PERIODIC SOLUTIONS OF A MODEL OF LOTKA-VOLTERRA WITH VARIABLE STRUCTURE AND IMPULSES

We consider a generalized version of the classical Lotka Volterra model with differential equations. The version has a variable structure (discontinuous right hand side) and the solutions are subjected to the discrete impulsive effects. The moments of right hand side discontinuity and the moments of impulsive effects coincide and they are specific for each solution. Using the Brouwer fixed point theorem, sufficient conditions for the existence of periodic solution are found. Provide examples of relevant academic for this E.g.,


INTRODUCTION
We assume that the predator-prey community is subjected to the external effects.These effects are discrete in time and relate to:

DESCRIPTION OF THE MODEL
Consider the following initial value problem for systems of differential equations with variable structure and moments of impulsive effects , , , ; , , , ; ... ;  5) is defined as follows: coincides with the solution of the system (1), ( 2) for coincides with the solution of system (1), (2) for ; and so on.
Further, we use the notations , where the initial points   0 0 0 ,, is a trajectory of system ( 1), ( 2).This trajectory is closed; Let c and C be arbitrary constants such that 0 cC  .We assume that for 1, 2,..., i  the phase space of system (1), ( 2) is defined as 8. The switching sets of system ( 1)-( 4) satisfy the inclusions , .
, , ln 1 ln , respectively.The constants This means that they are compact, connected and convex sets.
Let the conditions H1 and H2 hold.O c t o b e r 2 6 , 2 0 1 5 Then the system ( 1)-( 4) has at least one periodic solution with initial point   , ,min 0 0 1 .The solution of initial value problem (1)-( 5) is defined for 0 0 t t T    .In fact, for 01 0 t t t    , the solution of considered problem with variable structure and impulses coincides with the solution of initial value problem ( 6), ( 7), ( 8), where ,, m M m M  (see equality (9)).From condition H2, it is seen that the point   , ,min . For 1 ii t t t   , the solution of ( 1)- (5)   coincides with the solution of initial value problem ( 6), ( 7), ( 8), where ,, . In other words, for ..., .


For concreteness, we assume that tt  is fulfilled.The remaining case is considered analogously.According to the property 13 of Remark 2, it follows that for each sufficiently small constant 0  , we have From the theorem of continuous dependence (see Theorem 7.1, § 7, Ch.I, [4]), it follows that tt  ), we reach the following inequalities t t t t        .According to the property 11 of Remark 2, we have: ;0, , ;0, , ;0, , ; ;0, , ;0, , ;0, , .
From the above two inequalities and (12), we find 10) and ( 14), we conclude that Thus, we have proved that the mapping Similarly as in Part 2, it can be established that this mapping is continuous on its domain.
According to condition H2, the mapping The theorem is proved.
are specific growth factors, relevant to the first species (prey) and second (predator), respectively; -are the coefficients indicating interspecies competition.In the common case, they are different for the prey and predator; the prey and predator biomasses at the initial moment 0M M t m M of the initial value problem (1)-(

Remark 5 .
Pay attention that for .It can be shown that just then the victim's biomass is maximum.Consequently, the withdrawal of biomass from the victim in these moments   12 , ,... tt is justified.

Remark 6 .
From condition H1, we obtain the equalities:

Part 1 .
For convenience, the proof will be presented in several parts.Let the initial point  

1 .
For this purpose, it is sufficient to establish that

Part 5 .
Let the mapping c

i  is continuous. Part 6 .Part 7 .
Let us introduce the mapping connected and convex set.Using the Brouwer fixed point theorem, we reach the conclusion that Taking into account the definitions of functions i M M t m M is a periodic solution of the initial value problem (1)-(5).The period of solution is 0 k Tt  .
Further, we denote by ,max It can be seen immediately that the sets