A new insight into complexity from the local fractional calculus view point: modelling growths of populations

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1 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)

2 .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 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.

3 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

4 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

5 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 The solution for the CDRE. The solution for the FDRE. The solution for the LFRE. 3.5 M(τ) 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 (τ) τ Figure 3. The close solution of local fractional logistic equation when & D ln2=ln3.

6 Table IV. LEs of ifferent types. LEs TSs./ D./.1.//./ D./.1.// 1 P &./ & D./.1.// LE, logistic equation..0/.0/c.1.0//e.0/1.0/ ie.n /.0/.0/C.1.0//E &. & / 6. Conclusions This work presente a new application of LFC to moel the complexity of linear an nonlinear PD. The LFRE an LFLE of the growth in human population an their close solutions with non-ifferentiable graphs were use to explain the PD. Comparative results of two moels with CD, FD, an LFD were iscusse. It is shown that LFC is a relevant tool to analyze the PD. References 1. Malthus TR. Malthus Population: The First Essay (1798). University of Michigan Press: nn rbor, MI, Royama T.nalytical Population Dynamics. Springer: Netherlans, Verhulst PF. Notice sur la loi que la population sint ons son accroissement. Mathematical Physics 1838; 10: Webb GF. Theory of Nonlinear ge-epenent Population Dynamics.CRC Press:New York, West BJ. Exact solution to fractional logistic equation. Physics 2015; 429: Klafter J, Lim SC, Metzler R. Fractional Dynamics: Recent vances. Worl Scientific: Lonon, Povstenko Y. Linear Fractional Diffusion-wave Equation for Scientists an Engineers. Birkhäuser: Basel, Mainari F. Fractional Calculus an Waves in Linear Viscoelasticity: n Introuction to Mathematical Moels. Worl Scientific: Lonon, Tarasov VE. Fractional Dynamics: pplications of Fractional Calculus to Dynamics of Particles, Fiels an Meia. Springer: Heielberg, Ortigueira MD. Fractional Calculus for Scientists an Engineers. Springer: New York, Jesus IS, Machao JT. Fractional control of heat iffusion systems. Nonlinear Dynamics 2008; 54(3): Machao JT, Galhano M, Trujillo JJ. On evelopment of fractional calculus uring the last fifty years. Scientometrics 2014; 98(1): bbas S, Banerjee M, Momani S. Dynamical analysis of fractional-orer moifie logistic moel. Computers & Mathematics with pplications 2011; 62(3): El-Saye M,El-Mesiry EM,El-Saka H.On the fractional-orer logistic equation. pplie Mathematics Letters 2007; 20(7): Momani S, Qaralleh R. Numerical approximations an Paé approximants for a fractional population growth moel. pplie Mathematical Moelling 2007; 31(9): Xu H. nalytical approximations for a population growth moel with fractional orer. Communications in Nonlinear Science 2009; 14(5): Suat Erturk V, Yilirim, Momanic S, Khan Y. The ifferential transform metho an Pae approximants for a fractional population growth moel. International Journal of Numerical Methos for Heat an Flui Flow 2012; 22(6): Yang XJ, Srivastava HM. n asymptotic perturbation solution for a linear oscillator of free ampe vibrations in fractal meium escribe by local fractional erivatives.communications in Nonlinear Science an Numerical Simulation 2015; 29(1): Yang XJ, Srivastava HM, He JH, Baleanu D. Cantor-type cylinrical-coorinate metho for ifferential equations with local fractional erivatives. Physics Letters 2013; 377(28): Yang XJ, Baleanu D, Srivastava HM. Local fractional similarity solution for the iffusion equation efine on Cantor sets. pplie Mathematics Letters 2015; 47: Liu HY, He JH, Li ZB. Fractional calculus for nanoscale flow an heat transfer. International Journal of Numerical Methos for Heat an Flui Flow 2014; 24(6): Yan SP, Jafari H, Jassim HK. Local fractional omian ecomposition an function ecomposition methos for Laplace equation within local fractional operators. vances in Mathematical Physics 2014; 2014, 1-7:rticle ID Chen ZY, Cattani C, Zhong WP. Signal processing for nonifferentiable ata efine on Cantor sets: a local fractional Fourier series approach. vances in Mathematical Physics 2014; 2014, 1-7:rticle ID Yang XJ, Baleanu D, Srivastava HM. Local Fractional Integral Transforms an Their pplications. caemic Press: New York, Yang XJ. vance Local Fractional Calculus an its pplications. Worl Science: New York, Yang XJ, Srivastava HM, Cattani C. Local fractional homotopy perturbation metho for solving fractal partial ifferential equations arising in mathematical physics. Romanian Reports in Physics 2015; 67(3): Yang XJ, Baleanu D, Baleanu MC. Observing iffusion problems efine on cantor sets in ifferent coorinate systems. Thermal Science 2015; 19(S1): Yang XJ, Baleanu D, Lazarevic MP, Cajic MS. Fractal bounary value problems for integral an ifferential equations with local fractional operators. Thermal Science 2015; 19(3):