DDNS Discrete Dynamics in Nature and Society 1607-887X 1026-0226 Hindawi 10.1155/2017/7893049 7893049 Research Article Convergence of a Logistic Type Ultradiscrete Model http://orcid.org/0000-0003-4632-3385 Sekiguchi Masaki 1 Ishiwata Emiko 2 http://orcid.org/0000-0001-5727-5855 Nakata Yukihiko 3 Villatoro Francisco R. 1 Tokyo Metropolitan Ogikubo High School 5-7-20 Ogikubo Suginami-ku Tokyo 167-0051 Japan 2 Department of Applied Mathematics Tokyo University of Science 1-3 Kagurazaka Shinjuku-ku Tokyo 162-8601 Japan sut.ac.jp 3 Department of Mathematics Shimane University 1600 Nishikawatsu-cho Matsue-shi Shimane 690-8504 Japan shimane-u.ac.jp 2017 2782017 2017 05 05 2017 16 07 2017 2782017 2017 Copyright © 2017 Masaki Sekiguchi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We derive a piecewise linear difference equation from logistic equations with time delay by ultradiscretization. The logistic equation that we consider in this paper has been shown to be globally stable in the continuous and discrete time formulations. Here, we study if ultradiscretization preserves the global stability property, analyzing the asymptotic behaviour of the obtained piecewise linear difference equation. It is shown that our piecewise linear difference equation has a threshold property concerning global attractivity of equilibria, similar to the stable logistic equations with time delay.

Japan Society for the Promotion of Science JP26400212 16K20976
1. Introduction

Ultradiscretization is proposed as a procedure to obtain a discrete system, where unknown variables also take discretized values . The discrete systems are a class of piecewise-defined difference equations [2, 3]. Specifically, ultradiscretization converts addition, multiplication, and division for two numbers in a discrete system into max operator, addition, and subtraction for other two numbers in the ultradiscrete model. Ultradiscrete models are related to the continuous and discrete models via formal solutions and conserved quantities [1, 4]. See also  for the application of ultradiscretization to the traffic flow.

In this paper, we consider the following difference equation:(1)xn+1=maxxn,b+xn-ω-maxc,b+xn-ω,where b is a real number and c is a real positive number, with the following initial condition:(2)x-jRfor  j0,1,,ω.

We derive the difference equation (1) from a nonlinear difference equation studied in [8, 9]. The difference equation is an extension of a discrete logistic map and can be seen as a discrete analogue of a disease transmission dynamics model studied in . In Section 2, we briefly introduce the disease transmission dynamics model formulated as a scalar delay differential equation. Subsequently we derive the difference equation (1) from the differential equation via discretization and ultradiscretization. It is known that the applied discretization gives stable numerical solutions . Nonstandard finite difference schemes are used, from a continuous dynamical system, to derive a dynamically consistent discrete system, which preserves qualitative and quantitative properties of the solution of the original continuous differential equation such as positivity, stability of equilibria, and conservation laws; see [12, 13] and references therein. The ultradiscrete model (1) is related to two delay equations: delay difference equation studied in [8, 9] and delay differential equation studied in .

Those delay equations are an extension of a discrete logistic map and the logistic equation, respectively. For the nondelay case, three equations are related to each other, sharing the qualitative property that every solution converges to an equilibrium . It is known that the solution of the discrete model exactly follows the continuous solution of the logistic equation . In the two delay equations, the corresponding equilibria are globally asymptotically stable; thus the discretization preserves the global stability property as well as in the nondelay case. In this paper, we study if the difference equation (1) derived from the stable difference and differential equations has a similar property. The convergence property of the difference equation (1) is analyzed in detail.

The paper is organized as follows. In Section 2, we summarize stability of differential and difference logistic equations in  and derive our ultradiscrete model from the difference equation. Our objective for this section is clarifying qualitative correspondence between the differential and difference equations. In Section 3, we discuss the convergence property of (1). We prove that the model exhibits the threshold behaviour, similar to the differential equation studied in  and the difference equation studied in [8, 9]. We find here that a subsequence of the solution has a monotone property and this monotonicity is used for the proof. We then summarize our results in Section 4.

2. Differential and Difference Logistic Equations

In this section, we summarize the previous studies related to (1).

We start with a logistic equation:(3)dytdt=ytλ-δ-λyt,where λ and δ are real positive constants. The reason for the parameterization becomes clear, when we introduce time delay. It is well known that, for the positive initial conditions, the trivial equilibrium, y=0, is globally asymptotically stable if λ-δ0, while the positive equilibrium, y=(λ-δ)/λ, is globally asymptotically stable if λ-δ>0.

By an applied discretization , the following discrete analogue can be derived from the logistic equation (3):(4)yn+1=yn+βyn1+γ+βyn.Let h be a sufficiently small step size. Then the parameters β and γ are related to λ and δ via β=hλ and γ=hδ. The difference equation (4) captures the continuous solution of the differential equation (3); that is, the solution shows the logistic curve . See also .

The author in  obtains the following piecewise linear difference equation from (4) by ultradiscretization:(5)xn+1=b-c-max-xn,b-c,where b and c are constants that satisfied b-c0. Ultradiscretization is proposed as a procedure to obtain the discrete system, where unknown variables also take discretized values . In , it is shown that the three models (3), (4), and (5) share the qualitative property that every solution converges to an equilibrium.

Following [14, 1719], let us derive (5) from (4). For ε>0, we introduce a variable x via(6)yn=exn/εand parameters b and c>0 through(7)β=eb/ε,1+γ=ec/ε.Then we have(8)exn+1/ε=exn/ε1+eb/εec/ε+eb+xn/ε;thus(9)xn+1=xn+εlog1+eb/ε-εlogec/ε+eb+xn/ε.Letting ε+0 and assuming b>0, we get (5) by the following manipulations:(10)xn+1=xn+max0,b-maxc,b+xn=b-maxc-xn,b=b-c-max-xn,b-c.The key relation used here is the following limit:(11)limε+0εlogeA/ε+eB/ε=maxA,Bfor A,B>0.

An epidemic model considered in  is an extension of the logistic equation (3). The model is formulated as the following delay differential equation:(12)dytdt=λyt-τ1-yt-δyt,where τ is a real positive constant. The global stability condition for (12) is the same as the condition for the nondelay case (3): for the positive initial conditions, the trivial equilibrium, y=0, is globally asymptotically stable if λ-δ0, while the positive equilibrium, y=(λ-δ)/λ, is globally asymptotically stable if λ-δ>0. Different logistic equations with time delay have the instability property; see .

In order to ensure positivity of the solution in discrete analogues of the differential equation (11), we use Mickens nonstandard finite difference scheme  to discretize (12) as follows:(13)yt+h-yth=λyt-τ1-yt+h-δyt+h,where h>0 is a step size. Equation (13) can be written by the following explicit form:(14)yt+h=yt+hλyt-τ1+hδ+hλyt-τ;thus (13) is equivalently written as the following difference equation with hδ=γ,hλ=β, y(t)=yn, and y(t+h)=yn+1:(15)yn+1=yn+βyn-ω1+γ+βyn-ω,where β and γ are positive constants and ω is a nonnegative integer. It is obvious that the delay equation (15) is reduced to (4) when ω=0. Equation (15) is a special case of the model considered in [8, 9]. For some specific and general cases, the authors in  show global asymptotic stabilities of the zero and positive equilibria. The zero equilibrium of (15) is globally asymptotically stable when βγ. The unique equilibrium of (15) is globally asymptotically stable when β>γ. From those results, the difference equation (15) can be seen as a discrete analogue that preserves the global stability property of (12).

Let us now derive the difference equation (1) from (15). For ε>0, we introduce the variable x and the parameters b and c in the same way as the derivation of (5); then we get(16)xn+1=εlogexn/ε+eb+xn-ω/ε-εlogec/ε+eb+xn-ω/ε.Letting ε+0 and using the key relation (11), we get (1). Finally, we note that (5) is a special case of (1). In fact, let ω=0 and b>0 in (5). Then(17)xn+1=maxxn,b+xn-maxc,b+xn=xn+max0,b-xn-maxc-xn,b=b-c-max-xn,b-c.In the following section, we study the convergence of the solution of (1).

3. Global Properties of the Solution

In this section, we elucidate that the three models (12), (15), and (1) have the same qualitative properties. To do that, we study the asymptotic behaviour of the solutions of (1).

Lemma 1.

For any solution, there exists n¯N+ such that xn0 for nn¯.

Proof.

Let us assume that xnc for some n0. Then(18)maxxn,b+xn-ω=xn-c+maxc,-xn+c+b+xn-ωxn-c+maxc,b+xn-ω.Using this estimation in (1), we get(19)xn+1xn-c+maxc,b+xn-ω-maxc,b+xn-ω=xn-c.This implies that xn is decreasing with respect to n as long as xnc. Therefore, there exists k such that xk-1c and xk<c. Then from (1) with n=k, it follows that(20)xk+1maxc,b+xk-ω-maxc,b+xk-ω=0.Inductively we get that xm0 for all mk+1=n¯.

From Lemma 1, without loss of generality, we can set the initial condition as(21)x-j0,j0,1,,ω.Note that Lemma 1 implies that(22)xRx0is an invariant set.

To discuss global attractivity of equilibria of the scalar difference equation (1), it seems to be convenient to consider an equivalent two-dimensional system. From (22) and c>0, one has(23)b+xn-ωxn-c+b+xn-ω,and then we can write(24)maxxn,b+xn-ω=maxxn,xn-c+b+xn-ω,b+xn-ω.Now we define wn+1=-max0,-c+b+xn-ω; then from (1) and (24) one has(25)xn+1=maxxn,xn-c+b+xn-ω,b+xn-ω+wn+1-c=maxxn-wn+1,b+xn-ω+wn+1-c=maxxn,b+xn-ω+wn+1-c,where we use wn+10 in (22). Therefore we can get the following system:(26a)wn+1=-max0,-c+b+xn-ω,(26b)xn+1=maxxn,b+xn-ω+wn+1-c.The initial condition is given as (22).

To discuss global attractivity of equilibria, we now consider (26a) and (26b) in the set given as in (22).

Theorem 2.

Let one assume that b<c holds. Then(27)wn=0for  n1,limnxn=-.

Proof.

Since for any n0 one has that -c+b+xn-ωb-c<0 from Lemma 1, it follows that(28)wn+1=-max0,-c+b+xn-ω=0.Therefore it follows that wn+1=0 for any n0 from (26a). From (26b), we get(29)xn+1=maxxn-c,xn-ω+b-c.

Let(30)x^mmax0jωxmω+1-jfor  mN+.Note that xm(ω+1)-jx^m for j0,1,,ω. We show that(31)x^m+1x^m+max-c,b-c.From (29), we have(32)xm+1ω+1-ω=xmω+1+1=maxxmω+1-c,xmω+1-ω+b-cmaxx^m-c,x^m+b-c=x^m+max-c,b-c.

For some j1,2,,ω, suppose that(33)xm+1ω+1-jx^m+max-c,b-c.Then using (29) and (33), we obtain(34)xm+1ω+1-j+1=maxxm+1ω+1-j-c,xmω+1-j+1+b-cx^m+max-c,b-c.By mathematical induction, it holds that x(m+1)(ω+1)-jx^m+max-c,b-c for any j0,1,,ω. Therefore we get(35)x^m+1x^m+max-c,b-c.Now it is obvious that limmx^m=- and hence limnxn=-. We thus obtain the conclusion.

If b>c, wn and xn converge to a unique equilibrium. First we show that system (26a) and (26b) has a nontrivial equilibrium.

Proposition 3.

Let one assume that b>c holds. Then system (26a) and (26b) has an equilibrium -b+c,0.

Proof.

Let b>c>0 holds. We show that system (26a) and (26b) has the constant solution -b+c,0. From direct computations, one can see(36)-max0,-c+b=-b+c,max0,c-c=0.

Proposition 4.

Let one assume that b>c holds. It follows that(37)xn+1=0if  xn-ω-b+cmaxxn-c,xn-ω+b-cif  xn-ω<-b+c.

Proof.

Assume that xn-ω-b+c. Then it is straightforward to get wn+1=c-b-xn-ω from (26a). Since we have xn0 (see Lemma 1), we get(38)xn+1=maxxn,c-c=0.On the other hand, assume that xn-ω<-b+c. Then wn+1=0 follows from (26a). Thus we immediately obtain the conclusion from (26b) with wn+1=0.

We now show that every solution converges to the nontrivial equilibrium.

Theorem 5.

Let us assume that b>c. Then(39)limnwn=-b+c,limnxn=0.

Proof.

Let(40)xlxlω+1,xlω+1-1,,xlω+1-ωfor lN+. From Proposition 4, one can see that(41)xlω+1-kb-c+xl-1ω+1-kfor k0,1,2,,ω if x(l-1)(ω+1)-k<-b+c. Therefore,(42)limlxlω+1-k=0,k0,1,2,,ω;that is, each component of xl converges to the equilibrium as l. Then, there exists a sufficiently large integer m such that xn==xn-ω=0 for nm. For nm, we obtain(43)wn+1=-max0,-c+b=-b+c.

Theorems 2 and 5 show that b<c and b>c are, respectively, the criteria of the global divergence to - and the convergence to the unique equilibrium. Equation (1) also has the threshold dynamics as in (12) and (15).

For Y=y1,y2,,yω+1Rω+1, we define Y=j=1ω+1yj2. Let(44)Xnxn,xn-1,,xn-ω.Here we show that the equilibrium is stable. Assume that xn-ω>0. If Xn<c, then from (1) we obtain xn+1=0. Thus(45)XnXn+1follows. On the other hand, assume that xn-ω0. If Xn<b-c, then, from Proposition 4, we obtain xn+1=0. Thus (45) follows. Consequently, if Xn<minc,b-c, then we obtain (45). Thus the equilibrium is stable in Rω+1.

In Figure 1, we plot xn with respect to n. For ω=6, the initial condition is chosen as(46)x-6,x-5,x-4,x-3,x-2,x-1,x0=-10,-11,-20,-13,-20,-15,-16.We set the parameters as b=1 and c=3 in Figure 1(a), while b=5 and c=2 in Figure 1(b). As in Theorems 2 and 5, one can see that x tends to - as n in Figure 1(a) and that x tends to 0 as n in Figure 1(b).

Numerical illustration of a solution behaviour (ω=6).

b < c

b > c

4. Conclusion

In this paper, we consider an ultradiscrete model with time delay. In Theorems 2 and 5, we show that the ultradiscrete model also has the threshold property concerning global attractivity of equilibria, similar to the difference equation  and differential equation . For the proof of global attractivity of the nontrivial equilibrium in Theorem 5, we reduce system (26a) and (26b) to the scalar difference equation in Proposition 4 and then use a certain monotone property of the solution.

In a different study, the scalar difference equation system (26a) and (26b) also appears, where we derive an ultradiscrete model from an SIR type epidemic model. In the SIR type epidemic model, no reinfection is assumed after the recovery , differently from the assumption of the SIS type epidemic model. Although those model structures are different, we encounter the same difference equation system (26a) and (26b) in the ultradiscrete level. The implication shall be explored in the future study.

In this paper, we study qualitative properties of the ultradiscrete model (1). In , it is shown that simple ultradiscrete models can capture disease transmission dynamics. Cellular automata have been used to model complex phenomena including disease transmission dynamics. Since cellular automata are computational models, in general, it is not straightforward to perform a mathematical analysis, in order to provide theoretical basis for the simulation studies. Our analytical study for the ultradiscrete model could be complement for numerical simulation studies for some cellular automaton models.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The second author was supported by JSPS Grant-in-Aid for Scientific Research (C) JP26400212. The third author was supported by JSPS Grant-in-Aid for Young Scientists (B) 16K20976.

Tokihiro T. Takahashi D. Matsukidaira J. Satsuma J. From soliton equations to integrable cellular automata through a limiting procedure Physical Review Letters 1996 76 18 3247 3250 2-s2.0-0347518863 10.1103/PhysRevLett.76.3247 Kent C. M. Toni B. Williamson K. Ghariban N. Haile D. Xie Z. Piecewise-defined difference equations: open problem 24 Bridging Mathematics, Statistics, Engineering and Technology 2012 New York, NY, USA Springer Springer Proceedings in Mathematics & Statistics Tikjha W. Lapierre E. Sitthiwirattham T. The stable equilibrium of a system of piecewise linear difference equations Advances in Difference Equations 2017 10.1186/s13662-017-1117-2 MR3615156 Nagai A. Tokihiro T. Satsuma J. Conserved quantities of box and ball system Glasgow Mathematical Journal 2001 43A 91 97 10.1017/S0017089501000088 MR1869688 Zbl1045.37046 2-s2.0-33747577436 Nishinari K. Takahashi D. Analytical properties of ultradiscrete Burgers equation and rule-184 cellular automaton Journal of Physics A 1998 31 24 5439 5450 10.1088/0305-4470/31/24/006 Zbl0981.37028 2-s2.0-0032546711 Nishinari K. Takahashi D. A new deterministic CA model for traffic flow with multiple states Journal of Physics A 1999 32 1 93 104 10.1088/0305-4470/32/1/010 Zbl0917.90126 2-s2.0-0033534271 Matsuya K. Kanai M. Exact solution of a delay difference equation modeling traffic flow and their ultra-discrete limit https://arxiv.org/abs/1509.07861 Chatterjee E. On the global character of the solutions of Xn + 1 = (α + βXn + γXn − k)/(A + Xn − k) International Journal of Applied Mathematics 2013 26 1 9 17 MR3098189 Zhang C. Li H.-X. Dynamics of a rational difference equation of higher order Applied Mathematics E-Notes 2009 9 80 88 MR2496034 Zbl1241.39008 Cooke K. L. Stability analysis for a vector disease model The Rocky Mountain Journal of Mathematics 1979 9 1 31 42 10.1216/RMJ-1979-9-1-31 MR517971 Zbl0423.92029 2-s2.0-0003100558 Mickens R. E. Discretizations of nonlinear differential equations using explicit nonstandard methods Journal of Computational and Applied Mathematics 1999 110 1 181 185 10.1016/S0377-0427(99)00233-2 MR1715559 Zbl0940.65079 2-s2.0-0033338885 Anguelov R. Lubuma J. M. Shillor M. Dynamically consistent nonstandard finite difference schemes for continuous dynamical systems Discrete and Continuous Dynamical Systems. Series A 2009 34 43 MR2641378 Roeger L. W. Dynamically consistent discrete-time SI and SIS epidemic models Discrete and Continuous Dynamical Systems. Series A 2013 653 662 10.3934/proc.2013.2013.653 MR3462410 Willox R. Modelling natural phenomena with discrete and ultradiscrete systems Proceedings of the RIAM Symposium Held at Chikushi Campus, Kyushu Universiy 22AO-S8 13 22 Morisita M. The fitting of the logistic equation to the rate of increase of population density Researches on Population Ecology 1965 7 1 52 55 2-s2.0-51249194763 10.1007/BF02518815 Elaydi S. An Introduction t o Difference Equations 2005 New York, NY, USA Springer MR2128146 Zbl1071.39001 Willox R. Grammaticos B. Carstea A. S. Ramani A. Epidemic dynamics: discrete-time and cellular automaton models Physica A 2003 328 1-2 13 22 10.1016/S0378-4371(03)00552-1 MR2012462 Satsuma J. Willox R. Ramani A. Grammaticos B. Carstea A. S. Extending the SIR epidemic model Physica A 2004 336 3-4 369 375 10.1016/j.physa.2003.12.035 2-s2.0-1642632895 Ramani A. Carstea A. S. Willox R. Grammaticos B. Oscillating epidemics: a discrete-time model Physica A 2004 333 1-4 278 292 10.1016/j.physa.2003.10.051 MR2100220 Cooke K. van den Driessche P. Zou X. Interaction of maturation delay and nonlinear birth in population and epidemic models Journal of Mathematical Biology 1999 39 4 332 352 10.1007/s002850050194 MR1727839 Zbl0945.92016 2-s2.0-0033209987 Huang G. Liu A. Forys U. Global stability analysis of some nonlinear delay differential equations in population dynamics Journal of Nonlinear Science 2016 26 1 27 41 10.1007/s00332-015-9267-4 MR3441272 Ruan S. Delay differential equations in single species dynamics Delay Differential Equations and Applications 2006 205 477 517 10.1007/1-4020-3647-7_11 MR2337823 Kulenović M. R. Ladas G. Prokup N. R. A rational difference equation Computers & Mathematics with Applications 2001 41 5-6 671 678 10.1016/S0898-1221(00)00311-4 MR1822594 Camouzis E. Ladas G. Dynamics of third-order rational difference equations with open problems and conjectures 2008 5 Boca Raton, FL, USA Chapman & Hall/CRC Press Advances in Discrete Mathematics and Applications MR2363297 Li W.-T. Sun H.-R. Dynamics of a rational difference equation Applied Mathematics and Computation 2005 163 2 577 591 10.1016/j.amc.2004.04.002 MR2121812 2-s2.0-13544252478 Kocić V. L. Ladas G. Global attractivity in nonlinear delay difference equations Proceedings of the American Mathematical Society 1992 115 4 1083 1088 10.2307/2159359 MR1100657 Hethcote H. W. Qualitative analyses of communicable disease models Mathematical Biosciences 1976 28 3/4 335 356 10.1016/0025-5564(76)90132-2 MR0401216 Zbl0326.92017 2-s2.0-0017228276