Rubens' "A Hippopotamus and Crocodile Hunt" depicts a chaotic scene of men battling a fierce hippopotamus and crocodile amidst a turbulent waterscape.

Cliodynamics and Mathematical Models in History. Part 2

Historical Processes and the Lotka-Volterra equation

(Continued. See Part 1)

Peter Turchin employs the predator-prey equation to model the complex dynamics of medieval agrarian states, framing them as oscillatory systems influenced by geopolitical variables and military success.

One of the prime examples, illustrating the reduction of a dynamic process to the “rise-fall” form, Turchin designated the well-known Lotka-Volterra equation, or as it is also known, the “predator-prey equation”. The equation was initially proposed in 1910 to illustrate the challenge of comparing the dynamics of a certain renewable resource and its consumption.

The Lotka-Volterra equation or the “predator-prey equation”.

The main variables in the equation were given as the population volume of predators and prey, which serve as predators' primal food sources. In practice, this could be represented in the form of population dynamics in a certain area of foxes and hares, the processes of catching fish by people, or other comparable examples. If we select balance values for one of the populations, the system will always be in oscillatory equilibrium, and none of the populations will be depleted as a result of predator famine or full prey annihilation. As additional parameters of the equation, “fertility rates” and “extinction rates” of representatives of chosen populations are presented. It is evident that in the lack of a sufficient number of predators, the prey population would develop unrestrictedly, and if the number of predators surpasses certain thresholds, the prey population will be entirely exterminated since it will not have the time to reproduce in adequate numbers. This, in turn, will lead to the annihilation of the predator population, which will be left without the essential food intake to sustain life processes. Thus, an imbalance in at least one of the populations will inevitably lead either to uncontrolled growth or to the complete destruction of the ecosystem in question.

According to Turchin, such a notion could be used to analyse the dynamics of historical processes in medieval agrarian states, which can likewise be represented as an ecosystem. To form the variables and coefficients of the system of differential equations, Turchin opted to use the provisions of the geopolitical theory of Randall Collins. The fundamental variable in this model is the “territory size”, which increases or decreases based on the “military success”. These variables will be influenced by parameters such as "Geopolitical resources" - the amount of taxes collected and the recruiting potential for the army,  "Logistic loads" (or as they are sometimes called "the burden of the empire") - the maintenance of defence capability, police functions, and extraction of resources, and finally - “Peripheral position”, that is, the extent to which the state is geographically surrounded by enemy countries. It is evident that a state totally encircled by possible foes within the continent would have more defensive challenges than a state located on the shore of a sea or ocean or surrounded by mountain ranges from its geopolitical competitors.

All five variables in the equation system are in positive or negative feedback with one another. Territory size is favourably connected with Military Success, Geopolitical Resource Loads, and Logistics Loads, but negatively correlated with Periphery, because as territory grows, it will unavoidably begin to border on a significant number of neighbouring states. Military Success, on the other hand, is in positive feedback with Geopolitical Resources and Periphery, while with Logistics, the feedback is negative.

The use of the Collins model resulted in the conclusion that the relationship between the size of the territory and the pace of change is non-linear. The reason for this non-linearity lies in the understanding of the fact that once a territory is conquered, the frontier moves further away from the centre, and the state begins to bear the higher cost of extending its military power to the new frontier. An additional burden is also created by ensuring order in the conquered territories. This suggests that mathematically there is a certain equilibrium point (in the graph below, where the abscissa shows the size of the territory, and the ordinate shows the rate of change of the territory, it is A2), after which the expansion of the territory becomes a very expensive and inefficient process for the state.

To be continued.

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