Correction of the error in the paper: "Global exponential stability for a delay differential neoclassical growth model" by Wei Chen and Wentao Wang Leonid Shaikhet Department of Higher Mathematics, Donetsk State University of Management, Chelyuskintsev 163-a, 83015 Donetsk, Ukraine, Email: leonid.shaikhet@usa.net

Leonid Shaikhet, Donetsk State University of Management

21 January 2015

In the paper [1] the global exponential stability of the unique positive equilibrium of the delay differential neoclassical growth model

ẋ(t) = βx^{γ}(t − τ)e^{−δx(t−τ)} − αx(t) (1)

is studied. Here α, β, γ, δ are positive parameters, τ ≥ 0.

The model (1) has a very important applications in economics and is very popular in researches (see, for instance, [2,3]). In particular, for γ = 1 the model (1) is well-known Nicholson's blowflies model (see [1, 4, 5] and the references therein). So, it is very important to avoid errors and inaccuracies in publications.

It is known [1, 3] that if γ ∈ (0, 1) then the model (1) has the unique positive equilibrium. This equilibrium is globally exponentially stable and is defined by the algebraic equation

βx^{γ}e^{−δx} = αx or x^{1−γ}e^{δx} =β/α. (2)

In [1] the numerical simulation of the model (1) is given with the values of the parameters

α = 2.8, β = 4, γ = 0.5, δ = 1, τ = 1. (3)

It is easy to see that the equation (2) with the parameters (3) takes the form, √xe^{x} = 4/2.8 = 1.42857 and has the solution X = 0.606610.

But in the paper [1] another result is mistakenly given: X = 0.802906. Besides in the given in [1] Figure 1 the different solutions of the equation (1), (3) with the different initial functions

ϕ(s) = 0.1, ϕ(s) = 1, ϕ(s) = 3, s ∈ [−1.0]. (4)

converge to the error equilibrium X=0.802906 Maybe here mistakenly the numerical simulation of some another situation is shown.

The right convergence of the solutions of the equation (1), (3) with the initial functions (4) to the equilibrium X = 0.606610 is shown below in Figure 1: Numerical solutions x(t)to the model (1), (3) with the initial functions (4).

1. Chen W, Wang W: Global exponential stability for a delay differential neoclassical growth model. Advances in Difference Equations 2014, 2014:325

2. Matsumoto, A, Szidarovszky, F: Delay differential neoclassical growth model. J. Econ. Behav. Organ. {\bf 78}, 272-289 (2011)

3. Matsumoto, A, Szidarovszky, F: Asymptotic behavior of a delay differential neoclassical growth model. Sustainability, {\bf 5}, 440-455 (2013)

4. Shaikhet L: Lyapunov Functionals and Stability of Stochastic Difference Equations. Springer, London, Dordrecht, Heidelberg, New York (2011)

5. Shaikhet L: Lyapunov functionals and stability of stochastic functional differential equations. Springer, Dordrecht, Heidelberg, New York, London (2013)

## Correction of the error in the paper: "Global exponential stability for a delay differential neoclassical growth model" by Wei Chen and Wentao Wang Leonid Shaikhet Department of Higher Mathematics, Donetsk State University of Management, Chelyuskintsev 163-a, 83015 Donetsk, Ukraine, Email: leonid.shaikhet@usa.net

Leonid Shaikhet, Donetsk State University of Management

21 January 2015

In the paper [1] the global exponential stability of the unique positive equilibrium of the delay differential neoclassical growth model

ẋ(t) = βx

^{γ}(t − τ)e^{−δx(t−τ)}− αx(t) (1)is studied. Here α, β, γ, δ are positive parameters, τ ≥ 0.

The model (1) has a very important applications in economics and is very popular in researches (see, for instance, [2,3]). In particular, for γ = 1 the model (1) is well-known Nicholson's blowflies model (see [1, 4, 5] and the references therein). So, it is very important to avoid errors and inaccuracies in publications.

It is known [1, 3] that if γ ∈ (0, 1) then the model (1) has the unique positive equilibrium. This equilibrium is globally exponentially stable and is defined by the algebraic equation

βx

^{γ}e^{−δx}= αx or x^{1−γ}e^{δx}=β/α. (2)In [1] the numerical simulation of the model (1) is given with the values of the parameters

α = 2.8, β = 4, γ = 0.5, δ = 1, τ = 1. (3)

It is easy to see that the equation (2) with the parameters (3) takes the form,

√xeand has the solution^{x}= 4/2.8 = 1.42857X = 0.606610.But in the paper [1] another result is mistakenly given:

X = 0.802906. Besides in the given in [1] Figure 1 the different solutions of the equation (1), (3) with the different initial functionsϕ(s) = 0.1, ϕ(s) = 1, ϕ(s) = 3, s ∈ [−1.0]. (4)

converge to the error equilibrium X=0.802906 Maybe here mistakenly the numerical simulation of some another situation is shown.

The right convergence of the solutions of the equation (1), (3) with the initial functions (4) to the equilibriumto the model (1), (3) with the initial functions (4).

X = 0.606610is shown below in Figure 1: Numerical solutionsx(t)Please click here to access Figure 1: https://backup.filesanywhere.com/FS/v.aspx?v=8d696288605f70aa71a1

References

1. Chen W, Wang W: Global exponential stability for a delay differential neoclassical growth model. Advances in Difference Equations 2014, 2014:325

2. Matsumoto, A, Szidarovszky, F: Delay differential neoclassical growth model. J. Econ. Behav. Organ. {\bf 78}, 272-289 (2011)

3. Matsumoto, A, Szidarovszky, F: Asymptotic behavior of a delay differential neoclassical growth model. Sustainability, {\bf 5}, 440-455 (2013)

4. Shaikhet L: Lyapunov Functionals and Stability of Stochastic Difference Equations. Springer, London, Dordrecht, Heidelberg, New York (2011)

5. Shaikhet L: Lyapunov functionals and stability of stochastic functional differential equations. Springer, Dordrecht, Heidelberg, New York, London (2013)

## Competing interests