Stochastic metapopulation models and watershed estimates for playas on the Southern High Plains

Date

2009-08

Journal Title

Journal ISSN

Volume Title

Publisher

Texas Tech University

Abstract

Stochastic models incorporate the variability inherent in natural systems. Two types of stochastic modelling formats are studied and applied to populations: continuous time Markov chain (CTMC) and Itô stochastic differential equations (SDEs). CTMC models, which have discrete state space, are common in population biology. Recently, Itô SDEs, with continuous state space, have also been applied. SDE models have advantages over CTMC models, in that numerical simulations of SDE models are generally much faster than simulations of CTMC models, especially for large population sizes. In addition, the drift term in the SDE model relates directly to the population growth rate in a deterministic population model.

Differential equations for the moments of the distributions corresponding to the two types of stochastic models are derived and compared. These equations are not closed: each differential equation depends on higher-order moments. Closing the system to make a finite, solvable system requires assumptions about higher-order moments. Conditions on "closure assumptions" are derived under which the CTMC and SDE models will have either identical stationary states for the moments or identical moments for t < 0. Close agreement between CTMC and SDE models is illustrated in several numerical examples. The examples include birth, immigration, and death processes and patch occupancy models.

A simple patch occupancy model, referred to as a metapopulation model by Levins, a deterministic, scalar differential equation, follows the dynamics of the proportion of habitable patches of landscape occupied by a species. Another patch occupancy model is an extension of Levins' model, allowing for dynamic patch quality: a patch may be habitable and either empty or occupied, or may not be habitable. This extended metapopulation model is also a deterministic model but consists of a system of two differential equations. We formulate CTMC and SDE models for these patch occupancy models and derive differential equations for the moments of their distributions.

The patch occupancy models have potential application to the playa lakes on the Southern High Plains. The playa lakes (patches) are breeding habitat for amphibians. Knowing whether the playas are suitable habitat for breeding and whether they are occupied are important for conservation management purposes. To apply these metapopulation models to the playas and amphibians, information is required about playa hydroperiod, the length of time a playa contains water. Hydroperiod depends primarily on rainfall and the size of the watershed surrounding the playa. Two computational methods, the minimal distance and weighted minimal distance methods, are developed to efficiently estimate watershed sizes for large landscapes. It is shown that estimates from the weighted method are not significantly different from two other methods, based on ground surveys and topographical maps. But the survey and topographical map methods are more time consuming and costly than the two computational methods.

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Keywords

Southern High Plains, Playas, Watershed estimates

Citation