Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 3 / 13E / f / Solution

With response rates and elasticity effectively fixed, changing ${\lambda }_2$ changes $m_2$ and hence intrinsic noise but not the environmental term. Observation (i) therefore indicates extrinsic noise dominates, making the mean-dependent intrinsic contribution negligible. For a general feedback law whose elasticity changes substantially with the mean, compensation between terms is another possibility; independence from ${\lambda }_2$ alone would then not logically identify the dominant term.

Observation (ii) measures the local relaxation rate ${\beta }_{2}H$, which is larger under negative feedback. Observation (iii) is not explained by faster adjustment alone. At fixed susceptibility, increasing the response rate increases ${\beta }_{2}H/({\beta }_1+{\beta }_{2}H)$, reducing time averaging and increasing transmitted environmental noise. Feedback lowers variance primarily by reducing susceptibility and the intrinsic term. Thus rapid adjustment and low noise share a feedback mechanism but are not a direct causal equivalence.