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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.

Ultradiscretization is proposed as a procedure to obtain a discrete system, where unknown variables also take discretized values [

In this paper, we consider the following difference equation:

We derive the difference equation (

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 [

The paper is organized as follows. In Section

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

We start with a logistic equation:

By an applied discretization [

The author in [

Following [

An epidemic model considered in [

In order to ensure positivity of the solution in discrete analogues of the differential equation (

Let us now derive the difference equation (

In this section, we elucidate that the three models (

For any solution, there exists

Let us assume that

From Lemma

To discuss global attractivity of equilibria of the scalar difference equation (

To discuss global attractivity of equilibria, we now consider (

Let one assume that

Since for any

Let

For some

If

Let one assume that

Let

Let one assume that

Assume that

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

Let us assume that

Let

Theorems

For

In Figure

Numerical illustration of a solution behaviour (

In this paper, we consider an ultradiscrete model with time delay. In Theorems

In a different study, the scalar difference equation system (

In this paper, we study qualitative properties of the ultradiscrete model (

The authors declare that they have no conflicts of interest.

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.

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