Open access peer-reviewed chapter

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: 229

Abstract

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.

Keywords

  • 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

fxfx0xx0

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:

dqfx=fqxfx

and

dqx=qxx=q1x.

Then the q-derivative is defined as follows:

dqfxdqx=fqxfxq1x.

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

Fx=fxdqx=1q0xqjfxqj.

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

fσtfsfΔtσtsϵσts

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

fsfρtftsρtϵsρt

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 that f:TRis a function and tTκ. Then, we have the following:

  1. If fis delta differentiable at t,then fis continuous at t.

  2. If fis continuous at a right scattered t, then fis delta differentiable at twith

    fΔt=fσtftμt.

  • If tis right dense, then fis delta differentiable at tif and only if the limit

    limstftfsts
    exists as a finite number. In this case,
    fΔt=limstftfsts.
  • If fis delta differentiable at t,then

    fσt=ft+μtfΔt.

  • [23] If a,b,c,dT,αR, and f,g:TRare rd-continuous, then

    • abft+gtΔt=abftΔt+abgtΔt;

    • abαftΔt=αabftΔt;

    • abftΔt=baftΔt;

    • abftΔt=acftΔt+cbftΔt;

    • aaftΔt=0;

    • abftgΔtΔt=fgbfgaabfΔtgσtΔt;

    • abfσtgΔtΔt=fgbfgaabfΔtgtΔt.

    [23] If a,bT,αR, and f:TRare rd-continuous, then

    • If T=R,then

    abftΔt=abftdt,
    where the integral on the right is the Riemann integral from calculus.
    • If Tconsists of only isolated points and a<b,then

    tabftμt.

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

    1. For each λ01, any y satisfying Ly=λCyis not on δΩ,i.e., yδΩ

    2. For each yδΩKerL,VCy0and the Brouwer degree degJVCδΩKerL00.Then, Ly=Cyhas at least one solution lying in DomLδΩ.

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

    [4] Let the time scale Tincluding a fixed number t0Twhere Tbe a non-empty subset of T,such that there exist operators δ±:t0T×TTwhich satisfy the following properties:

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

    S0v,S0sD±{uvt0T×T:δ±uvT},
    then

    S0v<simplies δ±S0v<δ±S0s,

    P.2 If S1s,S2sDwith S1<S2, then δS1s>δS2s,, and if S1s,S2sD+with S1<S2, then δ+S1s<δ+S2s,

    P.3 If vt0T, then vt0D+and δ+vt0=s.Moreover, if vT,then t0vD+and δ+t0v=vholds

    P.4 If uvD±, then uδ±uvD±and δuδ±uv=v,respectively.

    P.5 If uvD±and sδ±uvD±, then uδsvD±and

    δsδ±uv=δ±uδsv,respectively

    Then the backward operator is δ, and the forward operator is δ+which are associated with t0T(called the initial point). Shift size is the variable ut0Tin δ±uv. The values δ+uvand δ+uvin Tindicate u unit translation of the term vTto the right and left, respectively. The sets D±are the domains of the shift operators δ±, respectively.

    [4] Let Tbe a time scale with the shift operators δ±associated with the initial point t0T. The time scale Tis said to be periodic in shifts δ±if there exists a qt0Tsuch that qtD±for all tT.Furthermore, if

    Qinfqt0T:qtD±foralltTt0
    then P is called the period of the time scale T.

    [4] (Periodic function in shifts δ+and δ). Let Tbe a time scale that is periodic in shifts δ+and δwith the period Q. We say that a real valued function g defined on Tis periodic in shifts if there exists a T˜QTsuch that

    gδ±T˜t=gt.

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

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

    [24]

    Notation 1 δ+2Tκ=δ+Tδ+Tκ,

    δ+3Tκ=δ+Tδ+Tδ+Tκ,
    δ+nTκ=δ+(T,δ+Tδ+Tδ+..

    [24] Let our time scale Tbe periodic in shifts, and for each tT,δ+nTtΔis constant. Then, κδ+TκutΔtmesδ+Tκis also constant κT,

    where κ=δ±mTt0for mNand mesδ+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).

    κ1δ+Tκ1utΔtmesδ+Tκ1=κ2δ+Tκ2utΔtmesδ+Tκ2.

    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

    κ1δ+Tκ1utΔtmesδ+Tκ1=δ+nTκ1δ+Tδ+nTκ1utΔtmes(δ+(T,δ+n(T,κ1))).

    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.

    δ+nTκ1δ+n+1Tκ1usΔs=c˜κ1δ+Tκ1utΔt,δ+nTκ1δ+n+1Tκ11Δt=c˜κ1δ+Tκ11Δt,
    and
    κ1δ+Tκ1utΔtmesδ+Tκ1=c˜κ1δ+Tκ1utΔtc˜mesδ+Tκ1.

    Hence, proof follows. □

    [24] It is obvious that if T=0qZ,then mesδ+Ttis equal for each tin 0qZ.

    The equation that we investigate is

    xΔt=atbtexpxtctexpytαt+βtexpxt+mtexpyt,yΔt=dt+ftexpxtαt+βtexpxt+mtexpyt,E2.1

    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 mN.mt>0and ct,ft,bt>0αt0,βt>0.Each function is from CrdTR.

    [24] Let t1,t2κδ+Tκand t0qZ. κis defined as in Lemma 1. If g:0qZRis periodic function in shifts, then

    gtgt1+κδ+TκgΔsΔsandgtgt2κδ+TκgΔsΔs.

    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

    gtgt1gtgt1=t1tgΔsΔst1tgΔsΔsκδ+TκgΔsΔs.
    Therefore,gtgt1+κδ+TκgΔsΔs.
    Ift<t1
    gt1gtgt1gt=tt1gΔsΔstt1gΔsΔsκδ+TκgΔsΔs,
    that gives gtgt1+κδ+TκgΔsΔs.
    The proof is complete.□

    [14] Consider the following equation:

    x˜'t=atx˜tbtx˜2tcty˜tx˜tαt+βtx˜t+mty˜t,y˜'t=dty˜t+ftx˜ty˜tαt+βtx˜t+mty˜t.E2.2

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

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

    xt+1xt=atbtexpxtctexpytαt+βtexpxt+mtexpyt,yt+1yt=dt+ftexpxtαt+βtexpxt+mtexpyt

    Here, again by taking expxt=x˜tand expyt=y˜t,we obtain

    x˜t+1=x˜texpatbtx˜tcty˜tαt+βtx˜t+mty˜t,y˜t+1=y˜texpdt+ftx˜tαt+βtx˜t+mty˜t,E2.3
    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), expxtand expytdenote the density of prey and the predator. Therefore, xtand ytcould be negative. By taking the exponential of xtand yt,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, xtand ytcould be negative.

    For Eqs. (2.2) and (2.3), since expxt=x˜tand expyt=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 scale T=0qZ, while TqZ, mesδ+Ttis equal for each tT.In addition to conditions on coefficient functions and

    Lemma 1 if κδ+TκatΔtκδ+TκctmtΔt>0and

    κδ+TκatΔtκδ+TκctmtΔtκδ+TκbtΔtexpκδ+TκatΔt+κδ+TκatΔt.(κδ+TκftΔtβuκδ+TκdtΔtαuκδ+TκdtΔt>0
    are satisfied, then there exist at least one δ±-periodic solution.

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

    uv=maxtt0δ+Tt0Tutvt

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

    uv=maxtt0δ+Tt0Tutvt

    Let us define the mappings Land Cby L:DomLXYsuch that

    Luv=uΔvΔ
    and C:XYsuch that
    Cuv=atbtexputctexpvtαt+βtexput+mtexpvtdt+ftexputαt+βtexput+mtexpvt

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

    ImL=uv:κδ+TκutΔtκδ+TκvtΔt=00.

    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

    κδ+TκgtΔt=κδ+TκutΔtI=0.

    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

    Uuv=1mesδ+Tκκδ+TκutΔtκδ+TκvtΔt
    and
    Vuv=1mesδ+Tκκδ+TκutΔtκδ+TκvtΔt.

    The generalized inverse KU=ImLDomLKerUis given:

    KUuv=κtusΔs1mesδ+Tκκδ+TκκtusΔsκtvsΔs1mesδ+Tκκδ+TκκtvsΔs.
    VCuv=1mesδ+Tκκδ+Tκasbsexpuscsexpvsαs+βsexpus+msexpvsΔsκδ+Tκds+fsexpusαs+βsexpus+msexpvsΔs

    Let

    atbtexputctexpvtαt+βtexput+mtexpvt=C1dt+ftexputαt+βtexput+mtexpvt=C21mesδ+Tκκδ+Tκasbsexpuscsexpvsαs+βsexpus+msexpvsΔs=C¯1
    and
    1mesδ+Tκκδ+Tκds+fsexpusαs+βsexpus+msexpvsΔs=C¯2KUIVCuv=KUC1C¯1C2C¯2=κtC1sC¯1sΔs1mesδ+Tκκδ+TκκtC1sC¯1sΔsκtC2sC¯2sΔs1mesδ+Tκκδ+TκκtC2sC¯2sΔs.

    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:

    xΔt=λatbtexpxtctexpytαt+βtexpxt+mtexpytyΔt=λdt+ftexpxtαt+βtexpxt+mtexpytE3.1

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

    κδ+TκatΔt=κδ+Tκbtexpxt+ctexpytαt+βtexpxt+mtexpytΔt,κδ+TκdtΔt=κδ+Tκftexpxtαt+βtexpxt+mtexpytΔt,E3.2

    From (3.1) and (3.2), we get

    κδ+TκxΔtΔtλκδ+TκatΔt+κδ+Tκbtexpxt+ctexpytαt+βtexpxt+mtexpytΔt,λκδ+TκatΔt+κδ+TκatΔtκδ+TκatΔt+κδ+TκatΔtM1E3.3
    κδ+TκyΔtΔtλκδ+TκdtΔt+κδ+Tκftexpxtαt+βtexpxt+mtexpytΔtλκδ+TκdtΔt+κδ+TκdtΔtκδ+TκdtΔt+κδ+TκdtΔtM2E3.4

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

    xξ1=mint[tκδ+Tκxt,xη1=maxt[tκδ+Tκxt,yξ2=mint[tκδ+Tκyt,yη2=maxt[tκδ+TκytE3.5

    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)

    κδ+TκatΔtκδ+Tκbtexpxη1+ctmtΔt=expxη1κδ+TκbtΔt+κδ+TκctmtΔt.

    Since κδ+TκbtΔt>0, so we get

    xη1lnκδ+TκatΔtκδ+TκctmtΔtκδ+TκbtΔtl1

    Using the second inequality in Lemma 2, we have

    xtxη1κδ+TκxΔtΔtxη1κδ+TκatΔt+κδ+TκatΔt=l1M1H1E3.6

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

    κδ+TκatΔtκδ+Tκbtexpxξ1Δt=expxξ1κδ+TκbtΔt.

    Then, we get

    xξ1lnκδ+TκatΔtκδ+TκbtΔtl2

    Using the first inequality in Lemma 2, we have

    xtxξ1+κδ+TκxΔtΔtxξ1+κδ+TκatΔt+κδ+TκatΔt=l2+M1H2E3.7

    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

    κδ+TκdtΔtκδ+Tκftexpxtβlexpxt+mlexpytΔtκδ+TκfteH2βleH2+mlexpyξ2Δt=eH2βleH2+mlexpyξ2κδ+TκftΔt.

    Therefore,

    expyξ21mleH2κδ+TκftΔtκδ+TκdtΔtβleH2

    By the assumption of the Theorem 5, we get,

    κδ+TκftΔtβlκδ+TκdtΔt>0 and
    yξ2ln1mleH2κδ+TκftΔtκδ+TκdtΔtβleH2L1

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

    ytyξ2+κδ+TκyΔtΔtyξ2+κδ+TκdtΔt+κδ+TκdtΔtL1+M2H3.E3.8

    Again, using the second equation of system (3.2), we obtain

    κδ+TκdtΔtκδ+Tκftexpxtαu+βuexpxt+muexpytΔtκδ+TκfteH1αu+βueH1+muexpyη2Δt=eH1αu+βueH1+muexpyη2κδ+TκftΔt,
    expyη21mueH1κδ+TκftΔtκδ+TκdtΔtβueH1αu.

    Using the assumption of the Theorem 5, we obtain

    eH1κδ+TκftΔtβuκδ+TκdtΔtαuκδ+TκdtΔt>0
    and
    yη2ln1mueH1κδ+TκftΔtκδ+TκdtΔtβueH1αuL2.

    By using the second inequality in Lemma 2

    ytyη2κδ+TκyΔtΔtyη2κδ+TκdtΔt+κδ+TκdtΔt=L2M2H4.E3.9

    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,

    VCxy=κδ+Tκasbsexpxcsexpyαs+βsexpx+msexpyΔtκδ+Tκds+fsexpxαs+βsexpx+msexpyΔt00
    JVCxy=VCxy

    where J:ImVKerLis the identity operator.

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

    Gxy=κδ+TκasbsexpxΔtκδ+Tκdsfsexpxαs+βsexpx+msexpyΔt

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

    [exκδ+TκbsΔt0κδ+Tκexfsαs+βsex+mseyΔt+κδ+Tκex2fsβsαs+βsex+msey2Δtκδ+Tκexeyfsmsαs+βsex+msey2Δt]

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

    degJVCΩKerL0=degGΩKerL0=xyG100signDJGxy0.

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

    Let T=0qZ.δ±qtis the shift operator and t0=1.

    xΔt=1lntlnq+41lntlnq+0.5expxtexpytexpxt+2expyt,yΔt=0.3+1lntlnq+7expxtexpxt+2expyt,E3.10

    Each function in system (12) is δ±q2tperiodic and satisfies Theorem 1; then, the system has at least one δ±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.

    Acknowledgments

    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].

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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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