To describe population growth, we adopt the linear form used by Lotka and Volterra in their original predator-prey formulations. The first component is a base net fertility rate, n, that would apply in the absence of a resource stock (and would presumably be negative in our environment). In addition, there is a second term that, from our point of view, captures the Malthusian idea that net fertility rises when per capita consumption rises. This implies that proportional net fertility should rise when, other things equal, the resource stock is larger. The effect follows because a higher stock (for a given population) would normally give rise to higher per capita consumption. This reasoning leads to the following population growth function:
dP/dt = nP + bPS, (2)
where b is a parameter that reflects economic conditions. It can be seen from Eq. (2) that proportional population growth rate (dP/dt)/P equals the base rate plus some increment related to the per capita resource stock. The functional form could be changed to include a ‘‘congestion effect,’’ similar to the congestion effect that arises in the logistic growth function that is used for the resource stock, but
Eq. (2) should allow a reasonable approximation to actual events.
Equations (1) and (2) form an interdependent dynamic system similar to the Lotka-Volterra predator-prey model. Equation (1) shows the evolution of prey (the forest stock) and has the key property that the prey tends to diminish more quickly when there are more predators (humans). Equation (2) shows the evolution of the predator and has the property that the predator increases more rapidly when prey are more numerous. Figure 2 illustrates the relationship between the principal ‘‘stocks’’ and ‘‘flows’’ in the model using a formal flow diagram.
It is possible to examine the properties of this model analytically. Solutions for steady states can be determined and the dynamic path leading from any initial conditions can be characterized. Depending on parameter values, three steady states are possible, one with no humans and the forest at environmental carrying capacity, one with no humans and no resource stock, and one ‘‘interior’’ steady state with positive stocks of humans and forest biomass. The dynamic evolution of the model can be either monotonic or cyclical, once again depending on parameter values.