Homogenization analysis of invasion dynamics in heterogeneous landscapes with differential bias and motility.

Animal movement behaviors vary spatially in response to environmental heterogeneity. An important problem in spatial ecology is to determine how large-scale population growth and dispersal patterns emerge within highly variable landscapes. We apply the method of homogenization to study the large-scale behavior of a reaction-diffusion-advection model of population growth and dispersal. Our model includes small-scale variation in the directed and random components of movement and growth rates, as well as large-scale drift. Using the homogenized model we derive simple approximate formulas for persistence conditions and asymptotic invasion speeds, which are interpreted in terms of residence index. The homogenization results show good agreement with numerical solutions for environments with a high degree of fragmentation, both with and without periodicity at the fast scale. The simplicity of the formulas, and their connection to residence index make them appealing for studying the large-scale effects of a variety of small-scale movement behaviors.

Homogenization of a Directed Dispersal Model for Animal Movement in a Heterogeneous Environment.

The dispersal patterns of animals moving through heterogeneous environments have important ecological and epidemiological consequences. In this work, we apply the method of homogenization to analyze an advection-diffusion (AD) model of directed movement in a one-dimensional environment in which the scale of the heterogeneity is small relative to the spatial scale of interest. We show that the large (slow) scale behavior is described by a constant-coefficient diffusion equation under certain assumptions about the fast-scale advection velocity, and we determine a formula for the slow-scale diffusion coefficient in terms of the fast-scale parameters. We extend the homogenization result to predict invasion speeds for an advection-diffusion-reaction (ADR) model with directed dispersal. For periodic environments, the homogenization approximation of the solution of the AD model compares favorably with numerical simulations. Invasion speed approximations for the ADR model also compare favorably with numerical simulations when the spatial period is sufficiently small.

Modeling the effects of developmental variation on insect phenology.

Phenology, the timing of developmental events such as oviposition or pupation, is highly dependent on temperature; since insects are ectotherms, the time it takes them to complete a life stage (development time) depends on the temperatures they experience. This dependence varies within and between populations due to variation among individuals that is fixed within a life stage (giving rise to what we call persistent variation) and variation from random effects within a life stage (giving rise to what we call random variation). It is important to understand how both types of variation affect phenology if we are to predict the effects of climate change on insect populations.We present three nested phenology models incorporating increasing levels of variation. First, we derive an advection equation to describe the temperature-dependent development of a population with no variation in development time. This model is extended to incorporate persistent variation by introducing a developmental phenotype that varies within a population, yielding a phenotype-dependent advection equation. This is further extended by including a diffusion term describing random variation in a phenotype-dependent Fokker-Planck development equation. These models are also novel because they are formulated in terms of development time rather than developmental rate; development time can be measured directly in the laboratory, whereas developmental rate is calculated by transforming laboratory data. We fit the phenology models to development time data for mountain pine beetles (MPB) (Dendroctonus ponderosae Hopkins [Coleoptera: Scolytidae]) held at constant temperatures in laboratory experiments. The nested models are parameterized using a maximum likelihood approach. The results of the parameterization show that the phenotype-dependent advection model provides the best fit to laboratory data, suggesting that MPB phenology may be adequately described in terms of persistent variation alone. MPB phenology is simulated using phloem temperatures and attack time distributions measured in central Idaho. The resulting emergence time distributions compare favorably to field observations.

Modeling the evolution of insect phenology.

Climate change is likely to disrupt the timing of developmental events (phenology) in insect populations in which development time is largely determined by temperature. Shifting phenology puts insects at risk of being exposed to seasonal weather extremes during sensitive life stages and losing synchrony with biotic resources. Additionally, warming may result in loss of developmental synchronization within a population making it difficult to find mates or mount mass attacks against well-defended resources at low population densities. It is unknown whether genetic evolution of development time can occur rapidly enough to moderate these effects. We present a novel approach to modeling the evolution of phenology by allowing the parameters of a phenology model to evolve in response to selection on emergence time and density. We use the Laplace method to find asymptotic approximations for the temporal variation in mean phenotype and phenotypic variance arising in the evolution model that are used to characterize invariant distributions of the model under periodic temperatures at leading order. At these steady distributions the mean phenotype allows for parents and offspring to be oviposited at the same time of year in consecutive years. Numerical simulations show that populations evolve to these steady distributions under periodic temperatures. We consider an example of how the evolution model predicts populations will evolve in response to warming temperatures and shifting resource phenology.

Homogenization techniques for population dynamics in strongly heterogeneous landscapes.

An important problem in spatial ecology is to understand how population-scale patterns emerge from individual-level birth, death, and movement processes. These processes, which depend on local landscape characteristics, vary spatially and may exhibit sharp transitions through behavioural responses to habitat edges, leading to discontinuous population densities. Such systems can be modelled using reaction-diffusion equations with interface conditions that capture local behaviour at patch boundaries. In this work we develop a novel homogenization technique to approximate the large-scale dynamics of the system. We illustrate our approach, which also generalizes to multiple species, with an example of logistic growth within a periodic environment. We find that population persistence and the large-scale population carrying capacity is influenced by patch residence times that depend on patch preference, as well as movement rates in adjacent patches. The forms of the homogenized coefficients yield key theoretical insights into how large-scale dynamics arise from the small-scale features.