new insight into complexity from the local fractional calculus view point: moelling growths of populations Xiao-Jun Yang anj..tenreiromachao Communicate by. Debbouche In this paper, we moel the growths of populations by means of local fractional calculus. We formulate the local fractional rate equation an the local fractional logistic equation. The exact solutions of local fractional rate equation an local fractional logistic equation with the Mittag-Leffler function efine on Cantor sets are presente. The obtaine results illustrate the accuracy an efficiency for moeling the complexity of linear an nonlinear population ynamics (PD). Copyright 2015 John Wiley & Sons, Lt. Keywors: exact solution; logistic equation; population ynamics; local fractional erivative 1. Introuction The rate equation (RE) of the population ynamics (PD), structure by T. R. Malthus in 1798 [1, 2], is characterize by the orinary ifferential equation (ODE): M./ D M./, (1) where M./ represents the number of elements of the population in time, an. 0/ is the Malthus constant. With the initial population M.0/, the solution for the RE in time is M./ D M.0/ e, which expresses extinction or growth of the Malthus population. The logistic equation (LE) of the PD, evelope by P. R. Verhulst in 1838 [3, 4], is as follows:./ D./ 1./, (2) where M./ =M max, for the population M./ an the value of the maximum attainable population M max, is the intrinsic growth parameter, an is the environmental carrying capacity. The exact solution of the LE can be written in the form [4]: where.0/ D M.0/=M max,an D lim./.!1 t the limit of environmental carrying capacity D 1, the LE becomes [5] an its exact solution is [5]./, (3) 1 C.=.0/ 1/ e D./.1.//, (4)
.0/, (5).0/ C.1.0// e where.0/ D M.0/=M max. Fractional calculus (FC) was successfully use to escribe the ynamics systems in engineering an applie science [6 12]. Recently, the applications of FC to the population growth moels [5, 13 17] were propose. The fractional-orer RE an LE for fractional population ynamics in the sense of Caputo fractional erivative (FD) were etermine by the expressions [5] M./ D M./.0 <<1/ (6) an./ D./.1.//.0 <<1/, (7) respectively, where M./ is the number of the element of the fractional population in time,./ D M./ =M max,. 0/ is the west constant of fractional population, an is orer of FD. The solutions of the ifferentiable types is as follows [5]: an where the Mittag-Leffler function (MLF) is efine by using the series M./ D M.0/ E, (8) 1X.0/ 1 i E n, (9).0/ E./ D nx i.1 C i/. (10) More recently, local fractional calculus (LFC) was presente to explain the complexity of ynamics systems in the fiels of mathematical physics, such as the oscillator of free ampe vibrations [18], heat transfer [19], iffusion on Cantor sets [20], Navier-Stokes flow [21], Laplace equation (LE) [22], signal processing [23], an others [24 28]. The purpose of this article is to present for moelling complexity of the growth of populations using LFC. The structure of this article is esigne as follows. In Section 2, the recent results for the local fractional erivative (LFD) are presente. In Section 3, the local fractional RE (LFRE) for the PD an its solution of non-ifferentiable type are iscusse. In Section 4, the local fractional LE (LFLE) for the PD an its solution are propose. In Section 5, the iscussions for LFRE an LFLE are given. Finally, the conclusions are outline in Section 6. 2. Preliminaries In this section, the concept an properties of LFD an local fractional Laplace transform (LFLT), which is applie in this article, are presente. Let C &.a, b/ be a set of the non-ifferentiable functions (NDFs). The LFD of }./ of orer &.0 <&<1/ at the point D 0 is efine as follows ([18 28]): where with }. 0 / 2 C &.a, b/. The properties of the LFD are liste as follows [24, 25]: D.&/ }. 0 / D & }. 0 / & D lim! 0 &.}./ }. 0 //. 0 / &, (11) &.}./ }. 0 // Š.1 C &/Œ}./ }. 0 /, (12) 1. D.&/ Œˆ././D D.&/ˆ./ D.&/./,proveˆ./,./2 C &.a, b/; 2. D.&/ Œˆ./ =./ D D.&/ˆ././ ˆ./ D.&/./ = 2./,provieˆ./,./2 C &.a, b/ an./ 0. The Mittag-Leffler function efine on Cantor sets (MLFCS) with the fractal imension & is efine as follows [18,24,25]: E & & 1X i& D.1Ci&/. (13) The formulas of the LFD of NDFs [18, 24, 25] are liste in Table I.
Table I. The formulas of the LFD of the NDFs. NDFs LFDs E &. & / E &. & / & =.1 C &/ 1 LFD, local fractional erivative; NDF, non-ifferentiable function. Table II. The operations of LFLTs of the NDFs. }./ } &.s/ E &.a & / 1=.s & a/ & =.1 C &/ 1=s 2& LFLT, local fractional Laplace transform; NDF, non-ifferentiable functions. The LFLT of }./, enote by [24] is applie to the LFD of the NDFs, namely, = & Œ}./ D } &.s/, (14) i = & h}.&/./ D s & } &.s/ }.0/. (15) The LFLTs of the NDFs (see, for example, [24]) are shown in Table II. 3. The growth rate for population within local fractional erivative The conventional an fractional-orer REs are ifferentiable for the linear PD by using the ODEs. The LFRE of the non-ifferentiable PD is characterize by using the local fractional ODE: & M./ D M./.0 <&<1/, (16) & where M./ is the number of the fractal population in time, is the parameter of the fractal population, an & is the fractal imension. Using the LFLT of Eq.(16), we have the following: s & M &.s/ M.0/ D M &.s/, (17) which yiels M &.s/ D M.0/ s &. (18) Thus, using the inverse LFLT, the non-ifferentiable solution for the LFRE is written in close form: 4. The growth for population within local fractional erivative M./ D M.0/ E & &. (19) Similarly to the Section 3, conventional an fractional-orer LFLEs are ifferentiable for the nonlinear PD by using the nonlinear ODEs. The LFLE of the fractal PD is formulate as follows: &./ & D./.1.//.0 <&<1/, (20) where M./ =M max an is the intrinsic growth parameter for the fractal population. The LFLE of the fractal PD may be etermine by the following: &./ D./ 1./, (21) & where M./ =M max, is the intrinsic growth parameter for the fractal population, an is the environmental carrying capacity, which satisfies with D lim./.!1
With the help of Eq.(5), we can formulate a general solution in the form./ D B C C E &. & /, (22) where,,an are the parameters for the fractal population ynamics. Using the properties of the LFD (b), we fin that &./ & D & & BC CE &. & / D CE&.& / Œ BC CE &. & / 2 D BE&.& / Œ BC CE &. & / 2 C E&.& / BC CE &. & /, (23) which leas to the following: where an &./ & B D./ 1 B./, (24) 2./ D B E &. & / Œ B C C E &. & / 2, (25)./ D E &. & / B C C E &. & /. (26) Making use of = B D 1an.0/ D =. B C C /, the non-ifferentiable solution for Eq.(20) may be written in the form:.0/.0/ C.1.0// E &. & /, (27) where.0/ D M.0/=M max. In view of the expressions = B D an.0/ D =. B C C /, the exact solution for Eq.(21) is of non-ifferentiable type: where.0/ D M.0/=M max. 5. Discussion 1 C.=.0/ 1/ E &. & /, (28) Using the limit of the MLFCS lim E & & D e, (29) &!1 κ>0 κ<0 M(τ) Figure 1. The non-ifferentiable solution for the local fractional rate equation when & D ln2=ln3. τ
the solution of the RE for the PD is obtaine [1,2]. We remark the non-ifferentiable solution for the LFRE as follows. Figure 1 shows that the solution of the extremely large fractal population, when <0, is the extinction of Mittag-Leffler function efine on Cantor sets, an that the solution of the extremely small fractal population, when >0, is the growth of the Mittag-Leffler function efine on Cantor sets. The RE within conventional erivative (CDRE) [1, 2] presents the population in time; The fractional-orer RE within fractional erivative (FDRE) [5] epicts the fractional population in time; The LFRE in our manuscript isplays the fractal population in time. The CDRE, FDRE, an LFRE an their solutions (TSs) are liste in Table III. The solutions for the CDRE, FDRE, an LFRE, when > 0an M.0/ D 1, are represente in Figure 2. Similarly, using Eq.(29), Eq.(27), an Eq.(28) are rewritten as Eq.(5) an Eq.(3), respectively. The LE within conventional erivative (CDLE) [1, 2] epicts the PD in time; The fractional-orer LE within fractional erivative (FDLE) [5] exhibits the fractional PD in time; The LFLE in our manuscript shows the fractal PD in time. The exact solutions of the LFLE for nonlinear PD in ifferent initial-values are illustrate in Figure 3. The CDLE, FDLE, an LFLE an TSs are liste in Table IV. Table III. REs of ifferent types. REs TSs M./ D M./ M./ D M.0/ e M./ D M./ M./ D M.0/ E. / & M./ D M./ M./ D M.0/ E & &. & / RE, rate equation. 4.5 4 The solution for the CDRE. The solution for the FDRE. The solution for the LFRE. 3.5 M(τ) 3 2.5 2 1.5 1 0 0.2 0.4 0.6 0.8 1 τ Figure 2. The solutions for the RE within conventional erivative, RE within fractional erivative, an local fractional rate equation when >0 an M.0/ D 1. 0.6 0.55 0.5 0.45 0.4 (τ) 0.35 0.3 0.25 0.2 0.15 0.1 0 0.2 0.4 0.6 0.8 1 τ Figure 3. The close solution of local fractional logistic equation when & D ln2=ln3.
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