# Persistence and spread of stage-structured populations in heterogeneous landscapes

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## Abstract

Conditions for population persistence in heterogeneous landscapes and formulas for population spread rates are important tools for conservation ecology and invasion biology. To date, these tools have been developed for unstructured populations, yet many, if not all, species show two or more distinct phases in their life cycle. We formulate and analyze a stage-structured model for a population in a heterogeneous habitat. We divide the population into pre-reproductive and reproductive stages. We consider an environment consisting of two types of patches, one where population growth is positive, one where it is negative. Individuals move randomly within patches but can show preference towards one patch type at the interface between patches. We use linear stability analysis to determine persistence conditions, and we derive a dispersion relation to find spatial spread rates. We illustrate our results by comparing the structured population model with an appropriately scaled unstructured model. We find that a long pre-reproductive state typically increases habitat requirements for persistence and decreases spatial spread rates, but we also identify scenarios in which a population with intermediate maturation rate spreads fastest.

## Keywords

Structured population model Reaction–diffusion equation Spatial heterogeneity Persistence condition Critical patch size Traveling periodic waves Spread speed## Mathematics Subject Classification

35K37 35Q92 92D40## Notes

### Acknowledgements

We are grateful for inspiring discussions with Gabriel Maciel and Jeffrey Musgrave. We also thank Brian Yurk and Christina Cobbold to share their work on homogenization in the early stages of their work. Finally, we deeply thank two anonymous reviewers who pointed out the necessity for the eigenfunctions in Propositions 3 and 5 to be positive. We also thank editor Vincent Calvez for suggestions in the proof of Proposition 4.

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