with fluid would be an example of a system in transient state, because the volume
of fluid contained in it changes with time.
3.3
Eigenvalue and Stability
Detection of stability in these models is not that simple as in one-variable
models. Let's consider a predator-prey model with two variables: (1) density of prey and
(2) density of predators. Dynamics of the model is described by the system of 2
differential equations:
⎧ dN
⎪⎪ dT = f ( N , P )
⎨
⎪ dP = g ( N , P )
⎪⎩ dT
This is the 2-variable model in a general form. Here, N is the density of prey,
and P is the density of predators. The first step is to find equilibrium densities of prey
N * and predator P * . We need to solve a system of equations:
⎧⎪ f ( N , P ) = 0
⎨
⎪⎩ g ( N , P ) = 0
(
)
The second step is to linearize the model at the equilibrium point N = N * , P = P* by
estimating the Jacobian matrix:
32
⎡ df
⎢ dN
J =⎢
⎢ dg
⎢ dN
⎣
df ⎤
dP ⎥
⎥
dg ⎥
dP ⎥⎦
Third, eigenvalues of matrix J should be estimated. The number of eigenvalues
is equal to the number of state variables. In our case there will be two eigenvalues.
Eigenvalues are generally complex numbers. If real parts of all eigenvalues are
negative, then the equilibrium is stable. If at least one eigenvalue has a positive real
part, then the equilibrium is unstable.
Eigenvalues are used here to reduce a 2-dimensional problem to a couple of 1dimensional problem problems. Eigenvalues have the same meaning as the slope of a
line in phase plots. Negative real parts of eigenvalues indicate a negative feedback. It is
important that all eigenvalues have negative real parts. If one eigenvalue has a positive
real part then there is a direction in a 2-dimensional space in which the system will not
tend to return back to the equilibrium point.
3.4
Ruth-Hurwitz Stability Criteria
The Routh-Hurwitz criterion is a method for determining whether a linear
system is stable or not by examining the locations of the roots of the characteristic
equation of the system. In fact, the method determines only if there are roots that lie
outside of the left half plane; it does not actually compute the roots.
33
In this section we briefly touch on methods for gaining insight into models for k
species interacting in a community, where k > 2 . The models we have seen thus take
the form
dN 1
= f ( N1 , N 2
dt
)
dN 2
= g ( N1 , N 2
dt
)
More generally a system comprising of k species with populations N1 , N 2 ,KK , N k is
being governed by k equations
dN 1
= f 1 ( N 1 , N 2 ,KK , N k ),
dt
dN 2
= f 2 ( N 1 , N 2 ,KK , N k ),
dt
M
dN k
= f k ( N 1 , N 2 ,KK , N k ).
dt
Since it is cumbersome to carry this longhand version, one often sees the shorthand
notation
dN i
= f i ( N 1 , N 2 ,KK , N k )
dt
(i = 1, 2 ,KK , k )
or, better still, the vector notation
dN
= F (N ) ,
dt
(3.1)
34
for N = ( N 1 , N 2 ,KK , N k ) , F = ( f1 , f 2 ,KK , f k ) , where each of the functions
f1 , f 2 ,KK , f k may depend on all or some of the species populations N 1 , N 2 ,KK , N k .
We shall now suppose that it is possible to solve the equation (or set of
equations)
F(N ) = 0
So as to identify one (or possibly several) steady-state points, N = ( N 1 , N 2 ,KK , N k ) ,
( )
satisfying F N = 0 . The next step, as the diligent reader might have guessed, would be
to determine stability properties of this steady solution. In linearizing equation (3.1) we
find, as before, the Jacobian of F(N ) . This is often symbolized
J=
( )
∂F
N
∂N
Recall that this really means
⎛ ∂f1
⎜
⎜ ∂N 1
⎜
J =⎜ M
⎜
⎜ ∂f k
⎜ ∂N
⎝ 1
∂f 2
∂N 2
∂f k
∂N 2
∂f k ⎞
⎟
∂N k ⎟
⎟
⎟ ,
⎟
∂f k ⎟
L
∂N k ⎟⎠ N
L
so that J is now a k × k matrix. Population biologists frequently refer to J as the
community matrix. Eigenvalues λ of this matrix now satisfy
det (J − λI ) = 0
We should arrive at the conclusion that λ must satisfy a characteristic equation of the
form
35
λk + a1 λk −1 + a 2 λk − 2 + K K + a k = 0
(3.2)
In general, the characteristic equation is a polynomial whose degree k is equal
to the number of species interacting. Although for k = 2 the quadratic characteristic
equation is easily solved, for k > 2 this is no longer true.
While we are unable in principle to find all eigenvalues, we can still obtain
information about their magnitudes. Suppose λ1 , λ2 ,KK , λk are all (known) eigenvalues
of the linearized system
dN
= J ⋅ N.
dt
All eigenvalues must have negative real parts since close to the steady states each of the
species population can be represented by a sum of exponentials in λi t as follows:
N i = N i + a1e λ1t + a 2 e λ2t + LL + a k e λk t
If one or more eigenvalues have positive real parts, N i − N i will be an increasing
function of t , meaning that N i will not return to its equilibrium value N i . Thus the
equation of stability of a steady state can be settled if it can be determined whether or
not all eigenvalues λ1 , λ2 ,KK , λk have negative real parts. This can be done without
actually solving for these eigenvalues by checking certain criteria. For k > 2 these
conditions are known as the Ruth-Hurwitz criteria.
Given the characteristic equation (3.2) define k matrix as follows:
36
⎛a
H 2 = ⎜⎜ 1
⎝ a3
H1 = (a1 ) ,
⎛ a1
⎜
⎜ a3
Hj =⎜
a
⎜ 5
⎜a
⎝ 2 j −1
1
a2
a4
0
a1
a3
a 2 j −2
a 2 j −3
1⎞
⎟,
a2 ⎟⎠
⎛ a1
⎜
H 3 = ⎜ a3
⎜a
⎝ 5
1
a2
a4
0 L0⎞
⎛ a1
⎟
⎜
1 L0⎟
⎜ a3
LL H k = ⎜
⎟
a2 L 0
M
⎟
⎜
⎜0
a 2 j −4 L a j ⎟⎠
⎝
0⎞
⎟
a1 ⎟ , LLL
a3 ⎟⎠
1
a2
0 L
a1 L
M
0
M
L
⎞
⎟
⎟
⎟,
⎟
a k ⎟⎠
0
0
0
where the (l , m ) term in the matrix H j is
a 2l −m
for 0 < 2l − m < k
1
for 2l = m
0
for 2l < m or 2l > k + m
Then all eigenvalues have negative real parts: that is, the steady state N is stable if and
only if the determinants of all Hurwitz matrices are positive:
det H j > 0
( j = 1, 2 ,KK , k ) .
CHAPTER 4
INTRAGUILD PREDATION
4.1
Intraguild Predation Model
Let the densities of IG predator, IG prey and basal resources be p , n and z
respectively, then the IGP model proposed by Holt and Polis (1997) is as below
dp
dt
=
dn
dt
= n (abZ − dp − m )
dz
dt
⎡ ⎛
z⎞
= z ⎢r ⎜1 − ⎟ − an − a'
K⎠
⎣ ⎝
p (a' b' z + cd n − m' )
(4.1)
⎤
p⎥
⎦
38
a and a' are predation rates on the common resource for the IG prey and IG predator
respectively. IG predation rates on n is d and the density independent mortality rates
are m for IG prey and m' for IG predator. b , b' and c are the conversion rates of
predation into offsprings. r and K are the birth rate and the carrying capacity for the
common resource respectively.
To analyze the system, we non-dimensionalize the equation using
T = rt
β=
P=
cd
a
a'
p
r
ρ=
N=
abK
r
a
n
r
χ=
d
a'
Z=
z
K
δ=
α=
m'
r
a' b' K
r
τ=
m
r
and obtain
4.2
dP
dT
= P (α Z + β N − δ )
dN
dT
=
dZ
dT
= Z [1 − Z − N − P ]
N (ρ Z − χ P − τ )
(4.2)
Stability Analysis
A steady state is a situation in which the system does not appear to undergo any
change. We solved the system by letting the derivatives equal to zero, which mean that
39
the populations remain unchanged with respect to time, and then we got several
equilibrium points. The equilibrium points for the system (4.2) above are
1.
The trivial equilibrium point:
E0 ( P , N , Z ) = E0 ( 0,0,0
2.
)
Only basal resource exists:
E1 ( P , N , Z ) = E1 ( 0, 0,1 )
3.
The semi-trivial equilibrium point with IG predator excluded:
⎛ ρ−τ τ
E 2 ( P , N , Z ) = E 2 ⎜⎜ 0,
,
ρ ρ
⎝
4.
The semi-trivial with IG prey excluded:
E3 ( P , N ,Z
5.
⎞
⎟⎟
⎠
α−δ
δ ⎞
,0, ⎟
α ⎠
⎝ α
) = E 3 ⎛⎜
Coexistence equilibrium point:
(
E 4 P* , N * ,Z *
)= E
4
⎛ βρ + ατ − βτ − δρ χα + ατ − δρ − δχ βχ + βτ − δχ ⎞
,
,
⎜
⎟
C
C
C
⎝
⎠
where C = βρ + βχ − αχ .
The Jacobian for the system (4.2) is
⎡ αZ + β N − δ
⎢
J =⎢ −χN
⎢
⎣ −Z
βP
ρZ − χ P − τ
−Z
⎤
⎥
ρN
⎥
⎥
1 − 2Z − N − P ⎦
αP
The eigenvalues for the equilibrium point E0 are
λ1 = −δ
λ2 = − τ
λ3 = 1 .
Since we assume all the parameters in the system (4.2) are positive, the equilibrium
point E0 is unstable.
40
For the equilibrium point E1 , which only the basal resource exists, the
eigenvalues are
λ1 = α − δ
λ2 = ρ − τ
λ3 = −1 .
This equilibrium point is locally asymptotically stable if α < δ and ρ < τ .
The eigenvalues for the equilibrium point E2 are
βρ + ατ − βτ − δρ
λ1 =
ρ
λ2 ,3
− τ ± τ 2 − 4τρ( ρ − τ )
=
.
2ρ
We can say that the equilibrium point E2 is stable if βρ + ατ − βτ − δρ < 0 and ρ > τ .
For the equilibrium point E2 to be in the first quadrant, ρ > τ . If so, the equilibrium
point E1 is unstable.
α > δ is the necessary condition required for the equilibrium point E3 to be
positive. The eigenvalues for this equilibrium point are
δρ + δχ − αχ − ατ
λ1 =
α
λ2.3 =
− δ ± δ 2 − 4δα(α − δ )
.
2α
So, this equilibrium point E3 is stable if δρ + δχ − αχ − ατ < 0 and α > δ .
For the coexistence equilibrium point E4 , if C > 0 , it exists when
41
P*
> 0 ⇒
βρ + ατ − βτ − δρ > 0
N*
> 0 ⇒
χα + ατ − δρ − δχ
Z*
> 0 ⇒
βχ + βτ − δχ
> 0
> 0
If C < 0 , then the inequality sign above are reversed. At the equilibrium point E4 the
Jacobian evaluated is
⎡ 0
⎢
J = ⎢ χ N*
⎢
⎢⎣ − Z *
β P*
0
− Z*
α P* ⎤
⎥
− ρ N* ⎥
⎥
− Z * ⎥⎦
The characteristic polynomial is λ3 + A1 λ2 + A2 λ + A3 = 0 , with
A1
= Z*
A2
=
A3
= N * P* Z *
ρ N*Z*
+
β χ N * P*
[ β ( χ + ρ)
+ α P* Z *
− αχ
]
According to Routh-Hurwitz criterion [27], equilibrium point E5 is stable if Ai > 0 ,
i = 1, 2 ,3 and A1 A2 − A3 > 0 .
Assuming that P * , N * and Z * are positive, we can see that A2 > 0 . When we
substitute P * , N * and Z * in A1 and A3 ,
A1 =
β χ −δ χ + βτ
and
C
A3 > 0
iff
β χ −δ χ + βτ > 0
42
A1 , A2 and A3 are positive iff C > 0 and P * , N * and Z * are
which mean that
positive. To satisfy the second condition of the Routh-Hurwitz criteria, we must have
B 0 ρ 3 + B1 ρ 2 + B 2 ρ + B3 > 0
with
B0 = − β δ ( β − δ)
B1 = − χ δ 2 − β 2 χ δ + β δ τ + β 2 δ τ + β 2 χ α + β χ δ + β δ 2 χ − χ δ 2 α + β 2 α τ − 2 β α τ δ
B2 = − α 2 χ 2 β + χ δ τ β + α χ 2 δ + β 2 χ δ τ + χ 2 β δ − 2α β χ δ + α β 2 τ + α β 2 χ − α β τ δ
+ α χ δ 2 + α 2 χ 2 δ − α χ 2 β + α β χ 2 δ − 2α β τ δ − α β 2 τ 2 − α τ 2 β + 2α 2 χ τ δ + α 2 β τ 2
− 2β α χ τ − α β 2 χ τ − α χ 2δ 2 − χ 2δ 2 + α τ χ δ
(
B3 = α τ (α − γ) β χ + β τ − χ 2 α − α χ τ − χ δ + χ 2 δ
)
Fixing τ = δ = 0.4 , χ = 1.0 and α = 0.5 , we have the stability region for
equation (4.2) in Figure 4.1. When ρ is low, which means that IG prey is not superior
in exploiting the shared resources, it will be excluded from the system (region I). When
IG prey is superior in exploiting the shared resources and the benefit of predation on IG
prey is small for IG predator, then the IG predator will be excluded from the system
(region II). In this region, IGP by the IG predator is too low to offset the competition. In
region III, the coexistence rest point is stable. Further increase of ρ and β result in the
coexistence rest points losing its stability such in region IV. In region V, both semitrivial rest points E2 and E3 are stable. In this region, there is a priority effect, in which
the outcome will depend on the initial condition.
43
IV
III
I
II
V
Figure 4.1:
The stability region in a ρ and β parameter space when τ = δ = 0.4 ,
χ = 1.0 and α = 0.5 .
In this predator prey with IGP system, we can have either two stable semi-trivial
rest points or a stable coexistence point. The additional new feature is the possibility of
oscillatory coexistence, reminiscent of a Lotka-Volterra predation prey situation.
CHAPTER 5
INTRAGUILD PREDATION WITH DISEASE
5.1
Intraguild Predation Model with Disease
In the Case I, n I is the density of the infected IG prey, and n S is the density of
the susceptible IG prey. The model is as follow
dp
dt
=
p (a' b' z + cdnS − m' − hn I )
dn I
dt
= n I ( λ n S − ep − γ )
dn S
dt
= n S (abz − dp − m − λ n I ) + γ n I
dz
dt
⎡ ⎛
⎤
z⎞
= z ⎢r ⎜1 − ⎟ − a ( n S + n I ) − a' p ⎥
K⎠
⎣ ⎝
⎦
(5.1)
The habitat for the preys is assumed to be unlimited, so that in absence of IG predator
the IG prey will reproduce exponentially. Infected IG consumes basal resource but they
assumed do not reproduce. All the coefficients in the model will be positive real
45
numbers. In this model, all letters are denote the same explanations as in the IGP model
stated above with the exception of parameter n which is divided into two different
species. They are nS represents the susceptible IG prey and n I represents the infected
IG prey. The coefficients related to the disease represented by e , denotes the value of
predation upon infected prey, γ represents the recovery rate from infective to
susceptible and we assume that the predation rates on the common resources among the
infected and susceptible IG preys remain the same. h is the conversion rates of
predation upon infected prey into offspring. The constant λ is leading to the simple
mass action among the IG prey.
Using the substitutions below,
T = rt
P=
α=
a' b' K
r
θ=
a'
p
r
β=
γ
r
ρ=
NS =
cd
a
abK
d
χ=
r
a'
a
nS
r
υ=
NI =
h
a
ε=
δ=
a
nI
r
λ
a
m'
r
Z=
ω=
τ=
z
K
e
a'
m
r
we non-dimensionalized the system into
dP
dT
= P (α Z + β N S − δ − υ N I )
dN I
dT
=
dN S
dT
=
dZ
dT
N I (ε N S − ω P − θ )
(5.2)
N S (ρ Z − χ P − τ − ε N I ) + θ N I
= Z (1 − Z − N S − P − N I )
46
We will consider at first only the SI model which mean that infective IG prey will not
recover from the disease, i.e suppose γ = 0 , then θ =
γ
= 0 if a ≠ 0 . Thus the system
a
(5.2) becomes
dP
dT
= P (α Z + β N S − δ − υ N I )
dN I
dT
=
dN S
dT
=
dZ
dT
N I (ε N S − ω P )
(5.3)
N S (ρ Z − χ P − τ − ε N I )
= Z (1 − Z − N S − P − N I )
The equilibrium points for the system (5.3) are
1.
The trivial point:
E0 ( P , N I , N S , Z
2.
)
= E0 ( 0,0,0,0
)
The semi trivial points:
E1 ( P , N I , N S , Z
⎛
ρ−τ τ
,
= E1 ⎜⎜ 0,0,
ρ ρ
⎝
)
E2 ( P , N I , N S , Z
= E 2 ( 0, N I ,0,0
)
E3 ( P , N I , N S , Z
)
= E3 ( 0, N I , 0,1 − N I
E4 ( P , N I , N S ,Z
)
=
E5 ( P , N I , N S ,Z
with
)
)
=
⎞
⎟⎟
⎠
)
δ ⎞
⎛ α−δ
E4 ⎜
,0 ,0 , ⎟
α ⎠
⎝ α
∧
∧
⎛ ∧
⎞
E 5 ⎜ P ,0 , N S , Z ⎟
⎝
⎠
47
ρ ( β − δ ) + τ (α − β )
B
∧
P =
∧
=
NS
∧
β (χ + τ) −
B
=
Z
δ ( χ + ρ) − α ( χ + τ )
B
where B =
3.
χδ
β ( χ + ρ) −
χα
The coexistence point
E6 ( P , N I , N S , Z )
=
(
*
*
E6 P* , N I , N S , Z *
)
where
=
P*
NI
*
=
*
=
NS
Z*
=
with D*
ε [ δ ( ρ + ε ) + υ ( ρ − τ ) − α (τ + α ) ]
D
ω [ τ (α − β ) +
ρ (β − δ ) ] + ε [ α ( χ + τ ) − δ (ρ + χ ) ]
D
ω [ δ ( ρ + ε ) + υ ( ρ − τ ) − α (τ + α ) ]
D
ω [ τ ( β + υ) + ε ( β − δ ) ] + ε [ δ ( χ − ε ) − υ ( χ + τ ) ]
D
=
ω [ ρ ( β + υ) + ε ( β − α ) ] + ε [ α ( χ − ε ) + υ ( ρ + χ ) ] .
48
5.2
Stability Analysis
The Jacobian for system (5.3) is
⎡ αZ + βN S − δ − υN I
⎢
ωN I
⎢
J =⎢
− χN S
⎢
⎢
−Z
⎣
− υP
βP
εN S − ωP
εN I
− ε NS
ρZ − χP − τ − ε N I
−Z
⎤
⎥
0
⎥
⎥
ρN S
⎥
⎥
1 − 2Z − N S − P − N I ⎦
αP
−Z
The trivial equilibrium point E1 is positive if and only if ρ > τ . The
eigenvalues are
λ1 =
ρ ( β − δ ) + τ (α − β )
ρ
λ2 =
ε(ρ − τ )
ρ
λ3,4 =
− τ ± τ 2 − 4τρ( ρ − τ )
2ρ
According to its’ eigenvalues, trivial point, E1 is stable if ρ ( β − δ ) < τ ( β − α ) and
ρ < τ . But, if ρ < τ , contradiction occur when we stated that the trivial point have to
be positive. Thus, this point is unstable.
The eigenvalues for the semi trivial point, E 2 are
λ1 = − ( δ + υ N I )
λ 2 = − (τ + ς N I )
λ3 = 0
λ4 = 1 − N I
49
All eigenvalues have to be negative real part in order to state an equilibrium point to
stable. Since there is a value of λ does not satisfied the condition, this equilibrium point
E2 is unstable.
Proceed to the semi trivial point E3 . This equilibrium point is positive if and
only if N I < 1 and all the eigenvalues that we get are
λ1 = α − δ − N I (α + υ )
λ2 = 0
λ3 = ρ − τ − N I ( ρ + ε )
λ4 = N I − 1
At this point, again exist λ2 = 0 thus we said this semi trivial point E3 is unstable for
the same reason as previous point.
The semi trivial point, E 4 is positive if α > δ . The eigenvalues for the semi
trivial point E 4 are
ω(δ − α )
λ1 =
α
δ ( ρ + χ ) − α (τ + χ )
λ2 =
α
λ3,4 =
−δ ±
(δ )2 − 4αδ(α − δ )
2α
At this equilibrium point, we can say that this equilibrium point is stable if all the
eigenvalues are negative real part satisfying two conditions, which are
δ ( ρ + χ ) − α (τ + χ ) < 0
and
α >δ.
For equilibrium point E5 , if B > 0 then the equilibrium point is positive if
50
∧
ρ ( β − δ ) + τ (α − β ) > 0
P > 0 ⇒
∧
> 0 ⇒ δ( χ + ρ ) − α( χ + τ ) > 0
NS
∧
β( χ + τ ) − δ χ
> 0 ⇒
Z
> 0
The inequality signs above is reversed if B < 0 . Since this semi trivial point seems like
much tedious and the eigenvalues are not easy to analyze, we analyze this point
according to its characteristic polynomial. The Jacobian evaluated at this equilibrium
point is
⎡
⎢
⎢
⎢
J5 = ⎢
⎢
⎢
⎢
⎢
⎣⎢
∧
∧
⎤
αP ⎥
⎥
0 ⎥
⎥
⎥
∧
ρ NS ⎥
⎥
∧ ⎥
− Z ⎦⎥
∧
−υP
0
βP
*
−
0
∧
NI D
B
0
∧
− ε NS
χ NS
∧
0
∧
−Z
∧
−Z
−Z
∧
∧
∧
∧
The characteristic polynomial is λ4 + A1 λ3 + A2 λ2 + A3 λ + A4 = 0 where
∧
A1
∧
A2
∧
A3
∧
A4
*
=
NI D
B
=
NI D Z
B
=
NI D ⎛ ∧ ∧
⎜ρZ N
B ⎝
*
∧
+ Z
∧
+
∧ ∧
ρZ N
*
*
∧ ∧ ∧
= NI D Z P N
+
+
∧ ∧
β χPN
∧ ∧
β χPN
∧ ∧
+ αZ P
∧ ∧
∧ ∧ ∧
⎞
+ α Z P⎟ + B Z N P
⎠
51
∧
According to Ruth-Hurwitz criteria, equilibrium points are stable if Ai > 0 , i = 1,3, 4
∧
∧
∧
∧ 2
∧ 2
∧
and A1 A2 A3 − A3 − A1 A4 > 0 . Since the characteristic polynomial is very tedious,
thus we let numerical examples in order to analyze it. In order to do so, we set two
different numerical examples. For the first example, we let
α = 0.5 , β = 1.5 , χ = 0.6 , ω = 0.6 , τ = 0.4 , δ = 0.4 , ε = 0.9 , υ = 0.8 ,
ρ = 0.4
By fixing those values, we can see that the semi trivial equilibrium point
E 4 (P , N I , N S , Z ) = E 4 (0.2, 0,0, 0.8)
and its eigenvalues are
λ1 = −0.12 , λ2 = −0.2 , λ3 = −0.117 and λ4 = −0.683
which are all negative, thus this point is stable since we satisfied the condition ρ > 0.65
52
I
Figure 5.1:
II
Stability region in ρ and β parameter space when the other parameter is
fixed.
In region I, when ρ is low which means that susceptible IG prey is not superior in
exploiting shared resources, then it can be exclude from the system. In region II,
although susceptible IG prey is superior in exploiting the shared resources, independent
on benefit of predation on IG susceptible prey, β , IG predator still can survive because
the parameter related to the disease for IG prey is high.
53
time
Figure 5.2:
Density of each species versus time when α = 0.5 , β = 1.5 , χ = 0.6 ,
ω = 0.6 , τ = 0.4 , δ = 0.4 , ε = 0.9 , υ = 0.8 , ρ = 0.4
From the Figure 5.2, we can see that when the system is steady, only IG predator can
survive where as IG prey can be excluded.
For the semi trivial point E5 , we let new fixed parameter which are
α = 0.3 , β = 0.8 , χ = 0.7 , ω = 1 , τ = 0.4 , δ = 0.4 , ε = 0.5 , υ = 0.5 , ρ = 1
and we analyze this point according to Ruth-Hurwitz criteria. This equilibrium point is
^
^
^
stable if it satisfies both two conditions. The first condition, A1 , A3 , A4 > 0 and the
∧
∧
∧
∧ 2
∧ 2
∧
second condition we have A1 A2 A3 − A3 − A1 A4 > 0 .
54
With those fixed parameter values, we get the semi trivial equilibrium
E5 ( P , N I , N S , Z
)
= E5 ( 0.174, 0,0.304,0.522
)
and its eigenvalues are
λ1 = −0.209 , λ2 = −0.022 , λ3 = −0.154 + 0.357i and λ4 = −0.154 − 0.357i .
We can see that those eigenvalues are positive real parts. Thus we can say that this point
is stable. According to Ruth-Hurwitz criteria, the first criterion is satisfied since
^
^
^
A1 = 0.54 , A3 = 0.0364 and A4 = 0.000696
are all positive. For the second condition, we have
∧
∧
∧
∧ 2
∧ 2
∧
A1 A2 A3 − A3 − A1 A4 = 0.00296 .
Since both conditions are satisfied, thus, with those fixed parameters, this semi trivial
E5 is stable, as shown in Figure 5.3.
55
time
time
Figure 5.3:
Density of each species versus time when α = 0.3 , β = 0.8 ,
ω = 1 , τ = 0.4 , δ = 0.4 , ε = 0.5 , υ = 0.5 , ρ = 1
For the coexistences point E6 , fixed parameter chosen are
α = 0.4 , β = 0.7 , χ = 0.5 , ω = 0.3 , τ = 0.4 , δ = 0.3 , ε = 0.3 , υ = 0.9 ,
ρ = 0.5
From those fixed parameter we get coexistence point
E6 ( P , N I , N S , Z )
= E 6 ( 0.0267 , 0.0749,0.0267 ,0.872
)
and its eigenvalues are
λ1 = −0.844 , λ2 = 0.00246 , λ3 = −0.0241 + 0.026i and λ4 = −0.0241 − 0.026i .
56
Since there exist one eigenvalue that is positive thus we can simply state that this
coexistence is unstable.
CHAPTER 6
CONCLUSIONS AND DISCUSSIONS
6.1
Introduction
This chapter summarizes the analysis that was done on the stability of the
equilibriums throughout the entire project. Recommendations are further given for
consideration for other researchers to be done in the future.
6.2
Conclusions and Discussions
Ecosystem stability is an important issue in conservation of biodiversity. The
stability of predator-prey systems or competitive systems has been studied extensively.
However, natural communities are far more complex than those simple ecosystems.
58
Intraguild predation (IGP) presents a sound example of a complex ecosystem
with both competition and predation. The 3-species ecosystem with IGP is the simplest
such complex ecosystem.
In chapter 4, we analyzed IGP model without disease. From the region of the
stability in the ρ and β parameter space in Figure 4.1, we can see that
1.
Coexistence occurs when the IG prey is superior at exploiting competition for
the shared resource. In region III, the coefficient which represents competition
for the shared resource for IG prey is β , which is greater than α = 0.5 .
2.
Potential for alternative stable states in the IGP system are most likely if the IG
predator gains little benefit from consuming the IG prey. This can be seen in
region V, where the IG predator gains little benefit from consuming the IG prey;
ρ is low.
3.
As before, IG predator removal will lead to a depression in resource levels.
In chapter 5, we have constructed a new model of IGP by adding a disease into
predator-prey with IGP system. The model seems much complicated but the stability
analysis was worst. The model analyzed in this research is
59
dP
dT
= P (α Z + β N S − δ − υ N I )
dN I
dT
=
N I (ε N S − ω P )
dN S
dT
=
N S (ρ Z − χ P − τ − ε N I )
dZ
dT
= Z (1 − Z − N S − P − N I )
This model has six equilibrium points. E1 , E2 , and E3 are theoretically proved
that they are unstable by analyzing their eigenvalues. E4 , E5 and E6 are stable under
certain condition. We analyzed the last three equilibrium point by applying RuthHurwitz criteria. Since the stability analysis for these equilibrium points was very
complicated, we try to fix the values of parameters.
In this chapter, we introduced different numerical examples so that we can
found any point that the equilibrium points are stable. In this system no limit cycle
occur. For the first example, only E4 is stable. This equilibrium point depends on value
of ρ . This point only stable when ρ > 0.65 and there exist IG predator and the basal
resource only. Even though susceptible IG prey superior in exploiting basal resource,
they cannot survive. IG predator still can survive because the conversion rate into
offspring by consuming IG prey is high and also the parameter related to disease for IG
prey is high. This leads to the extinction of infected IG prey.
For the second example, only E 5 is stable. IG prey superior exploiting the basal
resource. IG predator can survive because the benefit of predation on IG prey is high. In
this example, ω , the predation rate upon infected IG prey is high, thus infected IG prey
cannot survive since we have the assumption that infected IG prey do not reproduce
when consuming the basal resource and the contact rate among IG prey is lower.
60
In the last example, we get E4 is stable since ρ > 0.65 . But we want to focus at
the coexistence equilibrium point E6 . The point E6 are unstable at any example
introduced. It is so difficult to find the stability region for this point because this
coexistence point is so complicated. With this fixed parameter, E6 is unstable since
there exist eigenvalue which positive.
6.3
Recommendations
For future research, we can try to sketch the stability region for the model
proposed in this research so that we can easily determine the stability point for each
equilibrium point. On the other hand, we also can extend this research by introducing a
new parameter which is recovery rate by considering the SIS model.
61
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