# Global exponential stability for a delay differential neoclassical growth model

- Wei Chen
^{1}and - Wentao Wang
^{2}Email author

**2014**:325

https://doi.org/10.1186/1687-1847-2014-325

© Chen and Wang; licensee Springer. 2014

**Received: **30 September 2014

**Accepted: **5 December 2014

**Published: **16 December 2014

## Abstract

This paper aims to analyze the stability of the positive equilibrium of a delay differential neoclassical growth model. We prove the existence, positivity and permanence of solutions which help to deduce the global exponential stability of the unique positive equilibrium for this model. Our method relies upon the differential inequality technique and the Lyapunov functional. Moreover, we give an example with numerical simulations to demonstrate theoretical results.

## Keywords

## 1 Introduction

*et al.*[7]. Due to the principle of economics, to describe the long-run behavior of the economy, the neoclassical growth model is based on the two assumptions: one is the full-employment of labor and capital and the other is instantaneous adjustment in the output market. However, it is unreasonable in real world since nonlinearities of production functions and a production delay are inevitable. Matsumoto and Szidarovszky [8] have tried to overcome the above-mentioned difficulties and have provided a special neoclassical model with a mound-shaped production function

where *x* is the capital per labor, $s\in (0,1)$ is the average propensity to save and $\alpha =n+s\mu $ with *μ* being the depreciation ratio of capital and *n* being the growth rate of labor; the mound-shaped production function $F(x)=A{x}^{a}{(1-x)}^{b}$ (*a*, *b* and *A* are positive parameters) is of Cobb-Douglas type, and the factor ${(1-x)}^{b}$ reflects the influence of pollution on *per capita* output.

where *α*, *γ*, *δ*, $\beta =s\epsilon $ are positive parameters, *τ* is the delay in the production process, *γ* can be thought of as a proxy for measuring returns to scale of the production function, *δ* reflects a strength of a ‘negative effect’ caused by increasing concentration of capital and is determined by a damaging degree of natural environment or energy resources. From the analysis of [9], when $\gamma >1$, model (1.2) has one (two) positive equilibrium(s) provided $g(\frac{\gamma -1}{\delta})=0$ (>0), where $g(x)=-\alpha +\beta {x}^{\gamma -1}{e}^{-\delta x}$, and when $0<\gamma <1$, model (1.2) has a unique positive equilibrium. In particular, if $\gamma =1$, then model (1.2) is well-known Nicholson’s blowflies model [10] and it has a unique positive equilibrium under the condition $\beta >\alpha $. Recently, there have been a lot of results on Nicholson’s blowflies model, and we refer readers to [11–19] and the references cited therein for more details. However, to the best of our knowledge, there is not much work on model (1.2) when $\gamma \ne 1$. Although they have considered the local asymptotical stability of the positive equilibrium of model (1.1) and (1.2) in [8, 9], the results on the permanence and global stability of the delay differential neoclassical model do not appear heretofore.

Motivated by the above discussions, in this paper we mainly study the global exponential stability of the unique positive equilibrium of model (1.2) with $\gamma \in (0,1)$.

Throughout this paper, let ${R}^{+}$ denote a nonnegative real number space, $C=C([-\tau ,0],R)$ be the Banach space of the set of all continuous functions from $[-\tau ,0]$ to *R* equipped with supremum norm $\parallel \cdot \parallel $ and ${C}_{+}=C([-\tau ,0],{R}^{+})$. Furthermore, for a continuous function *x* defined on $[-\tau ,\sigma )$ with $\sigma >0$, we define ${x}_{t}\in C$ by ${x}_{t}(\theta )=x(t+\theta )$ for $\theta \in [-\tau ,0]$.

Then *f* is a locally Lipschitz map with respect to $\phi \in {C}_{+}$, which ensures the existence and uniqueness of the solution of (1.2) with admissible initial condition (1.4).

We write ${x}_{t}(\phi )$ (*i.e.* $x(t;\phi )$) for an admissible solution of admissible initial value problem (1.2) and (1.4). Also, let $[0,\eta (\phi ))$ be the maximal right interval of the existence of ${x}_{t}(\phi )$.

An outline of this paper is as follows. We devote Section 2 to some lemmas and definitions on the existence, positivity and permanence of solutions, which play an important role in Section 3 to establish the global exponential stability of the positive equilibrium. In Section 4, an example and its numerical simulations are provided to illustrate our results obtained in the previous sections. The last section concludes the paper.

## 2 Preliminary results

In this section, we present some lemmas and definitions to prove our main results in Section 3.

**Definition 2.1**Let $\overline{x}$ be the unique positive equilibrium of model (1.2). If there exist constants $\lambda >0$, $\tilde{K}>0$ and $T>0$ such that every solution $x(t;\phi )$ to the initial value problem (1.2) and (1.4) always satisfies

then $\overline{x}$ is said to be globally exponentially stable.

**Definition 2.2**All solutions are uniformly permanent if there exist positive constants

*m*and

*M*such that for any solution $x(t)$, we have

**Lemma 2.1** *There exists a unique positive global solution of model* (1.2) *and* (1.4) *on the interval* $[0,+\mathrm{\infty})$.

*Proof*Because of $\phi \in {C}_{+}$, we have ${x}_{t}(\phi )\in {C}_{+}$ by using Theorem 5.2.1 in [20]. Set $x(t)=x(t;\phi )$ for $t\in [0,\eta (\phi ))$. From the variation of constants formula and initial condition $\phi (0)>0$, we obtain

for all $t\in [0,\eta (\phi ))$.

It remains to show $\eta (\phi )=+\mathrm{\infty}$.

which excludes the possibility that ${lim}_{t\to \eta {(\phi )}^{-}}x(t)=\mathrm{\infty}$. Hence it violates Theorem 2.3.1 in [21]. So we obtain the existence of the unique global positive solution of (1.2) and (1.4) on $[0,+\mathrm{\infty})$. Therefore Lemma 2.1 is proved. □

**Lemma 2.2**

*Suppose that there exists a positive constant*$K\in (\kappa ,\tilde{\kappa})$

*such that*

*then solutions of*(1.2)

*and*(1.4)

*are uniformly permanent with*

*Proof*Let $x(t)=x(t;\phi )$. By Lemma 2.1, $x(t)>0$ for $t\ge 0$. From Theorem 1.6.1 in [22] and (2.1), we get that the solution $x(t)$ to the initial value problem (1.2) and (1.4) is not greater than the solution to the initial value problem

which obviously contradicts with (2.2). Thus, we have proved that $l>0$.

which contradicts with (2.2). This proves that $l>\frac{\kappa}{\delta}$. Hence the proof of Lemma 2.2 is completed. □

## 3 Main results

In this section, we establish sufficient conditions on the global exponential stability for the unique positive equilibrium of model (1.2).

**Theorem 3.1**

*Suppose that all conditions in Lemma*2.2

*are satisfied and assume that*

*Then the unique positive equilibrium*$\overline{x}$

*of model*(1.2)

*is globally exponentially stable*,

*that is*,

*there exist constants*$\lambda >0$, $\tilde{K}>0$

*and*$T>0$

*such that*

*Proof*Define a continuous function $\mathrm{\Gamma}(u)$ as

This completes the proof of Theorem 3.1. □

## 4 An example

In this section, we give an example and its numerical simulations in order to support the results obtained in previous sections.

**Example 4.1**Consider the following delay differential neoclassical growth model:

Obviously, $\alpha =2.8$, $\beta =4$, $\gamma =0.5$, $\delta ,\tau =1$.

**Remark 4.1** To the best of our knowledge, few authors have considered the global exponential stability of the unique positive equilibrium for model (1.2). It is clear that all the results in [8, 9] and the references therein cannot be applicable to prove that all solutions to model (4.1) with initial value (1.4) converge exponentially to the positive equilibrium. So the results of this paper are essentially new and complement the work of Matsumoto and Szidarovszky [8, 9].

## 5 Conclusion

In the paper, we have studied a special delay differential neoclassical model. A mound-shaped production for capital growth and the delay in the production process are assumed in the dynamic equation. We have established some criteria to ensure the global exponential stability of the unique positive equilibrium for model (1.2) with $\gamma \in (0,1)$. In the future, we will investigate model (1.2) with $\gamma >1$ since it has more complex dynamic behavior and there possibly exist two positive equilibria.

## Declarations

### Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11301341, 11201184) and Innovation Program of Shanghai Municipal Education Commission (Grant No. 13YZ127).

## Authors’ Affiliations

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