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On Historical Dynamics by P. Turchin

A Mathematical Review

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Abstract

The book Historical Dynamics: Why States Rise and Fall by P. Turchin contains one of the first efforts to apply basic system dynamics (SD) to the analysis of the past human history. The present review paper focuses on the consistency and solidity of the mathematical modelling advanced in the book. It is found that all the models used in the book are based on human-centred drivers only, and lack any connection with the physical world. Moreover, all of the models are flawed, some by omission (i.e. the discussion presented in the book omits relevant trends that can be tested against historical data) and some by structure (i.e. they are mathematically wrong). These findings imply that the mathematical conclusions of the book do not support the interpretation of historical events, and this paper wants to alert the researchers about the pitfalls that must be faced when dealing with the application of SD to human history modelling.

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Acknowledgements

The author wishes to thank Dr. Charles A. S. Hall and the anonymous referee for their valuable comments which helped to improve the manuscript. The author wishes to thank Dario Faccini, Dr. Francesco D’Eugenio and Dr. Dimitri Douchin who assisted in the proof-reading of the manuscript. The author wishes to thank Dr. Roberto Iaconi for the help on the numerical simulations. The author wishes to thank Macquarie University for the kind hospitality provided during the writing of the manuscript. None of these people or the Institution necessarily reflect the opinions of the author about the content of this paper.

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Correspondence to Alessandro Maini.

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Appendices

Appendices

Appendix A: Asabiya-Territory Model (ATM)

System

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle {\overset{\hbox{\tiny$\ \bullet$}}{A}} = c_0\,A\,S\,\left( 1 - \frac{A}{\ h\ }\right) - a \\ \displaystyle {\overset{\hbox{\tiny$\ \bullet$}}{S}} = r_0\,\left( 1 - \frac{A}{\ 2\,b\ }\right) \,S\,(1 - S) \end{array}\right. } \end{aligned}$$

Zeroclines

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle {\overset{\hbox{\tiny$\ \bullet$}}{A}} = 0 \qquad \Rightarrow \qquad S_{\text{ zero }} = \frac{a\,h}{\ c_0\,A\,(h - A)\ } \\ \displaystyle {\overset{\hbox{\tiny$\ \bullet$}}{S}} = 0 \qquad \Rightarrow \qquad A_{\text{ zero }} = 2\,b, \quad S_{\text{ zero }} = 0, \quad S_{\text{ zero }} = 1 \end{array}\right. } \end{aligned}$$

Mathematical Properties of the \({\varvec{S}}_{\text{ zero }}\) Zerocline

$$\begin{aligned}&\displaystyle \frac{\ \partial S_{\text{ zero }}\ }{\partial A} = \frac{\ a\,h\ }{c_0}\,\left[ \frac{-1}{\ A^2\,(h - A)\ } + \frac{1}{\ A\,(h - A)^2\ }\right] \\&\quad = \frac{\ a\,h\ }{c_0}\,\frac{2\,A - h}{\ A^2\,(h - A)^2\ }\\&\lim _{\ A \rightarrow 0^+} S_{\text{ zero }} = +\infty \qquad \qquad \lim _{\ A \rightarrow h^-} S_{\text{ zero }} = +\infty \end{aligned}$$

Equilibrium Point

$$\begin{aligned} \displaystyle P_{\text{ eq }} = \left( A_{\text{ eq }}, S_{\text{ eq }}\right) = \left( 2\,b, \frac{a\,h}{\ c_0\,2\,b\,(h - 2\,b)\ }\right) \end{aligned}$$

Eigenvalues of the System at the Equilibrium Point

$$\begin{aligned}&\displaystyle \frac{\ \partial {\overset{\hbox{\tiny$\ \bullet$}}{A}}\ }{\partial A} = c_0\,S\,\frac{\ h - 2\,A\ }{h} \quad \displaystyle \frac{\ \partial {\overset{\hbox{\tiny$\ \bullet$}}{S}}\ }{\partial A} = -\frac{r_0}{\ 2\,b\ }\,S\,(1 - S) \\&\displaystyle \frac{\ \partial {\overset{\hbox{\tiny$\ \bullet$}}{A}}\ }{\partial S} = c_0\,A\,\frac{\ h - A\ }{h} \quad \displaystyle \frac{\ \partial {\overset{\hbox{\tiny$\ \bullet$}}{S}}\ }{\partial S} = r_0\,\frac{\ 2\,b - A\ }{2\,b}\,(1 - 2\,S)\\&\det { \left( \mathbb {J} - \lambda \,\mathbb {I}\right) }_{P_{\text{ eq }}} \\&\quad = \begin{vmatrix} \displaystyle \,c_0\,S\,\frac{\ h - 2\,A\ }{h} - \lambda&\displaystyle \ \ \ c_0\,A\,\frac{\ h - A\ }{h} \\ \displaystyle \,-\,\frac{r_0}{\ 2\,b\ }\,S\,(1 - S)&\displaystyle \ \ \ r_0\,\frac{\ 2\,b - A\ }{2\,b}\,(1 - 2\,S) - \lambda \, \end{vmatrix}_{P_{\text{ eq }}} \\&\quad = \begin{vmatrix} \displaystyle \,c_0\,\frac{a\,h}{\ c_0\,2\,b\,(h - 2\,b)\ }\,\frac{\ h - 4\,b\ }{h} - \lambda&\ \ \ \displaystyle c_0\,2\,b\,\frac{\ h - 2\,b\ }{h} \\ \displaystyle \,-\,\frac{r_0}{\ 2\,b\ }\,\frac{a\,h}{\ c_0\,2\,b\,(h - 2\,b)\ }\,\left[ 1 - \frac{a\,h}{\ c_0\,2\,b\,(h - 2\,b)\ }\right]&\ \ \ \displaystyle -\lambda \, \\ \end{vmatrix} \\&\quad = \lambda ^2 - \frac{a}{\,2\,b\,}\,\frac{\,h - 4\,b\,}{h - 2\,b}\,\lambda \\&\qquad + \frac{\,a\,r_0\,}{2\,b}\,\left[ 1 - \frac{a\,h}{\,c_0\,2\,b\,(h - 2\,b)\,}\right] = 0\\&\displaystyle \lambda _{+,-} = \frac{\,1\,}{2}\,\left\{ \frac{a}{\ 2\,b\ }\,\frac{\ h - 4\,b\ }{h - 2\,b} \pm \sqrt{\ \left[ \frac{a}{\ 2\,b\ }\,\frac{\ h - 4\,b\ }{h - 2\,b} \right] ^2 - 4\,\frac{\ a\,r_0\ }{2\,b}\,\left[ 1 - \frac{a\,h}{\ c_0\,2\,b\,(h - 2\,b)\ }\right] \ } \right\} \end{aligned}$$

Appendix B: Basic Demographic-Fiscal Model (BDFM)

System

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle {\overset{\hbox{\tiny$\ \bullet$}}{N}} = r_0\,N\,\left( 1 - \frac{N}{\ k(S)\ }\right) \\ \displaystyle {\overset{\hbox{\tiny$\ \bullet$}}{S}} = N\,\left( 1 - \beta - \frac{N}{\ k(S)\ }\right) \end{array}\right. } \end{aligned}$$

with:

$$\begin{aligned}&\displaystyle k(S) = k_0 + (k_{\text{max}} - k_0) \times \frac{1}{\ 1 + \frac{\ s_0\ }{S}\ }\\&\displaystyle \lim _{\ S \rightarrow 0^+} k(S) = k_0 \qquad \lim _{S \rightarrow +\infty } k(S) = k_{\text{max}} \end{aligned}$$

Zeroclines

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle {\overset{\hbox{\tiny$\ \bullet$}}{N}} = 0 &{}\quad \Rightarrow \quad N_{\text{ zero }} = 0 \ \text {(trivial)}, \quad N_{\text{ zero }} = k(S) \\ \displaystyle {\overset{\hbox{\tiny$\ \bullet$}}{S}} = 0 &{}\quad \Rightarrow \quad N_{\text{ zero }} = 0 \ \text {(trivial)}, \quad N_{\text{ zero }} = (1 - \beta )\,k(S) \end{array}\right. } \end{aligned}$$
Fig. 10
figure 10

Schematic 2D-representation of the zeroclines for Systems A.15 and A.16, as they appear on planes with \(S = const\). For Sys. A.15, the green line and its intersections do not exist. The two black dots on the apices of the blue and green parabolas are the maxima reached by those parabolas, with common abscissa \(\frac{\,1\,}{2}\) and ordinates expressed by the black formulae on the y-axis (\(E^{\text{max}}(E_{\text{ zero }})\) and \(E^{\text{max}}(S_{\text{ zero }})\), from top to bottom). The half points at the edges of the zeroclines are the intersections of those lines with the Cartesian axes, and their coordinates are expressed by the coloured formulae along the axes (colour as per colour of the corresponding zerocline). Finally, the black-circled red points at the intersections between the zeroclines (labelled with 1 and 2) are the equilibrium points for System A.15 (only point 1) and System A.16 (points 1 and 2). See text for the coordinates of these points (Color figure online)

Appendix C: Demographic-Fiscal Models with Class Structure (DFM)

The discussion about the DFM with Class Structure (Sys. A.15 of HD) and with Class Structure + State Feedback (Sys. A.16 of HD) will be carried out on System A.16, as System A.15 is a specific case of it (absence of state feedback).

Systems

$$\begin{aligned}&{\left\{ \begin{array}{ll} \displaystyle {\overset{\hbox{\tiny$\ \bullet$}}{P}} = \frac{\ \beta _1\,\rho _0\,(1 - P)\,P\ }{1 + E} - \delta _1\,P \\ \displaystyle {\overset{\hbox{\tiny$\ \bullet$}}{E}} = \frac{\ \beta _2\,E\,\rho _0\,(1 - P)\,P\ }{1 + E} - \delta _2\,E \\ \displaystyle {\overset{\hbox{\tiny$\ \bullet$}}{S}} = \gamma \,{\overset{\hbox{\tiny$\ \bullet$}}{E}} - \alpha E \end{array}\right. }\\&{\left\{ \begin{array}{ll} \displaystyle {\overset{\hbox{\tiny$\ \bullet$}}{P}} = \frac{\ \beta _1\,\rho _0\,(1 - P)\,P\ }{1 + E} - \delta _1\,P \\ \displaystyle {\overset{\hbox{\tiny$\ \bullet$}}{E}} = \frac{\ \beta _2\,E\,\rho _0\,(1 - P)\,P\ }{1 + E} - \frac{\delta _2}{\ 1 + S\ }\,E \\ \displaystyle {\overset{\hbox{\tiny$\ \bullet$}}{S}} = \gamma \,{\overset{\hbox{\tiny$\ \bullet$}}{E}} - \alpha E \end{array}\right. } \end{aligned}$$

Zeroclines

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle {\overset{\hbox{\tiny$\ \bullet$}}{P}} = 0 \qquad \Rightarrow \qquad P_{\text{ zero }} = 0 \ \text {(trivial)}, \qquad P_{\text{ zero }} = 1 - \frac{\delta _1}{\ \beta _1\,\rho _0\ }\,(1 + E) \\ \displaystyle {\overset{\hbox{\tiny$\ \bullet$}}{E}} = 0 \qquad \Rightarrow \qquad E_{\text{ zero }} = 0 \ \text {(trivial)}, \qquad E_{\text{ zero }} = -1 + \frac{\ \beta _2\,\rho _0\ }{\delta _2}\,(1 - P)\,P\,(1 + S) \\ \displaystyle {\overset{\hbox{\tiny$\ \bullet$}}{S}} = 0 \qquad \Rightarrow \qquad E_{\text{ zero }} = 0 \ \text {(trivial)}, \qquad S_{\text{ zero }} = -1 + \frac{\gamma \,\delta _2\,(1 + E)}{\ \gamma \,\beta _2\,\rho _0\,(1 - P)\,P - \alpha \,(1 + E)\ } \end{array}\right. } \end{aligned}$$

Mathematical Properties of the Zeroclines

  • \(P_{\text{ zero }} = 1 - \frac{\delta _1}{\ \beta _1\,\rho _0\ }\,(1 + E)\)

    This zerocline is common for Sys. A.15 and A.16. It does not depend on S, and it is a plane of constant slope which crosses the plane \(P = 0\) on the straight line \(E = \frac{\ \beta _1\,\rho _0\ }{\delta _1} - 1\) \(\ \forall \,S\), and the plane \(E = 0\) on the straight line \(P = 1 - \frac{\delta _1}{\ \beta _1\,\rho _0\ }\) \(\ \forall \,S\).

  • \(E_{\text{ zero }} = -1 + \frac{\ \beta _2\,\rho _0\ }{\delta _2}\,(1 - P)\,P\,(1 + S)\)

    This zerocline is the same for Sys. A.15 and Sys. A.16 only on the plane \(S = 0\). For whatever \(S = const\), this zerocline is a concave downwards parabola ( \(\frac{\ \partial E\ }{\partial P} = \frac{\ \beta _2\,\rho _0\ }{\delta _2}\,(1 - 2\,P)\,(1 + S)\), \(\frac{\ \partial ^2 E\ }{\partial P^2} = -2\,\frac{\ \beta _2\,\rho _0\ }{\delta _2}\,(1 + S) < 0\) \(\ \forall \,S\) ). The maximum of each parabola is at \(P = \frac{1}{\ 2\ }\), where \(E^{\text{max}}(E_{\text{ zero }}) = -1 + \frac{\ \beta _2\,\rho _0\ }{4}\,(1 + S)\,\frac{1}{\ \delta _2\ }\). This family of parabolas crosses the plane \(E = 0\) along the lines:

    $$\begin{aligned}&P^2 - P + \frac{\delta _2}{\ \beta _2\,\rho _0\ }\,\frac{1}{\ 1 + S\ } = 0 \\&\quad \Rightarrow \qquad P^{E_{\text{ zero }}}_{+,-} \\&\qquad = \frac{\ 1\ }{2}\,\left[ 1 \pm \sqrt{\ 1 - 4\,\frac{\delta _2}{\ \beta _2\,\rho _0\ }\,\frac{1}{\ 1 + S\ }\ }\ \right] \end{aligned}$$

    The condition for having at least one crossing point is: \(\frac{\delta _2}{\ \beta _2\,\rho _0\ }\,\frac{1}{\ 1 + S\ } \le \frac{\ 1\ }{4}\).

  • \(S_{\text{ zero }} = -1 + \frac{\gamma \,\delta _2\,(1 + E)}{\ \gamma \,\beta _2\,\rho _0\,(1 - P)\,P - \alpha \,(1 + E)\ }\)

    This zerocline exists only for Sys. A.16. Solving for E (\(E = -1 + \frac{\ \gamma \,\beta _2\,\rho _0\,(P - P^2)\,(1 + S)\ }{\gamma \,\delta _2 + \alpha \,(1 + S)}\)), it is possible to see that also this zerocline, for whatever \(S = const\), is a concave downwards parabola (\(\frac{\ \partial E\ }{\partial P} = \frac{\ \gamma \,\beta _2\,\rho _0\,(1 - 2\,P)\,(1 + S)\ }{\gamma \,\delta _2 + \alpha \,(1 + S)}\), \(\frac{\ \partial ^2 E\ }{\partial P^2} = -2\,\frac{\gamma \,\beta _2\,\rho _0\,(1 + S)}{\ \gamma \,\delta _2 + \alpha \,(1 + S)}\ < 0\) \(\forall \,S\)). The maximum of each parabola is at \(P = \frac{\ 1\ }{2}\), where \(E^{\text{max}}(S_{\text{ zero }}) = -1 + \frac{\ \gamma \,\beta _2\,\rho _0\,\frac{\,1\,}{4}\,(1 + S)\ }{\gamma \,\delta _2 + \alpha \,(1 + S)} = -1 + \frac{\ \beta _2\,\rho _0\ }{4}\,(1 + S)\,\frac{1}{\ \delta _2 + \frac{\alpha }{\gamma }\,(1 + S)\ }\). This family of parabolas crosses the plane \(E = 0\) along the lines:

    $$\begin{aligned}&P^2 - P + \frac{1}{\ \gamma \,\beta _2\,\rho _0\ }\,\left( \frac{\gamma \,\delta _2}{\ 1 + S\ } + \alpha \right) = 0 \\&\quad \Rightarrow \qquad P^{S_{\text{ zero }}}_{+,-} \\&\qquad = \frac{\,1\,}{2}\,\left[ 1 \pm \sqrt{\ 1 - 4\,\frac{\delta _2}{\ \beta _2\,\rho _0\ }\,\frac{1}{\ 1 + S\ } - 4\,\frac{\alpha }{\ \gamma \,\beta _2\,\rho _0\ }\ }\ \right] \end{aligned}$$

    The condition for having at least one crossing point is: \(\frac{1}{\ \gamma \,\beta _2\,\rho _0\ }\,\left( \frac{\gamma \,\delta _2}{\ 1 + S\ } + \alpha \right) \le \frac{\ 1\ }{4}\).

Notice how \(E^{\text{max}}(S_{\text{ zero }}) \le E^{\text{max}}(E_{\text {zero}})\), and \(P^{E_{\text {zero}}}_- \le P^{S_{\text {zero}}}_- \le P^{S_{\text {zero}}}_+ \le P^{E_{\text {zero}}}_+\), for any values of the variables involved. In other words, for any fixed \(S = const\), the parabola \(S_{\text {zero}}\) has always smaller values than the parabola \(E_{\text {zero}}\) (see Fig. 10 for a representation of the zeroclines on planes with \(S = const\), and Fig. 11 for the zeroclines on the 3D phase space).

Fig. 11
figure 11

Schematic 3D-representation of the zeroclines for System A.16. The red plane is the zerocline \({\overset{\hbox{\tiny$\ \bullet$}}{P}} = 0\), the green surface is the zerocline \({\overset{\hbox{\tiny$\ \bullet$}}{S}} = 0\), and the blue surface is the zerocline \({\overset{\hbox{\tiny$\ \bullet$}}{E}} = 0\) (Color figure online)

‘Equilibrium Points’

Eq. 2 and 3 of Sys. A.16 contain the same functional form (\({\overset{\hbox{\tiny$\ \bullet$}}{E}}\)), differing only for a scaling factor (\(\gamma \)) and a shift (\(-\alpha \,E\)). The parabolic surfaces described by their zercolines (\(E_{\text {zero}}\) and \(S_{\text {zero}}\)), therefore, do not cross each other except along lines lying in the unphysical area of the phase space (\(E < 0\), see Fig. 11), and the analysis of these lines is skipped. Here reported are only the physically meaningful (\(E > 0\)) intersections of zerocline \(E_{\text {zero}}\) with zerocline \(P_{\text {zero}}\) (both Sys. A.15 and A.16) and of zerocline \(S_{\text {zero}}\) with zerocline \(P_{\text {zero}}\) (Sys. A.16 only):

  • \(P_{\text {zero}}\,\text {(plane)}\ \cap \ E_{\text {zero}}\,\text {(parabolic downwards surface)}\)

    Intersection line common for Sys. A.15 and A.16. On planes with \(S = const\) it is represented by point \(\textcircled {1}\) on Fig. 10.

    $$\begin{aligned}&{\left\{ \begin{array}{ll} \displaystyle 1 - P_{\text {zero}} = \frac{\delta _1}{\ \beta _1\,\rho _0\ }\,(1 + E_{\text {zero}}) \\ \displaystyle 1 + E_{\text {zero}} = \frac{\ \beta _2\,\rho _0\ }{\delta _2}\,(1 - P_{\text {zero}})\,P_{\text {zero}}\,(1 + S) \\ \end{array}\right. }\nonumber \\&\quad \Rightarrow \quad \textcircled {1} = \left( P^{1}_{\text{eq}}, E^{1}_{\text{eq}}\right) \nonumber \\&\qquad = \left( \,\frac{\ \beta _1\,\delta _2\ }{\ \beta _2\,\delta _1\ }\,\frac{1}{\ (1 + S)\ }, \,-1 + \frac{\ \beta _1\,\rho _0\ }{\delta _1}\,\frac{\ \beta _2\,\delta _1\,(1 + S) - \beta _1\,\delta _2\ }{\beta _2\,\delta _1\,(1 + S)\,}\,\right) \end{aligned}$$
    (9)
  • \(P_{\text {zero}}\,\text {(plane)}\ \cap \ S_{\text {zero}}\,\text {(parabolic downwards surface)}\)

    Intersection line that exists only for Sys. A.16. On planes with \(S = const\) it is represented by point \(\textcircled {2}\) on Fig. 10.

    $$\begin{aligned}&{\left\{ \begin{array}{ll} \displaystyle 1 - P_{\text {zero}} = \frac{\delta _1}{\,\beta _1\,\rho _0\,}\,(1 + E_{\text {zero}}) \\ \displaystyle 1 + S_{\text {zero}} = \frac{\gamma \,\delta _2\,(1 + E_{\text {zero}})}{\,\gamma \,\beta _2\,\rho _0\,(1 - P_{\text {zero}})\,P_{\text {zero}} - \alpha \,(1 + E_{\text {zero}})\,} \end{array}\right. }\nonumber \\&\quad \displaystyle \Rightarrow \quad \textcircled {2} = \left( P^{2}_{\text{eq}}, E^{2}_{\text{eq}}\right) \nonumber \\&\qquad = \left( \frac{\ \beta _1\,\delta _2\ }{\ \beta _2\,\delta _1\ }\,\frac{1}{\ (1 + S)\ }\,\left( \,1 + \frac{\alpha }{\ \gamma \,\delta _2\ }\,(1 + S)\,\right) , -1 \right. \nonumber \\&\qquad \left. + \frac{\ \beta _1\,\rho _0\ }{\delta _1}\,\left[ 1 - \frac{\ \beta _1\,\delta _2\ }{\ \beta _2\,\delta _1\ }\,\frac{1}{\ (1 + S)\ } \times \left( \,1 + \frac{\alpha }{\ \gamma \,\delta _2\ }\,(1 + S)\,\right) \right] \,\right) \end{aligned}$$
    (10)

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Maini, A. On Historical Dynamics by P. Turchin. Biophys Econ Sust 5, 3 (2020). https://doi.org/10.1007/s41247-019-0063-x

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