Download chapter 1

Chapter 1
GENERAL INTRODUCTION
Community ecology is often perceived as a “mess", given the seemingly vast number of
process that can underlie the many patterns of interest, and the apparent uniqueness of
each study system. However, at the most general level, patterns in the composition and
diversity of species - the subject matter of community ecology – are influenced by only
four classes of process: selection, drift, speciation, and dispersal.
Vellend (2010)
1.1 INTRODUCTION AND BASIC CONCEPTS
Ecology describes the relationships among organisms, and between them and the
environment, thus determining aspects such as species abundance, compositions of
biological community that represents a group of organisms living in a specified place and
time. The word ‘Ecology’ has attracted attention of scientists and philosophers from the
ages of human civilization. It was first defined by the German biologist Ernst Haeckel in
1869; According to him ecology is the science of interrelation between living organisms
and their environment. The word ‘ecology’ owes its origin to the Greek word ‘Oikos’
meaning ‘house’ or ‘place to live’ (Tansley, A.G.,1946).
In ecology, biological interactions or the relationships between two species in an
ecosystem can be categorized into many different classes based either on the effects or on
the mechanism of the interaction and can be classified as:
i.
Neutralism describes as the relationship between two species which do interact
but do not affect each other.
1
ii.
Competition is an interaction between two species that is mutually detrimental.
Competition between organisms occurs in case of limited supply of essential
resource, such as food, in case of animals and water, nutrients and light in the case
of plants.
iii.
Predation is the consumption of one organism by another where the consumed
organism (prey) was alive when the predator first attacked it. Predators may or
may not kill their prey prior to feeding on them, but the act of predation always
results in the death of the prey.
iv.
Parasitism is a relationship between two species in which one benefits at the
expense of the other. A parasitoid is an insect that parasitizes and kills other
insects.
Predation and parasitism are an example of antagonistic ecological interactions in
which one species take advantage of another species. Predators use their prey as a
source of food only, whereas parasites use their hosts both as a food and as a
habitat.
v.
Mutualism is a relationship between two species that have developed a positive,
reciprocal dependency and both populations benefit from this association.
Through the relationship both the species strengthen their chances for survival,
fitness or growth. A facultative Mutualist is a species that benefit from interaction
with other species, but does not absolutely require the interaction, whereas an
obligate Mutualist is a species that cannot survive without the Mutualist species.
2
vi.
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.
The application of mathematical biology has an immense impact towards the
study of ecology and in development of commonly used biological resources like
fisheries, wildlife and forestry. Recently Scientists and researchers give emphasis on the
interaction between mathematics and biology which initiate a new research area. Most of
our knowledge of biological systems is comparatively recent, and most biological
theories evolve rapidly; therefore it is necessary to develop some useful mathematical
models to describe the consequences of these biological systems. It is observed that these
newly developed mathematical models are significantly influenced through the biological
theories in the past and the consequent expansion of those theories in recent time. The
traditional mathematical model of predator-prey interactions consists of the following
system of two differential equations:
dx
 P( x)  eg ( x, y ) y ;
dt
dy
 eg ( x, y ) y  D( y ) ;
dt
x(0)  x0  0,
(i)
y (0)  y 0  0.
Where x and y represent the prey and the predator population sizes respectively, and
functions P( x), D( y ) describe the intrinsic growth rate of the prey and the mortality rate
of the predator, respectively. The function D( y ) is assumed to be linear
while the function P (x ) may have a linear
P( x)  ax
D( y)  dy  ,

x 

or logistic  P( x)  ax1   
 K 

expression. The function g ( x, y ) is called “functional response” or “feeding rate” and
3
represents the prey consumption per unit time. The model assumes a linear
correspondence between the prey consumption and the predator production through the
positive constant e. In the literature, there are various forms for functional responses
which are classified into the following categories:
Prey Dependent: Prey dependent functional response is unaffected by predator density.
This includes:
Holling type I: in which the attack rate of the individual predator (consumers) increases
linearly with prey density, however when the consumers are satisfied, it consumes prey at
a constant rate.
x
g ( x, y)  g ( x)  
c
for 0  x  K
for
xK
Here K is level of prey at which predator is satisfied at c .
Holling type II: in which the attack rate increases with decreasing prey density until it
becomes constant at satiation. This response is a representation of the typical behaviors of
predators that specialize on one or a few prey. It is also called hyperbolic functional
response.
g ( x, y)  g ( x) 
ax
Ax
 1
1  Ahx b1  x
Here A is search rate, h is handling time, a1 is a maximum consumption rate and b1 is
half saturation level.
4
Holling type III: in which attacks rate accelerate at first and then decelerates towards
satiation. It is also called sigmoid functional response. Sigmoid functional responses are
representation of typical behavior of generalist’s natural enemies which readily switch
from one food species to another and or which concentrates their feeding in areas where
certain resources are not abundant.
a1 x 2
Ax 2
g ( x, y )  g ( x ) 

.
1  Ahx 2 b1  x 2
Holling type IV: When the nutrient concentration reaches a high level, an inhibitory
effect on the specific growth rate may occur. To model such an inhibitory effect,
Andrews (2006) proposed the response function , called the Monod-Haldane function,
and also called a Holling type-IV function. The Holling type IV functional response is of
the form.
g ( x, y)  g ( x) 
a1 x
.
b1  x 2
Ivelev-Type: Ivelev (1955) derived the functional response of a consumer by arguing
that the rate of consumption should depend on how hungry it is. This functional response
is of the form.
g ( x, y)  g ( x)  1  e  a1 x
The prey dependent functional responses fail to model the interference amongst predators
so we study the predator dependent response in which effect of both predator and prey
populations are considered. Predator dependent functional responses may be of following
kinds :
5
Beddington-De Angelis Type: This functional response was introduced by Beddington
(1975) and DeAngelis et al. (1975), independently. It is similar to Holling type II
functional response but contains an extra term describing mutual interference by
predators.
g ( x, y) 
a1 x
.
m  x  b1 y
Here a1 and m are positive parameters, b1 is a predator interference parameter. b1  0
is the case where predator benefit from co-feeding, m is saturation constant and a1 is
the maximum consumption rate.
Hassell-Varley Type: This is an elaboration of Holling type II, using a variable power
(sigma) of the amount of predators. It is of the form.
g ( x, y ) 
a1 x
b1 x  y 
Where a1  0, b1  0 and   0 .
Crowley–Martin type: This type of functional response was introduced by P.H.
Crowley and E. K. Martin (1989). This is defined as
g ( x, y) 
a1 x
.
1  b1 x  b2 y  b1b2 xy
Here constants ai , bi (i  1,2) are positive, stands for the effect of capturing rate, handling
time and the magnitude of interference among predators respectively.
6
Ratio-dependent Functional Response:
This function response is predator – dependence response , in which it only depends on
the ratio of prey population size to predator population size, not on the absolute number
 x
of species. In particular g ( x, y)  g   . There is growing evidences that in some
 y
situations when predators have to search for food (and therefore have to search or
compete for food), the per capita growth rate should be function of the ratio of prey to
predator abundance. It is observed that ratio – dependent models provide richer dynamics
in comparison to the predator–dependent models. This is strongly supported by numerous
fields and laboratory experiments and observations (Arditi and Ginzburg, 1989; Arditi et
al., 1991). It is given by
x
A 
 y   a1 x .
g ( x, y ) 
 x  b1 y  x
1  Ah 
 y
In the natural world, many species have a life history that takes their individual
members through two stages: immature and mature with a time lag. In ecology, more
realistic models should include some of the past states of the system: that is, ideally, a
real system should be modeled by differential equations with time-delay (Freedman and
Gopalasmay, 1986). The time delay is the inherent property of the dynamical systems and
plays an important role in almost all branches of science and particularly in the biological
sciences (e.g., population dynamics, epidemiology, etc.). Delay differential equations
differ from ordinary differential equations in that the derivative at any time depends on
the solution at prior time. The simplest constant time delay equations have the form
7
y (t )  f (t , y(t ), y(t   1 ), y(t   2 ),.......... y(t   k )),
where the time delays (lags) ,  j , are positive constants. More generally, state dependent
delays may depend on the solution, that is  i   i (t , y(t )) . The objective of studying
delay differential equations is to assess the qualitative or quantitative results for models
with delays representing different biological mechanisms (Maynard Smith, 1974). The
delay in the prey-predator model was first included by Volterra, 1926, who took into
account time taken for pollutants produced by a population to build up, eventually
increasing the death rate of the population. Delay may also be due to development time of
the population itself or of its resources. Kuang, 1993 mentioned that animals must take
time to digest their food before further activities and responses take place and hence any
model of species dynamics without delays is an approximation at best. Detailed
arguments on the importance of time delays in realistic models may be found in classical
books of Gopalsamy, 1992; Macdonald, 1989 and Kuang, 1993.
1.2 MATHEMATICAL TECHNIQUES
The following qualitative methods, lemmas, definitions and theorems have been used to
analyze various nonlinear dynamical models proposed in the thesis
1.2.1 REGION OF ATTRACTION
A part of the phase space including the attractor (sets of population states that are
invariant, they can be equilibria) itself, as well as all initial conditions from which a
trajectory will converge to the attractor.
8
1.2.2 EQUILIBRIUM POINT AND STABILITY
Equilibrium is a point at which a variable (or variables) remains unchanged over
time. A particular value of a variable is called an equilibrium value if, value of variable
never changes when it start at this value. It is the graph of a constant solution. In
mathematics, the point x 0   n is an equilibrium point for the differential equation
dx
 f ( x), if f ( x 0 )  0 for some value x 0 . Equilibria are sometimes called fixed points
dt
or steady states.
Stability is a very important issue in the study of nonlinear systems. Stability
means, that a small change (a small disturbance) of a physical system at some instant
time, changes the behavior of the system only slightly at all future time t. Various
stability concepts frequently associated with a dynamical systems are: local stability,
global stability, absolute stability and Lyapunov stability of equilibrium points. An
equilibrium point is said to be locally stable if small perturbations remain close to the
equilibrium, and locally asymptotically stable if small perturbations eventually returns to
the equilibrium. As such it does not predict the overall behavior of the system. On the
other hand, when small perturbations continue to move away from the equilibrium, the
equilibrium point is said to unstable. Equilibrium is globally stable if a system
approaches the equilibrium regardless of its initial position,
For example consider the following system of differential equations for n- interacting
species:
dx
 f ( x,  ).
dt
(1.2.1)
9
Here x  ( x1 ,..., x n ) t is a state variable and  is control parameter. The function
f captures the interaction amongst species. The equilibrium point x e (say) of system
(1.2.1) is a point, satisfying f ( xe ,  )  0 .
Linear stability is used to study the stability of the steady state
xe in a small
neighborhood. The system (1.2.1) can be linearized at xe , as
dx
 J x,
dt
(1.2.2)
Where, J is the Jacobian matrix (Variational matrix) of the system (1.2.2) whose
elements are given by
 f
a ij   i
 x
 j

 . The characteristic equation corresponding to the

 xe
system (1.2.2) is obtained as
 n  A1  n 1  A2  n  2  ...  An  0
(1.2.3)
The stability of equilibrium point x e can be determined by looking at the signs of
eigenvalues i of the Variational matrix J at each of the equilibrium points of the
system. The equilibrium point is asymptotically stable if the real parts of all eigenvalues
of characteristic equation corresponding to this equilibrium point are negative and
strongly stable if every eigenvalue of characteristic equation is pure imaginary and of
simple type. An equilibrium point is hyperbolic if none of the eigenvalues have zero real
part. If one or more of the real parts of the eigenvalues are positive then the state is
unstable.
10
The Routh-Hurwitz Stability Criterion provides the necessary and sufficient conditions
under which all roots of the characteristic equation (1.2.3) lie in the left half of complex
plane: A1  0,
A1
A3
1
A2
A1
A3
A5
... 0
0
A1
A3
A5
1
A2
A4
... 0
0
 0, 1
A2
A1
A3
... 0
0  0.
0
A1
A4  0,..., 0
A3
0
0
...
0
... 0
An
In particular, the Routh – Hurwitz criterion for n  2, n  3 and n  4 are
n2
A1  traceJ  0 ; A2  det J  0 .
n3
A1  0 ; A3  0 and A1 A2  A3  0.
n4
A1  0 ; A2  0 ; A3  0 ; A4  0 ; A1 A2  A3  0 and A3  A1 A2  A3   A12 A4 .
1.2.3 BIFURCATION ANALYSIS
The term bifurcation generally refers to something “splitting apart”. With regard
to differential equations or systems involving a parameter, it refers to abrupt changes in
the character of the solutions as the parameter is changed continuously. Bifurcation
analysis of nonlinear models addresses the effects of parameter changes on the qualitative
nature of the attractor (stable vs. unstable, fixed point vs. cycle, etc.). Bifurcations are
said to occur when small quantitative parameter changes produce large qualitative
changes in the dynamics. In one sense, bifurcation analysis is the next step beyond local
stability analysis, in that it describes the fact of large perturbations from equilibrium. In
the mathematical theory of bifurcations, a Hopf bifurcation is a local bifurcation in which
a fixed point of a dynamical system loses stability as a pair of complex conjugate
11
eigenvalues of the linearization around the fixed point cross the imaginary axis of the
complex plane resulting in a stable limit cycle (oscillations). The Hopf bifurcation break
the stability of system and give rise to oscillation that are uniform in space and periodic
in time.
1.2.4 PERSISTENCE
Biologically, persistence means the survival of all populations in future time.
Mathematically, persistence of a system means that strictly positive solutions do not have
omega limit points on the boundary of the non – negative cone. Persistence may be
defined mathematically as Freedman and Waltman (1984, 1985). A population P (t ) is
said to persist (sometimes called strongly persist) if P (0)  0 implies P(t )  0 and
lim inf P(t )  0. More related definitions in both ecological and abstract situations are
t 
given in Butler, et al., 1986; Butler and Waltman, 1986; Freedman and Moson (1990). A
population
P (t ) is said to persist uniformly if
P (t ) persists and there exists
P(t )   . Finally, we say that a system
  0 independent of P(0)  0, such that limt inf

persists (uniformly) whenever each component persists (uniformly).
To prove the persistence of populations, we have also used Butler – McGehee
lemma. Before the statement of Butler – McGehee lemma, we would like here to give
some definitions in regard of this lemma.
Isolated Hyperbolic Equilibrium: Equilibrium point ( x10 , x 20 , x 30 ,...x n0 ) is isolated if
there exists a neighborhood N of ( x10 , x 20 , x 30 ,...x n0 ) such that the only equilibrium of
12
x i  f i in N is ( x10 , x 20 , x 30 ,...x n0 ), and a point is hyperbolic if none of the eigenvalues of
the variational matrix at equilibrium point have zero real part.
Omega Limit Set (   Limit Set): If x(t ) is the solution of the system x (t )  f ( x) and if
there exists a number t 0 such that x(t ) is defined for all t  t 0 , then an omega limit point
of solution x(t ) is a point x 0 such that there exists a sequence of real numbers {tn } with
lim t n   and lim x(t n )  x 0 . If S denote the solution x(t ), then the set of omega limit
n
n
points of S will be denoted by  (S ) or ( x(t )).
Orbit: the orbit C ( x 0 ) of x (t )  f ( x) through x 0 is defined by
C ( x 0 )  {x  R n : x  x(t , x 0 ), t  R}.
Statement of Butler – McGhee: Let E be an isolated hyperbolic equilibrium in the
  limit set ( X ) of an orbit  ( X ). Then either ( X )  E or there exists points
E  and E  in ( X ) with E   M  (E ) and E   M  (E ).
1.2.5 PONTRYAGIN’S MAXIMUM PRINCIPLE
The maximum principle is an optimization technique that was first proposed in
1956 by Pontryagin and its associates for various types of time-optimizing continuous
processes. This is powerful method for the computation of optimal control, which has the
crucial advantage that it does not require prior evaluation of the infimal cost function and
its proof is based on maximizing the Hamiltonian. It has as a special case the EulerLagrange equation of the calculus of variations.
13
We assume from now on that b, c and C are differentiable in t and x with continous
derivaties, and that the stopping set D is a hyperplane, thus D  {y}   ,
for
some y  R d and some vector subspace  of R d .
Define for   R d the Hamiltonian
H (t , x, u,  )  T b(t , x, u)  c(t , x, u )
Pontryagin's maximum principle states that, if ( xt , u t )   is optimal, then there exist
adjoint paths ( t ) t  in R d and (  t ) t  in R with the following properties: for all t   ,
i.
H (t , xt , u, t )   t , has maximum value 0 , achieved at u  u t ,
ii.
Tt  Tt b(t , x t , u t )  c(t , x t , u t ) ,
iii.
 t  Tt b(t, xt , ut )  c(t , xt , u t ) ,
iv.
x t  b(t , xt , u t ) .
Moreover the following transversality conditions hold:
v.
(T  C (t , x ))  0 for all    ,
and, in the time-unconstrained case,
vi.
  C (t , x )  0 .
Note that, in the time-unconstrained case, if b, c and C are time-independent, then
 t  0 for all t .
14
The Hamiltonian serves as a way of remembering the first four statements, which could
be expressed alternatively as
(i)
0
H
,
u
(ii)   
H
,
t
(iii)   
H
,
t
(iv) x 
H
.

1.2.6 CHAOS
Apparently simple deterministic systems with only a few elements can generate
random behavior and this randomness is fundamental to the systems. This randomness
cannot be removed by gathering more and more information about the elements of the
system. This fundamental randomness is called chaos. Thus dynamical systems whose
behavior is not stationary, nor periodic nor quasi periodic are called chaotic.
Definition: Let J be a set. Then a flow defined by the vector function f : J  J
defined in a certain region / volume of space V is said to be chaotic if
i.
f has sensitive dependence on initial condition.
ii.
f is topologically transitive.
iii.
f has periodic points which are dense in V .
A function f : J  J is said to have sensitive dependence on initial conditions, if there
exists a   0 such that for any x  J and any neighborhood N of x , there exists y  N
and n  0 such that f n ( x)  f n ( y )   .
A function f : J  J is said to be topologically transitive, if for any pair of open sets,
U , V  J there exists K  0 such that f
K
U  V   .
15
A chaotic system is unpredictable because of sensitive dependence on initial conditions.
It cannot be broken down or decomposed into two subsystems (two invariant subsets)
which do not interact under f because of topological transitivity. And in the midst of
this random behavior we nevertheless have an element of regularity, namely the periodic
points which are dense.
If f
f
k
be a map on R then we call ' p ' to be a periodic point of period k , if
 p   p,
k being the smallest such positive integer.
In other words, we can say that, ‘Chaos is qualitatively more or less random behavior
which is intrinsic to a system and not due to externally imposed noise”.
1.3 LITERATURE SURVEY
Mathematical population models have been used to study the dynamics of preypredator systems. Lotka (1925) and Volterra (1927) proposed a simple model known as
Lotka-Volterra model of prey-predator interactions. Since then, many mathematical
models have been constructed based on more realistic explicit and implicit biological
assumptions. Modeling is a frequently evolving process, to gain a deep understanding of
the mathematical aspects of the problem and to yield non trivial biological insights, one
must carefully construct biologically meaningful and mathematically tractable population
models (Kuang, 2002). Some of the aspects that need to be critically considered in a
realistic and plausible mathematical model include; carrying capacity which is the
maximum number of prey that the ecosystem can sustain in the absence of predator,
competition among prey and predators which can be intra specific or inter specific,
harvesting of prey or predators and functional responses of predators .
16
Continuous predator–prey models have been studied mathematically. The
principles of Lotka (1925) and Volterra (1927)
model, conservation of mass and
decomposition of the rates of change into birth and death processes, remain valid until
today and many theoretical ecologists adhere to these principles. Modifications were
limited to replacing the Malthusian growth function, the predator per capita consumption
of prey or the predator mortality by more complex functions such as the logistic growth,
Holling type I, II, III and IV functional responses or density-dependent mortality rates.
The mentioned functional responses all depend on prey-abundance X only, but soon it
became clear that predator abundance Y can influence this function (Curds and
Cockburn, 1968; Hassell and Varley, 1969; Salt, 1974) by direct interference while
searching or by pseudo-interference [in the sense of Freedman et al. (1977)] and models
were developed incorporating this effect (Hassell and Varley, 1969; DeAngelis et al.,
1975; Beddington, 1975). However, these models usually require more parameters and
their analysis is complex. Therefore, they are, on one side, rarely used in applied ecology
and, on the other side, have received little attention in the mathematical literature. A
simple way of incorporating predator dependence into the functional response was
proposed by Arditi and Ginzburg (1989) who considered this response as a function of
the ratio X / Y . Interesting properties of this approach have emerged that are in contrast
with predictions of models where the functional response only depends on prey
abundance (e.g., Arditi et al., 1991; Ginzburg and Akcakaya, 1992; Arditi and Michalski,
1995). The interaction of multiple predators, predation rate and prey selection are also
modified by the habitat complexity (Hughes and Grabowski, 2006). For aquatic predatory
insects, rice fields, ponds, and temporary pools are habitats with ample heterogeneity
17
with regard to spatial structures and prey species abundance (Bambaradeniya et al.,
2004). When ample prey species are available structural complexity is more important in
determining predator success. Zhang et al. (2006) studied the stability of three species
population model consisting of an endemic prey (bird), an alien prey (rabbit) and an alien
predator (cat). It may be pointed out here that all the above studies are based on the
traditional prey dependent models. Recently, it has been observed that in some situations,
especially when a predator have to search for food and have two different choice of food,
a more suitable predator-prey theory should be based on the so-called ratio-dependent
theory, in which the per-capita growth rate should be function of the ratio of prey to
predator abundance, and should be the so-called predator functional response (Abrams
and Ginzburg, 2000; Akcakaya et al., 1995; Arditi et al., 1991; Arditi and Saiah, 1992).
Kesh et al. (2000) proposed and analyzed a mathematical model of two competing prey
and one predator species where the prey species follow Lotka-Volterra dynamics and
predator uptake functions are ratio dependent. They derived conditions for the existence
of different boundary equilibria and discussed their global stability. They also obtain
sufficient conditions for the permanence of the system. Hsu et al. (2001) studied the
qualitative properties of a ratio dependent predator-prey model. They showed that the
dynamic outcome of interactions depends upon parameter values and initial data. Three
general forms of functional response are commonly used in ecological models: linear,
hyperbolic, and sigmoidal. How predators respond to changes in prey availability
(functional response) is an issue of particular importance. There is evidence from several
models that the type of functional response specified can greatly affect model predictions
(Gao et al., 2000; Kar and Chaudhury, 2004).Many authors (Rijn and Sabelis 1993; Rijn
18
and Tanigoshi 1999; Eubanks and Denno 2000; Srinivasu and Prasad 2011) studied that
the alternative food has no dynamics of its own, that is, alternative food is always
available in constant amounts, unaffected by consumption. This simplification is justified
for many arthropod predators because they can rely on plant-provided alternative food
sources such as pollen or nectar, the availability of which is unlikely to be influenced by
the predator's consumption.
The theory of harvesting is important in natural resource management and
bioeconomics. Most species have a growth rate which more or less maintains a constant
population equal to the carrying capacity of the environment K (this of course depend
on the population). In this case the growth and death rates are nearly equal. The
harvesting of species affects their mortality rates and if the harvesting is not too much the
population will adjust to a new equilibrium X *  K . It has been evident that there is
need to develop ecologically acceptable strategies for harvesting any renewable resources
such as fish, plants, animals etc. It is interesting to note that even if the excess harvest
does not threaten extinction, it can cause damage to the resource in the long run. Massive
fruit collection from the forest has an adverse effect on regeneration. The problem then is
to determine a strategy which ensures steady harvest year after year without a progressive
decline in the abundance of the resource. The problem here is how to maximize the
sustainable yield (SY) by determining the population growth dynamics so as to obtain a
harvesting rate which keeps the population at its maximum growth. We consider a
logistic population growth model in which the mortality rate is enhanced by harvesting;
by a term that is proportional to the existing population N . We consider a logistic growth
19
X

model F  X   rX 1   , where r is the intrinsic growth rate and K the carrying
K

capacity of the environment. Considering the effort E of harvesting, we assume that the
harvest is proportional to the stock level as well as the effort i.e. h  qEX , where q is
the constant of proportionality called the catchability constant. The effort is measured in
man days. If grass is cut with strokes of a sickle, the harvest depends on the number of
strokes E on the grass density. If the effort is constant then the harvest as a function of the
stock is a straight line passing through the origin. The intersection of the line or the
growth curve gives the sustainable yield. The net growth rate after harvest is given
by
dX
X

 rX 1    qEX . There are numerous studies on the effects of harvesting on
dt
K

population growth. In the context of predator-prey interaction, some studies that treat the
populations being harvested as a homogeneous resource include those of Dai and Tang,
1998; Myerscough et al., 1992; Chaudhuri (1996) and Leung (1995). Theoretical ecology
remained silent about the astonishing array of dynamical behaviors of three species
models for a long time. Of course, the increasing number of differential equations and the
increasing dimensionality raise considerable additional problems both for the
experimentalist and theoretician. Freedman and Waltman (1984) considered three level
food webs – two competing predators feeding on a single prey and a single predator
feeding on two competing prey species. They obtain criteria for the system to be
persistent. Kar and Chaudhuri (2004) considered a two-prey one-predator harvesting
model with interference. The model is based on Lotka-Volterra dynamics with two
competing species which are affected not only by harvesting but also by the presence of a
20
predator, the third species. Optimal harvesting policy and the possibility of existence of a
bioeconomic equilibrium is discussed. Dubey and Upadhyay (2004) proposed a two
predator one prey system with ratio dependent predator growth rate. Criteria for local
stability, instability and global stability of the non-negative equlibria are obtained. They
also discussed about the permanent co-existence of the three species. Braza (2008)
considered a two predator; one prey model in which one predator interferes significantly
with the other predator is analyzed. The analysis centers on bifurcation diagrams for
various levels of interference in which the harvesting is the primary bifurcation
parameter.
Bioeconomic modelling of the exploitation and management of biological
resources, like fisheries and the techniques associated with the exploitation of these
resources have been discussed by Clark (1990) in models of single species fisheries
submitted to harvest, but it is difficult to construct and study a realistic model of a
multispecies interaction because the model may not be analytically tractable. It is also
difficult to determine the optimal harvest policy of a n-dimensional model having more
than one control variable. The studied problem deals with the sustainable exploitation of
two interacting species, one is the prey and the other is the predator involving combined
or nonselective harvesting of the two populations considering that the fishing effort is the
same for both species by using the Schaefer’s hypothesis (CPUE) (Schaefer, 1957). The
dynamical behavior of the exploited model has already been studied. The optimal policy
of exploitation is derived using Pontryagin’s maximal principle (Pontryagin et al.,1964).
On the other hand, the functional response of predators or consumption rate function
refers to the change in the density prey attacked per unit time per predator as prey density
21
changes. In most predator–prey models considered in the ecological literature, the
predator response to prey density is assumed to be monotonic increasing; this inherent
assumption means that the more prey population in the environment the better for the
predator population. The problem of harvesting of competitive species has drawn the
attention of bio – economists. Bhattacharya and Begum (1996) obtained the feasible
dynamic equilibrium to obtain a limit of cost per unit effort and maximum value of effort
for the joint harvesting of a logistic growth model of two competitive systems. Kar et al.
(2006) has studied the bio-economic model of a ratio – dependent prey – predator system
with optimal harvesting. They proved that optimal equilibrium populations lead to a
situation where total user’s cost of harvest per unit effort equals the discounted value of
the future profit. Kar et al. (2010) proposed a biological economic model based on a
prey-predator dynamics where prey species are continually harvested and predation is
considered type II functional response. Palma and Olivares (2012) have studied a
predator-prey model with Allee effect and sigmoid functional response. In general a
dynamic model consists of a biological (or biophysical) model that describe the behavior
of a living system and an economic model that relates the biological system to market
prices, resource and institutional constraints. Bio-economic models often contain a single
equation to represent biological process. Nowadays, the biological resources in the prey –
predator ecosystem are commercially harvested and sold with the aim of achieving
economic interest.
However, these models are less realistic. To make model realistic one should
include some of the past states of these systems; ideally a real system may modeled by
differential equations with time-delays. Time-delays occur so often in almost every
22
situation, that to ignore them is to ignore reality. The time delay is the inherent property
of the dynamical systems and plays an important role in almost all branches of science
and particularly in the biological sciences (e.g., population dynamics, epidemiology, etc.)
see (Yan and Zhang, 2008; Yang, 2009; Meng et al. 2011). The importance derives from
the fact that many of the phenomena around us do not act instantaneously from the
moment of their occurrence. For example, a change in the resources or environment does
not affect the survival of existing populations immediately. There is always a time lag
between the moment an action takes place and its effect is observed. In ecology, more
realistic models should include some of the past states, i.e., a real system should be
modeled by differential equations with time delays. Kuang (1993) mentioned that animals
take some time to digest their food before further activities and responses take place and
hence any model of species dynamics without delays is an approximation at best. Now it
is beyond doubt that in an improved analysis, the effect of time-delay due to the time
required in going from egg stage to the adult stage, gestation period, et cetera, has to be
taken into account. Detailed arguments on the importance and usefulness of time-delays
in realistic models may be found in the classical books of Macdonald, 1989; Gopalsamy,
1992; and Kuang, 1993.
Recently, it is of interest to investigate the possible existence of chaos in
biological population. The subjects of chaos and chaos control are growing rapidly in
many different fields such biological systems, structural engineering, ecological models,
aerospace science, and economics .Food chain modeling provides challenges in the fields
of both theoretical ecology and applied mathematics. Determining the equilibrium states
and bifurcations of equilibria in a nonlinear system is also an important problem in
23
mathematical models. Two-species continuous time models have only two basic patterns:
approach to equilibrium or to a limit cycle. In contrast, simple discrete time models of
even a single species can exhibit chaotic behavior. But research of the last 25 years
(Gilpin, 1979; Gucken-Heimer and Holmes, 1983; Schaffer, 1985) demonstrates the very
complex dynamics that can arise in model systems (in continuous time) with three or
more species. Several of the early mathematical investigations of chaos were of
ecological models (May, 1974). An investigation by Gilpin (1979) showed that a system
of one predator and two competing prey can exhibit chaotic behavior. Schaffer and
Collaborators (1986 a) have been especially persuasive in their view that chaos may be a
much more important phenomenon than ecologists had earlier believed. May (1974)
argued that the reason why field ecologists have not been able to get any reliable proof of
existence of chaos is that they have preoccupied notions and that they are observing what
they want. It was argued by many authors that poor data quality (short and noisy
character of the time-series) makes the techniques of nonlinear dynamics an unsuitable
tool for the analysis of ecological data, and thus, leads to failure of such attempts. One
common hindrance (in the study of nature populations) to understand the underlying
dynamical process has been the non-availability of data of suitable length and precision.
Another difficulty is that there is no method which can fix these parameters a priori .On
the other hand, the existence of chaos in almost all the physical systems motivates one to
critically study the same in natural populations. Hastings and Powell (1991) have studied
three species food chain with double Holling type II functional responses; they found that
there is ‘tea-cup’ attractor in the system. Gakkhar and Naji (2003) investigated a three
species ratio dependent food chain (Holling- Tanner Type) model, they also found that
24
there is ‘tea-cup’ attractor in the system .Wang et al., (2005) studied the three food chain
model with impulsive effects on top predator and found that the impulsive perturbations
bring to these simple systems chaos solutions. Wang and Pang (2008) studied a model of
a hybrid ratio dependent three species food chain model and they also found chaotic
attractor in the system. Ranjit Kumar Upadhyay (1997) studied why chaotic dynamics are
rarely observed in natural populations. Many papers studied a predator-prey model with
the Michaelis–Menten functional response. Naji and Balasim (2007) studied dynamical
behavior of a three species food chain model with Beddington-DeAngelis functional
response and investigated bifurcation and chaotic behavior at conversion rate of prey
from predator The control of chaotic dynamics with various ecological factors, namely,
migration, predation, conversion, refuge, omnivory, habitat-heterogeneity is the most
challenging task in such studies and thus it receives a good deal of attention from various
scientists .Vandermeer (1993) investigated a three species ratio-dependent food chain
model and show that there is chaotic attractor. Hastings and Powell (1991) have studied
three species food chain with double Holling type II functional responses,they found that
there is ‘tea-cup’ attractor in the system. Gakkhar and Naji (2003) investigated a three
species ratio dependent food chain (Holling- Tanner Type) model, they also found that
there is ‘tea-cup’ attractor in the system. In the last few years, researchers have shown
keen interest to investigate the direction and stability of Hopf bifurcation arising from a
delay-induced neural network. Study of the bifurcation and its stability, however, is
relatively new in the ecological system. Recently, Song and Wei, (2004); Yang and Tian,
(2005); Qu and Wei, 2007; Celik (2008, 2009) have studied the direction and stability of
Hopf bifurcation of delay-induced ecological systems. In most of the studies, delay
25
occurs in a first degree term. Sun et al. (2006) investigated the direction and stability of a
delay-induced eco-epidemiological system with Type I response function, where delay
occurred in the term of degree two.
1.4 SUMMARY AND SCOPE OF WORK DONE
Mathematical modeling as a tool in the study of population dynamics has a long
and diverse history, spanning at least three centuries. Although multitude of models has
been put forward and many challenging problems have been solved, the endeavor to
explain and formulate general principles underlying the dynamics of populations in space
and time is far from over. The pros and cons of particular models and concepts are still
subject of lively debates, which makes population dynamics a wide-open field for
researchers interested in mathematical modeling. This thesis attempts to provide new
insight into some population-dynamical problems and also proposes a new perspective on
certain models of ecological communities. The specific research questions have been
classified according to the level of modeling complexity on which they are situated and
the organization of this thesis as outlined below intends to reflect this classification.
Chapter 2. In this chapter, a nonlinear mathematical model is proposed and analyzed to
see the effect of harvesting on food chain model. We have also studied the effect of the
habitat complexity and gestation delay on the stability of a food chain model. It is
observed that there are stability switches, and Hopf bifurcation occurs when the delay
crosses some critical value. It is observed that the quantitative level of abundance of
system populations depends crucially on the delay parameter if the gestation period
exceeds some critical value. We have studied a delay-induced food-chain model in
presence of habitat complexity of space. Using stability theory of differential equations,
26
we have obtained conditions for the existence of different equilibria and discussed their
stabilities. Using differential inequality, conditions have been obtained under which
system persists.
Chapter 3. In this chapter we have discussed how additional food to predator changes
the structure of prey and predator system. Criteria for local stability, instability and global
stability of the equilibria are obtained. Using differential inequality, we have obtained
sufficient conditions that ensure the persistence of the system. The presence of additional
food incorporated in the shape of the functional response, it may also increase
equilibrium predator densities, which, in turn, will lead to a reduction in equilibrium prey
density. The effects of additional food may help to devise strategies for biological control
involving a supply of additional food. To complement our analytical findings numerical
simulations are carried out.
Chapter 4. In this chapter two time delays ( 1 and  2 ) are considered in the model to
describe the time that juveniles of prey and predator take to mature. The stability analysis
of the proposed model is carried out. The Hopf bifurcation conditions of the interior
equilibrium point are established. We have obtained the sufficient conditions for stability
of the interior equilibrium point. Moreover, the conditions for the existence of Hopf
bifurcation of the cases  1  0,  2  0 and  1   2    0 , respectively are determined.
We have found critical value of delay and observed that under certain conditions, system
bifurcates from the steady states of the system at some critical values of   1 ,  2  . It is
concluded from analysis that the stability properties of the system could switch with the
time delay that is incorporated on different densities in the model.
27
Chapter 5. In this chapter, we have studied the effect of harvesting on dynamics of preypredator model with Holling Type III functional response. A combined harvesting policy
for prey and predator species is discussed by using the Pontryagin’s Maximum principle.
Using differential inequality, we have obtained sufficient conditions that ensure the
persistence of the system. It has been shown that the total user’s cost of harvest per unit
effort is equal to the present value of marginal revenue of effort at the optimal
equilibrium level. It has also been noted that the increase in discount rate decreases the
economic rent and even it tend to zero if the discount rate tends to infinity. Finally,
numerical simulations is done to support the analytical findings.
Chapter 6. This chapter aims to study the effect of Harvesting on predator species with
time-delay on a prey-predator model with Holling type-IV functional response. Two
delays are considered in the model to describe the time that juveniles of prey and predator
take to mature. The dynamics of the system are studied in terms of local and Hopf
bifurcation analysis. We have concluded from our analysis that the stability properties of
the system could switch with the Harvesting with time delay that is incorporated on
different densities in the model.
Chapter 7. In this chapter, we have considered Beddington-De Angelis response in a
food chain model and investigated the Hopf bifurcation and Chaos of the model. The
mortality rate of predator has ability to control the chaotic oscillations. We have also
studied the effect of Harvesting on prey species. Our analysis shows that, using the
parameter mortality rate as control, it is possible to break the stable behavior of the
28
system and drive it in an unstable state. Also it is possible to keep the levels of the
populations in a stable state using the above control.
Chapter 8. In this chapter we have formulated and analyzed prey-predator model with
one prey dispersal in a two patch environment. Each patch is supposed to be
homogeneous. Patch two constitutes a reserved area of prey and no predation is permitted
in this zone whereas patch one is an open access predation zone. Criteria for local
stability, instability and global stability of the non-negative equilibria are obtained. Using
differential inequality, we have obtained sufficient conditions that ensure the persistence
of the system. We have found that prey reserve and team approach are most important
factor of ecology.
A deep insight into mathematical biology has opened up different areas of
research in the last few decades. Mathematical modeling of the dynamics of ecological
system has played a pivotal role in attempting an inter-relationship between a predatorprey system in an environment and the consequences of interaction of interspecies as well
as intraspecies. A great deal of mathematical modeling has been accompanied by a rich
theory of differential equations. For example in chapter 2, if we assume that predators are
impulsively released with some impulsive period, then model will be more realistic. We
leave it for future studies. Similarly a model in chapter 3 and 6 can be modified. In
chapter 5, we have used ‘tax’ as only control instrument, but there are other instruments
also like a catch- quotes and subsidies etc to apply to the problem of populations from
over exploitations. All these factors may be used in future research.
29