Derivatives of the Stochastic Growth Rate

TL;DR

This paper extends the mathematical framework for analyzing how the long-term growth rate of populations in stochastic environments responds to changes in demographic parameters and environmental transition probabilities, providing new formulas and error bounds.

Contribution

It introduces formulas for derivatives of the stochastic growth rate with respect to Markov chain parameters, expanding previous work on demographic parameters.

Findings

Derived new formulas for derivatives with respect to Markov chain parameters.

Provided rigorous bounds on computational estimation errors.

Validated formulas with theoretical derivations.

Abstract

We consider stochastic matrix models for population driven by random environments which form a Markov chain. The top Lyapunov exponent $a$, which describes the long-term growth rate, depends smoothly on the demographic parameters (represented as matrix entries) and on the parameters that define the stochastic matrix of the driving Markov chain. The derivatives of $a$ -- the "stochastic elasticities" -- with respect to changes in the demographic parameters were derived by \cite{tuljapurkar1990pdv}. These results are here extended to a formula for the derivatives with respect to changes in the Markov chain driving the environments. We supplement these formulas with rigorous bounds on computational estimation errors, and with rigorous derivations of both the new and the old formulas.