Model Description
Most models of Malthusian population dynamics specify logistic growth to a carrying capacity, but the historical record of agrarian societies strongly suggests that repeated cycles of overshoot and collapse (so-called "Malthusian crises") are endogenous to population dynamics (see Nefedov, 2013 for details and citations). In this model, Nefedov (2013) explicitly models harvest surplus production as the carrying capacity of an agricultural population. When population pressure drives the agricultural surplus to zero, a devastating famine occurs, which kills a large proportion of the population. This "Malthusian crisis" then enables renewed population growth in repeated cycles -- producing dampened oscillations. The eventual equilibrium is conceived of as a very rare case, as historical factors (economic, social, and political change) exogenous to the model are expected to repeatedly provoke recurring growth-collapse dynamics.
The model has two state variables -- population (N) in number of households, and surplus grain (K) in units of minimum annual rations per household. Population dynamics are logistic, where 'r' is the maximum growth rate (~0.007 < r < 0.02 for pre-industrial societies), and the carrying capacity is the the other state variable, K. As such, the population size of K corresponds to the number of minimum annual rations in storage. Annually, 'N' rations are consumed, so the annual change in K will be harvest minus consumption. Harvest is a function of N and the parameter 'q', which is conceptualized as how many households one farming household can support (~1.1 < q < 2.0 for pre-industrial societies). Depending on the values of q and r, the period of oscillations can range from >400 to 63 years.
$dN_{t}/dt = r N_{t} (1 - N_{t}/K_{t})$ |
| Logistic population growth |
$P_{t} = \dfrac{a N_{t}}{N_{t} + d}$ |
| 'P' is an auxiliary variable for annual agricultural production |
$dK_{t}/dt = P_{t} - N_{t}$ |
| Change in surplus equals total agricultural production minus total consumption |
$dK_{t}/dt = \dfrac{a N_{t}}{N_{t} + d} – N_{t}$ |
| Change in surplus expressed as a function of N, where a and d are arbitrary parameters |
$q = a/d, d = \bar{N}/(q – 1), a = q \bar{N}/(q – 1)$ |
| parameters 'a' and 'd' are re-scaled to 'q' (the maximum number of households that one farming household can support with their annual agricultural production), and the equilibrium value of 'N' (N-bar) is re-scaled to 1 |
$dK_{t}/dt = \dfrac{ q N_{t} }{ (q - 1)(N_{t}+ \frac{1}{q-1})} - N_{t}$ |
| substituting 'q' for 'a' and 'd', setting N-bar at 1, and simplifying algebraically, we have the final equation for change in K (in terms of N and q only). |
## Nefedov's (2013) model of malthusian population dynamics in agrarian societies ## In contrast to other malthusian models, this model includes "malthusian crises" ## where the the scarcity of surplus food (K) leads to population collapse (N) ## in repeated dampened oscillations ## There are two key parameters: ## r is the maximal rate of natural increase 0.004 < r < 0.02 ## q is how many households one farming family can support 1 < q < 2 ## the equilibrium population value is scaled to 1 par r=0.01,q=1.2 ## Initial conditions for the model init N=0.1, K=0.1 ## Differential equations dN/dt=r*N*(1-N/K) dK/dt=N*q/((q-1)*(N+1/(q-1)))-N @ dt=0.5, total=1000, xplot=N,yplot=K,axes=2d @ xmin=0,xmax=3,ymin=0,ymax=10 @ xlo=0,ylo=0,xhi=3,yhi=10 done
Cesaretti R, Arizona State University.
. 2013. Modeling Malthusian Dynamics in Pre-Industrial Societies. Cliodynamics. 4(2):229-240.
