Impulsive predator-prey model

In this paper we considered a predator-prey model with state-dependent impulse. We determine periodic solutions for the model without impulses and we prove the existence of nontrivial periodic solution in the case of impulse depending on the state of the model.


Introduction
In this work we consider a prey-predator model with impulse effects depending on the state of the model.The impulsive differential equations appear generally in the description of phenomena submitting jumps in the state variables for short time, these big changes are modeled by discrete equations called impulse effects (see [1][2][3] and [5]).
In this work, we consider a model inspired from [4] where two populations of insects denoted by x and y, respectively, are treated by chemical spray.The evolution of the two populations is governed by a predator-prey model and the effect of the chemical spray is described by impulse effects.The impulse effects in this model are depending on the size of the population density x, when it reaches some threshold h 2 the treatment is used in order to reduce x insects, but the effect of the treatment eliminate also a fraction of the population y, the reduction by treatment on x and y are px and py, respectively.
More specifically, we consider the following model where x and y represent the population densities at time t, the parameters a, b, , h and d are positive constants and p, q ∈ (0, 1).

Analysis of the model without impulses
In the case without impulse effects system (1) is reduced to the following predator-prey model ẋ = x(a − by), where x(t): density of the prey at time t, y(t): density of the predators at time t, a: growth rate of the prey in the absence of predator, b: predation rate of the predator on the prey per unit of time, d: death rate of the predator in the absence of prey, : time to search for prey, h: time to capture prey.

Existence of solutions of (2)
Let z = x y , the system (2) is equivalent to where .
The function f is locally Lipschitz.By using the Cauchy-Lipschitz theorem we obtain the local existence of solutions.

Positivity of solutions of (2)
From system (2) we have ds.

Global existence
For global existence, we use the following proposition.

Proposition 2.3:
Then H is the first integral for the system (2), i.e. if (x, y) is a solution of ( 2), then , for x = x * and y = 0. Then
2. The nontrivial equilibrium point E * is stable. Proof: We have The eigenvalues of this matrix are 1 = a > 0 and 2 = −d < 0. Therefore the equilibrium point (0, 0) is unstable.

2.
We have Df Since > dh, the matrix admits two eigenvalues 1 = i ad( b−dbh) and 2 = −i ad( b−dbh) with real part equal to zero.In this case we can't deduce the stability of E * .Let V be a Lyapunov function defined by , a b = 0 and V (x, y) = 0. Therefore, we conclude neutral stability of the equilibrium E * .
Proof: From Fig. 1, it shows four zones, denoted by I, II, III and IV, in which x and y are monotone.
Our proof consists to follow a trajectory through these areas to show its periodicity (see Fig. 2).We can prove easily that all trajectories pass through the four zones successively.We can prove that x 0 (t 4 ) = x 0 (t 0 ) (see Fig. 2), which prove that trajectories are periodic. 01008-p.4 WMLS 2014

Numerical simulations
When x(t) and y(t) are plotted individuals versus t, we see that periodic variation of the predator population y(t) lags slightly behind the prey population x(t) (see Fig. 4).

Proof:
Consider the autonomous system with impulse effects where P (x, y) and Q(x, y) are continuous differential functions defined on R 2 , and is a sufficiently smooth function with grad (x, y) = 0. 01008-p.6

Analysis of exponential stability of (x s , 0)
Let be the flow associated to (1), we have where X 0 = X(0).We assume that the flow applies up time T 1 .So, X(T ) = (T , X 0 ).
We have We have the following results

Theorem 3.3:
The semi trivial periodic solution (x s , 0) is not exponentially stable.

Conclusion
In this work we have studied the stability of some periodic solutions of an impulsive prey-predator model.We have found sufficient conditions for orbital stability of semi trivial periodic solution and proved that it is not exponentially stable.It will be interesting to see the eventual bifurcation of periodic solutions to study their stability and to give some numerical simulations to illustrate the results obtained. 01008-p.9

Figure 2 .
Figure 2. Trajectory the solution in phase space.