The economic value of ecological stability

Authors: by Armsworth and Roughgarden, 2003.

The paper considers a toy system consisting of a wooded area $A$ of which a portion $R$ is to be into a reserve and the remainder is to be clear-cut and the wood sold on the market. The question is how large to make the reserve, and the answer, predictably, depends on the variability that characterizes the population dynamics.

To put this idea on mathematical terms, the authors consider a fecundity function of the form $f(N_t, R, a)$, where $a$ is a parameter that characterizes variability. Under discounting factor $\delta$, the optimal reserve size satisfies

$\delta L = \frac{\partial E[\pi]}{\partial R}$,

where $\pi$ is the in-situ value of the birds and $E$ is the expectation operator. By the chain rule, this can be decomposed as

$\delta L = \frac{\partial E}{\partial N} \frac{dN}{dR} + \frac{\partial E}{\partial \sigma^2} \left(\frac{\partial \sigma^2}{\partial \lambda} \frac{d \lambda}{dR} + \frac{\partial \sigma^2}{\partial \nu} \frac{d \nu}{dR}\right)$

where $N$ is the equilibrium population size, $\sigma^2$ is the population variance, $\lambda$ is the stability level of the equilibrium, and $\nu$ is the sensitivity of the population to environmental conditions. This paper supposes that per capita fecundity decreases linearly with population density, i.e. $latex f$ has the form

$f = aN(1-N/(RK))$,

and that the revenue function decreases with each additional bird, i.e.

$\pi = p(1 - e^{-\alpha N})$.

Under these assumptions, the authors show that, for appropriate parameter choices, increased variability results in decreased expected in-situ revenues, but that the optimal reserve size can actually be greater than that for a constant environment. Matlab simulation code can be found here.