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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.

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-ratio- dependent and Holling-type have been studied. [

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 [

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 [

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 [

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-DeAn- glis type functional response and impulse to general time scales.

Below informations are from [

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. Suppose

(a) For each

(b)

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

Lemma 1. Let

The equation that we investigate is:

if

Lemma 2. If

Proof. If we using the first equation of (1) we obtain,

Since

Similarly

Theorem 2. In addition to conditions on coefficient functions

If

and

then there exist at least a w-periodic solution.

Proof.

and

with the norm:

Let us define the mappings

and

Then

There exist continuous projectors

and

where

The generalized inverse

Let

and

Clearly,

Ascoli theorem we can find

ally,

To apply the continuation theorem we investigate the below operator equation.

Let

From (2) and (3) we get

where

where

Note that since

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)

Therefore

By the assumption of the theorem we can show that

where

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

pendent of

where

Define the homotopy

Take

All the functions in jacobian of G is positive then

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 (1) and

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), (3) and (6)

By (3)

And using the second inequality in Lemma 1 we have

By the first equation of (3) and (6)

Then we get

Using the first inequality in Lemma 1 we have

By (12) and (13)

Therefore

Since

Hence, by using the first inequality in Lemma 1 and the second equation of (3),

By the assumption of Theorem 3 there exists

is true. We need to get

that

If such t, s does not exists then

By using first inequality in Lemma 1, we have

Using the second equality in (3) and the assumption of the Theorem 4, we obtain

This implies

Hence, according to the above discussion we have

in Lemma 1 we have

Thus

(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

Corollary 1. If

is satisfied then the system (1) has at least one w-periodic solution.

Example 1.

Impulse points:

Example 1 satisfies all the conditions of Theorem 2, thus it has at least one periodic solution.

Example 2.

Impulse points:

Example 2 satisfies all the conditions of Theorem 3, thus it has at least one periodic solution.

Theorem 4. If all the coefficient functions in system (1) is positive, w-periodic, from

is satisfied then there exist at least a w-periodic solution.

Proof. First part of the proof is similar to Theorem 2, only difference is the zero impulses. If the assumption of Theorem 4 is true then there exists

is satisfied. Suppose there exist

If such s, t does not exist

Thus we get

Then

If

with the maximum of the solution. Let

Then

If

Thus

Using (3) and (7) above results we obtain

This implies

Hence, according to the above discussion we have

Lemma 1 we have

is similar to Theorem 2.

Corollary 2. In Theorem 4 if we take

Example 3.

Example 3 satisfies all the conditions of Theorem 4, thus it has at least one periodic solution.

All the graphs that we see in Figures 1-3 are obtained by Mathlab.

In this paper, the impulsive predator-prey dynamic systems on time scales calculus are studied. We investigate when the system has periodic solution. Furthermore, 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. Also by using graphs, we are able to show that the conditions that are found in Theorem 2, 3

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

Ayşe FezaGüvenilir,BillurKaymakçalan,Neslihan NesliyePelen, (2015) Impulsive Predator-Prey Dynamic Systems with Beddington-DeAngelis Type Functional Response on the Unification of Discrete and Continuous Systems. Applied Mathematics,06,1649-1664. doi: 10.4236/am.2015.69147