Dynamics of populations with age-difference and diffusion: localization

A Stage-Structured Continuous-/Discrete-Time Population Model: Persistence and Spatial Spread

Population persistence and spatial propagation and their dependence on demography and dispersal are of great importance in spatial ecology. Many species with highly structured life cycles invade new habitats through the dispersal of organisms in their early life stages (e.g., seeds, larvae, etc.). We develop a stage-structured continuous/discrete-time hybrid model to describe the spatiotemporal dynamics of such species, in which a reaction-diffusion equation describes the random movement of dispersing individuals, while two difference equations describe the demography of sedentary individuals. We obtain a formula for the spreading speed of the population in terms of model parameters. We show that the spreading speed can be characterized as the slowest wave speed of a class of traveling wave solutions. We provide an explicit formula for the critical domain size that separates population persistence from extinction. By comparing our stage-structured model with a physically unstructured model, we find that the structured model reduces to the unstructured one in some special cases. Accordingly, the results about the spreading speed and the critical domain size for the unstructured model represent some special cases of those for the structured one. This highlights the significance of including stage structure in studying the spatial dynamics of species with complex life cycles.

Persistence and spread of stage-structured populations in heterogeneous landscapes

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.