Probabilistic foundation of nonlocal diffusion and formulation and analysis for elliptic problems on uncertain domains

2011 Summer.Includes bibliographical references.In the first part of this dissertation, we study the nonlocal diffusion equation with so-called Lévy measure ν as the master equation for a pure-jump Lévy process. In the case ν ∈ L1(R), a relationship to fractional diffusion is established in a limit of vanishing nonlocality, which implies the convergence of a compound Poisson process to a stable process. In the case ν ∉ L1(R), the smoothing of the nonlocal operator is shown to correspond precisely to the activity of the underlying Lévy process and the variation of its sample paths. We introduce volume-constrained nonlocal diffusion equations and demonstrate that they are the master equations for Lévy processes restricted to a bounded domain. The ensuing variational formulation and conforming finite element method provide a powerful tool for studying both Lévy processes and fractional diffusion on bounded, non-simple geometries with volume constraints. In the second part of this dissertation, we consider the problem of estimating the distribution of a quantity of interest computed from the solution of an elliptic partial differential equation posed on a domain Ω(θ) ⊂ R2 with a randomly perturbed boundary, where (θ) is a random vector with given probability structure. We construct a piecewise smooth transformation from a partition of Ω(θ) to a reference domain Ω in order to avoid the complications associated with solving the problems on Ω(θ). The domain decomposition formulation is exploited by localizing the effect of the randomness to boundary elements in order to achieve a computationally efficient Monte Carlo sampling procedure. An a posteriori error analysis for the approximate distribution, which includes a deterministic error for each sample and a stochastic error from the effect of sampling, is also presented. We thus provide an efficient means to estimate the distribution of a quantity of interest via a Monte Carlo sampling procedure while also providing a posteriori error estimates for each sample

The exit-time problem for a Markov jump process

The purpose of this paper is to consider the exit-time problem for a finite-range Markov jump process, i.e, the distance the particle can jump is bounded independent of its location. Such jump diffusions are expedient models for anomalous transport exhibiting super-diffusion or nonstandard normal diffusion. We refer to the associated deterministic equation as a volume-constrained nonlocal diffusion equation. The volume constraint is the nonlocal analogue of a boundary condition necessary to demonstrate that the nonlocal diffusion equation is well-posed and is consistent with the jump process. A critical aspect of the analysis is a variational formulation and a recently developed nonlocal vector calculus. This calculus allows us to pose nonlocal backward and forward Kolmogorov equations, the former equation granting the various moments of the exit-time distribution.Comment: 15 pages, 7 figure

The Utility of Transient Sensitivity for Wildlife Management and Conservation: Bison as a Case Study

Developing effective management strategies is essential to conservation biology. Population models and sensitivity analyses on model parameters have provided a means to quantitatively compare different management strategies, allowing managers to objectively assess the resulting impacts. Inference from traditional sensitivity analyses (i.e., eigenvalue sensitivity methods) is only valid for a population at its stable age distribution, while more recent methods have relaxed this assumption and instead focused on transient population dynamics. However, very few case studies, especially in long-lived vertebrates where transient dynamics are potentially most relevant, have applied these transient sensitivity methods and compared them to eigenvalue sensitivity methods. We use bison (Bison bison) at Badlands National Park as a case study to demonstrate the benefits of transient methods in a practical management scenario involving culling strategies. Using an age and stage-structured population model that incorporates culling decisions, we find that culling strategies over short time-scales (e.g., 1–5 years) are driven largely by the standing population distribution. However, over longer time-scales (e.g., 25 years), culling strategies are governed by reproductive output. In addition, after 25 years, the strategies predicted by transient methods qualitatively coincide with those predicted by traditional eigenvalue sensitivity. Thus, transient sensitivity analyses provide managers with information over multiple time-scales in contrast to the long time-scales associated with eigenvalue sensitivity analyses. This flexibility is ideal for adaptive management schemes and allows managers to balance short-term goals with long-term viability

Survival and Breeding Transitions for a Reintroduced Bison Population: A Multistate Approach

The iconic plains bison (Bison bison) have been reintroduced to many places in their former range, but there are few scientific data evaluating the success of these reintroductions or guiding the continued management of these populations. Relying on mark–recapture data, we used a multistate model to estimate bison survival and breeding transition probabilities while controlling for the recapture process. We tested hypotheses in these demographic parameters associated with age, sex, reproductive state, and environmental variables. We also estimated biological process variation in survival and breeding transition probabilities by factoring out sampling variation. The recapture rate of females and calves was high (0.78 ± 0.15 [SE]) and much lower for males (0.41 ± 0.23), especially older males (0.17 ± 0.15). We found that overall bison survival was high (\u3e0.8) and that males (0.80 ± 0.13) survived at lower rates than females (0.94 ± 0.04), but as females aged survival declined (0.89 ± 0.05 for F ≥15 yr old). Lactating and non-lactating females survived at similar rates. We found that females can conceive early (approx. 1.5 yr of age) and had a high probability (approx. 0.8) of breeding in consecutive years, until age 13.5 years, when females that were non-lactating tended to stay in that state. Our results suggest senescence in reproduction and survival for females. We found little support for the effect of climatic covariates on demographic rates, perhaps because the park\u27s current population management goals were predicated from drought-year conditions. This reintroduction has been successful, but continued culling actions will need to be employed and an adaptive management approach is warranted. Our demographic approach can be applied to other heavily managed large-ungulate systems with few or no natural predators