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Archived Comments for: Global exponential stability for a delay differential neoclassical growth model

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  1. 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) = βxγ(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

                                                        βxγe−δx = αx      or      x1−γeδx =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, √xex = 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).

    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

    No Competing Interests

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