The relation between the time scale calculus and quantum calculus and the δ ± -periodicity in quantum calculus with the notion is considered. As an application, in two-dimensional predator–prey system with Beddington-DeAngelis-type functional response on periodic time scales in shifts is used.
- predator prey dynamic systems
- Beddington-DeAngelis-type functional response
- δ±-periodic solutions on quantum calculus
- periodic time scales in shifts
The traditional infinitesimal calculus without the limit notion is called calculus without limits or quantum calculus. After the developments in quantum mechanics, -calculus and -calculus are defined. In these calculi, h is Planck’s constant and q stands for the quantum. These two parameters and are related with each other as This equation is the reduced Planck’s constant. -calculus can also be seen as the calculus of the differential equations, and this was first studied by George Boole. Many other scientists also made some studies on -calculus, and it was shown that it is useful in a number of fields, among them, combinatorics and fluid mechanics. The -calculus is more useful in quantum mechanics, and it has an intimate connection with commutative relations . In the following, the main notions and its relation to the time scale calculus will be discussed.
In , in classical calculus when the equation
is considered and as tends to the differentiation notion is obtained. When the differential equations are considered, the difference of a function is defined as In quantum calculus, the -differential of a function is equal to the following:
Then the -derivative is defined as follows:
The differentiation in time scale calculus is given in Theorem 1, and if the differentiation notion in this theorem is applied when one can easily see that the same -derivative is obtained.
As an inverse of -derivative, one can get -integral that is also very significant for the structure of this calculus. A function is a -antiderivative of if is satisfied where
This is also called the Jackson integral . When the definition of the antiderivative of a function in time scale calculus is considered, it can be easily seen that when 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 -periodicity notion in time scale calculus is defined in Definition 1 in  for the application. In this study, by using time scale calculus, the application of -periodicity notion of 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  and originally derived by Pierre Françis Verhulst . 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 . In 1925, he obtained the equations to analyze predator–prey interactions in his book on biomathematics  arriving at the equations that we know today.
After the development of the equations for predator–prey systems, it becomes important to obtain the type of functional response. The first functional response was proposed by C. S. Holling in [11, 12]. Both the Lotka-Volterra model and Holling’s extensions have been used to model the moose and wolf populations in Isle Royale National Park . In addition to these, there are many studies that use the predator–prey dynamic systems with Holling-type functional responses. These studies especially analyze the permanence, stability, periodicity, and such different aspects of these systems. The papers , [15, 16] can be some of its examples.
Arditi and Ginzburg made some changes and extension on the functional response of Holling, and this new functional response is known as the ratio-dependent functional response. Also, from this functional response, the semiratio-dependent functional responses are also derived. Again, there are many studies that are about the several structures of the predator–prey dynamic systems such as [14, 17, 18, 19], [20, 21].
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 . By a time scale, denoted by , we mean a non-empty closed subset of 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 is defined by , and the set is defined by The forward jump operator is defined by The backward jump operator is defined by The forward graininess function is defined by The backward graininess function is defined by Here, it is assumed that and
For a function , we define the -derivative of at , denoted by for all There exists a neighborhood of such that
For the same function, the -derivative of at , denoted by , for all , is defined. There exists a neighborhood of such that
A function is rd-continuous if it is continuous at right-dense points in and its left-sided limits exist at left-dense points in The class of real rd-continuous functions defined on a time scale is denoted by If then there exists a function such that . The delta integral is defined by  Suppose that is a function and . Then, we have the following: If is delta differentiable at then is continuous at If is continuous at a right scattered , then is delta differentiable at with
 Suppose that is a function and . Then, we have the following:
If is delta differentiable at then is continuous at
If is continuous at a right scattered , then is delta differentiable at with
If is right dense, then is delta differentiable at if and only if the limit
If is delta differentiable at then
 If , and are rd-continuous, then
 If , and are rd-continuous, then
If consists of only isolated points and then
 (Continuation Theorem). Let L be a Fredholm mapping of index zero and C be L-compact on . Assume
For each , any y satisfying is not on i.e.,
For each and the Brouwer degree Then, has at least one solution lying in .
We will also give the following lemma, which is essential for this chapter.  Let the time scale including a fixed number where be a non-empty subset of such that there exist operators which satisfy the following properties: P.1 With respect to their second arguments, the functions are strictly increasing, i.e., if
 Let the time scale including a fixed number where be a non-empty subset of such that there exist operators which satisfy the following properties:
P.1 With respect to their second arguments, the functions are strictly increasing, i.e., if
P.2 If with , then , and if with , then
P.3 If , then and Moreover, if then and holds
P.4 If , then and respectively.
P.5 If and , then and
Then the backward operator is , and the forward operator is which are associated with (called the initial point). Shift size is the variable in . The values and in indicate u unit translation of the term to the right and left, respectively. The sets are the domains of the shift operators , respectively.
 Let be a time scale with the shift operators associated with the initial point . The time scale is said to be periodic in shifts if there exists a such that for all Furthermore, if
 (Periodic function in shifts and ). Let be a time scale that is periodic in shifts and with the period Q. We say that a real valued function g defined on is periodic in shifts if there exists a such that
The smallest number such that is called the period of f.
 Let our time scale be periodic in shifts, and for each is constant. Then, is also constant
where for and = Here, is a periodic function in shifts.
Proof. We get the desired result, if we can be able to show that for any ().
Since is a periodic time scale in shifts (WLOG ), there exits such that
Hence, it is also enough to show that
Because of the definition of the time scale and
, and for each By using change of variables, we get the result. If then by the assumption of the lemma When then , and when then
Hence, proof follows. □
 It is obvious that if then is equal for each in
The equation that we investigate is
In Eq. (2.1), let , , , , , , , and and , , , , and such that for and Each function is from
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 If then the first inequality is obviously true. If
 Consider the following equation:
This is the predator–prey dynamic system that is obtained from ordinary differential equations. Let . In (2.1), by taking and we obtain the equality (2.2), which is the standard predator–prey system with Beddington-DeAngelis functional response.
Let By using equality (2.1), we obtain
Here, again by taking and we obtain
For Eq. (2.1), and denote the density of prey and the predator. Therefore, and could be negative. By taking the exponential of and 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, and could be negative.
3. Application of -periodicity of Q-calculus
The following theorem is the modified version of Theorem 8 from . Assume that for the given time scale , while , is equal for each In addition to conditions on coefficient functions and Lemma 1 if and
Assume that for the given time scale , while , is equal for each In addition to conditions on coefficient functions and
Lemma 1 if and
Proof. with the norm:
with the norm:
Let us define the mappings and by such that
Then, , and are constants.
is closed in Its obvious that . To show , we have to prove that It is obvious that when we take an element from Ker L, an element from Im L, we find an element of Y by summing these two elements. If we take an element and WLOG taking , we have where is a constant. Let us define a new function Since is constant by Lemma 1, if we take the integral of from to we get
Similar steps are used for can be written as the summation of an element from Im L and an element from Ker L. Also, it is easy to show that any element in Y is uniquely expressed as the summation of an element Ker L and an element from Im L. So, is also 2, we get the desired result. Hence, is a Fredholm mapping of index zero. There exist continuous projectors and such that
The generalized inverse is given:
Clearly, and are continuous. Here, and are Banach spaces. Since for the given time scale while T is constant, is equal for each ; then, we can apply Arzela-Ascoli theorem, and by using Arzela-Ascoli theorem, we can find that is compact for any open bounded set Additionally, is bounded. Thus, is L-compact on with any open bounded set
To apply the continuation theorem, we investigate the below operator equation:
Since , then there exist and such that
If is the minimum point of on the interval because is a function that is periodic in shifts for any on the interval , the minimum point of is and We have similar results for the other points for and
Since , so we get
Using the second inequality in Lemma 2, we have
Then, we get
Using the first inequality in Lemma 2, we have
By the assumption of the Theorem 5, we get,
Again, using the second equation of system (3.2), we obtain
Using the assumption of the Theorem 5, we obtain
By using the second inequality in Lemma 2
where is the identity operator.
Let us define the homotopy such that where
Take as the determinant of the Jacobian of Since , then Jacobian of is
All the functions in Jacobian of is positive; then, is always positive. Hence,
Let is the shift operator and
Each function in system (12) is periodic and satisfies Theorem 1; then, the system has at least one periodic solution. Here,
The important results of this study are:
The definition of -periodicity notion is adapted to the quantum calculus.
The importance of time scale calculus is pointed out for the analysis of quantum calculus.
As an application, the -periodicity notion for quantum calculus is used for the predator–prey dynamic system whose coefficient functions are periodic.
As a result, it is seen that one can define a periodicity notion that is applicable to the structure of the quantum calculus. Additionally, it is shown that this notion is useful for different applications. One of its applications is analyzed in this study with an example.
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 -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 -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.
A major portion of the chapter is borrowed from the publication “Behavior of the solutions for predator-prey dynamic systems with Beddington-DeAngelis-type functional response on periodic time scales in shifts” .