Two species are competing in a region for control of a limited amount of a certain resource. Let the proportion of the resource controlled by species 1 and suppose has pdf
which is a uniform distribution on .
- In her book Ecological Diversity, E. C. Pielou calls this the “broken-stick” model for resource allocation, since it is analogous to breaking a stick at a randomly chosen point.
Then the species that controls the majority of this resource controls the amount$$ \begin{align} h\left( X \right) &= \max \left( X, 1 - X \right) \ &= \begin{cases} 1 - X & \text{if } 0 \leq X < \frac{1}{2} \ X & \text{if } \frac{1}{2} \leq X \leq 1 \end{cases} \end{align}
\begin{align} E\left[ h\left( X \right) \right] &= \int_{-\infty}^{\infty} \max \left( x, 1 - x \right) \cdot f\left( x \right) , dx \ &= \int_{0}^{1} \max \left( x, 1 - x \right) \cdot 1 , dx \ &= \int_{0}^{1/2} \left( 1 - x \right) \cdot 1 , dx + \int_{1/2}^{1} x \cdot 1 , dx \ &= \frac{3}{4} \end{align}