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Extended Solow–Swan model for capital and natural resources with the Allee effect

Volume 49 / 2022

Bartłomiej Kluczyński Applicationes Mathematicae 49 (2022), 59-77 MSC: Primary 34D23, 91B76; Secondary 34C60. DOI: 10.4064/am2424-5-2021 Published online: 28 October 2021


We study an extended Solow–Swan model for capital, denoted by $K$, and natural resources, denoted by $N$. We are interested in production, denoted by $ Y $, expressed by a simplification of the Cobb–Douglas function $ Y=f(K,L,A) $ with $L=A=1 $. We study the model described by the following system of ODEs: $$ \begin {cases} \dot {K}=s K^\alpha -\delta K,\\ \dot {N}=rN\left (\frac {N}{\mathsf {CT}}-1\right )\left (1-\frac {N}{\mathsf {CC}}\right )-\gamma _0 K^{\alpha n}. \end {cases} $$ We study the asymptotics of solutions; among other things, we find critical points, determine their stability and find their attraction basins. This article was inspired by [M. Kornafel, I. Telega, Int. J. Sustainable Economy 12 (2020), 1–24].


  • Bartłomiej KluczyńskiInstitute of Mathematics
    Łódź University of Technology
    Wólczańska 215
    90-924 Łódź, Poland

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