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

By Neslihan Nesliye Pelen, Ayşe Feza Güvenilir and Billur Kaymakçalan

Submitted: May 5th 2017Reviewed: February 6th 2018Published: May 30th 2018

DOI: 10.5772/intechopen.74952

Downloaded: 390


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

1. Introduction

The traditional infinitesimal calculus without the limit notion is called calculus without limits or quantum calculus. After the developments in quantum mechanics, q-calculus and h-calculus are defined. In these calculi, h is Planck’s constant and q stands for the quantum. These two parameters qand hare related with each other as q=eih=e2πih˜.This equation h˜=h2πis the reduced Planck’s constant. h-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 h-calculus, and it was shown that it is useful in a number of fields, among them, combinatorics and fluid mechanics. The q-calculus is more useful in quantum mechanics, and it has an intimate connection with commutative relations [1]. In the following, the main notions and its relation to the time scale calculus will be discussed.

In [2], in classical calculus when the equation


is considered and as xtends to x0,the differentiation notion is obtained. When the differential equations are considered, the difference of a function is defined as fx+1fx.In quantum calculus, the q-differential of a function is equal to the following:




Then the q-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 T=qN,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 Fxis a q-antiderivative of fxif DqFx=fxis satisfied where


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=qN0,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 qN,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.

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 [13]. 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 [14], [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 [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 Tκis defined by Tκ=T/ρsupTsupT, and the set Tκis defined by Tκ=T/infTσinfT.The forward jump operator σ:TTis defined by σtintT,fortT.The backward jump operator ρ:TTis defined by ρtsuptT,fortT.The forward graininess function μ:TR0+is defined by μtσtt,fortT.The backward graininess function ν:TR0+is defined by νttρt,fortT.Here, it is assumed that inf0/=supTand sup0/=infT.

For a function f:TT, we define the Δ-derivative of fat tTκ, denoted by fΔtfor all ϵ>0.There exists a neighborhood UTof tTκsuch that


for all sU.

For the same function, the -derivative of fat tTκ, denoted by ft, for all ϵ>0., is defined. There exists a neighborhood VTof tTκsuch that


for all sV.

A function f:TRis rd-continuous if it is continuous at right-dense points in Tand its left-sided limits exist at left-dense points in T.The class of real rd-continuous functions defined on a time scale Tis denoted by CrdTR.If fCrdTR,then there exists a function Ftsuch that FΔt=ft. The delta integral is defined by abfxΔx=FbFa.

[23] Suppose thatf:TRis a function andtTκ. Then, we have the following:

  1. Iffis delta differentiable att,thenfis continuous att.

  2. Iffis continuous at a right scatteredt, thenfis delta differentiable attwith


  • Iftis right dense, thenfis delta differentiable attif and only if the limit

    exists as a finite number. In this case,
  • Iffis delta differentiable att,then


  • [23] Ifa,b,c,dT,αR, andf,g:TRare rd-continuous, then

    • abft+gtΔt=abftΔt+abgtΔt;

    • abαftΔt=αabftΔt;

    • abftΔt=baftΔt;

    • abftΔt=acftΔt+cbftΔt;

    • aaftΔt=0;

    • abftgΔtΔt=fgbfgaabfΔtgσtΔt;

    • abfσtgΔtΔt=fgbfgaabfΔtgtΔt.

    [23] Ifa,bT,αR, andf:TRare rd-continuous, then

    • If T=R,then

    where the integral on the right is the Riemann integral from calculus.
    • IfTconsists of only isolated points anda<b,then


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

    1. For eachλ01, any y satisfyingLy=λCyis not onδΩ,i.e.,yδΩ

    2. For eachyδΩKerL,VCy0and the Brouwer degreedegJVCδΩKerL00.Then, Ly=Cyhas at least one solution lying inDomLδΩ.

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

    [4] Let the time scaleTincluding a fixed numbert0TwhereTbe a non-empty subset ofT,such that there exist operatorsδ±:t0T×TTwhich satisfy the following properties:

    P.1 With respect to their second arguments, the functionsδ±are strictly increasing, i.e., if



    P.2 IfS1s,S2sDwithS1<S2, thenδS1s>δS2s,, and ifS1s,S2sD+withS1<S2, thenδ+S1s<δ+S2s,

    P.3 Ifvt0T, thenvt0D+andδ+vt0=s.Moreover, ifvT,thent0vD+andδ+t0v=vholds

    P.4 IfuvD±, thenuδ±uvD±andδuδ±uv=v,respectively.

    P.5 IfuvD±andsδ±uvD±, thenuδsvD±and


    Then the backward operator isδ, and the forward operator isδ+which are associated witht0T(called the initial point). Shift size is the variableut0Tinδ±uv. The valuesδ+uvandδ+uvinTindicate u unit translation of the termvTto the right and left, respectively. The setsD±are the domains of the shift operatorsδ±, respectively.

    [4] LetTbe a time scale with the shift operatorsδ±associated with the initial pointt0T. The time scaleTis said to be periodic in shiftsδ±if there exists aqt0Tsuch thatqtD±for alltT.Furthermore, if

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

    [4] (Periodic function in shiftsδ+andδ). LetTbe a time scale that is periodic in shiftsδ+andδwith the period Q. We say that a real valued function g defined onTis periodic in shifts if there exists aT˜QTsuch that


    The smallest numberT˜QTsuch that is called the period of f.

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


    Notation 1δ+2Tκ=δ+Tδ+Tκ,


    [24] Let our time scaleTbe periodic in shifts, and for eachtT,δ+nTtΔis constant. Then,κδ+TκutΔtmesδ+Tκis also constantκT,

    whereκ=δ±mTt0formNandmesδ+Tκ= κδ+Tκ1Δt.Here, utis a periodic function in shifts.

    Proof.We get the desired result, if we can be able to show that for any κ1κ2(κ1,κ2T).


    Since Tis a periodic time scale in shifts (WLOG κ2>κ1), there exits nNsuch that

    κ2=δ+nTκ1.Hence, it is also enough to show that


    Because of the definition of the time scale and u,uκ1=uδ+nTκ1,

    uδ+Tκ1=uδ+n+1Tκ1, and for each tκ1δ+Tκ1,ut=uδ+nTt.By using change of variables, we get the result. If s=δ+nTt,then by the assumption of the lemma Δs=c˜Δt.When s=δ+nTκ1,then t=δnTs=κ1, and when s=δ+n+1Tκ1,then t=δnTs=δ+Tκ1.


    Hence, proof follows. □

    [24] It is obvious that ifT=0qZ,thenmesδ+Ttis equal for eachtin0qZ.

    The equation that we investigate is


    In Eq. (2.1), let at=aδ±Tt, bδ±Tt=bt, cδ±Tt=ct, dδ±Tt=dt, fδ±Tt=ft, αδ±Tt=αt, βδ±Tt=βt, and mδ±Tt=mt,and κδ+TκatΔt, κδ+TκbtΔt,κδ+TκdtΔt>0.βl=mintκδ+Tκβt, ml=mintκδ+Tκmt, βu=maxtκδ+Tκβt, and mu=maxtκδ+Tκmt,such that κ=δ±mTt0for>0and ct,ft,bt>0αt0,βt>0.Each function is from CrdTR.

    [24] Lett1,t2κδ+Tκandt0qZ. κis defined as inLemma 1. Ifg:0qZRis periodic function in shifts, then


    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 tκδ+Tκ.If t=t1then the first inequality is obviously true. If t>t1

    that gives gtgt1+κδ+TκgΔsΔs.
    The proof is complete.□

    [14] Consider the following equation:


    This is the predator–prey dynamic system that is obtained from ordinary differential equations. LetT=R. In(2.1), by takingexpxt=x˜tandexpyt=y˜t,we obtain the equality(2.2), which is the standard predator–prey system with Beddington-DeAngelis functional response.

    LetT=Z.By using equality(2.1), we obtain


    Here, again by takingexpxt=x˜tandexpyt=y˜t,we obtain

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

    ForEq. (2.1), expxtandexpytdenote the density of prey and the predator. Therefore,xtandytcould be negative. By taking the exponential ofxtandyt,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,xtandytcould be negative.

    ForEqs. (2.2) and (2.3), sinceexpxt=x˜tandexpyt=y˜t,the given dynamic systems directly depend on the density of the prey and predator.

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

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

    Assume that for the given time scaleT=0qZ, whileTqZ,mesδ+Ttis equal for eachtT.In addition to conditions on coefficient functions and

    Lemma 1 ifκδ+TκatΔtκδ+TκctmtΔt>0and

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

    Proof.XuvCrd0qZR2:uδ±Tt=ut,vδ±Tt=vtwith the norm:


    YuvCrd0qZR2:uδ±Tt=ut,vδ±Tt=vtwith the norm:


    Let us define the mappings Land Cby L:DomLXYsuch that

    and C:XYsuch that

    Then, KerL=uv:uv=c1c2, c1and c2are constants.


    ImLis closed in Y.Its obvious that dimKerL=2. To show dimKerL=codimImL=2, we have to prove that KerLImL=Y.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 uvY,and WLOG taking ut, we have κδ+TκutΔt=Iwhere Iis a constant. Let us define a new function g=uImesδ+Tκ.Since Imesδ+Tκis constant by Lemma 1, if we take the integral of gfrom κto δ+Tκ,we get


    Similar steps are used for v.uvYcan 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, codimImLis also 2, we get the desired result. Hence, Lis a Fredholm mapping of index zero. There exist continuous projectors U:XXand V:YYsuch that


    The generalized inverse KU=ImLDomLKerUis given:




    Clearly, VCand KUIVCare continuous. Here, Xand Yare Banach spaces. Since for the given time scale Twhile T is constant, mesδ+Ttis equal for each tT; then, we can apply Arzela-Ascoli theorem, and by using Arzela-Ascoli theorem, we can find that K¯UIVCΩ¯is compact for any open bounded set ΩX.Additionally, VCΩ¯is bounded. Thus, Cis L-compact on Ω¯with any open bounded set ΩX.

    To apply the continuation theorem, we investigate the below operator equation:


    Let xyXbe any solution of system (3.1). Integrating both sides of system (3.1) over the interval 0w, we obtain


    From (3.1) and (3.2), we get


    Since xyX, then there exist ηi,ξiand i=1,2such that


    If ξ1is the minimum point of xton the interval κδ+Tκbecause xtis a function that is periodic in shifts for any nNon the interval δ+nTκ1δ+n+1Tκ1, the minimum point of xtis δ+nTξ1and xξ1=xδ+nTξ1.We have similar results for the other points for ξ2,η1,and η2.

    By the first equation of systems (3.2) and (3.5)


    Since κδ+TκbtΔt>0, so we get


    Using the second inequality in Lemma 2, we have


    By the first equation of systems (3.2) and (3.5)


    Then, we get


    Using the first inequality in Lemma 2, we have


    By Eq. (3.6) and (3.7), maxtκδ+TκxtmaxH1H2B1.From the second equation of system (3.2) and the second equation of system (3.6), we can derive that




    By the assumption of the Theorem 5, we get,

    κδ+TκftΔtβlκδ+TκdtΔt>0 and

    Hence, by using the first inequality in Lemma 2 and the second equation of system (3.2)


    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


    By Eq. (3.8) and (3.9), we have maxtt0δ+Tt0ytmaxH3H4B2. Obviously, B1and B2are both independent of λ.Let M=B1+B2+1. Then, maxtt0δ+Tt0xy<M.Let Ω=xyX:xy<M; then, Ωverifies the requirement (a) in Theorem 4. When xyKerL∂Ω, xyis a constant with xy=M,; then,


    where J:ImVKerLis the identity operator.

    Let us define the homotopy such that Hν=νJVC+1νGwhere


    Take DJGas the determinant of the Jacobian of G.Since xyKerL, then Jacobian of Gis


    All the functions in Jacobian of Gis positive; then, signDJGis always positive. Hence,


    Thus, all the conditions of Theorem 4 are satisfied. Therefore, system (2.1) has at least a positive δ±-periodic solution. □

    LetT=0qZ.δ±qtis the shift operator andt0=1.


    Each function in system (12) isδ±q2tperiodic and satisfiesTheorem 1; then, the system has at least oneδ±q2tperiodic solution. Here,mesδ+q2t=2.

    4. Conclusion

    The important results of this study are:

    1. The definition of δ±-periodicity notion is adapted to the quantum calculus.

    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–prey dynamic system whose coefficient functions are δ±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.

    5. Discussion

    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.



    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” [24].

    © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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    Neslihan Nesliye Pelen, Ayşe Feza Güvenilir and Billur Kaymakçalan (May 30th 2018). Quantum Calculus with the Notion δ±-Periodicity and Its Applications, Advanced Technologies of Quantum Key Distribution, Sergiy Gnatyuk, IntechOpen, DOI: 10.5772/intechopen.74952. Available from:

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