<sup>2</sup>

Thus, all the conditions of Theorem 4 are satisfied. Therefore, system (2.1) has at least a

Each function in system (12) is <sup>δ</sup>� <sup>q</sup><sup>2</sup>; <sup>t</sup> � � periodic and satisfies Theorem 1; then, the system has at least

2. The importance of time scale calculus is pointed out for the analysis of quantum calculus.

3. As an application, the δ�-periodicity notion for quantum calculus is used for the predator–

3 7 5

<sup>e</sup>xeyf sð Þm sð Þ <sup>α</sup>ð Þþ <sup>s</sup> <sup>β</sup>ð Þ<sup>s</sup> ex <sup>þ</sup> m sð Þey � �<sup>2</sup> <sup>Δ</sup><sup>t</sup>

> x y � � � �

6¼ 0:

(3.10)

∈KerL, then Jacobian of G is

signDJG

exp x t ð Þþ ð Þ <sup>2</sup> exp y t ð Þ ð Þ ,

There are many studies about the predator–prey dynamic systems on time scale calculus such as [14, 19, 27, 28]. All of these cited studies are about the periodic solutions of the considered system on a periodic time scale. However, in the world, there are many different species. While investigating the periodicity notion of the different life cycle of the species, the w-periodic time scales could be a little bit restricted. Therefore, if the life cycle of this kind of species is appropriate to the Beddington-DeAngelis functional response, then the results that we have found in that study are becoming more useful and important.

In addition to these, the δ-periodic solutions for predator–prey dynamic systems with Holling-type functional response, semiratio-dependent functional response, and monotype functional response can be also taken into account for future studies. In that dynamic systems, delay conditions and impulsive conditions can also be added for the new investigations.

This is a joint work with Ayse Feza Guvenilir and Billur Kaymakcalan.
