F05434658

IOSR Journal of Engineering (IOSRJEN) ISSN (e): 2250-3021, ISSN (p): 2278-8719 Vol. 05, Issue 04 (April. 2015), ||V3|| PP 46-58

www.iosrjen.org

Study of predator switching in an eco-epidemiological model with disease in the prey Ala'a Abbas M.Rasheed1 , Raid Kamel Naji2, Basim N.Abood1 1.

Department of mathematics, College of science, University Technology, Baghdad, Iraq.

Department of mathematics, College of science, University Baghdad, Baghdad, Iraq.

2.

Abstract: - In this paper, the dynamical behavior of some eco-epidemiological models is investigated. Preypredator models involving infectious disease in prey population, which divided it into two compartments; namely susceptible population S and infected population I, are proposed and analyzed. The proposed model deals with SIS infectious disease that transmitted directly from external sources, as well as, through direct contact between susceptible and infected individuals. in addition to the infectious disease in a prey species, predator switching among susceptible and infected prey population . The model are represented mathematically by the set of nonlinear differential equations .The existence, uniqueness and boundedness of this model are investigated. The local and global stability conditions of all possible equilibrium points are established. Finally, using numerical simulations to study the global dynamics of the model .

Keyword: prey-predator model; local stability; global stability; switching.

I.

INTRODUCTION

A prey-predator model inovlving predator switching is received a lot of attention in literatures. It is well known that in nature there are different factors; such as stage structure, disease, delay, harvesting and switching; effect the dynamical behavior of the model. In addition, the existence of some type of disease divides the population into two classes known as susceptible and infected. Further, since the infected prey is weaker than the susceptible prey therefore catching the infected prey by a predator will be easer than catching the susceptible prey. Consequently most of the prey-predator models assume that the predator prefers to eat the infected prey, however when the infected prey species is rare or in significant defense capability with respect to predation then a predator switches to feed on the susceptible prey species. Number of mathematical models involving predator switching due to various reasons have been studied. Tansky [6]investigated a mathematical model of one predator and two prey system where the predator has switching tendency. Khan [3]studied a prey- predator model with predator switching where the prey species have the group defense capability. Malchow [4]studied an excitable plankton ecosystem model with lysogenic viral infection that incorporates predator switching using a Holling type-III functional response.

Keeping the above in view, in this paper, a prey-predator model involving SIS infectious disease in prey species with predator switching is proposed and analyzed. It is assumed that the disease transmitted within the prey population by direct contact and though an external sources. The existence, uniqueness and boundedness of the solution are discussed. The existence and the stability analysis of all possible equilibrium points are studied. Finally, the global dynamics of the model is carried out analytically as well as numerically.

The mathematical model: In this section an eco-epidemiological model describing a prey-predator system with switching in the predator is proposed for study, the system involving an SIS epidemic disease in prey population. In the presence of disease, the prey population is divided into two classes: the susceptible individuals S(t) and the infected individuals

I t  , here S t 

represents the density of susceptible individuals at time t while

I t  represents

the infected individuals at time t. The prey population grows logistically with intrinsic growth rate environmental carrying capacity K positive death rate

d1  0 .

 0.

r 0

The existence of disease causes death in the infected prey with

The predator species consumes the prey species (susceptible as well as infected)

according to predation rate represented by the switching functions [5]with switching rate constants and rates

P2  0

e1  0

and

P1  0

respectively, however it converts the food from susceptible and infected prey with a conversion and

e2  0

respectively. Finally, in the absence of prey species the predator species decay

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d2  0 .

exponentially with a natural death rate

Now in order to formulate our model, the following

assumptions are adopted: Consider a prey-predator system in which the density of prey at time



t

is denoted by

N t 

while

the density of predator species at time t is denoted by Y t . Let the following assumptions are adopted: 1. Only the susceptible prey can reproduce logistically, however the infected prey can't reproduce but still has a capability to compete with the other prey individuals for carrying capacity.

2. The susceptible prey becomes infected prey due to contact between both the species as well as external sources for the infection with the contact infection rate constant   0 and an external infection rate

3.

C  0 . However, the infected prey recover constant   0 .

P1SY P2 IY The functions

1



I 2 S

and

1



S 2 I

and return to the susceptible prey with a recover rate

mathematically characterize the switching property [34]

Biologically these functions signify the fact that the predatory rate decreases when the population of one prey species becomes rare compared to the population of the other prey species. Since logically the amount of food taken by a predator from one of its prey effected by the amount of food taken from the other prey species, consequently we will use the following modified switching functions, which are give by

P1SY

1

    I 2 S

P2 IY

and

S 2 I

1

    I 2 S

S 2 I

instead of the above functions.

According to the above hypothesis the dynamics of a prey-predator model involving an SIS epidemic disease in prey population can be describe by the following set of nonlinear differential equations:

dS  rS (1  dt

SI )  ( I K

 C ) S  I 

P1SY

1

P2 IY dI  ( I  C ) S  d1I  I  2 dt 1  SI 

SI 2  SI 2

 

(.1)

S 2 I

e P SY  e P IY dY  d 2Y  1 1 2 2 2 2 dt 1  SI  SI

 

   0 , I 0  0

here S 0 continuous



and Y 0  0 . Obviously the interaction functions of the system (1) are have continuous partial derivatives on the region.



and

R3  S , I , Y   R3 : S 0  0, I 0  0, Y 0  0 . .

Lipschitzian on

Therefore these functions are

R3 , and hence the solution of the system (1) exists and is unique. Further, in the following

theorem the boundedness of the solution of the system (1) in

R3

is established.

Theorem (1): All the solutions of the system (1) are uniformly bounded.

Proof: Let W  S  I  Y then we get

dW dt

 dS  dt



dY dt dS dI dY , , So, by substituting the values of and then simplifying the resulting terms we get dt dt dt dW  S r  1  S  d I  d Y 1 2 dt



Now since

S

dI dt



growth logistically in the absence of other species with internist growth rate

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r

and carrying

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Study of predator switching in an eco-epidemiological model with disease in the prey

K

capacity

S

then it is easy to verify that

dˆ  min 1, d1, d 2  dW  Kˆ r  1  dˆW

ˆ  max S 0 , K  . Further let is bounded above by K then

we

get: thus as

dt

Kˆ r 1 . Hence all solutions of system (1) are uniformly bounded and dˆ

t   it is obtain that 0  W 

■

therefore we have finished the proof.

The equilibrium points: In this section all possible equilibrium points of system (1) are presented: 1. Although, system (1) is not well defined at the origin and the vanishing equilibrium point

E0 0,0,0  is not

exist mathematically, the extinction of all species in any environment is still possible and hence the equilibrium point

E0

exists biologically and need to study.



E1  Sˆ , Iˆ,0

2. The predator free equilibrium point that denoted by

 B1  Sˆ 

B12  4 B 2



, where

CSˆ (2a) 2 d1    Sˆ Cr  r (  d1  K ) rK ( d1   )  Cd1 K 2 here B1   and B2  , exists uniquely in the Int.R of r r

Iˆ 

and

SI  plane provided that the following condition holds Cd Sˆ  d    1 1

r

3. The positive equilibrium point



E2  S  , I  , Y 

algebraic system has a unique positive solution given by





rS 1  SK I  I  C S  I 

I  C S  d1I  I   d 2Y 

exists uniquely in

S , I 

S I I S 2 2

4

P2 I 3 S 2Y S I I S 2 2

S I I S 4

and

Y

P1S 3 I 2Y

4

e1P1S 3 I 2Y  e2 P2 I 3 S 2Y 2 2



(2b)

4

4

4

Int.R3 provided that the follow

.

0

0

0

3.4 The stability analysis of system (1): In this section the stability of each equilibrium point of system (1) is studied and the results are summarized as follows: Since the Jacobian matrix of system (1) near the trivial equilibrium the dynamical behavior of system (1) around Arino [1]. Now, rewrite system (1) in

RN

E0

E0  0,0,0 

is not defined. Hence

is studied by using the technique of Venturino (2008) and

as follows:

dX  H  X t   Q X t  dt 1 in which H is C outside the origin and homogenous of degree 1, thus H sX   sH  X  International organization of Scientific Research

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Study of predator switching in an eco-epidemiological model with disease in the prey

s  0, X  R N , Q  X   o  X .

For all

Q

and

is a

C1

function such that in the vicinity of the origin we have

To study the behavior of the system at the origin point, we use

, 



that denotes the Euclidian norm on

RN ,

N  3 . Let X   x1 , x2 , x3   S , I , Y ; H  X   H1  X , H 2  X , H 3  X  ; Q X   Q1  X , Q2  X , Q3  X  .

Therefore, the functions H i and Qi i  1,2,3 are given by and

denotes the associated inner product, in the case of our model,

H1  X   rx1  Cx1  x2 ; H 2  X   Cx1  d1 x2  x2 ; H 3  X    d 2 x3

P1 x13 x22 x3  rx12  rx1 x2 Q1  X    x2 x1  2 2 K x1 x2  x24  x14 Q2  X   x2 x1  Q3  X  

Let

P2 x23 x12 x3 x12 x22  x24  x14

;

;

e1P1x13 x22 x3  e2 P2 x23 x12 x3

x12 x22  x24  x14 X t  be a solution of system (3). Assume that lim inf X (t )  0 t 

and

X

is bounded. Then it is

X t n  , t n   , from the family  X (t  ) t 0 X t n  .  0 locally uniformly on s  R .

X t n  s  Define y n s   (4) X t n  s  Recall that, Q  X   o X  in the vicinity of the origin. Then Q can be written as possible to extract the sequences

Q X   X O1 2

We have

.

dX t n  s   H  X t n  s   Q X t n  s . ds

such that

(5)

(6)

from (4) we have

X tn  s   yn s  X tn  s   yn s   X tn  s , X tn  s 

1 2

(7)

On the other hand we have

dX t n  s  d X t n  s , X t n  s   2 X t n  s , (8) ds ds So, by take the derivative of X t n  s  in Eq. (7) with respect to s, and using Eq. (8) we obtain dX t n  s  dyn s  y n s  dX t n  s   X t n  s   X t n  s , ds ds X t n  s  ds Therefore we have

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H ( X (t n  s ))  Q( X (t n  s ))   Now dividing by

dyn ( s ) X (t n  s ) ds

yn (s) X (t n  s ), H ( X (t n  s))  Q( X (t n  s )) X (t n  s)

X (t n  s)

and replacing

X (t n  s ) by X (t n  s )

y n s  , then simplifying the resulting terms

give

dyn ( s)  H ( yn ( s))  y n s  yn ( s), H  y n s   X t n  s  ds  Q X t  s  Q X t n  s    n   yn s  y n s , 2 2  X ( t  s X ( t  s )   n n Which is equivalent to

dyn  H  yn s   yn s  yn s , H  yn s   ds  X t n  s  Q yn s   yn s  yn s , Q yn s   Clearly, y n is bounded,

yn (s)  1, s,

theorem [2], one can extract from

y n (s )

and

dyn is bounded too. So, applying the Ascoli – Arzela ds

a subsequence-also denoted by

y n (s ) , which converges locally

R towards some function y, such that: X t n  s  Q ( yn ( s ))  y n s  y n s , Q y n s      0

uniformly on

t n 

and y satisfies the following system:

dy  H  yt   yt  yt , H  yt  dt Equation (9) is defined for all t  R .

;

yt   1, t

(9)

Let us, for a moment, focus on the study of equation (9). The steady states of

H V   V V , H V 

H

are vectors V satisfying

This is a so-called nonlinear eigenvalue. Note that the equation can be alternatively written as

H V   V

with

V  1 ; it then holds that

  V, H V 

(10) . These stationary solutions correspond to fixed directions

that the trajectories of equation (9) may reach asymptotically. Further more equation (10) can be written as

  r  C v1  v2  0 Cv1    d1   v2  0   d 2 v3  0

(11)

Now, we are in a position to discuss in detail the possibility of reaching the origin following fixed directions.

v3  0 v1  0 and v2  0 : in this case, there is a possibility to reach the origin following the I  axis when    d1 with   0 .

Case (1) (a)

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v1  0 and v 2  0 : in this case there is a possibility to reach the origin following S    r with C  0 . (c) v1  0 and v 2  0 : in this case, we obtain different results depending on the parameters:

(b)

axis when

Sub case (1): there is a possibility to reach the origin if

(d1    C  r ) 2  4r (d1   )  4Cd1

 



( d1    C  r ) 2

 12 (d1    C  r ) 2  (r (d1   )  Cd1 )



1 2

Sub case (2): we can not reach the origin otherwise.

Case (2)

v3  0

In this case, we obtain different results depending on the parameters as these given case 1 above with

  d 2 .

The Jacobian matrix of system (1) at the predator free equilibrium point

  ˆ ˆ  r 1  2 S  I  Iˆ  C K    J E1    Iˆ  C    0   









 rSˆ K

E1 can be written as:

   2 ˆ 2  Sˆ I 1 ˆ  ˆ  I S  ˆ  P2 I  2 ˆ 2 Sˆ I  1 ˆ  ˆ I S  ˆ ˆ e P S  e2 P2 I   d2  1 1 2 ˆ 2 Sˆ  I 1 ˆ  ˆ I  S  P1Sˆ

 Sˆ  

 

Sˆ  d1  

 

 

0

Consequently, the characteristic equation can be written as



2 1

here

  ˆ ˆ   e p S  e2 p2 I  T1  D  d 2  1 1   1 0 ˆ 2 ˆ 2 S I   1 ˆ  ˆ I S  



 

     (Iˆ  C)Sˆ  d     Iˆ  C D  r 1  ˆ ˆ

T  r 1  2SK I  Iˆ  C  Sˆ  d1   2Sˆ  Iˆ K

 rSˆ K

1

 Sˆ  



Clearly the eigenvalues of this Jacobian matrix satisfy the following relationships:

T  1S  1I , D  1S .1I , and 1Y  d 2  Accordingly the equilibrium point



ˆ ˆ





E1

r 1  2SK I  Iˆ  C



e1 p1Sˆ  e2 p2 Iˆ 1

    2 Iˆ Sˆ

2 Sˆ Iˆ

(12)

is locally asymptotically stable provided that:

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K  Sˆ r  K e1 p1Sˆ  e2 p2 Iˆ 1

   

2 Sˆ Iˆ

2 Iˆ S

(13b)

 d2

(13c)

However, it is unstable point otherwise.

The Jacobian matrix of coexistence equilibrium point

J ( E2 )  (cij )33

E2

can be written as:

where:



P1 S

2 2   2 2

I

  c11  r 1  2 S K I   M 3   



c12   rS  S     K c13 

c22

3 2

 P1S I M4

Y S  

I

S

4

 3I

4

    

M 42

3 4  4 2 P1S  I Y   S   I     2 M4

 0 , c21  M 3 

3 4  4 2 P2 S  I  Y   I   S     2 M4

2 2 2 2 4 4 P2 S  I  Y   S  I   I   3S   2 3   c   P2 S I  0   S   d1    23 2 M4 M4

2 2 2 2   Y  I  S   M 4 M 5  2S  M 6  I   2S        c31  M 42 2  2 2 2  Y  I  S   M 4 M 7  2 I  M 6  S   2 I      c32  2 M4 c33  0 .

M 3  I   C , M 4  S  I   I   S  2

where

2

4

4

.

M 5  3e1P1S   2e2 P2 I  , M 6  e1P1S   e2 P2 I 

M 7  2e1P1S   3e2 P2 I  .

Then the characteristic equation of J  E 2  can be written as: and

3  C12  C2  C3  0

here

C1  c11  c22  , C2  c11c22  c12 c21  c13c31  c23c32 C3  c31 c13c22  c12 c23   c32 c11c23  c13c21 

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  C1C2  C3  c11  c22 c12 c21  c11c22   c31 c11c13  c23c12 

According to Routh-Hurwitz

 c32 c22 c23  c13c21  criterion the equilibrium point E 2

is locally asymptotically stable

provided that C1  0 , C3  0 and   C1C 2  C 3  0 . Hence straightforward computation show that the positive equilibrium point

E2

is locally asymptotically stable provided that

K  2S   I  4

(14a)

4

4

I   S   3I 

(14b)

d 

K r  K

(14c)  S   1 2 2   2  2     2 2    2S  M 6  I   2S   2 I M 6  S  2 I        M 4  max .  ,  (14d) M5 M7     5 2 2 2 P1S  S   d1    P2 I M 42  2 P2 P1S  I  Y   I   3S   (14e)   P2 I  d1    S   M 3 P1S  M 24

 











2 2 4 4 2 3   S  I  Y   P22  S  I   I   3S   

Clearly, the condition (14a)-(14c) guarantee that However, condition (14d) guarantee that

c31

and

  2 P 2  I  4  S  4   (14f)   1    

c11 ,c12 and c 22 are negative while c21 is positive. c32 are positive and hence C1  0 . Further the conditions

(14a)-( 14d) with condition (14e) guarantee that C3  0 . Finally the condition (14a)-(14d) with condition (14e) guarantee that C1C 2  C 3  0 .

Theorem (3): Assume that

E1

is a locally asymptotically stable point in

asymptotically stable on the sub region of

G22  4G1G3



here

G1U 1  G3 U 2

G1  C  r 



2



R3

then

E1

is globally

that satisfy the following conditions:

Y SP1 N1  IP2 N 2  1   SI    SI  2

R3

2

(15a) (15b)

Kr   Iˆ  Kr Sˆ , G2    C  Kr   S  Iˆ

G3  d1     S , U1  S  Sˆ .

U 2  I  Iˆ , N1  Sˆ  e1 , N 2  Iˆ  e2 Proof: Consider the following positive definite function:

L1

2 2   S  Sˆ  I  Iˆ    Y

2

2

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Study of predator switching in an eco-epidemiological model with disease in the prey

L1 : R3  R

Clearly,

L1 S , I ,0   0

for all

S , I ,Y  R3 with S , I ,Y   Sˆ, Iˆ,0.

Therefore by differentiating this function with respect to the variable



dL1  S  Sˆ dt Substituting the value of



dSdt  I  IˆdIdt  dYdt

dS dt

,

t



L1 Sˆ , Iˆ,0  0

is continuously differentiable function so that

and

we get:

dI dY and in this equation and then simplifying the resulting terms we obtain: dt dt

dL1 Y SP1 N1  IP2 N 2   G1U12  G2U1U 2  G3U 2 2  2 2 dt 1 I  S

S   I 

So, by using condition (15a) we obtain that:



dL1   G1U1  G3U 2 dt

2  Y SP1NI 12 IPS2 N2 2  1

S    I 

Further according to condition (3.15b) it is easy to verify that Thus

E1

is globally asymptotically stable on the sub region of

Theorem (4): Assume that

E2

G5 2  4G4G6



G4 U 3  G6 U 4

G4  C  r 

 0 , and hence L1 is a Lyapnuov function.

R3

that satisfy the given conditions.

is a locally asymptotically stable point in

asymptotically stable on the sub region of

where

dL1 dt



2



R3

R3 ,

then

E2

(16a)



■

is globally

that satisfies the following condition:

Y SP1 N 3  IP2 N 4  1

    I 2 S

S 2 I





Y  P1S  N 5  P2 I  N 6 2

1   I     S  S   I  



2

(16b)

Kr   I   Kr S  , G5  Kr   S  I   C   ,

G6  d1     S , U 3  S  S  .

U 4  I  I  , N3  S   e1Y  .

N 4  I   e2Y  , N 5  S  e1Y  , N 6  I  e2Y  .

Proof: Consider the following positive definite function: 2 2 2  S  S   I  I   Y Y   L2   

2

Clearly,

2

L2 : R3  R

2

is continuously differentiable function so that





    S , I ,Y  R3 with S , I , Y   S , I , Y .

Therefore by differentiating this function with respect to the variable t we get: L2 S , I , Y   0 ;

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



L2 S  , I  ,Y   0

and

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Study of predator switching in an eco-epidemiological model with disease in the prey



dL2  S  S dt

dSdt  I  I  dIdt  Y  Y  dYdt

dI dY and in this equation and then simplifying the resulting terms we obtain: dt dt Y  P1S  N 5  P2 I  N 6 dL2 2 2 Y SP1 N 3  IP2 N 4  G4U 3  G5U 3U 4  G6U 4   2 2  2 dt S 2 I S     I 1 S  I 1        S   I 

Substituting the value of

dS dt

,







 



   2  Y SP1NI 32 IPS2 N2 4   Y P1S N52  P2 I N2 6 

So, by using condition (16a) we obtain that:



dL2   G4U 3  G6U 4 dt

1

S    I 

  1   I     S   S   I 

dL2 dt

Now according to condition (16b) it is easy to verify that function. Thus

E2

is globally asymptotically stable on the sub region of

 0 , and hence L2 R3

is a Lyapnuov

that satisfy the given conditions.

II. NUMERICAL SIMULATION In this section the dynamical behavior of system (1) is studied numerically for different sets of parameters and different sets of initial points. The objectives of this study are: to investigate the effect of varying the value of each parameter (especially the switching rate) on the dynamical behavior of system (1) and toconfirm our obtained analytical results. Now for the following set of hypothetical parameters values:

r  1, K  200,   0.2, C  0.1,   0.4, P1  1, P2  1,

(17)

d1  0.1, d 2  0.1, e1  0.5, e2  0.6 The trajectory of system (1) is drawn in the Fig.(1) for different initial point

16 14

initial point (9,4,4)

Y

12

initial point (10,5,5)

10 8 6 4 8

initial point (11,6,6)

Stable point (1.62,0.57,7.88) 6

15 4

10 2 I

5 0

0

S

Fig.(1): Phase plot of system (1) starting from different initial points.

In the above figure, system (1) approaches asymptotically to the stable coexistence equilibrium point starting from different initial points, which indicates the global stability of the positive equilibrium point.

Note that in time series figures, we will use throughout this section that: blue color for describing the trajectory of S , green color for describing the trajectory of I and red color for describing the trajectory of Y .

Now in order to discuss the effect of varying the intrinsic growth rate r on the dynamical behavior of system (1), the system is solved numerically for different values of parameter r  0.1, 2 keeping other

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Study of predator switching in an eco-epidemiological model with disease in the prey parameters fixed as given in Eq.(17) and then the solution of system (1) is drawn in Fig.(2a)-(2b) while their time series are drawn in Fig. (3a)-(3b) respectively. (a)

(b)

12

50

10 40

8 30

Y

Y

6

20

4

10

2 0 6

0 8

10

4

6

8 4 0

6

2

2

0

I

10 8

4

6

2

4 0

I

S

2

S

Fig.(2): Phase plots of system (1) for the data given by Eq.(17) (a) System (1) approaches asymptotically to

r  0.1 . (b) System (1) approaches asymptotically to coexistence equilibrium point when r=2

when

E0

(a) 12

(b) 45 40

10

35

8

Population

Population

30

6

4

25 20 15 10

2 5

0

0

1000

2000

3000

4000 Time

5000

6000

7000

0

8000

0

1000

2000

3000

4000 Time

5000

6000

7000

8000

Fig. (3): Time series for the solution of system (3.1). (a) Time series for the attractor in Fig.( 2a) (b) Time series for the attractor in Fig.(2b).

Clearly from the Figs. (2a) and (3a), the solution of system (1) approaches asymptotically to the vanishing equilibrium point E 0 , which confirm our analytical results regarding to possibility of approaching of the solution to the vanishing equilibrium point.

The effect of varying the switching rate

P2

of infected prey species on the dynamics of system (1) is

studied and the trajectories of system (1) are drawn in Fig. (4a)-(4b) for the values

P2  0.6, 0.36

respectively, while their time series are drawn in Fig. (5a)-(5b) respectively. (a)

(b)

25

60

20

50 40

Y

Y

15

30

10 20

5

10

0 20

0 15

12

15

10

I

12

10

10

8

10

5

6 5

8 6

4 2

S

I

0

4 2

S

Fig.(4): Phase plots of system (1) for the data given in Eq.(17). (a) System (1) approaches asymptotically to coexistence equilibrium point for

P2  0.36 .

P2  0.6 . (b) System (1) approaches to periodic attractor for

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Study of predator switching in an eco-epidemiological model with disease in the prey (b)

(a)

12

14

12

10

10

Population

8 Population

8

6

6

4 4

2

2

0 0

1000

2000

3000

4000 Time

5000

6000

7000

0

8000

0

1000

2000

3000

4000

5000 Time

6000

7000

8000

9000 10000

Fig.( 5): Time series for the solution of system (1). (a) Time series for the attractor in Fig. (4a) (b) Time series for the attractor in Fig.(4b).

According to the above figures the system loses it stability of positive equilibrium point and the

Int.R3 due to decreasing in the switching rate constant P2 .

The effect of varying death rate of predator d 2 on the dynamical behavior of system (1) is studied and the trajectories of system (1) are drawn in Fig. (6a)-( 6b) for the values d 2  0.8, 0.4 respectively, while solution approaches to periodic dynamics in

their time series are drawn in Fig.( 7a) -(7b) respectively. (a)

(b)

35

25

30 20

20

Y

Y

25 15

15 10 10 5 20

5 20 15

10

8

6

5

6

10

4 0

I

10

15

8

10

2

4 5

I

S

2

S

Fig. (6): Phase plots of system (1) for the data given in Eq. (17) (a) System (1) approaches asymptotically to predator free equilibrium point in the SI-plane for coexistence equilibrium point for

d 2  0.4.

d 2  0.8.

(b) System (1) approaches asymptotically to

(b)

(a)

12

10 9

10

8

8

6

Population

Population

7

5 4

6

4

3 2

2

1 0

0

1000

2000

3000

4000

5000 Time

6000

7000

8000

9000 10000

0

0

1000

2000

3000

4000 Time

5000

6000

7000

8000

Fig.( 7): Time series for the solution of system (1). (a) Time series for the attractor in Fig.(6a) (b) Time series for the attractor in Fig.( 6b) .

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Study of predator switching in an eco-epidemiological model with disease in the prey According to these figures, increasing the predator death rate causes extinction in predator species of system (1).

2.5 Discussion and conclusion: In this chapter, we proposed and analyzed an eco-epidemiological model that described the dynamical behavior of a prey-predator model with modify switching function. The model include three non-linear differential equations that describe the dynamics of three different populations namely predator Y susceptible prey S infected prey I. The boundedness of the system (1) has been discussed. The conditions for existence and stability of each equilibrium points are obtained. To understand the effect of varying each parameter, system (1) is solved numerically and then the obtained results can be summarized as follow: 1. Decreasing the intrinsic growth rate r causes extinction in all species and the solution of the system (1) approaches asymptotically to the vanishing equilibrium point E 0 for r  0.2 . However increasing the intrinsic growth rate causes persists of all species and the solution of system (1) approaches asymptotically to the positive equilibrium point

E2 .

2. Decreasing the values of predator switching rate coexistence equilibrium point periodic dynamics in

E2

P2

in the range

P2  0.39

causes destabilizing of the

3

in the Int .R and the solution of system (1) approaches asymptotically to

Int.R3 .

3. Increasing the values of death rate of predator

d2

causes extinction in predator species and the solution of

E1 for d 2  0.73 K ,  , P1 ,  , e1 , C , e2 , d1

the system (1) approaches asymptotically to the predator free equilibrium point 4. Finally it is observed that, varying each of the values of parameters

has no

effect on the dynamics of the system (1) and the system still approaches asymptotically to coexistence equilibrium point.

REFERENCES [1] [2] [3] [4] [5] [6]

Arino O., Abdllaoui A., Mikram J., and Chattopadhyay J., Infection in prey population may act as a biological control in ratio-dependent predator-prey models, Nonlinearity, 17, 1101-1116, (2004).

Friedman A., Foundations of Modern Analysis, New York, Dover Publications, Inc., (1982).

Khan, Q.A.J., Balakrishnan, E., Wake, G.C., Analysis of a predator-prey system with predator switching.

Bull. Math. Biol. 66,109-123,(2004).

Malchow, H., Hilker, F.M., Sarkar, R.R., Brauer, K., Spatiotemporal patterns in an excitable plankton system with lysogenic viral infections. Math. Comput. Model. 42, 1035-1048,(2005).

Mukhopadhyay. B. , Bhattacharyya .R. Role of Predator switching in an eco-epidemiological model with disease in the prey . Ecological Modelling , 931-939,(2009).

Tansky, M., Switching effects in prey-predator system. J. Theor. Biol. 70, 263-271, (1978).

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