[Paper Review] A General Formula for the Generation Time

François Bienvenu, Lloyd Demetrius+1 more|arXiv (Cornell University)|Jul 25, 2013

Gene Regulatory Network Analysis20 references20 citations

TL;DR

This paper proposes a general formula for the generation time in structured populations using a finite Markov chain framework, defining it as the mean first return time to reproductive transitions in a weighted directed graph. The key result is a simple closed-form expression: $ T = 1 / \sum_{[j\to i]\in R} e_{ij} $, where $ e_{ij} $ are elasticities of reproductive arcs, valid for any primitive matrix representing a life cycle.

ABSTRACT

We show that the generation time -- a notion usually described in a biological context -- can be defined in a general way as a return time in a conveniently constructed finite Markov chain. The simple formula we obtain agrees with previous results derived for structured populations projected in discrete time, and allows to define the generation time of any process described by a weighted directed graph whose matrix is primitive.

Motivation & Objective

• To generalize the concept of generation time beyond age-classified models to any structured population with a weighted directed graph representation.
• To define generation time as the mean first return time to reproductive transitions in a stochastic process.
• To derive a simple, universal formula for generation time applicable to any primitive matrix model of population dynamics.
• To unify and simplify existing complex formulas for generation time in demographic models.
• To establish a link between generation time and the elasticity of population growth rate with respect to reproductive parameters.

Proposed method

• Construct a finite Markov chain from the weighted directed graph of a life cycle, where nodes represent stages and arcs represent transitions.
• Identify reproductive arcs $ R $ as those leading to the creation of new individuals (e.g., gamete fusion in sexual organisms).
• Define the transition matrix $ \tilde{\mathbf{p}} $ of the Markov chain on the arcs, partitioned into reproductive ($ R $) and non-reproductive ($ S $) sets.
• Compute the stationary distribution $ \tilde{\boldsymbol{\pi}} $ of the arc-level Markov chain and scale it to obtain $ \boldsymbol{\varpi} $, a normalized distribution over arcs.
• Derive the first-passage time distribution to return to $ R $, leading to the generation time as the expected return time.
• Use matrix elasticity theory to express the sensitivity of the dominant eigenvalue $ \lambda $ to changes in reproductive and non-reproductive transitions.

Experimental results

Research questions

• RQ1 Can generation time be defined in a general way for any structured population model, not just age-classified ones?
• RQ2 What is the mathematical structure underlying the mean time between generations in complex life cycles?
• RQ3 How does the elasticity of population growth rate with respect to reproductive transitions relate to generation time?
• RQ4 Can a universal, simple formula for generation time be derived from the underlying graph structure of a population model?
• RQ5 What is the role of the stationary distribution on arcs in determining the mean return time to reproductive events?

Key findings

• The generation time is given by the simple formula $ T = 1 / \sum_{[j\to i]\in R} e_{ij} $, where $ e_{ij} $ are the elasticities of reproductive transitions.
• The formula generalizes previous results for age-classified models (e.g., Leslie models) and applies to any primitive matrix model with identifiable reproductive arcs.
• The derivation shows that the generation time corresponds to the mean first return time to reproductive transitions in a Markov chain on the arcs of the life cycle graph.
• The elasticity of the population growth rate $ \lambda $ with respect to a common multiplier $ c $ on all reproductive arcs is $ e_\lambda(c) = 1/T $, linking generation time directly to demographic sensitivity.
• For non-reproductive transitions, the elasticity sum is $ e_\lambda(d) = 1 - 1/T $, showing that long-lived species have lower sensitivity to juvenile survival and higher sensitivity to adult survival.
• The method provides a unified framework to compute generation time across diverse biological systems, including size-structured and metapopulation models.