‫بسم اهلل الرحمن الرحيم‬
‫وقل رب زدني علما‬
‫صدق هللا العظيم‬
‫الباحثة رحاب نوري‬
On the dynamics of some Prey-predator
Models with Holling type-IV functional
response
Abstract
This thesis deals with the dynamical
behavior of prey-predator models with
nonmonotonic functional response. Two
types of prey-predator models are
proposed and studied.
The first proposed models consisting of
Holling type-IV prey-predator model
involving intra-specific competition
However, the second proposed models
consisting of Holling type-IV prey-predator
model involving intra-specific competition
in prey population with stage structure in
predator.
Both
the
models
are
represented mathematically by nonlinear
differential equations. The existence,
uniqueness and boundedness of the
solution of these two models are
investigated. The local and global stability
conditions of all possible equilibrium
points are established.
The occurrence of local bifurcation
(such as saddle-node, transcritical
and pitchfork) and Hopf bifurcation
near each of the equilibrium points
are discussed. Finally, numerical
simulation is used to study the
global dynamics of both the
models.
Chapter one
Preliminary
Mathematical model; is mathematical construct,
often an equation or a relation ship, designed to study
the behavior of a real phenomena.
Definition: Any group of individuals, usually of a
single species, occupying a given area at the same
time is known as population.
The application of mathematical method to problem in
ecology has resulted in a branch of ecology know as
mathematical ecology,.
There are three different kinds of interaction between
any given pair of species, these are described as
bellow.
mutualisms or symbiosis (+,+), in which the
existence of each species effects positively on the
growth of the other species (i.e. increases in the
first species causes increases in the growth of the
second species )
Competition (-,-), in which the existence of each
species effect negatively on the growth of the other
species.
prey-predator (-,+) in which the existence of the first
species (prey) effects positively (increases) on the
growth of the second species (predator), while the
existence of the second species
(predator) effects negatively (decreases) on the growth
of the first species (prey).
Functional response; is that function, which refers to
the change in the density of prey attacked per unit at
time per predator as the prey density changes .the
functional response can be classified into three
classes with respect to its dependence on the prey or
predator ,those are :
prey-dependent functions : the term prey-dependent
means that the consumption rate by each predator is
a function of prey density only, that is . f ( x, y)  f ( x) In the
following, the most commonly used prey-dependant
types functional reasons are presented
(i) lotka-volterra type: in which the consumption rat of
individual predator increases linearly with prey. Hence
the lotka - volterra type of functional response can be
written as follows:
f ( x)  ax; x  0
(1)
Where a  0 is the consumption rate of prey by(1)
p
rate of prey by predator.
y
100
80
60
40
20
2
4
6
8
10
x value
(ii) Holling type-I
In which the consumption rate of individual predator
increases lineally with prey density and when the
predator satiate then it becomes constant. Hence the
Holling type-I functional response can be written as
follows:
y
100
x
f ( x) 

0  x 
 x
80
60
40
20
10
20
30
40
Where  is the value of prey density at which the
predator satiate at constant 
x value
(iii) Holling type-II: in which the consumption rate of
individual predator increases at a decreasing rate with
prey density until it becomes constant at satiation level.
1
a
It is a hyperbola the maximal value
h
asymptotically, and defined by
y
2
1.5
Ax
ax
f ( x) 

1  Ahx b  x
1
(3)
0.5
2
4
6
8
Here A is a search rate, h is the time spent on the
handling of one prey, a is the maximum attack rate
a
and b is the half saturation level  f (b)  2 


10
x value
(iv) Holling type-III:
In this type of functional response the consumption rate
of individual predator accelerates at first and then
decelerates towards satiation level. It is defined as:
f ( x) 
2
Ax
1  Ahx
2

2
ax
b x
(4)
2
y
2.5
2
1.5
1
0.5
2
4
6
8
10
x value
(V) Holling type-IV:
The type of functional response suggested by
Andrews. The Holling type IV functional response is of
the form.
f ( x) 
Mx
2
x
xa
i
(5)
2. Ratio-dependent function :
The prey-dependent functional responses do not
incorporate the predator abundance in it forms.
Therefore, Aridity and Ginsberg suggested using
new type of functional response called ratiodependent, in which the rate of prey consumption
per predator depends not only on prey density but
instead depends on the ratio between the prey and
predator density. The Aridity -Ginsberg ratio
dependent response can be written as:
A x 
y
Ax
ax

f ( x, y ) 


1  Ah x  y  Ahx by  x
 y
(6)
3.predator-dependent functions;
In this type, the functional responses are depending
on both prey and predator densities. It is observed
that high predator density leads to more frequent
encounters between predators.
The most commonly used predator dependent
functional response is that proposed by DeAngalies
and independently by Beddingnon, which is now
know as DeAnglis-Beddington functional response:
Ax
ax
f ( x, y ) 

By  Ahx  1 by  x  c
(7)
Chapter two
The dynamics of Holling type IV
prey predator model
with intra- specific competition
Mathematical models
Let x t  be the density of prey species at timet ,yt  be
the density of predator species at time t that consumes
the prey species according to Holling type IV functional
response then the dynamics of a prey–predator model
can be represented by the following system of ordinary
differential equation.

dx 
y
 x a  bx  2
  xf1 ( x, y)
dt
x  x   


dy 
ex
 y  d  2
 y   yf2 ( x, y)
dt
x  x  


(8)
with x(0)  0 and y (0)  0 Note that all the parameters of
system (8) are assumed to be positive constants and
can be described as follow:
is the intrinsic growth rate of the prey population; d is
the intrinsic death rate of the predator population;
the
a
parameter b is the strength of intra-specific competition
among the prey species; the parameter  can be
interpreted as the half-saturation constant in the
absence of any inhibitory effect; the parameter  is a
direct measure of the predator immunity from the prey;
is the maximum attack rate of the prey by a predator; e
represents the conversion rate; and finally  is the
strength of intra-specific competition among the
predator species .
a
e
Now I worked on my research and I finished to finding
and proofing the following
1. Existence and Local Stability analysis of system
(8) with persistence
(1) The trivial equilibrium point E0  (0,0) always exists.
(2) The equilibrium point
a 
E1   ,0 
b 
always
exists, as the prey population grows to the carrying
capacity in the absence of predation.
(3) There is no equilibrium point on y  axis as the
predator population dies in the absence of its prey.
(4) The positive equilibrium point E2  ( x*, y* )exists in
the interior of the first quadrant if and only if there is a
positive solution to the following algebraic nonlinear
equations:
We have the polynomial from five degrees
f ( x )  A5 x 5  A4 x 4  A3 x 3  A2 x 2  A1 x  A0  0
And its exists under the condition, and when I
investigated varitional matrix for my system and
a 

substitute E1   ,0  we get the eigenvalues are   a  0
1x
b 
abe
and  y  d 
Therefore, E is locally
1
1
2
a  ab  b
asymptotically stable if and only if
abe
a  ab  b  
2
2
d
While
a
E1  ( ,0)
b
is saddle point provided that
abe
a
2
 ab  b  
2
d
And when I investigated varitional matrix for my system
and substitute
E2  (x , y )
* *
we get E 2 is locally
asymptotically stable if and only if
x * y * (2x *  )  (bx* y*)R
and bR3  e 2 2 (  x*2 )  y* (2x*   ) R
2
And the persistence is proof by using the Gard and
Hallam method.
2.Global dynamical behavior of the system (8)
I proof the global dynamical behavior of system (8) and
investigated by using the Lyapunov method
3.The System(8) has a Hopf –bifurcation.
Near the positive equilibrium point at the parameter
value
*
2
(
bx

by
*)
R
* 
x * y * (2 x *  )
4.The system (8) is uniformly bounded
5. Numerical analysis
In this section the global dynamics of system (8) is studied
numerically. The system (8) is solved numerically for different sets
of parameters and for different sets of initial condition, by using
six order Runge-Kutta method
a 1
b  0.2   0.50   0.75
d  0.01 e  0.75
 1
  0.01
(8a)
(a)
b
4.8
Stable point
(4.71,3.99)
4.5
4.6
4.4
4.2
population
y
3.5
initial point
(6.0,3.0)
initial point
(3.0,3.0)
4
3.8
3.6
2.5
initial point
(4.0,2.0)
3.4
3.2
1.5
2.5
3
3.5
4.5
5.5
x
6.5
0
1
2
3
Time
4
5
x 10
Fig1: (a) Globally asymptotically stable positive point of
system (8) for the data starting from (8a) with
different initial values. (b) Time series of the attractor
given by Fig 1(a)
4
a
b
5.5
3
5
4.5
2.5
4
3.5
Population
Predator
2
1.5
3
2.5
2
1
1.5
1
0.5
0.5
0
3
3.5
4
Prey
4.5
5
0
0
2
4
6
Time
8
10
x 10
4
Fig. 2: (a) The system (8) approaches asymptotically to
stable point E1  5,0 with the data given in (8a)
with d  0.05 starting from (3, 3), (b) Time series of the
attractor given by Fig1 (a).
 
b
12
10
10
8
8
Predator
Predator
a
12
6
4
initial point
(6.0,4.0)
4
initial point
(3.0,3.0)
2
0
6
2
0
1
2
3
4
0
5
0
1
2
3
Prey
10
10
8
8
Population
population
12
6
4
2
2
0
2
4
6
Time
6
6
4
0
5
d
c
12
0
4
Prey
8
10
x 10
4
0
2
4
6
Time
8
10
x 10
4
7
Fig2: Globally asymptotically stable limit cycle of system
(8) for the data given by Eq. (8a) with   0.75
(a) The solution approaches to limit cycle from inside
(b) The solution approaches to limit cycle from outside
(c) Time series for Fig (1a) (d) Time series for Fig (1b).
Chapter three
Stability Analysis of a stage
structure
prey-predator model with
Holling type IV functional
response
1.The mathematical models

dx

y2
 x a  bx  2
dt
x  x  


  f1( x, y1, y2 )


dy1
exy 2
 (d1  D ) y1  2
 f 2 ( x, y1, y2 )
dt
x  x  
dy2
 Dy1  d 2 y2  f3 ( x, y1, y2 )
dt
(9)
1.All the solutions of system (9) which
3
initiate in  are uniformly bounded

2. Existence and Local Stability analysis of
system (9) are investigate
(2a) The trivial equilibrium point always exists.
(2b) It is well known that, the prey population grows
to the carrying capacity a
in the absence of
b
predator, while the predator population
dies in the absence of the prey, then the axial
equilibrium point E1   a ,0,0  always exists.
b 
(3) There is no equilibrium point in the
as the predator population dies in the
absence of its prey.
(4) The positive equilibrium point E2  ( x* , y1* , y2* )
3
Int
.

exists in the

Now we obtain the following third order polynomial
equation.
B3 x  B2 x  B1 x  B0  0
3
2
3. Global dynamical behavior of system (9) are
investigate with the help of Lyapunov function
4.The Bifurcation of system (9)
the occurrence of a simple Hopf bifurcation and local
bifurcation (such as saddle node,pitck fork and a
transcritical bifurcation) near the equilibrium points of
system (9) are investigated.
5.Numerical analysis
In this section the global dynamics of system (9) is studied
numerically. The system (9) is solved numerically for
different sets of parameters and for different sets of initial
condition, by using six order Runge-Kutta method
a  0.25, b  0.2,   1,   0.75,   2,
(9a)
d1  0.01, D  0.25, e  0.35, d 2  0.05
(a)
(b)
1.2
0.8
1
Initial point
(0.85,0.75,0.65)
Stable point
(0.38,0.08,0.44)
Initial point
(0.65,0.55,0.45)
0.5
0
0.8
Populations
Mature Predator
1.5
0.4
Initial point
(0.45,0.35,0.25)
1.5
0.6
0.4
Immature Predator
1
0.2
0.5
0
0
Prey
0
0
0.5
1
Time
1.5
2
x 10
5
Fig. (1): (a) The solution of system (9) approaches
asymptotically to the positive equilibrium point starting
from different initial values for the data given by Eq.
(9a). (b) Time series of the attractor in (a) starting at
(0.85, 0.75, 0.65).
(a)
Mature Predator
2
1
Initial point
(0.85,0.75,0.65)
Initial point
(0.85,0.75,0.65)
0
1
3
0.5
2
1
Immature Predator
0
0
Prey
(c)
2.5
2
2
Populations
Populations
(b)
2.5
1.5
1
0.5
0
1.5
1
0.5
0
2.5
5
Time
7.5
x 10
4
0
0
2.5
5
Time
7.5
x 10
4
Fig. (2): (a) Globally asymptotically stable limit cycle
of system (9) starting from different initial values for
the data given by Eq. (9a) with a  0.5
(b) Time series of the attractor in (a) starting at
(0.85, 0.75, 0.65). (c) Time series of the attractor in (a)
starting at (1.25, 0.25, 0.75).
6
5
Populations
4
3
2
1
0
0
2
4
Time
6
8
x 10
4
Fig. (3): The trajectory of system (9) approaches
asymptotically to stable point E1  5,0,0 for a  1
with the rest of parameter as in Eq. (9a).