Logistic equation with continuously distributed lag and application in economics

  • Vasily E. TarasovEmail author
  • Valentina V. Tarasova
Original Paper


In this paper, we consider nonlinear dynamics with continuously distributed lags. A generalization of the logistic equation, its solution and economic models of logistic growth are proposed by taking into account continuously distributed lags. The logistic integro-differential equations are considered for exponential and gamma distributions of delay time. The integro-differential equations of the proposed model of logistic growth with distributed lag are represented by differential equations with derivatives of integer orders. The solution of the logistic integro-differential equations with exponentially distributed lag is obtained. Characteristic properties of nonlinear dynamics with continuously distributed lags are described. The main difference between dynamics with lag from standard dynamics without delay lies in the existence of a cutoff threshold of growth. We propose the principle of growth clipping by distributed lag, which states that the distributed lag can lead to the emergence of the cutoff threshold, below which growth is replaced by decline. For economy, this means that for production growth, the starting production should exceed a certain minimum (critical) value of production.


Logistic equation Translation operator Distributed lag Memory Integro-differential operator Probability distribution Exponential distribution Gamma distribution 

Mathematics Subject Classification

91B55 Economic dynamics 47G20 Integro-differential operators 26A33 Fractional derivatives and integrals 


Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Skobeltsyn Institute of Nuclear PhysicsLomonosov Moscow State UniversityMoscowRussia
  2. 2.Faculty of EconomicsLomonosov Moscow State UniversityMoscowRussia
  3. 3.YandexMoscowRussia

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