##### Model Description

This is a simple model for resource dynamics under harvest. The authors added expected effects of investment in fixed structures on the dynamics of settlement. These sunk-cost effects are not included in this simple model.

The authors introduce a model of logictically regrowing resource exploited by a consumer. There is only one variable, which is level of local renewable sources. Parameters are: local renewable resources (R), settlement of humans (H), maximum growth rate (g), maximum level of resource (K), maximum per capita human resource consumption (C_max), resource level needed to reach 50% of the maximum consupmtion rate (h_R), and resupply of local resources from networks (i). Human population is treated as a parameter because of how slowly it changes over the given time frame.

This model shows the equilibrium level of local renewable resources as a function of human population size in the settlement. There are two levels of stable equilibria. One stable state has relatively high resource levels, and the other is an alternative overexploited state with low resources. The unstable equilibrium in the middle represents a border between the basins of attraction of the upper and lower stable states.

If resource levels are high, but human population continues to grow past the bifurcation point, the resource system collapses to the overexploited state. Recovery from here is only possible if human populaiton in the settlement falls below the other bifurcation point.

$\begin{equation}
\frac{dR}{dt}=gR(1-\frac{R}{K})-c_{max}\frac{R}{R+h_R}H+i
\end{equation}$ |

Parameters are: settlement density of humans (H), maximum growth rate (g), maximum level of resource (K), maximum per capita human resource consumption (C_max), resource level needed to reach 50% of the maximum consupmtion rate (h_R), and resupply of local resources from networks (i) |

#Sunk-Cost Effects and Vulnerability to Collapse in Ancient Societies # R is the resrouce base. H is the harvest. #parameters growth rate g, carring capacity,K, c_max, etc. par g=1,K=1,h_R=1,H=0,c_max=1,i=0 dR/dt = g*R*(1-(R/K)) - c_max*(R/R+h_R)*H + i

Baller K, Arizona State University.

Sunk-Cost Effects and Vulnerability to Collapse in Ancient Societies. Current Anthropology. 44(5)

. 2003.