On reducibility and ergodicity of population projection matrix models

Authors

  • Iain Stott,

    1. Centre for Ecology and Conservation, School of Biosciences, College of Life and Environmental Sciences, University of Exeter Cornwall Campus, Tremough, Treliever Road, Penryn, Cornwall TR10 9EZ, UK
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  • Stuart Townley,

    1. College of Engineering, Mathematics and Physical Sciences, Harrison Building, University of Exeter, North Park Road, Exeter, Devon EX4 4QF, UK
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  • David Carslake,

    1. Department of Biological Sciences, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK
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  • David J. Hodgson

    Corresponding author
    1. Centre for Ecology and Conservation, School of Biosciences, College of Life and Environmental Sciences, University of Exeter Cornwall Campus, Tremough, Treliever Road, Penryn, Cornwall TR10 9EZ, UK
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Correspondence author. E-mail: d.j.hodgson@exeter.ac.uk

Summary

1. Population projection matrices (PPMs) are probably the most commonly used empirical population models. To be useful for predictive or prospective analyses, PPM models should generally be irreducible (the associated life cycle graph contains the necessary transition rates to facilitate pathways from all stages to all other stages) and therefore ergodic (whatever initial stage structure is used in the population projection, it will always exhibit the same stable asymptotic growth rate).

2. Evaluation of 652 PPM models for 171 species from the literature suggests that 24·7% of PPM models are reducible (parameterized transition rates do not facilitate pathways from all stages to all other stages). Reducible models are sometimes ergodic but may be non-ergodic (the model exhibits two or more stable asymptotic states with different asymptotic stable growth rates, which depend on the initial stage structure used in the population projection). In our sample of published PPMs, 15·6% are non-ergodic.

3. This presents a problem: reducible–ergodic models often defy biological rationale in their description of the life cycle but may or may not prove problematic for analysis as they often behave similarly to irreducible models. Reducible–non-ergodic models will usually defy biological rationale in their description of the both the life cycle and population dynamics, hence contravening most analytical methods.

4. We provide simple methods to evaluate reducibility and ergodicity of PPM models, present illustrative examples to elucidate the relationship between reducibility and ergodicity and provide empirical examples to evaluate the implications of these properties in PPM models.

5. As a prevailing tool for population ecologists, PPM models need to be as predictive as possible. However, there is a large incidence of reducibility in published PPMs, with significant implications for the predictive power of such models in many cases. We suggest that as a general rule, reducibility of PPM models should be avoided. However, we provide a guide to the pertinent analysis of reducible matrix models, largely based upon whether they are ergodic or not.

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