Models for Persistence and Spread of Structured Populations in Patchy Landscapes
Creators: name: Alqawasmeh, Yousef
Publisher: name: Université d'Ottawa / University of Ottawa
Year: 2017
Subjects: Structured population model
Reaction-diffusion equation
Spatial heterogeneity
Persistence condition
Critical patch size
Traveling periodic waves
Spread speed
Marine protected area

description: In this dissertation, we are interested in the dynamics of spatially distributed populations. In particular, we focus on persistence conditions and minimal traveling periodic wave speeds for stage-structured populations in heterogeneous landscapes. The model includes structured populations of two age groups, juveniles and adults, in patchy landscapes. First, we present a stage-structured population model, where we divide the population into pre-reproductive and reproductive stages. We assume that all parameters of the two age groups are piecewise constant functions in space. We derive explicit formulas for population persistence in a single-patch landscape and in heterogeneous habitats. We find the critical size of a single patch surrounded by a non-lethal matrix habitat. We derive the dispersion relation for the juveniles-adults model in homogeneous and heterogeneous landscapes. 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. We apply sensitivity and elasticity formulas to the critical size of a single-patch landscape and to the minimal traveling wave speed in a homogeneous landscape. Secondly, we use asymptotic techniques to find an explicit formula for the traveling periodic wave speed and to calculate the spread rates for structured populations in heterogeneous landscapes. We illustrate the power of the homogenization method by comparing the dispersion relation and the resulting minimal wave speeds for the approximation and the exact expression. We find an excellent agreement between the fully heterogeneous speed and the homogenized speed, even though the landscape period is on the same order as the diffusion coefficients and not as small as the formal derivation requires. We also generalize this work to the case of structured populations of n age groups. Lastly, we use a finite difference method to explore the numerical solutions for the juveniles-adults model. We compare numerical solutions to analytic solutions and explore population dynamics in non-linear models, where the numerical solution for the time-dependent problem converges to a steady state. We apply our theory to study various aspects of marine protected areas (MPAs). We develop a model of two age groups, juveniles and adults, in which only adults can be harvested and only outside MPAs, and recruitment is density dependent and local inside MPAs and fishing grounds. We include diffusion coefficients in density matching conditions at interfaces between MPAs and fishing grounds, and examine the effect of fish mobility and bias movement on yield and fish abundance. We find that when the bias towards MPAs is strong or the difference in diffusion coefficients is large enough, the relative density of adults inside versus outside MPAs increases with adult mobility. This observation agrees with findings from empirical studies.