Abstract
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.
1. Introduction
Ultradiscretization is proposed as a procedure to obtain a discrete system, where unknown variables also take discretized values [1]. 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 [5–7] for the application of ultradiscretization to the traffic flow.
In this paper, we consider the following difference equation:where is a real number and is a real positive number, with the following initial condition:
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 [10]. 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 [11]. 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 [10].
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 [14]. It is known that the solution of the discrete model exactly follows the continuous solution of the logistic equation [15]. 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 [14] 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 [10] 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:where and 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, , is globally asymptotically stable if , while the positive equilibrium, , is globally asymptotically stable if .
By an applied discretization [16], the following discrete analogue can be derived from the logistic equation (3):Let be a sufficiently small step size. Then the parameters and are related to and via and . The difference equation (4) captures the continuous solution of the differential equation (3); that is, the solution shows the logistic curve [15]. See also [14].
The author in [14] obtains the following piecewise linear difference equation from (4) by ultradiscretization:where and are constants that satisfied . Ultradiscretization is proposed as a procedure to obtain the discrete system, where unknown variables also take discretized values [1]. In [14], it is shown that the three models (3), (4), and (5) share the qualitative property that every solution converges to an equilibrium.
Following [14, 17–19], let us derive (5) from (4). For , we introduce a variable viaand parameters and throughThen we havethusLetting and assuming , we get (5) by the following manipulations:The key relation used here is the following limit:for .
An epidemic model considered in [10] is an extension of the logistic equation (3). The model is formulated as the following delay differential equation:where 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, , is globally asymptotically stable if , while the positive equilibrium, , is globally asymptotically stable if . Different logistic equations with time delay have the instability property; see [20–22].
In order to ensure positivity of the solution in discrete analogues of the differential equation (11), we use Mickens nonstandard finite difference scheme [11] to discretize (12) as follows:where is a step size. Equation (13) can be written by the following explicit form:thus (13) is equivalently written as the following difference equation with , , and :where and are positive constants and is a nonnegative integer. It is obvious that the delay equation (15) is reduced to (4) when . Equation (15) is a special case of the model considered in [8, 9]. For some specific and general cases, the authors in [23–26] show global asymptotic stabilities of the zero and positive equilibria. The zero equilibrium of (15) is globally asymptotically stable when . The unique equilibrium of (15) is globally asymptotically stable when . 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 , we introduce the variable and the parameters and in the same way as the derivation of (5); then we getLetting and using the key relation (11), we get (1). Finally, we note that (5) is a special case of (1). In fact, let and in (5). ThenIn 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 such that for .
Proof. Let us assume that for some . ThenUsing this estimation in (1), we getThis implies that is decreasing with respect to as long as . Therefore, there exists such that and . Then from (1) with , it follows thatInductively we get that for all .
From Lemma 1, without loss of generality, we can set the initial condition asNote that Lemma 1 implies thatis 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 , one hasand then we can writeNow we define ; then from (1) and (24) one haswhere we use in (22). Therefore we can get the following system: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 holds. Then
Proof. Since for any one has that from Lemma 1, it follows thatTherefore it follows that for any from (26a). From (26b), we getLetNote that for . We show thatFrom (29), we haveFor some , suppose thatThen using (29) and (33), we obtainBy mathematical induction, it holds that for any . Therefore we getNow it is obvious that and hence . We thus obtain the conclusion.
If , and converge to a unique equilibrium. First we show that system (26a) and (26b) has a nontrivial equilibrium.
Proposition 3. Let one assume that holds. Then system (26a) and (26b) has an equilibrium .
Proof. Let holds. We show that system (26a) and (26b) has the constant solution . From direct computations, one can see
Proposition 4. Let one assume that holds. It follows that
Proof. Assume that . Then it is straightforward to get from (26a). Since we have (see Lemma 1), we getOn the other hand, assume that . Then follows from (26a). Thus we immediately obtain the conclusion from (26b) with .
We now show that every solution converges to the nontrivial equilibrium.
Theorem 5. Let us assume that . Then
Proof. Letfor . From Proposition 4, one can see thatfor if . Therefore,that is, each component of converges to the equilibrium as . Then, there exists a sufficiently large integer such that for . For , we obtain
Theorems 2 and 5 show that and 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 , we define . LetHere we show that the equilibrium is stable. Assume that . If , then from (1) we obtain . Thusfollows. On the other hand, assume that . If , then, from Proposition 4, we obtain . Thus (45) follows. Consequently, if , then we obtain (45). Thus the equilibrium is stable in .
In Figure 1, we plot with respect to . For , the initial condition is chosen asWe set the parameters as and in Figure 1(a), while and in Figure 1(b). As in Theorems 2 and 5, one can see that tends to as in Figure 1(a) and that tends to as in Figure 1(b).
(a)
(b)
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 [8] and differential equation [10]. 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 [27], 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 [19], 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.