Project Title:
Graph theoretic modeling of the population dynamics of Missouri bladderpod (Lesquerella filiformis)

Project Description (short):
(Longer description here.)

Skills needed:

Biology:

TBA

Mathematics:

TBA

Start Date:
January 2007

End Date:
December 2008

Mentors:
Prof. Michael Kelrick (Biology) mkelrick@truman.edu
Prof. Michael Adams (Mathematics), mjadams@truman.edu

Accomplishments:

• Beaulac, Dawn. "A Comparison of Habitat Suitability Models Developed at Two Spatial Scales for the Federally Threatened Plant Species, Missouri bladder-pod (Lesquerella filiformis)". Truman State University Student Research Conference. April 2004.
• Varney, L.*, M. I. Kelrick, M. Kim*, and H-J. Kim. 2005. Habitat suitability models for Missouri bladder-pod (Lesquerella filiformis): Comparison and application. Paper presented at the 9th biennial State of Missouri GIS Conference, Osage Beach, MO, February 22, 2005.
• Laura Larvey and Miju Kim. "Habitat Suitability Models for Missouri Bladder-pod (Lesquerella filiformis): Comparison and Application." Truman State University Student Research Conference. April 2005..
• Kim, M.* 2005. Development of habitat suitability models to test how spatial scale influences predictions of occurrence patterns of the federally threatened, rare plant species, Missouri bladder-pod (Lesquerella filiformis).
• Bobzien, E.* and W. Leeds*. 2005. Predicting occurrence of Missouri bladderpod (Lesquerella filiformis): Modeling over space and time. Paper presented at the 19th National Conference on Undergraduate Research, Washington & Lee University and Virginia Military Institute, Lexington, VA, April 21-23, 2005.
• Kim, H-J., E. Bobzien*, W. Leeds*, and M. I. Kelrick. 2005. Development of habitat suitability models of the federally threatened, rare plant species, Missouri bladder-pod (Lesquerella filiformis). Paper presented at Joint Statistical Meetings, Minneapolis, MN, August 7, 2005.
• Kim, H-J. "Temporal and Spatial analysis of the Missouri Bladder-pod: Ecological Statistics Undergraduate Research." Joint Statistical Meeting. August 2005.
• Bobzien, E.* and W. Leeds*. 2005. Predicting occurrence of Missouri bladderpod (Lesquerella filiformis): Modeling over space and time. Paper presented at the MAKO Undergraduate Mathematics Research Conference, Missouri State University, Springfield, MO, November 12, 2005.
• Kim, H-J., E. Bobzien*, W. Leeds*, and M. I. Kelrick. 2006. Temporal and spatial analysis of the Missouri bladder-pod: Ecological statistics in undergraduate research. Paper presented at Korean Science and Engineering Association South-Western Regional Technology Conference, Los Angeles, CA, March 4, 2006.
• Bobzien, E.*, W. Leeds*, M. I. Kelrick, and H-J. Kim. 2006. Predicting occurrence of Missouri bladderpod (Lesquerella filiformis): Modeling over space and time. Paper to be presented at the 21st Annual Symposium of the United States Regional Chapter of the International Association for Landscape Ecology, San Diego, CA, March 28-April 1, 2006.
• William B. Leeds* and Elizabeth R. Bobzien. "Predicting Occurence of Missouri Bladderpod: Modeling Over Space and Time." Truman's Student Research Conference. April 2006.
• Leeds, W.*, E. Bobzien*, M. I. Kelrick, and H-J Kim. 2006. Temporal and spatial analysis of the abundance of Missouri bladderpod (Lesquerella filiformis). Eastern North American Region/International Biometric Society, Tampa, FL, March 28, 2006.
• Bobzien, E.* and W. Leeds*. 2006. Predicting occurrence of Missouri bladderpod (Lesquerella filiformis): Modeling over space and time. Paper to be presented at the 20th National Conference on Undergraduate Research, University of North Carolina-Asheville, Asheville, NC, April 6-9, 2006.
• William Leeds and Corey Elledge. "The Federally Threatened Plant Species, Missouri Bladderpod: Model Selection and Validation." Joint Statistical Meeting. August 2006.
• William Leeds. Capstone paper.
• Bobzien, Elizabeth. "Predicting occurrence of Missouri bladderpod (Lesquerella filiformis): Modeling over space and time." The 21st Annual Symposium of the U.S. Regional Chapter of the International Association for Landscape Ecology. Sandiego, CA. 28-31 March 2006.
• Franklin, James and Elledge, Corey. "Habitat suitability modeling of a federally threatened plant species, Missouri bladderpod (Lesquerella filiformis): Evaluation of model selection criteria and model performance." Truman State University Student Research Conference. 2007.
• Presentations to Dept of Interior -- National Park Service, Biotic Resources Division of USGS

Current Students:

• James Franklin, 2006-present (Biology)
• Jonathan Vollmer, 2007-present (Computer Science)

Past Students:

• Corey Elledge, 2006 (Mathematics)
• William Leeds, 2005-2006 (Mathematics)
• Elizabeth Bobzien, 2005-2006 (Biology)
• Laura Varney, 2004 (Biology)
• Miju Kim, 2004 (Mathematics)

About Prof. Kelrick:
Biographical and Personal Sketch

About Prof. Adams:
Biographical and Personal Sketch

Project Description (kelrick and Adams) (long):

Understanding how populations change in size is central to ecology. For many species of conservation concern, estimating their population sizes over the long term is the sole focus of monitoring efforts; thus, knowledge of population dynamics takes on particular importance for land management decisions aimed at reducing extinction risks for these species, as well as for assessing consequences of such management actions. In ecology, there has been a decades-long history of using mathematical models of population dynamics, both to test the relevance of environmental factors for a population's fate, and to predict future abundance.

While mathematical methods of modeling population dynamics are well developed, it is only fairly recently that the role of spatial heterogeneity has been explicitly incorporated in these efforts. Given the pervasive influence of spatial heterogeneity in ecological processes and the acknowledged importance of spatial scale in determining and interpreting ecological data, spatially explicit population dynamics models are a welcome advance. The rise of Geographic Information System software has enabled development of these models, and the far-reaching effects of anthropogenic habitat fragmentation --- leading to subdivided populations in spatially disparate patches of suitable habitat --- have demanded it.

For rare species, and especially for endemic taxa, metapopulation dynamics modeling has emerged to address how assemblages of subdivided populations, potentially linked by dispersal, can be described as an ensemble. Here, demographic fates of each patch's individuals can be modeled separately, while the persistence of the entire patch ensemble may be understood in the context of its patterns of connectedness. For example, while populations in some patches may be flourishing, those in others may be on the wane. Flows of dispersing individuals from "successful", exporting patches may provide colonists to patches where local extinctions may have occurred. Thus, the population dynamical behavior of a species in both space and time can be modeled. This approach has become invaluable in conservation biology.

The species of interest in our proposed study is Missouri bladder-pod (Lesquerella filiformis), a winter annual closely affiliated with limestone glades developed on outcrops of the Mississippian-age Burlington-Keokuk formation in southwest Missouri (Hickey 1988). (Note that two occurrences in northern Arkansas have been recently reported by Logan [Arkansas Natural Heritage Commission; pers. comm.], one on a dolomitic glade.) Glades associated with the Burlington-Keokuk limestone are generally long and linear, following the structural trend of the underlying stratigraphy, and occurring intermittently where slope and aspect conditions coincide with appropriate bedrock horizons at the surface (Nelson and Ladd 1983). These glades are patchy at several overlapping spatial scales: from openings in forested areas (scale of 103 to 100 m2); to "islands" of exposed bedrock in a deeper-soiled matrix (scale of 101 to 10-2 m2); to mosaics composed of bare rock, very thin-soiled, pebble-clad patches supporting primarily annuals and some tolerant perennials, and deeper-soiled patches supporting herbaceous perennial and some woody vegetation (scale of 100 to 10-2 m2). The glade habitat of L. filiformis is characterized by abundant fragments of the parent carbonate bedrock, mixed with the thin, poorly developed soils, that are typically dry in summer and often saturated in winter and spring. Glades exhibit frequent soil disturbance (e.g., from sheet flow as well as cryoturbation), full insolation, wide diel temperature fluctuations and marked small-scale heterogeneity (Nelson 1985, Ware 1990, Thomas 1996, Akyz et al. 1998, Kelrick unpublished data).

Seeds of L. filiformis germinate primarily in late summer/early fall; established plants overwinter as diminutive rosettes and begin elongating indeterminate flowering stems in early spring. Seeds develop in April and May, and by mid-June of most years, only corpses of vegetative individuals remain on the glade. A substantial body of demographic data has been collected, identifying life history phases and environmental factors having important fitness consequences for individual plants and potentially influencing the dynamics of Missouri bladder-pod populations. Germination, establishment success, rosette survival, and estimates of fecundity have all been shown to vary significantly in both space and time. For example, L. filiformis seeds cycle into and out of dormancy under field conditions, and, in field experiments, ungerminated seeds have been shown to be persistent and viable through three fall recruitment seasons (Kelrick unpublished data). Fluctuating temperature conditions, like those measured at the surface of sparsely vegetated glade patches, are more stimulatory to seed germination than are temperature regimes measured 0.5 cm beneath perennial grass vegetation only centimeters away (Hanna and Kelrick unpublished manuscript). All phases of vegetative plant success can vary significantly within a single growing season (among cohorts; Harms 1992, Thomas 1996) and among years (Harms 1992, Thomas 1996), as well as among microhabitats on the glade (Harms 1992, Thomas 1996, Kelrick unpublished data). None of the factors responsible for this variation appears to act in a density-dependent manner (Harms 1992, Thomas 1996).

Anecdotal accounts, as well as repeated abundance estimates through time at particular sites, have indicated that the size of Missouri bladder-pod "populations" fluctuate markedly from year to year. Estimates of abundances of L. filiformis at the Bloody Hill Glade (BHG) site, collected annually by Thomas (National Park Service plant ecologist) at Wilson's Creek National Battlefield since 1988, ranged from more than 300,000 vegetative individuals (in 1991) to zero in two successive years (1993 and 1994) (Akyz et al. 1998). Observations of vegetative plants on Bloody Hill Glade in all years since 1994 support the inference (corroborated by seed dormancy information mentioned above) that L. filiformis seeds persist in the soil as a seed bank, and that the seed bank plays an important role in site persistence for the species.

Allozyme data from plants representing 17 sites across the geographic range of L. filiformis in Missouri indicated that the species possesses substantial genetic diversity for a "classically rare" (sensu Rabinowitz 1981), endemic annual. Significant genetic heterogeneity was observed both among (presumably due to drift) and within sites (Graham 1994). By comparison with genetic profiles of other species (e.g., Hamrick et al. 1979, Karron 1991), these data also indicated that the breeding system of L. filiformis involves outcrossing, and that breeding groups of plants may be of limited extent, relative to the size of a "site". A paternity analysis of L. filiformis seed sibships by Smart (1997) provided results consistent with a breeding system dominated by outcrossing. Westrich (1997) reported the existence of small-scale spatial structure in the pattern of L. filiformis genetic variation on BHG, despite the expected homogenizing effects of outcrossing. The inferred explanation for this pattern hinged on very limited seed dispersal leading to many, patchily distributed, local breeding groups of related individuals, with genetic differences among such groups.

Despite the wealth of information already available regarding Missouri bladderpod, especially at the BHG site, difficult questions remain. In 2005, a boom year occurred on BHG, and spatially explicit habitat suitability models have shown that the species maintained higher densities in some patches than in low abundance years, while also markedly broadening its use of certain other patch types. Indirect evidence of limited seed dispersal makes understanding this expansion of spatial occupation problematic. Metapopulation population dynamics modeling may help estimating and generating bounds for dispersal rates (apparently responsible for replenishing patches only intermittently supporting reproductive plants) that are otherwise extremely difficult to measure empirically.

The spatial distribution of these patches can be modeled using graphs, a mathematical structure consisting of a set of nodes, some of which may be joined by edges. In this setting, the nodes correspond to spatially distinguishable patches and two nodes will be joined by an edge in the graph if the patches are separated by no more than a given threshold distance. The structure of these graphs can reveal patterns of landscape connectivity and provide guidance in identifying patches which may play a critical role in the metapopulation dynamics (Urban, Keitt 2001). Recent work by Brooks (2006) suggests that changes in the structure of these landscape connectivity graphs as the distance threshold is varied can be used to identify biologically significant scales of aggregation related to the dispersal characteristics of an organism with respect to the landscape it inhabits.

Our goal is to apply these methods to the Missouri bladderpod data to generate estimates for dispersal rates and relate these estimates to the life history described using matrix population models (Caswell 2001).

This project is a follow-up to the project of Kelrick and Kim, described below: Efforts to inventory or monitor species of conservation concern are often hampered by a paucity of reliable geographic distributional information; even knowing where the organisms themselves occur locally can be problematic. Use of spatially explicit habitat attribute data to predict the distribution (often as presence/absence) of organisms of concern, or of resources crucial to them — so-called "habitat suitability models" — is increasingly prevalent in conservation biology and land management. Such predictive modeling approaches can provide valuable guidance regarding the relationships of creatures to their habitats, as well as the reliability of these relationships. Habitat suitability modeling and its supporting data structures are implemented within Geographic Information Systems (GIS), a virtual environment in which spatially registered data can be manipulated, analyzed and portrayed.

A critical implication using spatially explicit data is that the scale at which measurements are made determines what can and will be inferred from those data. Thus, data collected at any one scale will yield, at best, an incomplete picture of how the organism relates to its habitat. The goodness and utility of models aimed at defining these relationships are clearly tied to such considerations of scale; ideally, incorporating data collected at multiple scales would be desirable. Despite general acknowledgement of the importance of scale in ecology, there has been little rigorous comparison of alternative models describing the distribution of a single species, generated from multi-scale data. Such comparisons are especially warranted for species of conservation concern, since they would allow informed decisions on the most appropriate spatial scale at which to collect data needed for sound management and policy.

We propose to undertake a comprehensive habitat suitability model development and assessment, including comparing models derived from data collected at three different spatial scales. The system of interest addresses the ecological status of a federally threatened plant species, the Missouri bladder-pod Lesquerella filiformis, where it occurs on the property of a National Park, Wilson's Creek National Battlefield (Greene County, Missouri). We already possess data collected over a 6-yr period (1997, 1998, 2003), during which one of us (MIK) experimentally developed and wrote the National Park Service's monitoring protocol for the species. These data derive from a single population site, and were collected "on the ground" at two spatial scales (1m x 1m and 5m x 5m cells). We propose to extend this data set by collecting additional data from our primary site, as well as from a second site at Wilson's Creek. In addition, we will expand our ability to assess scale effects by adding a third more extensive spatial scale (15m x 15m), implemented via extensively ground-truthed digitized aerial photographs in GIS.

The qualities of this unique data set will provide numerous productive avenues to explore. 1)We will have numerous seasons' worth of data, so many combinations of data sets satisfying the needs of model development and then model validation are possible (see below). 2)We will have data at two alternative spatial scales collected in each of two years, and at three scales in other years, so that year-to-year differences can be controlled in comparisons of scale. 3)Unlike most studies examining how scale matters, our independent variables are identical at all spatial scales, as are their measurement modalities. Thus, values for the independent variables are perfectly commensurate across scales, and measurement errors are similar. These characteristics make our data particularly valuable for addressing issues of scale. 4)Data from two population sites in the same year will allow assessment of how well a model generated from one population's data predicts presence/absence patterns of the second population, when climate conditions are "held constant." This is especially valuable when the species of interest exhibits boom and bust population dynamics that are unsynchronized from one population to another. 5)Characteristics of the species of interest---its rarity, its conservation status, and its patchy distribution in space and time---make using predictive models to assess its occurrence particularly compelling.

We will pursue several model-building and -evaluation objectives: 1)comparing the qualities of habitat suitability models based on different statistical approaches; 2)evaluating various statistical metrics used to measure model qualities, as we identify the "best" model(s); 3)validating the "best" model(s), by comparing model predictions with empirical observations from a data set that is independent of the one used to construct the model(s); and 4)evaluating various statistical metrics used to measure the "goodness of fit" between predicted and observed values in the validation effort.

Logistic regression models are well established for the analysis of a binary response variable (such as presence/absence); they allow examination of the relationship of concomitant variables to a binary response. However, the habitat attribute data in this study (evaluated in the field as percent cover classes conventionally used in such plant ecological work) create attribute levels that are uneven, which often leads to a nonlinear relationship between the independent variable and the logit of the dependent variable. For these kinds of data, nonparametric logistic regression may be a favorable alternative to traditional parametric logistic regression, because it allows the logit of the dependent variable to be a nonlinear function of the independent variables. Nonparametric logistic regression models use ranks of predictor variables. Since we expect nonparametric logistic regression models to account for potential nonlinear relationships between habitat attribute data and the logit of the presence/absence response, they might more successfully predict patterns of presence/absence than more typically employed parametric logistic models. Spatially explicit ecological data often exhibit spatial autocorrelation that can compromise model quality; incorporating an autocovariate as an independent variable can successfully address this potential violation of model assumptions.

A number of model validation assessment methods for logistic regression have been proposed including classification tables (sometimes labeled "confusion matrices") and Receiver Operating Characteristic (ROC) plots. Numerous tests assessing the predictive performance of a model can be conducted based on classification tables. ROC plots are widely used in clinical chemistry; recently, their utility is achieving recognition in ecological studies involving habitat suitability models. ROC plots depict sensitivity (i.e., percentage of true positives (presences) correctly predicted) on the y-axis versus the quantity (1-specificity) on the x-axis (where specificity defines the percentage of true negatives (absences) correctly predicted) the steeper the ROC curve, the greater the predictive power.

In addition to the more generally applied classification table and ROC approaches, we will explore a validation method based on estimating Kullback information. Kullback's measures of directed and symmetric divergence serve to evaluate the difference between the generating or "true" model (i.e., the model which presumably gave rise to the data) and a fitted candidate model. The Akaike Information Criterion (AIC) and the Kullback Information Criterion (KIC) are asymptotic estimators of the directed and symmetric divergence respectively. Since a candidate model with smaller values of AIC and KIC is often considered to be closer to the true model, these criteria are commonly used for model selection among various candidate models. While AIC and KIC typically serve to determine which proposed model is most like the "idealized" generating model, in this project they will be used for both model development and model validation evaluation.

The culmination of these efforts will produce rigorously vetted habitat suitability models that can then be used to assess how a model generated from data collected at one spatial scale performs in predicting observations collected at another scale. Outcomes of these "scale-switching" comparisons can be evaluated using the "goodness of fit" approaches already mentioned. Thus, we will clarify which habitat attributes most effectively predict the occurrence of L. filiformis, as well as test whether habitat attributes maintain their explanatory value across spatial scales. To our knowledge, few such habitat suitability models have been developed for a rare plant species, and the explicit testing of the influence of scale is a currently emerging body of work. Beyond the theoretical relevance of our model development and scale assessment, this effort addresses monitoring goals for L. filiformis recently identified by the U.S. Fish and Wildlife Service.

References:

• Brooks, C. P. 2006. Quantifying population substructure: extending the graph-theoretic approach. Ecology 87: 864-872.
• Caswell, H. 2001. Matrix population models: construction, analysis, and interpretation. Second edition. Sinauer, Sunderland, Massachusetts, USA.
• de Kroon, H., J. van Groenendael, and J. Ehrlen. 2000. Elasticities: a review of methods and model limitations. Ecology 81: 607-618.
• Urban, D., and T. Keitt. 2001. Landscape connectivity: a graph-theoretic perspective. Ecology 82: 1205-1218.
• van Groenendael, H. de Kroon, S. Kalisz, and S. Tuljapurkar. 1994. Loop analysis: evaluating life history pathways in population projection matrices. Ecology 75: 2410-2415.
• Wardle, G. M. 1998. A graph theory approach to demographic loop analysis. Ecology 79: 2539-2549.