Available at Applications and Applied http://pvamu.edu/aam Mathematics: Appl. Appl. Math. ISSN: 1932-9466 An International Journal (AAM)
Vol. 6, Issue 2 (December 2011), pp. 529 – 551
The Dynamics of Stage Structured Prey-Predator Model Involving Parasitic Infectious Disease
Raid Kamel Naji and Dina Sultan Al-Jaf
Department of Mathematics College of Science University of Baghdad Baghdad, Iraq [email protected]
Received: August 20, 2010; Accepted: November 18, 2011
Abstract
In this paper a prey-predator model involving parasitic infectious disease is proposed and analyzed. It is assumed that the life cycle of predator species is divided into two stages immature and mature. The analysis of local and global stability of all possible subsystems is carried out. The dynamical behaviors of the model system around biologically feasible equilibria are studied. The global dynamics of the model are investigated with the help of Suitable Lyapunov functions. Conditions for which the model persists are established. Finally, to nationalize our analytical results, numerical simulations are worked out for a hypothetical set of parameter values.
Keywords: Stage structure, Prey-Predator model, Parasitic Infectious disease, Stability, Lyapunov function, Persistence
MSC 2010: 92D25, 92D30, 34D20
1. Introduction
It is well known that, in nature species do not exist in exclusion. In fact, any given habitat may contain dozens or hundreds of species, sometimes thousands. Since any species has at least the
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530 Raid Kamel Naji and Dina Sultan Al-Jaf
potential to interact with any other species in its habitat, the spread of a disease in a community may rapidly become astronomical as the number of infected species in the habitat increases. Therefore, it is more of a biological significance to study the effect of disease on the dynamical behavior of interacting species. In the last two decades, some prey- predator models with infectious disease have been considered. Freedman (1990) discussed a prey-predator model with parasitic infection and obtained conditions for persistence. Anderson and May (1986) remarked that a new strain of parasite could change the dynamics of a resident prey-predator or host- parasite system. Mukherjee (1998) studied a generalized prey-predator system with parasitic infection and derived a condition for persistence and impermanence. Xiao and Chen (2001) investigated a prey-predator system with disease in the prey. They showed boundedness of solutions, the nature of equilibria, permanence and global stability, and observed Hopf- bifurcation when the delay increases. Mukherjee (2003) considered delayed prey-predator model with disease in the prey and derived persistence conditions. Recently, Mukherjee (2006) analyzed a prey-predator model in which some members of a prey population and all predators are subjected to infection by a parasite. All these studies converged at one conclusion: that disease may cause vital changes in the dynamics of an ecosystem.
On the other hand, it is well known that, the age factor is importance for the dynamics and evolution of many mammals. The rate of survival, growth and reproduction almost depend on age or development stage and it has been noticed that the life history of many species is composed of at least two stages, immature and mature, and the species in the first stage may often neither interact with other species nor reproduce, being raised by their mature parents. Most of classical prey-predator models of two species in the literature assumed that all predators are able to attack their prey and reproduce, ignoring the fact that the life cycle of most animals consists of at least two stages (immature and mature). In the last three decades several of the prey-predator models with stage-structure are proposed and analyzed [Wang (1997), Wang and Chen (1997), Wang et al (2001), Xiao and Chen (2003), Georgescu and Hsieh (2007)].
In this paper, a prey-predator model involving both stage structure and parasitic infectious disease is proposed and analyzed. The effects of the parasite infection disease and the stage structure on the dynamical behavior of prey-predator model are considered analytically as well as numerically.
2. Mathematical Model
In this section, an eco-epidemiological model is proposed for study. The model consists of a prey, whose total population density is denoted by X (t) , interacting with predator whose total population density is denoted by Y(t) . It is assumed that some members of the prey population and all predators are subjected to infection by a parasite. In addition, the formulation of the proposed eco-epidemiological model depends on the following assumptions:
1. In the absence of parasites and predators, the prey population grows logistically with carrying capacity k (k 0) and an intrinsic birth rate constant r (r 0) . Then the evolution of the prey can be represented as:
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dX X rX 1 ; X (0) 0 . (1) dt k
2. In the presence of parasites, the total prey population is divided into two classes, namely, the susceptible prey s(t) and the infected prey i(t ) , so that at any time t we have X (t) s(t) i(t) . Further, susceptible prey becomes infected, due to existence of parasites,
at a specific infection rate of 0 (0 0 ) .
3. It is assumed that only susceptible prey are capable of reproducing logistically and that the infected prey population dies, with specific death rate constant c (c 0 ) , before having the chance to reproduce. However, the infected prey population still contributes along with susceptible prey population to population growth towards the carrying capacity. Hence, the evolution equations of prey population become
ds s i rs1 s ; s(0) 0 dt k 0 . (2) di s ci ; i(0) 0 dt 0
4. In the case of the existence predator, it is assumed that, the predator population is divided in to two classes, immature predator, whose total population density is denoted by y1(t) and the other mature, whose total population density is denoted by y2 (t) . Moreover, only the mature predator can attack prey population (without distinguish between infected i and healthy s prey) according to Holling type-П functional response with maximum attack rate a (a 0) and half saturation constant b (b 0 ) , and have reproduction ability. While, the immature predator does not attack prey and has no reproductive ability, instead of that, it depends completely on the food supplied by mature predator.
5. In addition to the above, it is assumed that, the immature predator becomes mature predator at a specific rate constant D (D 0 ) , and the predator populations (immature and mature)
decrease due to the natural death rates d1 (d1 0) and d2 (d2 0) respectively.
6. Finally, it is assumed that, susceptible prey individuals that infected by mature predators parasites are removed from the susceptible class at a specific rate proportional with mature predator population (i.e. 1y2 where 1 0 is infection rate due to predator) and an equivalent number of prey are added to the infected class.
Consequently, the dynamics of a stage structured prey-predator model with parasitic infection can be represented in the following set of equations:
532 Raid Kamel Naji and Dina Sultan Al-Jaf
ds s i asy2 rs 1 (0 1 y2 )s ; s(0) 0 dt k b s i di aiy ( y ) s ci 2 ;i(0) 0 dt 0 1 2 b s i (3) dy1 s i d1 y1 Dy1 m y2 ; y1(0) 0 dt b s i dy 2 Dy d y ; y (0) 0. dt 1 2 2 2
In order to simplifying the proposed model represented by system (3), the following dimensionless variables and parameters are used:
s i y a T rt, S , I , Y 1 , Y 1 y , w 0 , w , k k 1 k 2 r 2 1 r 2 k 1 . b c d D m Dk d w , w , w 1 , w , w , w 1 , w 2 3 4 5 6 7 8 2 9 k r r r 1k r r
Consequently, the proposed model can be written in the following dimensionless form:
dS w2SY2 S [1 (S I)] w1S SY2 S f1(S, I,Y1,Y2 ) dT w3 S I dI w IY w S SY w I 2 2 I f (S, I,Y ,Y ) dT 1 2 4 w S I 2 1 2 3 (4) dY S I 1 w5Y1 w6Y1 w7 Y2 Y1 f3 (S, I,Y1,Y2 ) dT w3 S I dY 2 w Y w Y Y f (S, I,Y ,Y ). dT 8 1 9 2 2 4 1 2
The initial condition for system (4) may be taken as any point in the region 4 4 R (S, I,Y1,Y2 ) R :S 0, I 0,Y1 0,Y2 0. Obviously, the interaction functions in the 4 right hand side of system (4) are continuously differentiable functions on R , hence they are Lipschitizian. Therefore the solution of system (4) exists and is unique. Further, all the solutions of system (4) with non-negative initial condition are uniformly bounded as shown in the following theorem.
4 Theorem 1. All the solutions of system (4), which initiate in R are uniformly bounded.
Proof: Let (S(T ),I(T ),Y1(T ),Y2 (T )) be any solution of system (4) with non negative initial condition (S0 ,I0 ,Y10 ,Y20 ) . Since we have AAM: Intern. J., Vol. 6, Issue 2 (December 2011) 533
dS S (1 S) . dT
Then according to the theory of differential inequality [Birkhoof and Rota (1982)], we have
Sup S(T) M , T 0 , where M max{S0 , 1 }.
Now, consider the function: W (T ) S(T ) I(T ) Y1(T ) Y2 (T ) . Then the time derivative of W (T) along the solution of the system (4) is: dW 2S S w I (w w w )Y w Y . dT 4 5 6 8 1 9 2
Note that since the parameters w5 and w6 stand for the natural death rate of immature predator and the grown up rate of immature predator to mature predator respectively. While, w8 represents the conversion rate from immature predator to mature predator. Therefore the following condition always satisfied.
w5 w6 w8 . (5)
Hence, we obtain:
dW m W 2M , dT
where m min{1,w4 ,w5 w6 w8 ,w9 }.
Again, due to the theory of differential inequalities we obtain
2M 2M W W (T ) 0 , where W (S(0), I(0),Y (0),Y (0 )). m memT emT 0 1 2
2M Thus, T 0 we have that 0 W (T ) M1 , where M1 max m ,W0 . Thus all solutions of system (4) are uniformly bounded, and then the proof is complete.
Keeping the above in view, since the dynamical system is said to be dissipative if and only if it is uniformly bounded, then system (4) is dissipative.
3. Stability Analysis of 2D Predator Free Subsystem
It is well known that according to the prey-predator interaction, the prey species can survive in the absence of predator, and since the prey population is divided into two classes namely
534 Raid Kamel Naji and Dina Sultan Al-Jaf
susceptible prey population S and the infected prey population I . Hence, the following 2D predator free subsystem is obtained:
dS S 1 S I w g (S, I) dT 1 1 . (6) dI w1S I w4 g2 (S,I) dT I
The analysis of local and global stability of subsystem (6) is carried out and the following results are obtained:
1. The vanishing equilibrium point P0 (0,0 ) always exists and its locally asymptotically stable provided that
w1 1 . (7)
ˆ ˆ 2. There is no axial equilibrium point such as P1 (S,0); S 0 on the S axis due to the fact that w1 0. However P1 (1,0) exists if we set w1 0 .
~ ~ 2 3. The interior equilibrium point P2 (S , I ) in the Int.R(SI ) , where
~ w (1 w ) ~ w (1 w ) S 4 1 ; I 1 1 , (8) w1 w4 w1 w4
exists under the following condition:
w1 1. (9)
0 Obviously, from a biological point of view the above condition w1( r ) 1, which means that the disease contact rate 0 is less than the reproduction rate r , is the necessary condition for the dS existence of (endemic) equilibrium point P2 otherwise (i.e. when 0 r ) dT 0 and hence the prey species will face extinction. In addition, the equilibrium point P2 is locally asymptotically stable whenever it exists. Further, the global dynamics of P2 is carried out in the next theorem.
~ ~ Theorem 2. The equilibrium point P2 (S , I ) is globally asymptotically stable in the 2 2 Int.R(SI ) (S,I) R ;S 0,I 0.
Proof: AAM: Intern. J., Vol. 6, Issue 2 (December 2011) 535
1 1 Consider the function H (S , I) SI ; clearly H (S , I ) is C function and is positive for all 2 (S ,I) Int.R(SI ) . Note that
(Hg ) (Hg ) (I w ) (S , I ) 1 2 1 0. S I I 2
2 Hence (S , I) does not change sign and is not identically zero in the Int.R(SI ) . Then according 2 to Bendixon-Dulic criterion, there is no periodic solution in the Int.R(SI ) . So, since all the solutions of the subsystem (6) are uniformly bounded and P2 is unique equilibrium point in the 2 Int.R(SI ) . Hence by using the Poincare-Bendixon theorem P2 is a globally asymptotically stable and hence the proof is complete.
4. Stability Analysis of 3D Disease Free Subsystem
In the absence of disease the prey population contains only susceptible individuals and the following 3D disease free subsystem of system (4) appears:
dS Y2 S 1 S w1 Y2 w2 S J1(S,Y1,Y2 ) dT w3 S
dY1 w7 SY2 Y1 (w5 w6 ) Y1J 2 (S,Y1,Y2 ) (10) dT (w3 S)Y1
dY2 w8Y1 Y2 w9 Y2 J 3 (S,Y1,Y2 ). dT Y2
Our analysis of subsystem (10) gave the following results:
1. The vanishing equilibrium point F0 (0, 0, 0) always exists and is unstable saddle point provided that condition (9) holds, while it is locally asymptotically stable provided that condition (7) holds.
2. The axial equilibrium point F1 (1 w1,0,0) , exists under condition (9) and is locally asymptotically stable in the R3 if and only if the following condition holds: (SY1Y2 )
w7w8 (1 w1) w9 (w5 w6 ) . (11a) w3 (1 w1)
536 Raid Kamel Naji and Dina Sultan Al-Jaf
While it is unstable saddle point provided that
w7w8 (1 w1) w9 (w5 w6 ) . (11b) w3 (1 w1)
3. The interior equilibrium point F2 (S ,Y1,Y2 ) , where
w3w9 (w5 w6 ) w9 (w3 S )(1 S w1) S , Y1 Y2 , Y2 . (12) w7w8 w9 (w5 w6 ) w8 w2 w3 S
exists in the Int.R3 (S,Y ,Y ) R3 :S 0,Y 0,Y 0 under the following (SY1Y2 ) 1 2 1 2 conditions:
w7w8 w9 (w5 w6 ) (13a)
(1 S w1) 0 S w1 1 . (13b)
In addition it is locally asymptotically stable in the Int.R3 provided that: (SY1Y2 )
2 w 2 2 Y min 1 , 9 2 3 1 2 3 , (14) 2 2 2 w2 w3w7w8w91 (w2 1)
2 2 where 1 w3 S , 2 S (1 w2Y2 ) and 3 w7 w8S w91 . Moreover, the global stability of F2 is investigated in the following theorem.
Theorem 3. Assume that F (S ,Y ,Y ) is locally asymptotically stable in the Int. R3 , and 2 1 2 (SY1Y2 ) let the following conditions are hold
a 0 , (15a)
w3w7Y2 a w2 , (15b) w7w8S 1 w9 1Y1 1w9Y1 Y2
w SY w Y w w S w 7 2 8 1 7 8 9 , (15c) Y1Y2 1w9Y1 Y2
where a (1 w2Y2 ) and (w3 S ) . Then F2 is globally asymptotically stable in the IntR3 . (SY1Y2 )
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Proof:
Consider the following function:
S Y1 Y2 U (S,Y1,Y2 ) S S S ln Y1 Y1 Y1 ln Y2 Y2 Y2 ln . S Y1 Y2
Clearly U : R3 R , and is a C1 positive definite function. Now, the derivative of U (SY1Y2 ) along the trajectory of subsystem (10) can be written as
dU w Y 2 w w 2 1 2 2 2 9 S S S S Y2 Y2 Y2 Y2 dT 21 2Y2
w Y 2 w w Y w w S 2 1 2 2 3 7 2 7 8 S S S S Y1 Y1 Y1 Y1 21 1Y1 21w9Y1
w w S 2 w SY w Y w 2 7 8 7 2 8 1 9 Y1 Y1 Y1 Y1 Y2 Y2 Y2 Y2 . 21w9Y1 Y1Y2 2Y2
Therefore, according to conditions (15a-15c) we obtain that:
dU 2 2 a w9 a w7w8S 2 S S 2Y Y2 Y2 2 S S 2 w Y Y1 Y1 dT 1 2 1 1 9 1
2 w7w8S w9 Y1 Y1 Y2 Y2 . 21w9Y1 2Y2
dU Hence, dT 0 , and then U is strictly Lyapunov function. Therefore, F2 is globally asymptotically stable in the Int. R3 . (SY1Y2 )
5. Local Stability Analysis of System (4)
In this section, the existence and local stability analysis of all possible equilibrium points of system (4) are discussed and the following results are obtained:
1. The vanishing equilibrium point E0 (0,0,0,0 ) always exists. ˆ ˆ 2. There is no axial equilibrium point such as E1 (S,0,0,0); S 0 on the S -axis due to the fact that w1 0. However E1 (1,0,0,0) exists if we set w1 0 . ~ ~ ~ ~ 3. The predator free equilibrium point E2 (S , I ,0,0 ) ; where S and I are given in Equation(8), exists under the condition (9) .
538 Raid Kamel Naji and Dina Sultan Al-Jaf
4. The disease free equilibrium point E3 (S ,0,Y1,Y2 ) ; where S , Y1 and Y2 are given in Equation(12), exists under the conditions (13a-13b). 5. The positive equilibrium point E4 (S , I ,Y1 ,Y2 ) ; where
* * * h(w41 w2Y2 ) * h1(w1 Y2 ) * w9 * * 11 S w1 S , I , Y1 Y2 , Y2 , (16) M1 M1 w8 M 2
* * * * with M1 1(w1 w4 Y2 ) w2Y2 , M 2 w2 1 , S (S I ) and S is given in 4 Equation (12), exists uniquely in the Int.R provided that conditions (13a-13b) hold.
In addition, it is observed that, the eigenvalues of the Jacobian matrix of system (4) at E0 , say J(E0 ) , are:
1 w , w 0, (w w ) 0, w 0 . (17) 0S 1 0I 4 0Y1 5 6 0Y2 9
Therefore, E is unstable saddle point with locally stable manifold in the R3 0 (IY1Y2 ) (i.e. dim( s ) 3) and with locally unstable manifold in the S direction (i.e. dim(u ) 1) provided that condition (9) holds. However, it is locally asymptotically stable provided that condition (7) holds.
The eigenvalues of Jacobian matrix of system (4) at E2 , say J (E2 ) , satisfy the following relations:
~ 2S 2I (S w4 ) 0 (18a)
~ 2S .2I (w1 w4 )S 0 (18b)
(w w w ) 0 (18c) 2Y1 2Y2 5 6 9
w7w8 (1w1) 2Y .2Y w9 (w5 w6 ) . (18d) 1 2 w3(1w1)
Note that, it is easy to verify that, according to Eqs. (18a-18d) all the eigenvalues of J (E2 ) have 4 negative real parts and hence E2 is locally asymptotically stable in the R if and only if 4 condition (11a) holds. However, E2 is an unstable saddle point in the R with locally unstable manifold of dimension one (i.e. dimu 1) and with locally stable manifold of dimension three (i.e.dim s 3 ) if condition (11b) holds.
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Now, the Jacobian matrix at the disease free equilibrium point E3 (S ,0,Y1,Y2 ) can be written as:
JE()[]344 bij ;, ij 1,2,3,4, (19)
2 (1 w2w3)Y2 w2Y2 w2 where b11 1 2S w1 2 , b12 S 1 2 , b13 0 , b14 S 1 0 , 1 1 1
w2Y2 w3w7Y2 b21 w1 Y2 0 , b22 w4 0 , b23 0 , b24 S 0 , b31 b32 2 0 , 1 1
w7S b33 (w5 w6 ) 0 , b34 0, b41 b42 0 , b43 w8 0 , b44 w9 0 . Then the 1 characteristic equation of the Jacobian matrix J (E3 ) is given by:
4 3 2 A1 A2 A3 A4 0 (20a)
Here, A1 (1 2 ) , A2 3 1 2 , A3 2 3 b31b43 4 , A4 b31b43( 5 6 ) with
1 b11 b22 , 2 b33 b44 , 3 b12b21 b11b22 , 4 b14 b24 , 5 b14b22 b12b24 and 6 b11b24 b14b21.
Note that, according to the elements of J (E3 ) , it is easy to verify that:
1 2 2 1 2 1 (1 2S w1) (1 w2w3 )Y2 (w41 w2Y2 )1, 1
2 (w5 w6 w9 ) 0 ,
1 2 2 2 3 2 S (w2Y2 1 )(w1 Y2 ) 1 (1 2S w1) (1 w2w3 )Y2 (w41 w2Y2 ), 1
w2S 4 0 , 1
S 2 5 2 (1 w2 )(w41 w2Y2 ) S (w2Y2 1 ), 1
S 2 6 2 1 (1 2S ) w1w21 w2SY2 . 1
Further
540 Raid Kamel Naji and Dina Sultan Al-Jaf
2 2 A1A2 A3 A3 A1 A4 2 2 3 1 2 3 b31b431 4 2 3 b31b43 2 4 1 2 (20b) 2 1 3 4 ( 2 3 b31b43 4 ) 4 (1 2 ) ( 5 6 ) b31b43.
Now, from the Routh-Hurwitz criterion all the eigenvalues of the J (E3 ) have roots with 4 negative real parts and hence E3 is locally asymptotically stable in the R , if and only if Ai (i 1,3,4) 0 and 0 .
Straight forward computations show that, 1 and 3 are negative while 5 and 6 are positive under the following condition:
(1 2S w ) 2 [(2S 1) w w ] 2 max 1 1 , 1 1 1 2 ,0 Y 1 . (21a) 2 w S 2 w 1 w2w3 2 2
Consequently, we obtain that Ai 0 for i 1,3,4 . Moreover, in addition to condition (21a), it is observed that 0 provided that:
1 2 1 ( 2 3 b31b43 4 ) 4 (1 2 ) ( 5 6 ) . (21b) 3 4
Hence, from to the above analysis, the following theorem can be proved easily.
4 Theorem 4. Assume that the disease free equilibrium point E3 (S ,0,Y1,Y2 ) exists in the R , then E3 is locally asymptotically stable if and only if conditions (21a-21b) are hold.
Finally, the Jacobian matrix of system (4) at the positive equilibrium point * * * * E4 (S , I ,Y1 ,Y2 ) can be written as:
JE(444 ) [ cij ] ; ij , 1, 2,3, 4, (22)
* * * * * w2Y2 S * w2Y2 I where c11 c12 S 1 2 , c13 0, c14 M 2 0, c21 w1 Y2 2 0 , 1 1 1 * * * * * * (w1Y2 )S w2Y2 I * w2I w3w7Y2 w7w8S c22 * 2 , c23 0, c24 S , c31 c32 2 0, c33 w 0 , I 1 1 1 9 1
w7S c34 0 , c41 c42 0 , c43 w8 0 , c44 w9 0 . Accordingly the characteristic 1 equation of J (E4 ) is given by:
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4 3 2 B1 B2 B3 B4 0 . (23)
Here, B1 (1 2 ) , B2 3 1 2 , B3 2 3 c31c43 4 , B4 c31c43( 5 6 ) , with 1 c11 c22 , 2 c33 c44 , 3 c12c21 c11c22 , 4 c14 c24 , 5 c14c22 c12c24 , 6 c11c24 c14c21 , and
*22 BBB123 B 3 B 1 B 4 22 3 1 2 3 cc 31 43 1 4 2 3 cc 31 43 2 4 1 2 2 13 4 ()()(). 2 3 cc 31434 4 1 2 5 6 cc 3143
So, due to the elements of J (E4 ), it is easy to verify that:
* * * * * S * 2 (w1Y2 )S w2I Y2 w7w8S 1 2 w2Y2 1 * 2 , 2 w w9 0 , 1 I 1 9 1
* * * * w2Y2 * (w1Y2 )S w2S 3 S 1 2 w1 Y2 * , 4 0, 1 I 1
* * * * * * * S M 2 w2Y2 I (w1Y2 )S * w2Y2 * w2I 5 2 * S 1 2 S , 1 1 I 1 1
* * * * * * * w2I w2Y2 S M 2 * w2I Y2 6 S S 1 2 w1 Y2 2 . 1 1 1 1
Therefore, in the following theorem, the local stability conditions for the positive equilibrium 4 point E4 in the Int.R is established.
* * * * 4 Theorem 5. Assume that E4 (S , I ,Y1 ,Y2 ) exists in the Int.R and the following conditions are satisfied:
w I * S * 2 , (24a) 1
2 * * 2 1 S 1[w1(w2 S ) w2 (I w1)] * 1 max ,0 Y2 , (24b) w2S 1M 2 w2
1 2 1 ( 2 3 b31b43 4 ) 4 (1 2 ) ( 5 6 ) . (24c) 3 4
542 Raid Kamel Naji and Dina Sultan Al-Jaf
4 Then, E4 is locally asymptotically stable in the Int.R .
Proof:
According to the Routh-Hurwitz criterion the proof follows if and only if Bi (i 1,3,4) 0 and * 0. Now, straightforward computations show that under the conditions (24a-24b) we obtain that 1 and 3 are negative while 5 and 6 are positive.
Consequently, due to the coefficients of Equation(23) and elements of J (E4 ), we get that Bi 0 for i 1,3,4 . Moreover, it is easy to verify that * 0 if and only if in addition to the conditions
(24a-24b), condition (24c) holds too. Hence, J (E4 ) have eigenvalues with negative real parts. 4 Therefore E4 is locally asymptotically stable in the Int.R and the proof is complete.
6. Global Dynamical Behavior of System (4)
In this section the global stability for the equilibrium points of system (4) is investigated using the Lyapunov method as shown in the following theorems.
Theorem 6. Assume that the vanishing equilibrium point E0 of system (4) is locally 4 4 asymptotically stable in the R . Then E0 is globally asymptotically stable in the R .
Proof:
Consider the function
V0 (S,I,Y1,Y2 ) c1S c2I c3Y1 c4Y2 . (25)
4 1 Clearly V0 :R R is C positive definite function, where ci ,(i 1,2,3,4 ) are nonnegative constants to be determined. Now, since the derivative of V0 along the trajectory of system (4) can be written as:
dV0 ()ccwcwScScSIccSY 2 () cw12 cw 37 SY dT 1112111122 2 cw12 cw 37 cwI24 IY 2 c 3() w 5 w 6 cw 48 Y 1 cwY 492 ,
w2 where (w3 S I) . Now choosing the positive constants c1 1, c2 0 , c3 and w1w7
w2 (w5 w6 ) c4 , yield that w1w7w8 AAM: Intern. J., Vol. 6, Issue 2 (December 2011) 543
dV0 w2 1 1 w1 S 1 SY2 IY2 . dT w1
dV0 Clearly dT 0 under the local stability condition (7), hence V0 is strictly Lyapunov function. 4 Therefore E0 is globally asymptotically stable in the R .
Now, in the following theorem we will study the global behavior of E2 .
Theorem 7. Assume that the predator free equilibrium point E2 is locally asymptotically stable 4 in the R , and let the following conditions:
w I w 1 2 4 , (26) I I
~ ~ S (w ) w I w w 2 2 2 9 (27) w7
4 hold, then E2 is globally asymptotically stable in the R .
Proof:
~ ~ S ~ ~ I w2 w2 4 1 Let V2 (S, I,Y1,Y2 ) S S S ln ~ I I I ln ~ Y1 Y2 . Clearly V2 : R R is C S I w7 w7 positive definite function. Now, according to conditions (26) it is easy to verify that:
2 ~ ~ ~ dV2 ~ w4 ~ ~ w2 (S I ) w2w9 ISY2 w2 (w5 w6 w8 ) S S I I S Y2 Y2. dT I w7 I w7
dV Therefore, 2 0 under condition (27), and hence V is strictly Lyapunov function. Therefore, dT 2 4 E2 is globally asymptotically stable in the R .
Theorem 8. Assume that the disease free equilibrium point E3 is locally asymptotically stable 4 in the R . Then E3 is globally asymptotically stable in the following region
(,,SIYY , ): S SI , ()wYS12 , Y YY , () w 212 Y . 12w41 1 12 Proof:
544 Raid Kamel Naji and Dina Sultan Al-Jaf
Consider the following function:
S Y1 Y2 V3 (S, I,Y1,Y2 ) c1 S S S ln c2 I c3 Y1 Y1 Y1 ln c4 Y2 Y2 Y2 ln . S Y1 Y2
4 1 Clearly V3 :R R is C positive definite function, where ci (i 1,2,3,4 ) are positive constants to be determined. Note that, simple mathematical manipulations give that:
2 2 dV3 cS S cIcw () Y S cScwI cwwS378 Y Y cw22 IY dT 11212124112wY911 2 cw49 ww212 cwwIY337 2 YY22 cY 121 Y 2 SS YY 11 Y2111 Y
cw37() 1 S SI cw 48 cwwY 3372 YYYY112 2 SSYY 11. 11YY 2 11 Y
1 So, choosing the positive constants as c1 w4 , c2 S , c3 and c4 1 gives: w7
2 2 dV3 wSS wISwY () S w8 S Y Y dT441211 w91 Y 2 w9 ww212 wIY32 YY22 wY 421 Y 2 SS YY 11 Y211 Y
()1281SSIY w Y wY 32 YYYY112 2 SSYY 11. YY12 Y 1
dV Clearly 3 0 in , and then V is a strictly Lyapunov function. Therefore E is globally dT 3 3 asymptotically stable in the region .
4 Obviously, in the above theorem represents the basin of attraction for E3 in R . Finally, in the following theorem, the conditions of global asymptotic stability of the positive equilibrium point E4 are established.
* * * * Theorem 9. Assume that the positive equilibrium point E4 (S , I ,Y1 ,Y2 ) is locally 4 asymptotically stable in the Int.R with
ei 0; i 1,2 (28a)
* * 1I 2w2Y2 (w1 Y2 ) 2 e1 e2 (28b) SI 3 * 1 3II 1
(S w I ) w ( w ) 2 2 9 2 (28c) e 3Y e I 2 2 1 * 3S 3II 1 AAM: Intern. J., Vol. 6, Issue 2 (December 2011) 545
w(SI )Y w Y w w Y * w w Y * w w S max 2 8 1 , 3 7 2 , 3 7 2 2 7 8 , (28d) w e e 3w Y Y Y 9 Y 1 Y 2 9 1 1 1 2 3Y 1 1 3 * 2 1 3II 1
* * * * * where e1 (1 w2Y2 ) and e2 1S (w1 Y2 ) w2II Y2 . Then E4 is globally 4 asymptotically stable in the Int.R .
Proof:
Consider the following function
VSIYY(,, , ) S S** S lnS I I ** I lnI 412SI**
YY** YlnYY12 YY ** Y ln . 11 1YY** 2 2 2 12
4 1 Clearly V4 : R R is C positive definite function. Since
22 22 dV4 ee12**wwS78 ** w 9 SS * II YY Y Y 19112II wY 11 Y 2 2 dT 1 ** 2()wY22 w 1 Y 2** w 2 * * 1 I SS II SS Y22 Y 1 SwI 2 IIYY**wSIYwY7281() YYYY ** IYY2212 11 22
** wwY372** wwY 372 ** SS YY11 II YY11 . 11YY 11
Therefore, according to conditions (28a-28d), it is easy to verify that: 2 2 dV4 e1 * e2 * e1 * w9 * 3 (S S ) * (I I ) 3 (S S ) 3Y (Y2 Y2 ) dT 1 3II 1 1 2 2 2 e2 * w9 * w7w8S * w9 * * (I I ) 3Y (Y2 Y2 ) 3w Y (Y1 Y1 ) 3Y (Y2 Y2 ) 3II 1 2 9 1 1 2 2 2 e1 * w7w8S * e2 * w7w8S * 3 (S S ) 3w Y (Y1 Y1 ) * (I I ) 3w Y (Y1 Y1 ) . 1 9 1 1 3II 1 9 1 1
dV4 So, dT 0 , and then V4 is strictly Lyapunov function. Therefore, E4 is globally asymptotically 4 stable in the Int.R .
In the following section, the persistence condition for system (4) is established.
546 Raid Kamel Naji and Dina Sultan Al-Jaf
7. Persistence Analysis
In this section, the persistence of system (4) is studied. It is well known that the system is said to persist if and only if each species persists. Mathematically, this means that, system (4) persists if the solution of the system with positive initial condition does not have omega limit set on the boundary of its domain [see Xiao and Chen (2003)]. In the following theorem the persistence conditions of system (4) are established.
Theorem 10. Assume that the equilibrium points E2 and E3 are globally asymptotically stable in the interior of R2 and R3 respectively. In addition, if conditions (9) and (11b) are (SI ) (SY1Y2 ) hold together with the following condition:
3I 0 . (29)
Here, 3I represents the eigenvalue of J (E3 ) that describe the dynamics in the positive direction of I . Then system (4) persists.
Proof:
4 Suppose that q is a point in the Int.R and o(q) is the orbit through q . Let (q) is the omega limit set of the o(q) . Note that (q) is bounded, due to the boundedness of system (4).
We first claim thatE0 (q ) . Assume the contrary, then since E0 is a saddle point due to condition (9), thus E0 cannot be the only point in (q ) , and hence by Butler-McGhee lemma [Freedman and Waltman (1984)] there is at least one other point, say p , such that p s (E ) (q) , where s (E ) is the stable manifold of E . Now, since s (E ) is the 0 0 0 0 space R3 and the entire orbit through p , which denoted by o( p ) , is contained in (q ) . (IY1Y2 ) Then, if p R3 (i.e. on the boundary axes of R3 ), we obtain that the positive (IY1Y2 ) (IY1Y2 ) specific axis (that containing p ) is contained in (q ) contradicting its boundedness. Now, let p Int.R3 . Since there is no equilibrium point in the Int.R3 , the orbit through p (IY1Y2 ) (IY1Y2 ) which is contained in (q) must be unbounded. Giving a contradiction too, this shows that E (q) . 0
Now, we show that E1 (1 w1,0,0,0) cannot be in (q ) . Since E1 is saddle point under s condition (11b). Then again by Butler-McGhee Lemma p1 (E1) (x) , Also, since s (E ) could be either the space R3 or the space R3 . Suppose that s (E ) is the 1 (SIY1 ) (SIY2 ) 1 space R3 (similar proof for the space R3 ). Note that, if p R3 , then we get (SIY1 ) (SIY2 ) 1 (SIY1) AAM: Intern. J., Vol. 6, Issue 2 (December 2011) 547
contradiction as in the first part of proof. Let now, p Int.R3 , again since there is no 1 (SIY1 ) equilibrium point in the Int.R3 , then the o( p ) (q) is unbounded, which gives a (SIY1) 1 contradiction to the boundedness of (q ) . Thus, E1 (q) .
4 Now, since the points E2 and E3 are saddle points in the R under the conditions (11b) and
(29) respectively. Then by using argument completely analogous to the above yields E2 , E3 4 cannot contained in (q ) . Thus (q ) must be in the Int.R , which proves persistence of the system (4).
8. Numerical Simulation
In this section the global dynamics of system (4) is studied numerically by solving it, for different sets of parameters and for different sets of initial conditions, using predictor-corrector method with six order Runge-Kutta method, and then the time series for the trajectories of system (4) are drown. Note that, we will use solid line type for S ; dash line type for I ; dot line type for Y1 and dash-dot line type for Y2 in the all of the following figures. Now, for the following set of hypothetical set of parameter values:
w1 0.35, w2 0.2, w3 0.2, w4 0.4, w5 0.1, w6 0.3, w7 0.1, w8 0.3, w9 0.1 (30)
It is observed that the equilibrium points E3 and E4 do not exist, while E2 is a global asymptotically stable point, as shown in Figure (1).
Figure 1: Time series of the trajectories of system (4), for data given in Equation (30), which shows Global asymptotically stable point (0.346, 0.3, 0, 0).
548 Raid Kamel Naji and Dina Sultan Al-Jaf
In order to investigate the effect of the infection rate (i.e., parameter w1 ) on the dynamics of system (4) in case of existence of E4 , the system is solved numerically for different values of w1 with w2 0.4 and w7 0.3, while the rest of parameters kept constant as given in Equation (30) and then the trajectories of system (4) are drown in the Figure (2a-2d).
(a) (b) 0.3 0.8
0.2 0.4 0.1 Population Population
0 0 1.6 1.8 2 0 4000 8000 Time 4 Time (c) x 10 (d) 0.8 0.8
0.4 0.4 Population Population
0 0 0 3000 6000 0 750 1500 Time Time
Figure. 2: Time series of the trajectories of system (4), for data given in
Equation (30) with w2 0.4 and w7 0.3, which shows that:
(a) Periodic attractor for w1 0.35 (b) Global asymptotically
stable point (0.08,0.078,0.069,0.2) for w1 0.5 . (c) Global asymptotically stable point (0.037,0.08,0,0) for w1 0.84 . (d) Global asymptotically stable point (0,0,0,0) for w1 1.
According to the above results, it is observed that the trajectory of system (4) approaches periodic dynamics for w1 0.39 as shown in the typical Figure(2a), while it approaches globally 4 asymptotically stable point in the Int.R for 0.39 w1 0.84 as shown in Figure (2b). However for the 0.84 w1 1 and 1 w1 , system (4) losses the persistence and the trajectory approaches ~ ~ 2 to the equilibrium point (S , I ,0,0) (0.037,0.08,0,0 ) in the Int.R(SI ) and the vanishing equilibrium point (0,0,0,0 ) respectively as shown in Figures (2c-2d) .
Finally, the effect of the conversion rate, at which the immature predator becomes mature (i.e. parameter w8 ), on the dynamics of system (4) is investigated. Here the system is solved numerically for the values w1 0.5, w2 0.4, w7 0. 3 while the rest of parameters kept constant as given in Equation (30) and then the trajectories of system (4) are drown in the
Figures (3) and (4a-4b) for w8 0.18, w8 0.34 and w8 0. 38 respectively. AAM: Intern. J., Vol. 6, Issue 2 (December 2011) 549
Figure 3: Time series of the trajectories of system (4) with w8 0.18 , which shows global asymptotically stable point (0.22, 0.27, 0, 0).
Figure 4: Time series for the trajectories of system (4) which shows that: (a) Periodic attractor for w8 0.34 . (b) Periodic attractor for w8 0.38 .
Obviously, as w8 increases, our numerical analysis shows that for w8 0. 18 the trajectory of ~ ~ 2 system (4) approaches to the equilibrium point (S , I ,0,0) (0.22,0.27,0,0 ) in the Int.R(SI ) as shown in Figure (3) and hence system (4) is not persists. However the trajectory of system (4)
550 Raid Kamel Naji and Dina Sultan Al-Jaf
approaches to globally asymptotically stable point for 0.18 w8 0. 33 as shown in Figure (2b), while it is approaches to periodic dynamics for w8 0.33 as shown in Figures (4a-4b).
9. Conclusions and Discussion
In this paper, an eco-epidemiological model has been proposed and analyzed. Our main objective is to understand the effect of parasite infection disease and stage structure on the dynamics of the prey-predator system. As such, the dynamical behavior of system (4) and all possible subsystems have been investigated analytically. The persistence conditions of system (4) have been derived. Moreover, the effects of changing the parameters on the dynamical behavior of system (4) are discussed numerically and the following results are obtained:
1. For the effect of varying w1 , keeping other parameters fixed as in Equation (30) with
w2 0.4, w7 0.3, it is obtained that, for small value of infection rate say ( w1 0.39 ) the 4 trajectory of system (4) approaches a periodic dynamics in the Int.R . As the infection rate increases 0.39 w1 0.84 the trajectory of system (4) approaches a globally asymptotically 4 stable point in the Int.R . Finally, system (4) losses persistence for w1 0.84 . Consequently, for
w1 0.84 the disease is under control and the system persists, for 0.84 w1 1 the disease is still under control but the system losses its persistence; finally for w1 1 the disease is uncontrolled. 2. For the effect of varying w8 , keeping other parameters fixed as, in Equation (30) with
w1 0.5, w2 0.4, w7 0.3, it is obtained that, for small value of conversion rate say w8 0. 18 system (4) losses the persistence, as the conversion rate increases 0.18 w8 0. 33 the 4 trajectory of system (4) approaches to globally asymptotically stable point in the Int.R finally 4 the trajectory of system (4) transfers to periodic dynamics in the Int.R for w8 0. 33.
Consequently, for w8 0. 18, system (4) is not persists, while for w8 0. 18 the system is persists.
3. Keeping the above in view, mathematically it is observed that, as the infection rate w1 decreases and passing through the value w1 0.39 the positive equilibrium points losses its 4 global stability and the trajectory of system (4) approaches periodic dynamics in the Int.R and hence a hopf bifurcation occurs at this value. Similarly as shown in Figure (4), a hopf bifurcation occurs as the value of conversion rate passes through w8 0. 33 .
Acknowledgements
The authors thank the referees for their valuable comments and suggestions that have significantly improved the presentation of this paper.
AAM: Intern. J., Vol. 6, Issue 2 (December 2011) 551
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