Impulsive Predator-Prey Dynamic Systems with Beddington-DeAngelis Type Functional Response on the Unification of Discrete and Continuous Systems

In this study, the impulsive predator-prey dynamic systems on time scales calculus are studied. When the system has periodic solution is investigated, and three different conditions have been found, which are necessary for the periodic solution of the predator-prey dynamic systems with Beddington-DeAngelis type functional response. For this study the main tools are time scales calculus and coincidence degree theory. Also the findings are beneficial for continuous case, discrete case and the unification of both these cases. Additionally, unification of continuous and discrete case is a good example for the modeling of the life cycle of insects.


Introduction
The relationships between species and the outer environment, and the connections between different species are the description of the predator-prey dynamic systems which is the subject of mathematical ecology in biomathematics.Various types of functional responses in predator-prey dynamic system such as Monod-type, semi-ratiodependent and Holling-type have been studied.[1] is an example for the study about Holling-type functional re-sponse.In this paper, we consider the predator-prey system with Beddington DeAngelis type functional response and impulses.This type of functional response first appeared in [2] and [3].At low densities this type of functional response can avoid some of the singular behavior of ratio-dependent models.Also predator feeding can be described much better over a range of predator-prey abundances by using this functional response.
In a periodic environment, significant problem in population growth model is the global existence and stability of a positive periodic solution.This plays a similar role as a globally stable equilibrium in an autonomous model.Therefore, it is important to consider under which conditions the resulting periodic nonautonomous system would have a positive periodic solution that is globally asymptotically stable.For nonautonomous case there are many studies about the existence of periodic solutions of predator-prey systems in continuous and discrete models based on the coincidence theory such as [4]- [12].
Impulsive dynamic systems are also important in this study and we try to give some information about this area.Impulsive differential equations are used for describing systems with short-term perturbations.Its theory is explained in [13]- [15] for continuous case and also for discerete case there are some studies such as [16].Impulsive differential equations are widely used in many different areas such as physics, ecology, and pest control.Most of them use impulses at fixed time such as [17] [18].By using constant functions, some properties of the solution of predator-prey system with Beddington-DeAnglis type functional response and impulse impact are studied in [19] for continuous case.
In this study unification of continuous and discrete analysis is also significant.To unify the study of differential and difference equations, the theory of Time Scales Calculus is initiated by Stephan Hilger.In [20] [21], unification of the existence of periodic solutions of population models modelled by ordinary differential equations and their discrete analogues in form of difference equations, and extension of these results to more general time scales are studied.
The unification of continuous and discrete case is a good example for the modeling of the life cycle of insects.Most of the insects have a continuous life cycle during the warm months of the year and die out in the cold months of the year, and in that period their eggs are incubating or dormant.These incubating eggs become new individuals of the new warm season.Since insects have such a continuous and discrete life cycle, we can see the importance of models obtained by the time scales calculus for the species that have unusual life cycle.Therefore, in this paper we try to generalize periodic solutions of predator-prey dynamic systems with Beddington-DeAnglis type functional response and impulse to general time scales.

Preliminaries
Below informations are from [20].Let X, Z be normed vector spaces, : L DomL X Z ⊂ → be a linear mapping, : N X Z → be a continuous mapping.The mapping L will be called a Fredholm mapping of index zero if dimKerL codimImL = < +∞ and ImL is closed in Z.If L is a Fredholm mapping of index zero and there exist continuous projections : P X X → and : We denote the inverse of that map by P K .If Ω is an open bounded subset of X, the mapping N will be called L-compact on Ω if

( )
QN Ω is bounded and

(
)  .We will also give the following lemma, which is essential for this paper.

Main Result
The equation that we investigate is: then there exist at least a w-periodic solution. Proof.
with the norm: Let us define the mappings L and N by : , 1 c and 2 c are constants. ( , therefore L is a Fredholm mapping of index zero.There exist continuous projectors :

( )
QN Ω is bounded.Thus, N is L-compact on Ω with any open bounded set .X Ω ⊂ To apply the continuation theorem we investigate the below operator equation.
be any solution of system (2).Integrating both sides of system (2) over the interval [ ] From ( 2) and (3) we get where ( ) ( ) ( ) where ( ) ( ) and there are q impulses which are constant, then there exist , { } By the second equation of ( 3) and ( 6) and the first assumption of Theorem 2, we have and ( ) Using the second inequality in Lemma 1 we have By the first equation of ( 3) and ( 6) we get ( ) using the first inequality in Lemma 1 and (4), we have By ( 8) and ( 9) , second equation of ( 3) and first equation of ( 7), we can derive that By the assumption of the theorem we can show that ( ) ( ) ( ) ( ) and ( ) Hence, by using the first inequality in Lemma 1 and the second equation of (3), We can also derive from the second equation of (3) that Again using second assumption of Theorem 2 we obtain and ( ) By using the second inequality in Lemma 1 and ( 5), we obtain By ( 10) and ( 11) we have and Ω verifies the requirement (a) in Theorem 1.When Thus all the conditions of Theorem 1 are satisfied.Therefore system (1) has at least a positive w-periodic solution.
Theorem 3. If same conditions are valid for the coefficient functions in system ( is satisfied then there exist at least a w-periodic solution. Proof.First part of the proof is very similar with the proof of Theorem 2. By (2), ( Then we get ( ) And using the second inequality in Lemma 1 we have By the first equation of ( 3) and ( 6) Then we get ( ) x l ξ ≤  where ( ) ( ) ( ) Using the first inequality in Lemma 1 we have By ( 12) and ( 13) From the second equation of (3) and the second equation of (7), we can derive that Hence, by using the first inequality in Lemma 1 and the second equation of (3), By the assumption of Theorem 3 there exists 0 n such that By using first inequality in Lemma 1, we have ( ) ( ) Using the second equality in (3) and the assumption of the Theorem 4, we obtain Hence, according to the above discussion we have ( ) { } then Ω verifies the requirement (a) in Theorem 1. Rest of the proof is similar to Theorem 2. Let there are two insect populations (one of them the predator, the other one the prey) both continuous while in season (say during the six warm months of the year), die out in (say) winter, while their eggs are incubating or dormant, and then both hatch in a new season, both of them giving rise to nonoverlapping populations.This situation can be modelled using the time scale Here impulsive effect of the pest population density is after its partial destruction by catching, poisoning with chemicals used in agriculture (can be shown by 1 0 k g − < < ) and impulsive increase of the predator population density is by artificially breeding the species or releasing some species ( ) 0 k p > .In addition to these, if the model assumes a BeddingtonDeAngelis functional response as in (1)    and 4 are enough for the periodic solution of the given system.In this work, since our system can model the life cycle of the such species like insects, what we have done new is finding necessary condition for the periodic solution of the given predator-prey system with sudden changes.In addition to these, according to the structure of the given time scale  , the conditions that are found in Theorem 2, 3 and 4 become useful.
there exists an isomorphism : J ImQ KerL → .The above informations are important for the Continuation Theorem that we give below.Theorem 1. (Continuation Theorem).Let L be a Fredholm mapping of index zero and N be L-compact on Ω. Nz = has at least one solution lying in DomL δΩ − are continuous.Since X and Y are Banach spaces, then by using Arzela-Ascoli theorem we can find ( y

Figure 2 .
Figure 2. Numeric solution of Example 2 shows the periodicity.

Figure 3 .
Figure 3. Numeric solution of Example 3 shows the periodicity. ) and if the assumptions in Theorem 2 or 3 ∈ ≤   Rest of the proof