The 25,000 playa wetlands on the Southern High Plains (SHP) serve as habitat for a diverse mixture of flora and fauna. Playas are shallow recharge wetlands with each individual playa existing within its own watershed. Amphibians on the SHP depend on the playas for breeding. Playas must contain water during the breeding period in order for eggs and larvae to mature. The timing and length of the playa hydroperiod (the period of time during which the playa contains water) are critical parameters in amphibian survival. The patchy occurrence of playa wetlands on the SHP, and on the Great Plains as a whole, suggest that wetland inhabitants live in multiple local populations sustained by occasional interaction among metapopulations. The mechanisms of how these wetland mosaics maintain their metapopulations are poorly understood. Yet clearly, mounting destruction (through sedimentation and chemical runoff) results in reduced wetland hydroperiod and wetland density, and increased isolation among wetlands. Indeed, sedimentation in playa wetlands decreases hydroperiod, increases the rate of water loss, and eventually fills the playa to such an extent that it is rendered non-functional as a wetland. Demographic and environmental variability, as well as global climate change, will have a significant impact on the reproduction and survival of amphibian populations over the long term. Current metapopulation theory is primarily based on deterministic mathematical models, i.e, systems of ordinary differential equations. The objectives of this project are (1) to extend standard deterministic metapopulation models to new stochastic differential equation models, (2) to develop new stochastic metapopulation models for populations that depend on a dynamic landscape, (3) to collect data on amphibian community composition in the playas, and (4) to apply these new stochastic metapopulation models to amphibian populations on the SHP based on the data collected in this project.
Stochastic differential equation modeling in ecology is a relatively new area but is an important and rapidly expanding area of interest in the mathematical and biological sciences. There have been no comprehensive mathematical models for the dynamics of species inhabiting the Southern High Plains (SHP). This project, in addition to advancing the theory and application of stochastic differential equations, will provide a greater understanding of amphibian populations and the ecology of the SHP. Graduate students will be trained and programs strengthened at Texas Tech University and Oklahoma State University in biological modeling, stochastic mathematics, wildlife management, and ecology. In addition, the results of this project will lead to recommendations for conservation of amphibian populations and for maintenance of playa integrity that will have a broad impact on future research on the SHP, the Great Plains as a whole, and on other similar semiarid environments.
Amphibian populations in the Southern High Plains (SHP) of Texas rely on playa wetlands as the primary source of breeding habitat. Playas are shallow, approximately circular, ephemeral wetlands that quickly fill and empty after rainfall events. More than 25,000 playas exist in the SHP of Texas, making it the densest region of playas in the United States. Because playas receive all their water from surface runoff from the surrounding uplands and row crop agriculture dominates these catchments, playas in cropland watersheds receive significant amounts of sediments. In this research, new stochastic metapopulation models are developed and data collected in field research on playas of the SHP. The intellectual merit of advancing the theory of stochastic modeling while promoting teaching, training, and learning has been achieved and has led to broader impacts that have increased our understanding of the ecology of the SHP and of agricultural practices and environmental changes that negatively impact amphibian survival and playa integrity. Selected physical and hydrological characteristics and resident amphibian community diversity were examined in playa wetlands surrounded by three dominant land use types (native grassland, cropland, and USDA Conservation Reserve Program CRP) in 2008 and 2009 in the SHP of Texas. Sediment depth and water loss rate differed among land use types. Playas surrounded by cropland had deeper sediments, deeper initial water depths due to greater runoff, faster water loss rates, and contrary to previous research findings, longer hydroperiods, than playas in native grassland watersheds. Playas surrounded by CRP were generally intermediate to cropland and grassland playas. Land use type did not influence cumulative amphibian richness, but did influence the frequency of occurrence of individual species. Tiger salamanders, Great Plains toads, Woodhouse’s toads, spotted chorus frogs, Plains leopard frogs, and spadefoot toads were observed most frequently across all land use types. In 2008, a relatively dry year, individual amphibian species were observed at a lower frequency in playas than in 2009. In addition, specific land use influences on individual species varied in 2008, as observed frequencies of certain species were lowest in native grassland (salamanders) and CRP (Great Plains toads), or highest in cropland (spadefoot toad larvae) and CRP (Plains leopard frogs and spotted chorus frogs) playas. In 2009, differences in species occurrence among land use types were also observed, with higher frequencies in cropland (Woodhouse’s toads, Plains leopard frogs) and CRP (spotted chorus frogs) playas. Overall these results differ from the published literature. However, consistent with previous studies, hydroperiod best explained species richness and predicted the probability of occurrence of the predominantly observed species relative to sediment depth, water loss rate, and percent native grassland. Playa characteristics and data on rainfall events were used in the development of a stochastic dynamic model to estimate average hydroperiod. Rainfall was modeled using probability distributions for the time between rainfall events, the magnitude of rainfall events, the duration of rainfall events, and the distribution of rain across a rainfall event. Evaporation, seepage, and runoff were combined in a water budget equation. It was found that playa shape has the greatest impact on hydroperiod. In addition, a computer simulation of a stochastic model representing playas and amphibian populations on the SHP was developed to test effects of climate change, agricultural practices, and amphibian movement patterns on long-term amphibian survival. Rainfall events, evaporation, playa density, playa sedimentation depth, amphibian reproduction, and amphibian movement patterns were included in the computational model. Increased sedimentation and prolonged periods of drought were shown to have a significant impact on amphibian survival. In a modeling study of the boreal toad, Bufo boreas boreas, a species listed as endangered in Colorado and New Mexico, the impact of a fungal pathogen, breeding onset, and breeding frequency on amphibian survival were investigated. Analytical and numerical investigations of the model showed that environmental or climate changes that increase the pathogen or delay breeding onset and frequency will further exacerbate this declining population. In research that advances stochastic modeling and methods, stochastic ordinary differential equations (SDEs) and stochastic partial differential equations (SPDEs) are derived from first principles and applied to models such as Levins metapopulation model, age- and size-structured models, reaction-diffusion equations, and correlated random walk models. Discrete stochastic models are constructed, carefully taking into account the inherent randomness in births, deaths, transitions, reactions, movements, and size changes. As the time interval decreases, the discrete stochastic models lead to systems of Ito SDEs. As the size and age intervals or spatial interval decrease, SPDEs are derived. Comparisons between numerical solutions of the SPDEs and independently formulated Monte Carlo calculations support the accuracy of the derivations. In addition, calculation of the moments of the distribution of SDE models are compared to the underlying Markov chain models. This work has been extended to other metapopulation models and to numerical procedures for simulating values of stochastic integrals.
