Dynamical Behavior of an eco-epidemiological Model involving Disease in predator and stage structure in prey

An eco-epidemic model is proposed in this paper. It is assumed that there is a stage structure in prey and disease in predator. Existence, uniqueness and bounded-ness of the solution for the system are studied. The existence of each possible steady state points is discussed. The local condition for stability near each steady state point is investigated. Finally, global dynamics of the proposed model is studied numerically.


The mathematical Formulation:
In this section, we , the food web model consists of two compartments of predator (susceptible and infected) and a stage-structure prey in which the prey species growth logistical without of predation, while the predator decay exponentially in the absence of prey species. It is assumed that the prey population separate into two compartments: ( )which represent the density of immature prey population at time T, and ( ) that denotes to the density of mature prey population at time T. Further the density of the susceptible predator at time T is denoted by ( ) while ( ) represents the density of infected predator population at time T. Now in order to formulate the dynamics of such system, the following hypotheses are adopted: 1. The immature prey depends completely in its feeding on the mature prey that grows logistically with intrinsic growth rate and carrying capacity The immature prey individual grow up and become mature prey with growth up rate . However the mature prey facing death with natural death rate 2. There is a kind of protection for the two stages of prey species from facing predation by the susceptible predator with refuge rate constant ( ) respectively. 3. The susceptible predator consumed the immature prey individuals according to Holling type-II functional response with predation rate and half saturation constant And consumed the mature prey individuals according to Holling type-II functional response with predation rate and contribute of portion of such food with conversion rate Moreover, the infected predator consumed the immature prey individuals according to lotka-volltera type of functional response with predation rate represent the disease transmission from susceptible predator to infected predator and contributes a portion of such food with conversion rate 4. Finally, in the absence of food the susceptible predator. Facing death with natural death rate but the infected predator facing death due to disease and natural death rate From above assumptions the system can be formulated mathematically with the following set of differential equations: Now, by simplifying the model (1), the number of parameters is reduced by using the following dimensionless variables and parameters: Accordingly, the dimensionless of system (1) becomes Clearly, the equations of system (2) are continuous and have continuous partial derivatives on the following positive 4th dim.space: *( ) ( ) ( ) ( ) ( ) + Therefore, these equations are Lipschizian on and hence the solution of system (2) exists and unique. Furthermore, each of the solutions of system (2) with positive initial condition is bounded as shown in the following. Theorem (1): Each of the solutions of system (2) which are initiated in arebounded. Proof: Let ( ( ) ( ) ( ) ( )) be a solution of system (2) with positive initial condition ( ) Now define the function ( ) ( ) ( ) ( ) ( ) and then taken the time derivative of ( ) along the solution of system (2).
So, due to the fact that the conversion rate constant from prey population to predator population cannot exceeding the maximum predation rate constant from predator population to prey population, hence from the biological point of view, always we get: represents prey specie which is growth logistically with carrying capacity(1), hence So that, Now, solve the differential equation with initial value ( ) we get: Then each the solution of system (2 )uniformly bounded.

3.Existence of the steady state points
In this part, the existence of all possible steady state points of system (2) is discussed. It is observed that, system (2) has only four steady state points, which are mentioned in the following:  The steady state point ( ) which is known as the varieshing point and is always exists.  The two species steady state point ( ̅ ̅ )where: Exists under the following condition ( )  The steady state point From the second equation of system (2) we have Now, with some simplification we have: Now, in order to determine the values of ̂ and ̂ , consider the two isoclines ( ) and ( ) as which gives: Consequently eq.(i) have a positive root say provided that one of the following conditions hold: However, equation ( )has just one positive root, say , provided that one of the following conditions hold: , we get by the above analysis, it is noted that the two isoclines (4d) and (4e) intersect at unique point ( ) iff the conditions (4f) , (4g) , (4h) and (4i) are satisfied, and hence the system (2) has only one positive steady state point if in addition to these conditions the following holds: From equation (5c) we obtain: Substituting equations (5e) and (5f) in equation (5a) we get:

Similarly from equation (5h), we noted
Note that and hence the isoclines (5h) is decreasing iff the following condition hold:

} ( )
Therefore the positive equilibrium point exists uniquely provided that in addition to the above conditions the following two conditions hold

The stability Conditions
In this part, the local conditions for stability near the steady state points of system (2) is investigated. It is to verify that the Jacobian matrix of system (2), at the general point ( ) Therefore, the Jacobian matrix of system (2) at the vanishing steady state point is: Thus the eigenvalues of ( )are Either and or which gives two eigenvalues √ where ( ) Therefore, is a saddle point. The Jacobian matrix of system (2) at is given by Accordingly the characteristic equation of ( )canbe written as: We get the eigenvalues of ( ) in the direction respectively as : Hence, we get the other two eigenvalues of ( ) in the direction as:

√ ( )
Then all the eigenvalues have negative real parts if the following conditions hold:

So, is a local stable in the . And it is unstable point on the other hand.
Thus Jacobain matrix of system (2) at is a given by: Then the eigenvalues of ( )are It is easy to verify that, the linearized system of system (2) can be written as: It is clearly that becontinuously differentiable function, So that ( ̆ ̆ ̆ ̆ ) ( ) otherwise so by differentiating with respect to time t, gives:

Therefore, is negative definite and hence is a Lyapunov faction with respect to in the sub region . So is asymptotically stable. Note that the faction is approaching to infant as any of its components to the same and its positive definite
,however its derivative is negative definite on the sub region due to the given sufficient conditions. Therefore is a globally asymptotically stable with in Theorem ( It is easy to see that and ( ) ( ̅ ̅ ) Furthermore, by the derivative with time and simplifying we get that: ) And then substituting in the above equation we get : Obviously for every initial point and then is a Lya.function provided that conditions (11a-11d) hold. Thus is global stable this completes the proof. Theorem (3): Assume that ( ̂ ̂ ̂ ) is a locally in , then it is global stable provided that the following conditions: Proof: Consider the following function It is easy to see that ( ) ( ) ( ) ( ) ( ̂ ̂ ̂ ) Furthermore by taking the derivative with time and simplifying we get that: Obviously for every initial point and then is a Lyap. function provided that conditions (11a-11f) hold. Thus 2 E is global stable this completes the proof.

Theorem (4):
Assume that is local stable in . Then, it is a global stable in sub region of that satisfies the following conditions ) Proof: consider the following function )then by find the derivative with time, also simplifying it we get: Clearly, , and then is a Lyap. function provided that the given conditions (13a-13c) hold.
Therefore, 3 E is global stable in the interior of a basin of attraction of 3 E and the proof is complete.

Numerical illustrate
In this section, the dynamical behavior of system (2) is studied numerically for different ets of initial values and different sets of parameters values.
It is observed that for the following set of hypothetical parameters system (2) has an asymptotical stable steady state point E 1 =(0.2,0.99,0.0) as shown in Figure- from values of parameters that given in Eq. (14) with u 6 = 0.001, u 9 = 0.0002.the solution of system (2) approaches to E 2 =(0.5,0.9,0.6,0) as shown in Figure-