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Global exponential stability for a delay differential neoclassical growth model
Advances in Difference Equations volume 2014, Article number: 325 (2014)
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.
To explain the emergence of complex behavior even under simple economic structure, Day [1, 2] has seminally studied a neoclassical growth model and a productivity and population growth model which are the generalization and development of the early works of Solow  and Swan . Tracing Day’s pioneering works [1, 2], a great deal of effort in nonlinear economic dynamics was presented in Day , Puu  and Bischi et al. . 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  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, is the average propensity to save and with μ being the depreciation ratio of capital and n being the growth rate of labor; the mound-shaped production function (a, b and A are positive parameters) is of Cobb-Douglas type, and the factor reflects the influence of pollution on per capita output.
Following their own works, Matsumoto and Szidarovszky  adopt the production function and consider the time delay inherent in the production process, then the delay differential neoclassical growth model is proposed as follows:
where α, γ, δ, 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 , when , model (1.2) has one (two) positive equilibrium(s) provided (>0), where , and when , model (1.2) has a unique positive equilibrium. In particular, if , then model (1.2) is well-known Nicholson’s blowflies model  and it has a unique positive equilibrium under the condition . 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 . 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 .
At the outset of our results, we introduce two positive constants that will be crucial. Since the function is decreasing with the range and increasing with the range , it is easy to see that there exists a unique such that
Furthermore, since increases on and decreases on , let be the unique number in such that
Throughout this paper, let denote a nonnegative real number space, be the Banach space of the set of all continuous functions from to R equipped with supremum norm and . Furthermore, for a continuous function x defined on with , we define by for .
It is economically reasonable to assume that only positive solutions of model (1.2) are meaningful and therefore admissible. Thus we consider admissible initial conditions and it is convenient to choose ‘0’ as initial time for autonomous model (1.2):
Define a continuous map by setting
Then f is a locally Lipschitz map with respect to , which ensures the existence and uniqueness of the solution of (1.2) with admissible initial condition (1.4).
We write (i.e. ) for an admissible solution of admissible initial value problem (1.2) and (1.4). Also, let be the maximal right interval of the existence of .
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 be the unique positive equilibrium of model (1.2). If there exist constants , and such that every solution to the initial value problem (1.2) and (1.4) always satisfies
then 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 , we have
Lemma 2.1 There exists a unique positive global solution of model (1.2) and (1.4) on the interval .
Proof Because of , we have by using Theorem 5.2.1 in . Set for . From the variation of constants formula and initial condition , we obtain
for all .
It remains to show .
For the sake of contradiction, assume that is bounded. Note the fact that , we obtain from (1.2) that
This leads to
which excludes the possibility that . Hence it violates Theorem 2.3.1 in . So we obtain the existence of the unique global positive solution of (1.2) and (1.4) on . Therefore Lemma 2.1 is proved. □
Lemma 2.2 Suppose that there exists a positive constant such that
then solutions of (1.2) and (1.4) are uniformly permanent with
Proof Let . By Lemma 2.1, for . From Theorem 1.6.1 in  and (2.1), we get that the solution to the initial value problem (1.2) and (1.4) is not greater than the solution to the initial value problem
In view of (2.2) and
We next show that . Otherwise, we assume that . For each , we define
It follows from that as and that
From the definition of , we know that or
which implies that
This yields that there exists such that
In view of (2.3), (2.4) and , for , we have
which obviously contradicts with (2.2). Thus, we have proved that .
Finally, we prove that . Again, by way of contradiction, we assume that . By the fluctuation lemma [, Lemma A.1], there exists a sequence such that
Since is bounded and equicontinuous, by the Ascoli-Arzelá theorem, there exists a subsequence, still denoted by itself for simplicity of notation, such that
Then recall that , and it follows from
that (taking limits)
which contradicts with (2.2). This proves that . 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 of model (1.2) is globally exponentially stable, that is, there exist constants , and such that
Proof Define a continuous function as
From (3.1), we obtain
which implies that there exist two constants and such that
For convenience of notation, denote by . Lemma 2.2 implies that there exists such that
Moreover, since the positive equilibrium is the special solution of model (1.2), Lemma 2.2 also implies that
Set , . For , we have
Next we consider the Lyapunov functional
For all , calculating the upper left derivative of along with the solution of (3.5), we have
Now, we claim that
If this is not true, then there must exist such that
Then (3.2), (3.3), (3.4) and (3.7) yield
which is a contradiction. Here we have applied the inequality
which can be easily obtained by (1.3) and the differential mean value theorem. Hence, (3.8) holds and
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, , , , .
Note that and . Let . Then
Hence, from Theorem 3.1, the unique positive equilibrium of model (4.1) is globally exponentially stable with exponential convergence rate . The numerical simulations in Figure 1 strongly support this conclusion.
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].
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 . In the future, we will investigate model (1.2) with since it has more complex dynamic behavior and there possibly exist two positive equilibria.
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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).
The authors declare that they have no competing interests.
The authors declare that the study was realized in collaboration with the same responsibility. Both authors read and approved the final version of the manuscript.
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