2. Preliminaries about time scale calculus

The main tool we have used, in this study, is time scale calculus, which was first appeared in 1990 in the thesis of Stephen Hilger [22]. By a time scale, denoted by T, we mean a non-empty closed subset of R: The theory of time scale calculus gives a way to unify continuous and discrete analysis.

The following informations are taken from [14, 23]. The set <sup>T</sup><sup>κ</sup> is defined by <sup>T</sup><sup>κ</sup> <sup>¼</sup> <sup>T</sup><sup>=</sup> <sup>r</sup> sup<sup>T</sup> ; sup<sup>T</sup> , and the set <sup>T</sup><sup>κ</sup> is defined by <sup>T</sup><sup>κ</sup> <sup>¼</sup> <sup>T</sup>=½ Þ infT; <sup>σ</sup>ð Þ inf<sup>T</sup> : The forward jump operator σ : T ! T is defined by σð Þt ≔in tð Þ ; ∞ <sup>T</sup>, for t∈ T: The backward jump operator r : T ! T is defined by rð Þt ≔supð Þ �∞; t <sup>T</sup>, for t∈ T: The forward graininess function μ : T ! R<sup>þ</sup> <sup>0</sup> is defined by μð Þt ≔σð Þ� t t, for t ∈T: The backward graininess function ν : T ! R<sup>þ</sup> <sup>0</sup> is defined by νð Þt ≔t � rð Þt , for t∈ T: Here, it is assumed that inf0= ¼ supT and sup0= ¼ infT:

For a function <sup>f</sup> : <sup>T</sup> ! <sup>T</sup>, we define the <sup>Δ</sup>-derivative of <sup>f</sup> at <sup>t</sup><sup>∈</sup> <sup>T</sup><sup>κ</sup>, denoted by <sup>f</sup> <sup>Δ</sup>ð Þ<sup>t</sup> for all <sup>e</sup> <sup>&</sup>gt; <sup>0</sup>: There exists a neighborhood U ⊂T of t∈T<sup>κ</sup> such that

$$|f(\sigma(t)) - f(s) - f^\Lambda(t)(\sigma(t) - s)| \le \epsilon |\sigma(t) - s|$$

for all s∈ U:

In [2], in classical calculus when the equation

184 Advanced Technologies of Quantum Key Distribution

Then the q-derivative is defined as follows:

and

calculus, the q-differential of a function is equal to the following:

F xð Þ¼

ð

f xð Þ� f xð Þ<sup>0</sup> x � x<sup>0</sup>

is considered and as x tends to x0, the differentiation notion is obtained. When the differential equations are considered, the difference of a function is defined as f xð Þ� þ 1 f xð Þ: In quantum

dqð Þ¼ f xð Þ f qx ð Þ� f xð Þ

dqð Þ¼ x qx � x ¼ ð Þ q � 1 x:

dqð Þ<sup>x</sup> <sup>¼</sup> f qx ð Þ� f xð Þ

The differentiation in time scale calculus is given in Theorem 1, and if the differentiation notion in this theorem is applied when T ¼ q<sup>N</sup>, one can easily see that the same q-derivative is obtained. As an inverse of q-derivative, one can get q-integral that is also very significant for the structure of this calculus. A function F xð Þ is a q-antiderivative of f xð Þ if DqF xð Þ¼ f xð Þ is satisfied where

f xð Þdqx <sup>¼</sup> ð Þ <sup>1</sup> � <sup>q</sup> <sup>X</sup><sup>∞</sup>

This is also called the Jackson integral [3]. When the definition of the antiderivative of a function in time scale calculus is considered, it can be easily seen that when T ¼ q<sup>N</sup><sup>0</sup> , these two definitions become equivalent. Therefore, to understand the quantum calculus, it is very important to understand the time scale calculus. In addition to these, the δ�-periodicity notion in time scale calculus is defined in Definition 1 in [4] for the application. In this study, by using time scale calculus, the application of δ�-periodicity notion of q<sup>N</sup>, which overlaps with the q-calculus, to a predator–prey system with Beddington-DeAngelis-type functional response is studied.

To understand this application in a much better sense, the following information about the predator–prey dynamic systems is given. Predator–prey equations are also known as the Lotka-Volterra equations. This model was initially proposed by Alfred J. Lotka in the theory of autocatalytic chemical reactions in 1910 [5, 6] which was effectively the logistic Equation [7] and originally derived by Pierre Françis Verhulst [8]. In 1920, Lotka extended this model to "organic systems" by using a plant species and a herbivorous animal species. The findings of this study were published in [9]. In 1925, he obtained the equations to analyze predator–prey interactions in his book on biomathematics [10] arriving at the equations that we know today.

ð Þ <sup>q</sup> � <sup>1</sup> <sup>x</sup> :

0 xqj f xqj � �:

dqð Þ f xð Þ

For the same function, the ∇-derivative of f at t∈Tκ, denoted by f <sup>∇</sup>ð Þ<sup>t</sup> , for all <sup>e</sup> <sup>&</sup>gt; <sup>0</sup>:, is defined. There exists a neighborhood V ⊂ T of t ∈T<sup>κ</sup> such that

$$|f(\mathbf{s}) - f(\rho(t)) - f^\nabla(t)(\mathbf{s} - \rho(t))| \le \mathbf{e}|\mathbf{s} - \rho(t)|$$

for all s∈ V:

A function f : T ! R is rd-continuous if it is continuous at right-dense points in T and its leftsided limits exist at left-dense points in T: The class of real rd-continuous functions defined on a time scale T is denoted by Crdð Þ T; R : If f ∈ Crdð Þ T; R , then there exists a function F tð Þ such that <sup>F</sup><sup>Δ</sup>ðÞ¼ <sup>t</sup> f tð Þ. The delta integral is defined by <sup>Ð</sup> <sup>b</sup> <sup>a</sup> f xð ÞΔx ¼ F bð Þ� F að Þ:

• If T ¼ R, then

compact on Ω. Assume

properties:

then

ðb a

where the integral on the right is the Riemann integral from calculus.

a. For each λ ∈ð Þ 0; 1 , any y satisfying Ly ¼ λCy is not on δΩ, i.e., y∉δΩ

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

Ly ¼ Cy has at least one solution lying in DomL ∩ δΩ.

S<sup>0</sup> ≤ v < s implies δ � ð Þ S0; v < δ � ð Þ S0;s ,

S1 < S2, then δþð Þ S1;s < δþð Þ S2;s ,

δþð Þ¼ t0; v v holds

• If T consists of only isolated points and a < b, then

f tð ÞΔt ¼

X t∈ ½ Þ a;b ðb a f tð Þdt,

Quantum Calculus with the Notion δ±-Periodicity and Its Applications

http://dx.doi.org/10.5772/intechopen.74952

187

f tð Þμð Þt :

Theorem 4. [14] (Continuation Theorem). Let L be a Fredholm mapping of index zero and C be L-

b. For each y∈ δΩ ∩KerL, VCy 6¼ 0 and the Brouwer degree degf g JVC; δΩ ∩KerL; 0 6¼ 0: Then,

Definition 1. [4] Let the time scale T including a fixed number t<sup>0</sup> ∈T<sup>∗</sup> where T<sup>∗</sup> be a non-empty subset of <sup>T</sup>, such that there exist operators <sup>δ</sup>� : ½ Þ <sup>t</sup>0; <sup>∞</sup> <sup>T</sup> � <sup>T</sup><sup>∗</sup> ! <sup>T</sup><sup>∗</sup> which satisfy the following

ð Þ <sup>S</sup>0; <sup>v</sup> , Sð Þ <sup>0</sup>;<sup>s</sup> <sup>∈</sup> <sup>D</sup>�≔fð Þ <sup>u</sup>; <sup>v</sup> <sup>∈</sup>½ Þ <sup>t</sup>0; <sup>∞</sup> <sup>T</sup> � <sup>T</sup><sup>∗</sup> : <sup>δ</sup> � ð Þ <sup>u</sup>; <sup>v</sup> <sup>∈</sup>T<sup>∗</sup>g,

P.2 If Sð Þ <sup>1</sup>;s , Sð Þ <sup>2</sup>;s ∈ D� with S<sup>1</sup> < S2, then δ�ð Þ S1;s > δ�ð Þ S2;s , , and if Sð Þ <sup>1</sup>;s , Sð Þ <sup>2</sup>;s ∈ D<sup>þ</sup> with

P.3 If v<sup>∈</sup> ½ Þ <sup>t</sup>0; <sup>∞</sup> <sup>T</sup>, then vð Þ ; <sup>t</sup><sup>0</sup> <sup>∈</sup> <sup>D</sup><sup>þ</sup> and <sup>δ</sup>þð Þ¼ <sup>v</sup>; <sup>t</sup><sup>0</sup> <sup>s</sup>: Moreover, if v<sup>∈</sup> <sup>T</sup><sup>∗</sup>, then tð Þ <sup>0</sup>; <sup>v</sup> <sup>∈</sup> <sup>D</sup><sup>þ</sup> and

P.1 With respect to their second arguments, the functions δ� are strictly increasing, i.e., if

P.4 If uð Þ ; v ∈ D�, then uð Þ ; δ�ð Þ u; v ∈ D� and δ<sup>∓</sup> ð Þ¼ u; δ�ð Þ u; v v, respectively.

P.5 If uð Þ ; v ∈ D� and sð Þ ; δ�ð Þ u; v ∈ D�, then uð Þ ; δ<sup>∓</sup> ð Þ s; v ∈ D� and

Theorem 1. [23] Suppose that f : <sup>T</sup> ! <sup>R</sup> is a function and t <sup>∈</sup>T<sup>κ</sup>. Then, we have the following:


$$f^{\Delta}(t) = \frac{f(\sigma(t)) - f(t)}{\mu(t)}.$$

3. If t is right dense, then f is delta differentiable at t if and only if the limit

$$\lim\_{s \to t} \frac{f(t) - f(s)}{t - s}$$

exists as a finite number. In this case,

$$f^\Delta(t) = \lim\_{s \to t} \frac{f(t) - f(s)}{t - s}.$$

4. If f is delta differentiable at t, then

$$f''(t) = f(t) + \mu(t)f^\Delta(t).$$

Theorem 2. [23] If a, b, c, d ∈T, α ∈ R, and f , g : T ! R are rd-continuous, then


Theorem 3. [23] If a, b∈ T, α∈ R, and f : T ! R are rd-continuous, then

• If T ¼ R, then

a time scale T is denoted by Crdð Þ T; R : If f ∈ Crdð Þ T; R , then there exists a function F tð Þ such

<sup>Δ</sup>ðÞ¼ <sup>t</sup> <sup>f</sup>ð Þ� <sup>σ</sup>ð Þ<sup>t</sup> f tð Þ

f tð Þ� f sð Þ t � s

> f tð Þ� f sð Þ <sup>t</sup> � <sup>s</sup> :

> > <sup>Δ</sup>ð Þ<sup>t</sup> :

<sup>μ</sup>ð Þ<sup>t</sup> :

Theorem 1. [23] Suppose that f : <sup>T</sup> ! <sup>R</sup> is a function and t <sup>∈</sup>T<sup>κ</sup>. Then, we have the following:

2. If f is continuous at a right scattered t, then f is delta differentiable at t with

f

3. If t is right dense, then f is delta differentiable at t if and only if the limit

f

f σ

<sup>a</sup> f tð ÞΔð Þþ <sup>t</sup> <sup>Ð</sup> <sup>b</sup>

<sup>c</sup> f tð ÞΔt;

a f

Theorem 3. [23] If a, b∈ T, α∈ R, and f : T ! R are rd-continuous, then

a f

<sup>Δ</sup>ð Þ<sup>t</sup> <sup>g</sup>ð Þ <sup>σ</sup>ð Þ<sup>t</sup> <sup>Δ</sup>t;

<sup>Δ</sup>ð Þ<sup>t</sup> g tð ÞΔt:

Theorem 2. [23] If a, b, c, d ∈T, α ∈ R, and f , g : T ! R are rd-continuous, then

<sup>a</sup> g tð ÞΔt;

lims!t

<sup>Δ</sup>ðÞ¼ <sup>t</sup> lim<sup>s</sup>!<sup>t</sup>

ðÞ¼ t f tð Þþ μð Þt f

<sup>a</sup> f xð ÞΔx ¼ F bð Þ� F að Þ:

that <sup>F</sup><sup>Δ</sup>ðÞ¼ <sup>t</sup> f tð Þ. The delta integral is defined by <sup>Ð</sup> <sup>b</sup>

186 Advanced Technologies of Quantum Key Distribution

1. If f is delta differentiable at t, then f is continuous at t:

exists as a finite number. In this case,

4. If f is delta differentiable at t, then

<sup>a</sup> ½ � f tð Þþ g tð Þ <sup>Δ</sup><sup>t</sup> <sup>¼</sup> <sup>Ð</sup> <sup>b</sup>

Ð b <sup>a</sup> f tð ÞΔt;

<sup>b</sup> f tð ÞΔt;

<sup>a</sup> f tð ÞΔ<sup>t</sup> <sup>þ</sup> <sup>Ð</sup> <sup>b</sup>

<sup>a</sup> f tð Þg<sup>Δ</sup>ð Þ<sup>t</sup> <sup>Δ</sup><sup>t</sup> <sup>¼</sup> fg bð Þ� fg að Þ� <sup>Ð</sup> <sup>b</sup>

<sup>a</sup> <sup>f</sup>ð Þ <sup>σ</sup>ð Þ<sup>t</sup> <sup>g</sup><sup>Δ</sup>ð Þ<sup>t</sup> <sup>Δ</sup><sup>t</sup> <sup>¼</sup> fg bð Þ� fg að Þ� <sup>Ð</sup> <sup>b</sup>

<sup>a</sup> αf tð ÞΔt ¼ α

<sup>a</sup> f tð ÞΔ<sup>t</sup> ¼ � <sup>Ð</sup> <sup>a</sup>

<sup>a</sup> f tð ÞΔ<sup>t</sup> <sup>¼</sup> <sup>Ð</sup> <sup>c</sup>

<sup>a</sup> f tð ÞΔðÞ¼ t 0;

• Ð <sup>b</sup>

• Ð <sup>b</sup>

• Ð <sup>b</sup>

• Ð <sup>b</sup>

• Ð <sup>a</sup>

• Ð <sup>b</sup>

• Ð <sup>b</sup>

$$\int\_{a}^{b} f(t)\Delta t = \int\_{a}^{b} f(t)dt,$$

where the integral on the right is the Riemann integral from calculus.

• If T consists of only isolated points and a < b, then

$$\sum\_{t \in [a,b)} f(t)\mu(t).$$

Theorem 4. [14] (Continuation Theorem). Let L be a Fredholm mapping of index zero and C be Lcompact on Ω. Assume


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

Definition 1. [4] Let the time scale T including a fixed number t<sup>0</sup> ∈T<sup>∗</sup> where T<sup>∗</sup> be a non-empty subset of <sup>T</sup>, such that there exist operators <sup>δ</sup>� : ½ Þ <sup>t</sup>0; <sup>∞</sup> <sup>T</sup> � <sup>T</sup><sup>∗</sup> ! <sup>T</sup><sup>∗</sup> which satisfy the following properties:

P.1 With respect to their second arguments, the functions δ� are strictly increasing, i.e., if

$$(\mathcal{S}\_0, \upsilon)\_\prime (\mathcal{S}\_0, s) \in D\_\pm := \{ (\mu, \upsilon) \in [t\_0, \infty)\_\mathbb{T} \times \mathbb{T}^\* : \delta \pm (\mu, \upsilon) \in \mathbb{T}^\* \}\_\prime.$$

then

S<sup>0</sup> ≤ v < s implies δ � ð Þ S0; v < δ � ð Þ S0;s ,

P.2 If Sð Þ <sup>1</sup>;s , Sð Þ <sup>2</sup>;s ∈ D� with S<sup>1</sup> < S2, then δ�ð Þ S1;s > δ�ð Þ S2;s , , and if Sð Þ <sup>1</sup>;s , Sð Þ <sup>2</sup>;s ∈ D<sup>þ</sup> with S1 < S2, then δþð Þ S1;s < δþð Þ S2;s ,

P.3 If v<sup>∈</sup> ½ Þ <sup>t</sup>0; <sup>∞</sup> <sup>T</sup>, then vð Þ ; <sup>t</sup><sup>0</sup> <sup>∈</sup> <sup>D</sup><sup>þ</sup> and <sup>δ</sup>þð Þ¼ <sup>v</sup>; <sup>t</sup><sup>0</sup> <sup>s</sup>: Moreover, if v<sup>∈</sup> <sup>T</sup><sup>∗</sup>, then tð Þ <sup>0</sup>; <sup>v</sup> <sup>∈</sup> <sup>D</sup><sup>þ</sup> and δþð Þ¼ t0; v v holds

P.4 If uð Þ ; v ∈ D�, then uð Þ ; δ�ð Þ u; v ∈ D� and δ<sup>∓</sup> ð Þ¼ u; δ�ð Þ u; v v, respectively.

P.5 If uð Þ ; v ∈ D� and sð Þ ; δ�ð Þ u; v ∈ D�, then uð Þ ; δ<sup>∓</sup> ð Þ s; v ∈ D� and

$$\delta\_{\mp}(s, \delta\_{\pm}(u, v)) = \delta\_{\pm}(u, \delta\_{\mp}(s, v)), respectively$$

Ð <sup>δ</sup>þð Þ <sup>T</sup>;κ<sup>1</sup> <sup>κ</sup><sup>1</sup> u tð ÞΔt mesð Þ δþð Þ T; κ<sup>1</sup>

þð Þ <sup>T</sup>; <sup>κ</sup><sup>1</sup> : Hence, it is also enough to show that

Ð <sup>δ</sup>þð Þ <sup>T</sup>;κ<sup>1</sup> <sup>κ</sup><sup>1</sup> u tð ÞΔt mesð Þ δþð Þ T; κ<sup>1</sup>

Because of the definition of the time scale and u, uð Þ¼ <sup>κ</sup><sup>1</sup> <sup>u</sup> <sup>δ</sup><sup>n</sup>

þð Þ <sup>T</sup>; <sup>κ</sup><sup>1</sup> , then <sup>t</sup> <sup>¼</sup> <sup>δ</sup><sup>n</sup>

ð<sup>δ</sup>nþ<sup>1</sup> <sup>þ</sup> ð Þ <sup>T</sup>;κ<sup>1</sup>

δn þð Þ <sup>T</sup>;κ<sup>1</sup>

Ð <sup>δ</sup>þð Þ <sup>T</sup>;κ<sup>1</sup> <sup>κ</sup><sup>1</sup> u tð ÞΔt mesð Þ δþð Þ T; κ<sup>1</sup>

<sup>y</sup><sup>Δ</sup>ðÞ¼� <sup>t</sup> d tð Þþ f tð Þexp x t ð Þ ð Þ

<sup>κ</sup> b tð ÞΔt, <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

ð<sup>δ</sup>nþ<sup>1</sup> <sup>þ</sup> ð Þ <sup>T</sup>;κ<sup>1</sup>

δn þð Þ <sup>T</sup>;κ<sup>1</sup>

<sup>þ</sup> ð Þ <sup>T</sup>; <sup>κ</sup><sup>1</sup>

change of variables, we get the result. If <sup>s</sup> <sup>¼</sup> <sup>δ</sup><sup>n</sup>

<sup>κ</sup><sup>2</sup> <sup>¼</sup> <sup>δ</sup><sup>n</sup>

<sup>u</sup>ð Þ¼ <sup>δ</sup>þð Þ <sup>T</sup>; <sup>κ</sup><sup>1</sup> <sup>u</sup> <sup>δ</sup><sup>n</sup>þ<sup>1</sup>

<sup>Δ</sup><sup>s</sup> <sup>¼</sup> <sup>~</sup>cΔt: When <sup>s</sup> <sup>¼</sup> <sup>δ</sup><sup>n</sup>

Hence, proof follows. □

The equation that we investigate is

ð Þ¼ T;s δþð Þ T; κ<sup>1</sup> :

and

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

<sup>κ</sup> a tð ÞΔt, <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

¼

Since T is a periodic time scale in shifts (WLOG κ<sup>2</sup> > κ1), there exits n∈ N such that

¼

δn

� �, and for each <sup>t</sup> <sup>∈</sup>½ � <sup>κ</sup>1; <sup>δ</sup>þð Þ <sup>T</sup>; <sup>κ</sup><sup>1</sup> , utðÞ¼ <sup>u</sup> <sup>δ</sup><sup>n</sup>

u sð ÞΔs ¼ ~c

1Δt ¼ ~c

¼ ~c

Remark 1. [24] It is obvious that if T ¼ f g0 ∪q<sup>Z</sup>, then mesð Þ δþð Þ T; t is equal for each t in f g0 ∪q<sup>Z</sup>:

<sup>α</sup>ð Þþ <sup>t</sup> <sup>β</sup>ð Þ<sup>t</sup> exp x t ð Þþ ð Þ m tð Þexp y t ð Þ ð Þ ,

In Eq. (2.1), let a tðÞ¼ að Þ δ�ð Þ T; t , bð Þ¼ δ�ð Þ T; t b tð Þ, cð Þ¼ δ�ð Þ T; t c tð Þ, dð Þ¼ δ�ð Þ T; t d tð Þ, fð Þ¼ δ�ð Þ T; t f tð Þ, αð Þ¼ δ�ð Þ T; t αð Þt , βð Þ¼ δ�ð Þ T; t βð Þt , and mð Þ¼ δ�ð Þ T; t m tð Þ, and

<sup>x</sup><sup>Δ</sup>ðÞ¼ <sup>t</sup> a tð Þ� b tð Þexp x t ð Þ� ð Þ c tð Þexp y t ð Þ ð Þ

Ð <sup>δ</sup>þð Þ <sup>T</sup>;κ<sup>2</sup> <sup>κ</sup><sup>2</sup> u tð ÞΔt mesð Þ δþð Þ T; κ<sup>2</sup>

Ð <sup>δ</sup><sup>þ</sup> <sup>T</sup>;δ<sup>n</sup> ð Þ þð Þ <sup>T</sup>;κ<sup>1</sup>

mesðδþðT, <sup>δ</sup><sup>n</sup>

þð Þ <sup>T</sup>;κ<sup>1</sup> u tð ÞΔ<sup>t</sup>

�ð Þ¼ <sup>T</sup>;<sup>s</sup> <sup>κ</sup>1, and when <sup>s</sup> <sup>¼</sup> <sup>δ</sup><sup>n</sup>þ<sup>1</sup>

u tð ÞΔt,

1Δt,

<sup>α</sup>ð Þþ <sup>t</sup> <sup>β</sup>ð Þ<sup>t</sup> exp x t ð Þþ ð Þ m tð Þexp y t ð Þ ð Þ ,

<sup>κ</sup> d tð ÞΔ<sup>t</sup> <sup>&</sup>gt; <sup>0</sup>: <sup>β</sup><sup>l</sup> <sup>¼</sup> min<sup>t</sup>∈½ � <sup>κ</sup>;δþð Þ <sup>T</sup>;<sup>κ</sup> <sup>β</sup>ð Þ<sup>t</sup> , <sup>m</sup><sup>l</sup> <sup>¼</sup> min<sup>t</sup>∈½ � <sup>κ</sup>;δþð Þ <sup>T</sup>;<sup>κ</sup>

ð<sup>δ</sup>þð Þ <sup>T</sup>;κ<sup>1</sup> κ1

ð<sup>δ</sup>þð Þ <sup>T</sup>;κ<sup>1</sup> κ1

Ð <sup>δ</sup>þð Þ <sup>T</sup>;κ<sup>1</sup> <sup>κ</sup><sup>1</sup> u tð ÞΔt ~c mesð Þ δþð Þ T; κ<sup>1</sup> :

þðT, <sup>κ</sup>1ÞÞÞ:

þð Þ <sup>T</sup>; <sup>κ</sup><sup>1</sup> � �,

þð Þ <sup>T</sup>; <sup>t</sup> , then by the assumption of the lemma

þð Þ <sup>T</sup>; <sup>t</sup> � �: By using

<sup>þ</sup> ð Þ <sup>T</sup>; <sup>κ</sup><sup>1</sup> , then <sup>t</sup> <sup>¼</sup> <sup>δ</sup><sup>n</sup>

�

189

(2.1)

:

Quantum Calculus with the Notion δ±-Periodicity and Its Applications

http://dx.doi.org/10.5772/intechopen.74952

Then the backward operator is <sup>δ</sup>�, and the forward operator is <sup>δ</sup><sup>þ</sup> which are associated with t<sup>0</sup> <sup>∈</sup>T<sup>∗</sup> (called the initial point). Shift size is the variable u∈½ Þ t0; ∞ <sup>T</sup> in δ�ð Þ u; v . The values δþð Þ u; v and <sup>δ</sup>þð Þ <sup>u</sup>; <sup>v</sup> in <sup>T</sup><sup>∗</sup> indicate u unit translation of the term v <sup>∈</sup>T<sup>∗</sup> to the right and left, respectively. The sets D� are the domains of the shift operators δ�, respectively.

Definition 2. [4] Let T be a time scale with the shift operators δ� associated with the initial point <sup>t</sup><sup>0</sup> <sup>∈</sup>T<sup>∗</sup> . The time scale <sup>T</sup> is said to be periodic in shifts <sup>δ</sup>� if there exists a q∈ð Þ <sup>t</sup>0; <sup>∞</sup> <sup>T</sup><sup>∗</sup> such that ð Þ <sup>q</sup>; <sup>t</sup> <sup>∈</sup> <sup>D</sup>� for all t<sup>∈</sup> <sup>T</sup><sup>∗</sup>: Furthermore, if

$$Q \coloneqq \inf \left\{ q \in (t\_0, \simeq)\_{\mathbb{T}^\*} \, : \, (q, t) \in D\_{\pm} \text{ } \begin{array}{c} \text{for} \ all \ t \in \mathbb{T}^\* \end{array} \right\} \neq t\_0$$

then P is called the period of the time scale T.

Definition 3. [4] (Periodic function in shifts δ<sup>þ</sup> and δ�). Let T be a time scale that is periodic in shifts <sup>δ</sup><sup>þ</sup> and <sup>δ</sup>� with the period Q. We say that a real valued function g defined on <sup>T</sup><sup>∗</sup> is periodic in shifts if there exists a <sup>T</sup><sup>~</sup> <sup>∈</sup>½ Þ <sup>Q</sup>; <sup>∞</sup> <sup>T</sup><sup>∗</sup> such that

$$\mathbf{g}\left(\delta\_{\pm}\left(\tilde{T},t\right)\right) = \mathbf{g}(t).$$

The smallest number <sup>T</sup><sup>~</sup> <sup>∈</sup>½ Þ <sup>Q</sup>; <sup>∞</sup> <sup>T</sup><sup>∗</sup> such that is called the period of f.

Definition 1, Definition 2, and Definition 3 are from [4].

[24]

Notation 1 δ<sup>2</sup> þð Þ¼ <sup>T</sup>; <sup>κ</sup> <sup>δ</sup>þð Þ <sup>T</sup>; <sup>δ</sup>þð Þ <sup>T</sup>; <sup>κ</sup> ,

$$
\delta\_+^3(T,\kappa) = \delta\_+(T, \delta\_+(T, \delta\_+(T, \kappa))), \dots
$$

$$
\delta\_+^\text{tr}(T, \kappa) = \delta\_+(\iota \delta\_+(T, \delta\_+(T, \delta\_+(\dots)))).
$$

Lemma 1. [24] Let our time scale T be periodic in shifts, and for each t∈ T<sup>∗</sup>, δ<sup>n</sup> þð Þ <sup>T</sup>; <sup>t</sup> � �<sup>Δ</sup> is constant. Then, Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> <sup>κ</sup> u tð ÞΔ<sup>t</sup> mesð Þ <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> is also constant <sup>∀</sup><sup>κ</sup> <sup>∈</sup>T,

where <sup>κ</sup> <sup>¼</sup> <sup>δ</sup><sup>m</sup> �ð Þ <sup>T</sup>; <sup>t</sup><sup>0</sup> for m <sup>∈</sup> <sup>N</sup> and mesð Þ <sup>δ</sup>þð Þ <sup>T</sup>; <sup>κ</sup> <sup>=</sup> <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> <sup>κ</sup> 1Δt: Here, u tð Þ is a periodic function in shifts.

Proof. We get the desired result, if we can be able to show that for any κ<sup>1</sup> 6¼ κ<sup>2</sup> (κ1, κ<sup>2</sup> ∈ T).

$$\frac{\int\_{\kappa\_1}^{\delta\_+(T,\kappa\_1)} \mu(t) \Delta t}{\mathrm{mes}(\delta\_+(T,\kappa\_1))} = \frac{\int\_{\kappa\_2}^{\delta\_+(T,\kappa\_2)} \mu(t) \Delta t}{\mathrm{mes}(\delta\_+(T,\kappa\_2))}.$$

Since T is a periodic time scale in shifts (WLOG κ<sup>2</sup> > κ1), there exits n∈ N such that <sup>κ</sup><sup>2</sup> <sup>¼</sup> <sup>δ</sup><sup>n</sup> þð Þ <sup>T</sup>; <sup>κ</sup><sup>1</sup> : Hence, it is also enough to show that

$$\frac{\int\_{\kappa\_1}^{\delta\_+(T,\kappa\_1)} u(t) \Delta t}{\mathrm{mes}(\delta\_+(T,\kappa\_1))} = \frac{\int\_{\delta\_+^n(T,\kappa\_1)}^{\delta\_+(T,\kappa\_1)} u(t) \Delta t}{\mathrm{mes}(\delta\_+(T,\delta\_+^n(T,\kappa\_1)))}.$$

Because of the definition of the time scale and u, uð Þ¼ <sup>κ</sup><sup>1</sup> <sup>u</sup> <sup>δ</sup><sup>n</sup> þð Þ <sup>T</sup>; <sup>κ</sup><sup>1</sup> � �,

<sup>u</sup>ð Þ¼ <sup>δ</sup>þð Þ <sup>T</sup>; <sup>κ</sup><sup>1</sup> <sup>u</sup> <sup>δ</sup><sup>n</sup>þ<sup>1</sup> <sup>þ</sup> ð Þ <sup>T</sup>; <sup>κ</sup><sup>1</sup> � �, and for each <sup>t</sup> <sup>∈</sup>½ � <sup>κ</sup>1; <sup>δ</sup>þð Þ <sup>T</sup>; <sup>κ</sup><sup>1</sup> , utðÞ¼ <sup>u</sup> <sup>δ</sup><sup>n</sup> þð Þ <sup>T</sup>; <sup>t</sup> � �: By using change of variables, we get the result. If <sup>s</sup> <sup>¼</sup> <sup>δ</sup><sup>n</sup> þð Þ <sup>T</sup>; <sup>t</sup> , then by the assumption of the lemma <sup>Δ</sup><sup>s</sup> <sup>¼</sup> <sup>~</sup>cΔt: When <sup>s</sup> <sup>¼</sup> <sup>δ</sup><sup>n</sup> þð Þ <sup>T</sup>; <sup>κ</sup><sup>1</sup> , then <sup>t</sup> <sup>¼</sup> <sup>δ</sup><sup>n</sup> �ð Þ¼ <sup>T</sup>;<sup>s</sup> <sup>κ</sup>1, and when <sup>s</sup> <sup>¼</sup> <sup>δ</sup><sup>n</sup>þ<sup>1</sup> <sup>þ</sup> ð Þ <sup>T</sup>; <sup>κ</sup><sup>1</sup> , then <sup>t</sup> <sup>¼</sup> <sup>δ</sup><sup>n</sup> � ð Þ¼ T;s δþð Þ T; κ<sup>1</sup> :

$$\int\_{\delta\_+^t(T,\kappa\_1)}^{\delta\_+^{n+1}(T,\kappa\_1)} u(s) \Delta s = \tilde{c} \int\_{\kappa\_1}^{\delta\_+(T,\kappa\_1)} u(t) \Delta t\_r$$

$$\int\_{\delta\_+^t(T,\kappa\_1)}^{\delta\_+^{n+1}(T,\kappa\_1)} 1 \Delta t = \tilde{c} \int\_{\kappa\_1}^{\delta\_+(T,\kappa\_1)} 1 \Delta t\_r$$

and

δ<sup>∓</sup> ð Þ¼ s; δ�ð Þ u; v δ�ð Þ u; δ<sup>∓</sup> ð Þ s; v , respectively

Then the backward operator is <sup>δ</sup>�, and the forward operator is <sup>δ</sup><sup>þ</sup> which are associated with t<sup>0</sup> <sup>∈</sup>T<sup>∗</sup> (called the initial point). Shift size is the variable u∈½ Þ t0; ∞ <sup>T</sup> in δ�ð Þ u; v . The values δþð Þ u; v and <sup>δ</sup>þð Þ <sup>u</sup>; <sup>v</sup> in <sup>T</sup><sup>∗</sup> indicate u unit translation of the term v <sup>∈</sup>T<sup>∗</sup> to the right and left, respectively. The sets

Definition 2. [4] Let T be a time scale with the shift operators δ� associated with the initial point <sup>t</sup><sup>0</sup> <sup>∈</sup>T<sup>∗</sup> . The time scale <sup>T</sup> is said to be periodic in shifts <sup>δ</sup>� if there exists a q∈ð Þ <sup>t</sup>0; <sup>∞</sup> <sup>T</sup><sup>∗</sup> such that

<sup>Q</sup>≔inf <sup>q</sup>∈ð Þ <sup>t</sup>0; <sup>∞</sup> <sup>T</sup><sup>∗</sup> : ð Þ <sup>q</sup>; <sup>t</sup> <sup>∈</sup> <sup>D</sup>� for all t∈T<sup>∗</sup> � � 6¼ <sup>t</sup><sup>0</sup>

Definition 3. [4] (Periodic function in shifts δ<sup>þ</sup> and δ�). Let T be a time scale that is periodic in shifts <sup>δ</sup><sup>þ</sup> and <sup>δ</sup>� with the period Q. We say that a real valued function g defined on <sup>T</sup><sup>∗</sup> is periodic in shifts if

<sup>g</sup> <sup>δ</sup>� <sup>T</sup><sup>~</sup> ; <sup>t</sup> � � � � <sup>¼</sup> g tð Þ:

þð Þ¼ <sup>T</sup>; <sup>κ</sup> <sup>δ</sup>þð Þ <sup>T</sup>; <sup>δ</sup>þð Þ <sup>T</sup>; <sup>δ</sup>þð Þ <sup>T</sup>; <sup>κ</sup> ,…

þð Þ¼ <sup>T</sup>; <sup>κ</sup> <sup>δ</sup>þð, <sup>δ</sup>þð Þ <sup>T</sup>; <sup>δ</sup>þð Þ <sup>T</sup>; <sup>δ</sup>þð Þ :… :

Proof. We get the desired result, if we can be able to show that for any κ<sup>1</sup> 6¼ κ<sup>2</sup> (κ1, κ<sup>2</sup> ∈ T).

þð Þ <sup>T</sup>; <sup>t</sup> � �<sup>Δ</sup> is constant.

<sup>κ</sup> 1Δt: Here, u tð Þ is a periodic function in

D� are the domains of the shift operators δ�, respectively.

ð Þ <sup>q</sup>; <sup>t</sup> <sup>∈</sup> <sup>D</sup>� for all t<sup>∈</sup> <sup>T</sup><sup>∗</sup>: Furthermore, if

188 Advanced Technologies of Quantum Key Distribution

there exists a <sup>T</sup><sup>~</sup> <sup>∈</sup>½ Þ <sup>Q</sup>; <sup>∞</sup> <sup>T</sup><sup>∗</sup> such that

[24]

Then,

shifts.

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> <sup>κ</sup> u tð ÞΔ<sup>t</sup>

where <sup>κ</sup> <sup>¼</sup> <sup>δ</sup><sup>m</sup>

Notation 1 δ<sup>2</sup>

then P is called the period of the time scale T.

The smallest number <sup>T</sup><sup>~</sup> <sup>∈</sup>½ Þ <sup>Q</sup>; <sup>∞</sup> <sup>T</sup><sup>∗</sup> such that is called the period of f.

δ3

δn

Lemma 1. [24] Let our time scale T be periodic in shifts, and for each t∈ T<sup>∗</sup>, δ<sup>n</sup>

�ð Þ <sup>T</sup>; <sup>t</sup><sup>0</sup> for m <sup>∈</sup> <sup>N</sup> and mesð Þ <sup>δ</sup>þð Þ <sup>T</sup>; <sup>κ</sup> <sup>=</sup> <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

Definition 1, Definition 2, and Definition 3 are from [4].

þð Þ¼ <sup>T</sup>; <sup>κ</sup> <sup>δ</sup>þð Þ <sup>T</sup>; <sup>δ</sup>þð Þ <sup>T</sup>; <sup>κ</sup> ,

mesð Þ <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> is also constant <sup>∀</sup><sup>κ</sup> <sup>∈</sup>T,

$$\frac{\int\_{\kappa\_1}^{\delta\_+(\vec{T},\kappa\_1)} u(t) \Delta t}{\text{mes}(\delta\_+(\vec{T},\kappa\_1))} = \frac{\tilde{c} \int\_{\kappa\_1}^{\delta\_+(\vec{T},\kappa\_1)} u(t) \Delta t}{\tilde{c} \text{ } \text{mes}(\delta\_+(\vec{T},\kappa\_1))}.$$

Hence, proof follows. □

Remark 1. [24] It is obvious that if T ¼ f g0 ∪q<sup>Z</sup>, then mesð Þ δþð Þ T; t is equal for each t in f g0 ∪q<sup>Z</sup>:

The equation that we investigate is

$$\begin{aligned} \mathbf{x}^{\Lambda}(t) &= a(t) - b(t) \exp\left(\mathbf{x}(t)\right) - \frac{c(t) \exp\left(\mathbf{y}(t)\right)}{a(t) + \beta(t) \exp\left(\mathbf{x}(t)\right) + m(t) \exp\left(\mathbf{y}(t)\right)} \\ \mathbf{y}^{\Lambda}(t) &= -d(t) + \frac{f(t) \exp\left(\mathbf{x}(t)\right)}{a(t) + \beta(t) \exp\left(\mathbf{x}(t)\right) + m(t) \exp\left(\mathbf{y}(t)\right)} \end{aligned} \tag{2.1}$$

In Eq. (2.1), let a tðÞ¼ að Þ δ�ð Þ T; t , bð Þ¼ δ�ð Þ T; t b tð Þ, cð Þ¼ δ�ð Þ T; t c tð Þ, dð Þ¼ δ�ð Þ T; t d tð Þ, fð Þ¼ δ�ð Þ T; t f tð Þ, αð Þ¼ δ�ð Þ T; t αð Þt , βð Þ¼ δ�ð Þ T; t βð Þt , and mð Þ¼ δ�ð Þ T; t m tð Þ, and Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> <sup>κ</sup> a tð ÞΔt, <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> <sup>κ</sup> b tð ÞΔt, <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> <sup>κ</sup> d tð ÞΔ<sup>t</sup> <sup>&</sup>gt; <sup>0</sup>: <sup>β</sup><sup>l</sup> <sup>¼</sup> min<sup>t</sup>∈½ � <sup>κ</sup>;δþð Þ <sup>T</sup>;<sup>κ</sup> <sup>β</sup>ð Þ<sup>t</sup> , <sup>m</sup><sup>l</sup> <sup>¼</sup> min<sup>t</sup>∈½ � <sup>κ</sup>;δþð Þ <sup>T</sup>;<sup>κ</sup>

m tð Þ, <sup>β</sup><sup>u</sup> <sup>¼</sup> max<sup>t</sup> <sup>∈</sup>½ � <sup>κ</sup>;δþð Þ <sup>T</sup>;<sup>κ</sup> <sup>β</sup>ð Þ<sup>t</sup> , and <sup>m</sup><sup>u</sup> <sup>¼</sup> max<sup>t</sup>∈½ � <sup>κ</sup>;δþð Þ <sup>T</sup>;<sup>κ</sup> m tð Þ, such that <sup>κ</sup> <sup>¼</sup> <sup>δ</sup><sup>m</sup> �ð Þ <sup>T</sup>; <sup>t</sup><sup>0</sup> for <sup>m</sup> <sup>∈</sup> <sup>N</sup>: m tð Þ > 0 and c tð Þ,f tð Þ,b tð Þ > 0 αð Þt ≥ 0, βð Þt > 0: Each function is from Crdð Þ T; R :

Lemma 2. [24] Let t1, t<sup>2</sup> ∈½ � κ; δþð Þ T; κ and t ∈f g0 ∪q<sup>Z</sup>. κ is defined as in Lemma 1. If g : f g0 ∪q<sup>Z</sup> ! R is periodic function in shifts, then

$$\mathbf{g}(t) \le \mathbf{g}(t\_1) + \int\_{\kappa}^{\delta\_+(T,\kappa)} |\mathbf{g}^\Lambda(s)| \Delta \mathbf{s} \qquad \text{and} \qquad \mathbf{g}(t) \ge \mathbf{g}(t\_2) - \int\_{\kappa}^{\delta\_+(T,\kappa)} |\mathbf{g}^\Lambda(s)| \Delta \mathbf{s}.$$

Proof. We only show the first inequality as the proof of the second inequality is similar to the proof of the other one. Since g is a periodic function in shifts, without loss of generality, it suffices to show that the inequality is valid for t ∈½ � κ; δþð Þ T; κ : If t ¼ t<sup>1</sup> then the first inequality is obviously true. If t > t<sup>1</sup>

$$|\lg(t) - \lg(t\_1)| \le |\lg(t) - \lg(t\_1)| = \left| \int\_{t\_1}^t \lg^\Lambda(s) \Delta s \right| \le \int\_{t\_1}^t \lg^\Lambda(s) |\Delta s| \le \int\_{\varkappa}^{\delta\_+(T,\varkappa)} |\lg^\Lambda(s)| \Delta s.$$

Therefore,

$$\mathbf{g}(t) \le \mathbf{g}(t\_1) + \int\_{\kappa}^{\delta\_+(T,\kappa)} |\mathbf{g}^\Delta(s)| \Delta s.t$$

If

$$|\lg(t\_1) - \lg(t) \ge -|\lg(t\_1) - \lg(t)| = -\left| \int\_t^{t\_1} \lg^\Lambda(s) \Delta s \right| \ge -\int\_t^{t\_1} |\lg^\Lambda(s)| \Delta s \le -\int\_\kappa^{\delta\_+(T,\kappa)} |\lg^\Lambda(s)| \Delta s \ge 0$$

t < t<sup>1</sup>

that gives g tð Þ ≤ g tð Þþ <sup>1</sup> Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> <sup>κ</sup> <sup>∣</sup>g<sup>Δ</sup>ð Þ<sup>s</sup> <sup>∣</sup>Δs:

The proof is complete. □

Remark 2. [14] Consider the following equation:

$$\begin{split} \ddot{\tilde{\mathbf{x}}}'(t) &= a(t)\ddot{\tilde{\mathbf{x}}}(t) - b(t)\ddot{\tilde{\mathbf{x}}}^2(t) - \frac{c(t)\ddot{\tilde{\mathbf{y}}}(t)\ddot{\tilde{\mathbf{x}}}(t)}{a(t) + \beta(t)\ddot{\tilde{\mathbf{x}}}(t) + m(t)\ddot{\tilde{\mathbf{y}}}(t)}, \\ \ddot{\tilde{\mathbf{y}}}'(t) &= -d(t)\ddot{\tilde{\mathbf{y}}}(t) + \frac{f(t)\ddot{\tilde{\mathbf{x}}}(t)\ddot{\tilde{\mathbf{y}}}(t)}{a(t) + \beta(t)\ddot{\tilde{\mathbf{x}}}(t) + m(t)\ddot{\tilde{\mathbf{y}}}(t)}. \end{split} \tag{2.2}$$

This is the predator–prey dynamic system that is obtained from ordinary differential equations. Let T ¼ R. In (2.1), by taking exp x t ð Þ¼ ð Þ ~x tð Þ and exp y t ð Þ¼ ð Þ ~y tð Þ, we obtain the equality (2.2), which is the standard predator–prey system with Beddington-DeAngelis functional response.

Let T ¼ Z: By using equality (2.1), we obtain

x tð Þ� <sup>þ</sup> <sup>1</sup> x tðÞ¼ a tð Þ� b tð Þexpð Þ� x tð Þ c tð Þexpð Þ y tð Þ

<sup>~</sup>x tð Þ¼ <sup>þ</sup> <sup>1</sup> <sup>~</sup>x tð Þexp a tð Þ� b tð Þ~x tð Þ� c tð Þ~y tð Þ

<sup>~</sup>y tð Þ¼ <sup>þ</sup> <sup>1</sup> <sup>~</sup>y tð Þexp �d tð Þþ f tð Þ~x tð Þ

αð Þþ t βð Þt expð Þþ x tð Þ m tð Þexpð Þ y tð Þ

<sup>α</sup>ð Þþ <sup>t</sup> <sup>β</sup>ð Þ<sup>t</sup> <sup>~</sup>x tðÞþ m tð Þ~y tð Þ � �,

which is the discrete time predator–prey system with Beddington-DeAngelis-type functional response and also the discrete analogue of Eq. (2.2). This system was studied in [25, 26]. Since Eq. (2.1) incorporates Eqs. (2.2) and (2.3) as special cases, we call Eq. (2.1) the predator–prey dynamic system

For Eq. (2.1), expð Þ x tð Þ and expð Þ y tð Þ denote the density of prey and the predator. Therefore, x tð Þ and y tð Þ could be negative. By taking the exponential of x tð Þ and y tð Þ, we obtain the number of preys and predators that are living per unit of an area. In other words, for the general time scale case, our equation is based on the natural logarithm of the density of the predator and prey. Hence, x tð Þ and y tð Þ could be negative.

For Eqs. (2.2) and (2.3), since exp x t ð Þ¼ ð Þ ~x tð Þ and exp y t ð Þ¼ ð Þ ~y tð Þ, the given dynamic systems

Theorem 5. Assume that for the given time scale T ¼ f g0 ∪q<sup>Z</sup>, while T ∈q<sup>Z</sup>, mesð Þ δþð Þ T; t is equal for

exp � <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

<sup>κ</sup> <sup>j</sup>a tð ÞjΔ<sup>t</sup> <sup>þ</sup> <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

<sup>κ</sup> d tð Þ � �Δ<sup>t</sup> <sup>&</sup>gt; <sup>0</sup>

h i � �

<sup>κ</sup> a tð ÞΔt

<sup>α</sup>ð Þþ <sup>t</sup> <sup>β</sup>ð Þ<sup>t</sup> <sup>~</sup>x tð Þþ m tð Þ~y tð Þ � �,

y tð Þ� <sup>þ</sup> <sup>1</sup> y tðÞ¼�d tð Þþ f tð Þexpð Þ x tð Þ

Here, again by taking expð Þ¼ x tð Þ ~x tð Þ and expð Þ¼ y tð Þ ~y tð Þ, we obtain

with Beddington-DeAngelis functional response on time scales.

directly depend on the density of the prey and predator.

3. Application of δ�-periodicity of Q-calculus

each t∈T: In addition to conditions on coefficient functions and

κ

κ

are satisfied, then there exist at least one δ�-periodic solution.

<sup>κ</sup> a tð ÞΔ<sup>t</sup> � <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

<sup>κ</sup> a tð ÞΔ<sup>t</sup> � <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> <sup>κ</sup> b tð ÞΔt

<sup>κ</sup> f tð ÞΔ<sup>t</sup> � <sup>β</sup><sup>u</sup> <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

Lemma 1 if Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

0

BB@

: Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

The following theorem is the modified version of Theorem 8 from [24].

c tð Þ

c tð Þ m tð ÞΔ<sup>t</sup>

<sup>κ</sup> d tð ÞΔt

m tð ÞΔ<sup>t</sup> <sup>&</sup>gt; <sup>0</sup> and

1

CCA

� � � <sup>α</sup><sup>u</sup> <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

<sup>α</sup>ð Þþ <sup>t</sup> <sup>β</sup>ð Þ<sup>t</sup> expð Þþ x tð Þ m tð Þexpð Þ y tð Þ ,

Quantum Calculus with the Notion δ±-Periodicity and Its Applications

http://dx.doi.org/10.5772/intechopen.74952

(2.3)

191

#### Quantum Calculus with the Notion δ±-Periodicity and Its Applications http://dx.doi.org/10.5772/intechopen.74952 191

$$\begin{aligned} x(t+1) - x(t) &= a(t) - b(t) \exp(\mathbf{x}(t)) - \frac{c(t) \exp(y(t))}{a(t) + \beta(t) \exp(\mathbf{x}(t)) + m(t) \exp(y(t))}, \\ y(t+1) - y(t) &= -d(t) + \frac{f(t) \exp(\mathbf{x}(t))}{a(t) + \beta(t) \exp(\mathbf{x}(t)) + m(t) \exp(y(t))} \end{aligned}$$

Here, again by taking expð Þ¼ x tð Þ ~x tð Þ and expð Þ¼ y tð Þ ~y tð Þ, we obtain

m tð Þ, <sup>β</sup><sup>u</sup> <sup>¼</sup> max<sup>t</sup> <sup>∈</sup>½ � <sup>κ</sup>;δþð Þ <sup>T</sup>;<sup>κ</sup> <sup>β</sup>ð Þ<sup>t</sup> , and <sup>m</sup><sup>u</sup> <sup>¼</sup> max<sup>t</sup>∈½ � <sup>κ</sup>;δþð Þ <sup>T</sup>;<sup>κ</sup> m tð Þ, such that <sup>κ</sup> <sup>¼</sup> <sup>δ</sup><sup>m</sup>

g : f g0 ∪q<sup>Z</sup> ! R is periodic function in shifts, then

g tð Þ� g tð Þ<sup>1</sup> ≤ ∣g tð Þ� g tð Þ<sup>1</sup> ∣ ¼

g tð Þ� <sup>1</sup> g tð Þ ≥ � ∣g tð Þ� <sup>1</sup> g tð Þ∣ ¼ �

Remark 2. [14] Consider the following equation:

~x 0

~y 0

Let T ¼ Z: By using equality (2.1), we obtain

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

<sup>κ</sup> <sup>∣</sup>g<sup>Δ</sup>ð Þ<sup>s</sup> <sup>∣</sup>Δs:

ðÞ¼ <sup>t</sup> a tð Þ~x tð Þ� b tð Þ~x<sup>2</sup>

ð<sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> κ

g tð Þ ≤ g tð Þþ <sup>1</sup>

190 Advanced Technologies of Quantum Key Distribution

is obviously true. If t > t<sup>1</sup>

that gives g tð Þ ≤ g tð Þþ <sup>1</sup>

Therefore,

If

m tð Þ > 0 and c tð Þ,f tð Þ,b tð Þ > 0 αð Þt ≥ 0, βð Þt > 0: Each function is from Crdð Þ T; R :

ðt t1

� � � �

g tð Þ ≤ g tð Þþ <sup>1</sup>

ðt1 t

� � � �

Lemma 2. [24] Let t1, t<sup>2</sup> ∈½ � κ; δþð Þ T; κ and t ∈f g0 ∪q<sup>Z</sup>. κ is defined as in Lemma 1. If

<sup>∣</sup>g<sup>Δ</sup>ð Þ<sup>s</sup> <sup>∣</sup>Δs and g tð Þ <sup>≥</sup> g tð Þ� <sup>2</sup>

Proof. We only show the first inequality as the proof of the second inequality is similar to the proof of the other one. Since g is a periodic function in shifts, without loss of generality, it suffices to show that the inequality is valid for t ∈½ � κ; δþð Þ T; κ : If t ¼ t<sup>1</sup> then the first inequality

<sup>g</sup><sup>Δ</sup>ð Þ<sup>s</sup> <sup>Δ</sup><sup>s</sup>

ð<sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> κ

t < t<sup>1</sup>

� � � � ≥ � ðt1 t

<sup>g</sup><sup>Δ</sup>ð Þ<sup>s</sup> <sup>Δ</sup><sup>s</sup>

The proof is complete. □

ð Þ� t

This is the predator–prey dynamic system that is obtained from ordinary differential equations. Let T ¼ R. In (2.1), by taking exp x t ð Þ¼ ð Þ ~x tð Þ and exp y t ð Þ¼ ð Þ ~y tð Þ, we obtain the equality (2.2), which

<sup>α</sup>ð Þþ <sup>t</sup> <sup>β</sup>ð Þ<sup>t</sup> <sup>~</sup>x tð Þþ m tð Þ~y tð Þ:

ðÞ¼� <sup>t</sup> d tð Þ~y tð Þþ f tð Þ~x tð Þ~y tð Þ

is the standard predator–prey system with Beddington-DeAngelis functional response.

� � � � ≤ ðt t1

<sup>g</sup><sup>Δ</sup>ð Þ<sup>s</sup> <sup>∣</sup>Δ<sup>s</sup> <sup>≤</sup>

<sup>∣</sup>g<sup>Δ</sup>ð Þ<sup>s</sup> <sup>∣</sup>Δ<sup>s</sup> <sup>≤</sup> �

c tð Þ~y tð Þ~x tð Þ <sup>α</sup>ð Þþ <sup>t</sup> <sup>β</sup>ð Þ<sup>t</sup> <sup>~</sup>x tð Þþ m tð Þ~y tð Þ,

<sup>∣</sup>g<sup>Δ</sup>ð Þ<sup>s</sup> <sup>∣</sup>Δs:

�ð Þ <sup>T</sup>; <sup>t</sup><sup>0</sup> for <sup>m</sup> <sup>∈</sup> <sup>N</sup>:

ð<sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> κ

ð<sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> κ

<sup>∣</sup>g<sup>Δ</sup>ð Þ<sup>s</sup> <sup>∣</sup>Δs:

<sup>∣</sup>g<sup>Δ</sup>ð Þ<sup>s</sup> <sup>∣</sup>Δs:

ð<sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> κ

<sup>∣</sup>g<sup>Δ</sup>ð Þ<sup>s</sup> <sup>∣</sup>Δs,

(2.2)

$$\begin{split} \tilde{\boldsymbol{x}}(t+1) &= \tilde{\boldsymbol{x}}(t) \exp\Big[\boldsymbol{a}(t) - \boldsymbol{b}(t)\tilde{\boldsymbol{x}}(t) - \frac{\boldsymbol{c}(t)\tilde{\boldsymbol{y}}(t)}{\boldsymbol{a}(t) + \beta(t)\tilde{\boldsymbol{x}}(t) + \boldsymbol{m}(t)\tilde{\boldsymbol{y}}(t)}\Big], \\ \tilde{\boldsymbol{y}}(t+1) &= \tilde{\boldsymbol{y}}(t) \exp\Big[-\boldsymbol{d}(t) + \frac{\boldsymbol{f}(t)\tilde{\boldsymbol{x}}(t)}{\boldsymbol{a}(t) + \beta(t)\tilde{\boldsymbol{x}}(t) + \boldsymbol{m}(t)\tilde{\boldsymbol{y}}(t)}\Big], \end{split} \tag{2.3}$$

which is the discrete time predator–prey system with Beddington-DeAngelis-type functional response and also the discrete analogue of Eq. (2.2). This system was studied in [25, 26]. Since Eq. (2.1) incorporates Eqs. (2.2) and (2.3) as special cases, we call Eq. (2.1) the predator–prey dynamic system with Beddington-DeAngelis functional response on time scales.

For Eq. (2.1), expð Þ x tð Þ and expð Þ y tð Þ denote the density of prey and the predator. Therefore, x tð Þ and y tð Þ could be negative. By taking the exponential of x tð Þ and y tð Þ, we obtain the number of preys and predators that are living per unit of an area. In other words, for the general time scale case, our equation is based on the natural logarithm of the density of the predator and prey. Hence, x tð Þ and y tð Þ could be negative.

For Eqs. (2.2) and (2.3), since exp x t ð Þ¼ ð Þ ~x tð Þ and exp y t ð Þ¼ ð Þ ~y tð Þ, the given dynamic systems directly depend on the density of the prey and predator.
