GayaThesisFinal

Why is the Grass Dying?
An Integral Projection Model For a Pseudoroegneria spicata (Bluebunch Wheatgrass)
Population in Eastern Washington
by
Heather Gaya
A thesis submitted in partial fulfillment of the requirements
for graduation in Environmental Studies- Biology
Whitman College
2016

ii
Table of Contents
Abstract...........................................................................................................................iii
Introduction......................................................................................................................4
Methods............................................................................................................................8
Data Collection............................................................................................................8
Generalized Model.....................................................................................................10
Survival and Growth Component...............................................................................11
Reproductive Component...........................................................................................12
Elasticity Calculations...............................................................................................14
Results............................................................................................................................15
Initial Equations.........................................................................................................15
Projection Models......................................................................................................15
Elasticity Analysis......................................................................................................16
Discussion......................................................................................................................17
Acknowledgements........................................................................................................24
Literature Cited ..............................................................................................................25
Appendix A....................................................................................................................27
Pre-Modeling Equations............................................................................................27
Integral Projection Model..........................................................................................34
Stable State and Reproductive Value Predictions .....................................................37
Elasticity ....................................................................................................................38
Appendix B....................................................................................................................40
Integral Projection Model Matlab Code....................................................................40
Elasticity Analysis Matlab Code................................................................................42
Chapter 2: Environmental Part.......................................................................................43

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Abstract
Bluebunch wheatgrass (Pseudoroegneria spicata), a bunchgrass native to the
Western United States and Canada, has declined over the past century from the
combined pressures of overgrazing, intensive agricultural production, and invasive
annuals. Site-specific soil moisture availability may further stress P. spicata
populations and lead to dramatic differences in population dynamics across small
spatial scales. Using a 4-year data set from a study site near Wallula, WA, I
parameterized two integral projection models to evaluate the differences in P. spicata
populations on north- and south-facing slopes and identify the demographic transitions
most important for population growth. Individuals on south-facing slopes were found
to have greater reproductive output and more culms on average than north-facing
slopes. Elasticity analysis suggested that survival and growth transitions were more
important than reproduction for both north- and south-facing slopes. These results
suggest that despite dramatic difference in moisture availability and reproductive
output, individuals on both slopes are likely to shrink in size in response to difficult
moisture conditions. These results suggest that plant communities will continue to
shrink in response to climate change, resulting in a reduction, or even an absence, of P.
spicata individuals within the species’ natural range.

4
Introduction
Within arid plant communities, bluebunch wheatgrass (Pseudoroegneria
spicata), a perennial bunchgrass native to the western United States and Canada, is
often considered one of the most important grass species for forage production for
wildlife and livestock (Zlatnik 1999; Tilley and St. John 2013). The bunchgrass grows
as numerous tightly packed ramets, or culms, connected by an extensive root system,
allowing older established plants to be resistant to drought and soil disturbances (Tilley
and St. John 2013). Though the leaves, stems, and flowers die each year, the plant can
regrow from the same basal material and root system each spring, expanding the
number and size of culms produced in a given year based on moisture and nutrient
availability. With favorable conditions, P. spicata bunches can grow more than 20 cm
in diameter and up to 2 m in height (Zlatnik 1999). Over the last century, P. spicata
populations have been declining from the combined pressures of overgrazing, intensive
agricultural production, and invasive annuals (Boyd et al. 2013; Humphrey and Schupp
2004; Harris 1967). As P. spicata populations decline, the invasive annual Bromus
tectorum (cheatgrass) has come to dominate much of the western United States that
formerly supported shrub-steppe communities. Once removed from an ecosystem, re-
establishing perennial grasses from seeds is often time consuming and unsuccessful
(Boyd et al. 2013).
In many respects, P. spicata is considered a more desirable species than B.
tectorum and other invasive annuals. Whereas B. tectorum grows in short, dense
stands, P. spicata tends to grow with space between individual plants, allowing for the
foraging and nesting of insects, birds and small mammals (O’Connor 2015). Initial P.

5
spicata growth occurs in September or October, taking advantage of early fall and
winter soil moisture (Harris 1967). Depending on the local climate, the reproductive
stage usually occurs in late spring or early summer, after which the plant falls dormant
until early fall. While growing or reproducing, P. spicata populations retain high
moisture content, decreasing the plant’s flammability during these life stages (Zlatnik
1999). If fire does occur, the relatively large spacing between P. spicata individuals
reduces the spread of fire across long distances (Tilley and St. John 2013), making it an
ideal species for ecosystems experiencing hot, dry summers. This bunchgrass also
provides a more reliable and nutritious food source for wildlife and domestic animals
than B. tectorum (Tilley and St. John 2013; Zlatnik 1999). Previous studies have shown
that B. tectorum yields can vary as much as 1000% between years (Harris 1967). Once
B. tectorum reaches maturity, the plant dries out rapidly, transforming into an
unpalatable, nonnutritive, and flammable material. Moreover, B. tectorum thrives in
disturbed areas and can effectively out-compete seedling perennial plants (Melgoza et
al. 1990; Aguirre and Johnson 1991; Humphrey and Schupp 2004) allowing B.
tectourm to dominate landscapes previously covered by P. spicata and other native
species. For these reasons, preserving established P. spicata populations is often
preferable to reseeding efforts.
One way that conservationists can maximize the efficacy of P. spicata
restoration efforts is by constructing projection models and determining which
demographic variables most strongly affect future population trends. For instance, the
integral projection model describes survival, growth, and reproduction as continuous
functions of an individual’s size (Easterling et al. 2000) making the model suitable for

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both plant and animal populations. Using similar assumptions to matrix models,
integral projection models predict a population’s growth rate, lambda, as well as the
associated eigenvectors, allowing for predictions of stable-size distributions and size-
specific reproductive vectors (Ellner and Rees 2006; Rees et al. 2014). These
projection models allow conservationists to identify the most effective restoration
strategies for the modeled species, thus minimize restoration cost (Dalgleish et al
2011). Additionally, using knowledge of the population state at a given time, integral
projection models can be used to find the expected population state at all subsequent
times, allowing for both short and long-term predictions of population growth or
survival (Easterling et al. 2000; Coulson 2012).
Previous research on P. spicata indicates that survival depends in part on plant
size, as evidenced by differing death rates between seedlings and established plants
(Humphrey and Schupp 2004). In a 30-year study at the US Sheep Experiment Station
in Idaho researchers used an integral projection model to evaluate the population
dynamics of a P. spicata population at the study site (Dalgleish et al. 2011). The
researchers observed that moisture, especially late-winter snow, had the largest effect
on P. spicata population growth. For demographic transitions, Dalgleish et al found
that survival and growth had a more significant impact than seedling establishment on
overall population growth. Similar ideas were proposed by researchers of a local P.
spicata population at Wallula Gap Biological Station in Southeast Washington. The
researchers noted that P. spicata density and reproductive output (recorded as the
average number of culms with spikes) differed between north- and south-facing slopes
(Oschrin 2013; Simonson 2014). While Oschrin and Simonson agreed that these

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differences were likely explained by differences in soil moisture, they proposed that
seedling establishment, rather than survival and growth of adult plants, was the most
important demographic transition for population growth. In order to better understand
these conflicting results, I decided to use my own integral projection model to assess
the population dynamics of the Wallula Gap Biological Station’s P. spicata population.
Based on these studies, I hypothesized that the difference in moisture between
the north- and south-facing slopes at the Wallula Gap Biological Station was the main
driver of the differences in P. spicata population structure. I hypothesized that the
moisture difference would lead to low seedling survival rates on south-facing slopes,
increasing the selection for south-facing P. spicata with high reproductive rates.
Incorporating the results from the 30-year study at the US Sheep Experiment Station, I
further hypothesized that despite increased reproductive rates, the survival and growth
of large, established plants would have a greater impact on population growth rates
than seedling establishment or seedling survival. Using my own integral projection
model, I predicted that the survival and growth patterns of plants on north-facing and
south-facing slopes would differ significantly. I expected that these differences would
lead to different lambda values on each hillside. I also predicted there would be
significant differences in the current and future projected reproductive outputs between
the north and south facing P. spicata populations, with south facing plants producing
more seeds per individual. Finally, I predicted that the highest elasticity values for P.
spicata would be found for the survival and growth equations of each population.

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Methods
Data Collection
I studied P. spicata at Spring Gulch in the Wallula Gap Biological Station near
Wallula, Washington (46°00'12" N, 118°54' 05" W). Study site elevations range from
255 to 400 m. The Columbia Basin was once characterized as a shrub-steppe
ecosystem due to the absence of trees and abundance of grass and shrub species
(O’Connor 2015). As a result of expanded human development and overgrazing, much
of the shrub-steppe ecosystem has been converted to farmland or altered by non-native
species such as B. tectorum and Centaurea solstitialis (yellow star thistle) (O’Connor
2015). The study site currently consists of an arid grassland ecosystem dominated by
native perennial grasses, Ericameria nauseosa (rabbitbrush), and invasive B. tectorum.
The density and species composition of vegetation varies between north and south-
facing hillsides. North-facing hillsides tend to be predominately covered by P. spicata
and other native perennials, whereas south-facing slopes tend to be dominated by B.
tectorum. These differences may be attributed to different moisture concentrations
between hillsides, as south facing slopes receive more direct sunlight, leading to
increased moisture loss (Oschrin 2013; Simonson 2014).
I and multiple other research students collected data from 14 long-term
demography plots each summer from 2012 to 2015. The plots were constructed in 2012
in order to mark a subset of P. spicata individuals for repeated study. We surveyed 14
demography plots, seven located on north-facing slopes and seven on south-facing
slopes. Plots were located approximately 200 meters apart at varying elevations along

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hillsides. Each plot consisted of a 1 m by 10 m transect containing ten 1 m by 1 m
quadrats. For each transect, we surveyed the plants in the odd numbered quadrats. Each
odd numbered quadrat was further divided to create a grid for mapping plant locations
within the quadrat.
To ensure accuracy in data collection, we recorded plant measurements as
systematically as possible. Each P. spicata found in the studied quadrat was tagged
with a unique number and measured for total number of culms, height of the tallest
culm and number of culms containing spikes. In 2014, we also collected random
samples of 5 spikes per plant from 30 plants from each hillside to calculate the average
number of seeds produced per spike for both north- and south-facing plants. To mark
each plant’s location, we indented aluminum tree tags with the plant’s unique number
and nailed these tags to the ground at the base of each plant. For plants with culms
containing spikes, we took a random sample of no more than 5 spikes and recorded the
number of spikelets per spike. Newly discovered plants were defined as any culms > 5
cm away from all tagged plants and not already considered part of another plant. Each
P. spicata’s location was then mapped within the quadrat to allow for easier
identification in following years. We considered plants dead after two consecutive
years without visible culms, though no plants were ever found to recover after being
recorded dead in a previous year.
One frequent problem with P. spicata data collection was frequent
misidentification of small, non-reproducing plants. In order to verify P. spicata
identification, we also recorded if plants without spikes had evidence of spikes from
past growing seasons. Plants that lacked spikes or evidence of spikes across multiple

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years were removed from the data set during modeling but left tagged in the plot to
allow for additional identification efforts in future years.
Generalized Model
I used an integral projection model to evaluate differences in the population
trends of P. spicata on north- and south-facing slopes in Spring Gulch from 2012 to
2015. The integral projection model describes how a size-structured population
changes throughout time (Easterling et al. 2000). The model can be used to understand
how changes in individual performance such as growth, survival or fecundity,
influence population dynamics and long term ecological trends (Rees et al. 2014). This
can be especially useful when comparing two populations to determine if and how the
populations differ from one another. Once constructed, integral projection models can
help predict theoretical stable size distributions or be used to determine the relative
impact of life stage transitions on population dynamics (Easterling et al. 2000).
The integral projection model for the proportion of individuals of size z’ at time
t +1 is given by:
(1) n(z',t+1) = k(z',z)n(z,t)dz
a
b
∫
where the kernel, k(z’,z) describes all possible size transitions from z to z’. The current
population at the time of data collection is represented by a probability density
function, n(z, t), which represents the proportion of individuals of size z at time t. In
general, the unit t =1 represents the time interval in which the data was measured. For
my model, I defined size as the total number of culms of a plant and used time intervals
of one year. The kernel is defined as:

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(2)
k(z',z) = P(z',z) + F(z',z)
= s(z)G(z',z)+pb(z)b(z)C0 (x)
where P(z’,z) and F(z’,z) are the survival-growth and reproductive components of the
kernel respectively. P(z’,z) is composed of the size-specific survival function, s(z), and
the growth distribution, G(z’z), which defines the probability of a plant transitioning
from size z to z’ during the t to t+1 time interval. F(z’z) incorporates three different
equations: the production of seeds, the survival of seedlings, and the size distribution of
seedlings produced between time t and time t+1. These were called pb(z), b(z) and
respectively. The integral was evaluated on the interval [1, (max
observation*1.1)] to ensure that a reasonable range of plant sizes was included in my
model.
Survival and Growth Component
To construct the survival equation s(z), each plant was assigned a value (1 for
survival, 0 for death) for each time transition from 2012 to 2015. The average survival
of each culm size was then plotted as a function of culm number. For instance, if three
plants of size 20 had values of 1, 1 and 0, the average survival would be plotted as (20,
0.67). The relationship between culm number and survival was then fitted with a
logarithmic regression equation in the form:
(3) z' = a*ln(z)+ b
where z’ represents the probability of survival to time t+1, and z represents the number
of culms.
C0 (x)

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For each plant of size z at time t, G(z’,z) represents the probability of that plant
reaching size z’ by time t+1. The growth function was assumed to be a normally
distributed probability density function of the form:
(4) G(z',z) =
1
2Πσ
*e
−1
2
(z'−µ
σ )2
where σ is the standard deviation in culm size and µ represents the expected size
changes of a plant between time t and time t+1 (Dalgleish et al. 2011, Briggs et al.
2010). For the growth equation µ, I plotted each plant’s culm number as a function of
its size the previous year and performed linear regression for each year’s size changes.
The overall function to describe the average growth across all transitions was found
using StatPlus for Excel’s multiple linear regression analysis software.
Reproductive Component
Since the production of seeds requires the presence of spikes, the equation pb(z)
was added to the fecundity kernel to represent the probability of having spikes. The
percentage of time each observed plant had spikes from 2012 to 2015 was plotted as a
function of culm number and fitted with a logarithmic regression in the form of Eq (3).
Seedling survival was estimated by:
(5) b(z) = ((1− d(z))S(z))−.33
where S(z) represents the number of seeds produced by a plant of size (z) and d(z) is
the survival probability (Briggs et al. 2010). Though some seedling survival data had
been collected using shaded and un-shaded plots on south-facing hillsides to represent
north-and south-facing hillsides, I felt this data set was not sufficiently representative

13
of seedling survival. Instead, I ran the model with different values of d(z) to observe
the effect of seedling survival rates on the model’s predicted stable size distributions
and reproductive values. The values chosen reflected low, medium, and high survival
rate estimates found in published P. spicata studies (Boyd et al. 2013, Aguirre and
Johnson 1991). The exact values can be seen in Table 1. Seed collection data from the
2014 random seed samples were used to estimate the average number of seeds
produced per plant on each hillside. Seed production was then plotted as a function of
culm size and the overall function to describe seed production was found using
StatPlus for Excel’s multiple linear regression analysis software. The product of
survival probability and seed production was then raised to the power of -0.33 to
represent the effects of density on seedling survival (Briggs et al. 2010).
Finally, the size distribution of the new seedlings, C0(x), was assumed to be a
normal distribution of the form of Eq (4). In this case, however, the mean was
calculated as a single point, rather than as an equation. This value represents the mean
seedling size for each hillside, as seedlings are only classified as seedlings for one year
before being counted as regular plants in subsequent years. Average seedling size
distributions for each slope were found using shaded and un-shaded experimental plots
setup on south-facing hillsides. The un-shaded plots were designed to simulate south-
facing moisture conditions, whereas the shaded plots approximated north-facing
moisture conditions. In the model, seedling size was assumed to be independent of
parental size. For the sake of simplicity, all future seedlings produced in the model are
assumed to be 2 culms large when first surveyed.

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Elasticity Calculations
Since the integrand proved too complicated to evaluate over an undefined time
frame, the kernel for each model was estimated as a 500 by 500 matrix, with each value
representing mesh points for different values of z and z’ (Rees et al 2014). I then
calculated the dominant eigenvalue for each hillside and found its corresponding left
and right eigenvectors using Matlab’s built-in eigenvalue commands. The dominant
eigenvalue represents the growth rate, lambda, of each hillside, whereas the left and
right eigenvectors, v(z) and w(z), correspond to the size-specific reproductive value
and stable size distribution respectively. Confidence intervals for lambda can be found
using jackknife estimations of variance (Zhou, Obuchowski, and McClish 2011).
In the integral projection model, elasticity describes the proportional effects of
kernel changes on lambda. In other words, elasticity seeks to determine how a
proportional change in a vital rate such as survival, growth, or reproduction will affect
the population growth rate (Benton and Grant 1999). This value can then be used to
determine where to focus management or conservation efforts, as the transitions with
the highest value indicate a high contribution to the fitness of the organism (Easterling
2000; Benton and Grant 1999) When considering density-dependent populations,
elasticity values can be found using the following relationship:
(6) e(x1,x2 ) =
k(x1,x2 )
λ
i s(x1,x2 ) =
k(x1,x2 )
λ
i
v(x1)w(x2 )
w,v
with w,v equal to the matrix approximation of w(z)v(z)dz∫ (Easterling 1998; Benton
and Grant 2000). In my model, these values were found using Matlab code included in
the appendix of Eastering et al. (2000).

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Results
Initial Equations
For individuals on both north- and south -acing slopes, survival, growth, and
reproduction increased with culm number (Table 1). Survival of plants with >150
culms approached 100% for both hillsides (Fig.1). There was no significant difference
between the survival equations for north- and south-facing hillsides (t = 0.47, d.f. =
698, p = 0.6, ANCOVA, Fig 1). Both north- and south-facing P. spicata growth
equations had positive slope values less than 1 indicating reversion to smaller size is
common as size increases (Table 1). P. spicata on south-facing slopes had a higher
probability of having spikes (t = 2.0, d.f. = 698, p = 0.05, ANCOVA, Fig 3) and tended
to produce more seeds (t = 18.2, d.f. = 698, p <0.001, ANCOVA, Fig 4) but there was
no significant difference in seedling size distributions between hillsides (t = -1.8, d.f. =
67, p = 0.08, two-tailed, Fig. 5). A one-tailed Wilcoxon Signed Ranks Test indicated
that between 2012 and 2015, individuals in the south-facing population tended to be
slightly larger than north-facing individuals (W+ = 302396.5, p < 0.001, Fig 6).
Projection Models
The projection models indicate that retrogression (shrinkage in culm number
between years) occurs on both hillsides, especially on the north-facing slope (Figs. 7
and 8). The lambda (growth rate) for north- and south-facing slopes was found to be
0.828 ± 0.083 and 0.644 ± 0.102 respectively, suggesting that both populations are
shrinking. However, 95% confidence intervals for these lambda values overlap,

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indicating the two population growth rates are not significantly different from one
another. For north-facing slopes, the effects of retrogression on the survival-growth
kernel can be most clearly seen for plants with fewer than 125 culms (Fig. 7). For
south-facing slopes, the survival-growth kernel noticeably decreases after 300 culms,
though the overall kernel values for south-facing slopes tend to be lower than north-
facing slopes (Fig 8). As expected from the size distribution patterns for 2012 to 2015,
the projected stable size distribution predicts that south-facing individuals will be
slightly larger than north facing individuals (t = -2.4, p = 0.008, one tailed, Fig 9). The
predicted reproductive output of south-facing plants will also be larger (Fig 9).
However, for both populations the largest proportions of individuals in the stable size
distributions are projected to have fewer than 50 culms. Running the two integral
models with higher values for seedling survival had no significant effects on the model
patterns or distribution projections (Table 2).
Elasticity Analysis
Perturbations of the survival-growth kernel had larger effects on lambda than
perturbations of recruitment for both hillsides, but south-facing elasticity values were
an order of magnitude greater than north-facing values (Fig. 10 and 11). For the north-
facing population, the greatest elasticity values were for individuals with 25 to 100
culms transitioning to fewer than 50 culms and for recruitment from individuals with
greater than 400 culms (Fig 10). Similar patterns were found for the south-facing
population (Fig 11).

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Discussion
These models indicate that survival and growth are the most important
demographic transitions affecting population growth for P. spicata. Though the kernel
values for reproduction increased slightly when the model was run with different
seedling survival rates, the survival-growth kernel consistently dominated the overall
kernel values. This suggests that reproductive differences between north- and south-
facing P. spicata populations are not the main contributors to the differing plant
densities on the two hillsides, challenging the findings of previous P. spicata research
students (Oschrin 2013; Simonson 2014). While the importance of the survival-growth
kernel initially appears consistent with other P. spicata integral projection models
(Dalgliesh et al. 2011), the highest kernel values were found for individuals between 25
and 100 culms rather than for adult plants with more than 150 culms. One possible
interpretation of the elasticity models is that once plants have been observed for more
than one summer, they are no longer counted as seedlings and instead are represented
in the population as very small plants. Perhaps reproduction is important, but only if
new seedlings can grow to a threshold size where they are large enough to access
deeper soil moisture and outcompete other plants in their vicinity. The elasticity
models suggest that successful P. spicata conservation strategies will incorporate
methods that protect small and medium sized individuals, ensuring their continuation
in the population and their eventual transformation into large, adult plants with low
mortality. Strategies could include protection from overgrazing, controlling P. spicata
competitors, or relocating large plants grown elsewhere into the desired location.

18
However, these elasticity results do not necessarily show an accurate depiction
the biology of the two P. spicata populations. Previous researchers working on tussock
bunchgrasses in the southern United States have argued that elasticity values change
with environmental variation (Vega and Montaña 2014). According to their studies,
lambda values were more sensitive to retrogression or stasis during dry years but were
more heavily influenced by fecundity and growth during times of moderate
precipitation. If these trends are true for the Wallula Gap P. spicata population, this
could explain both the strong regression patterns in the integral projection models as
well as the relatively low elasticity values for reproduction. Moreover, this could
potentially account for previous research students’ findings that seedling survival is
driving the differences in plant density between the two hillsides (Oschrin 2013;
Simonson 2014). Running my models again with data from multiple years separated
into separate graphs would likely reveal more variability in the size-transition elasticity
values.
Given that survival and growth appear to play an important role in population
growth dynamics, it is surprising that the north and south-facing populations had
similar survival and growth equations. Working under the assumption that moisture
was driving the differences in population densities and size distributions between the
two populations, I expected to see these differences reflected in the survival and
growth equations. One possible explanation is that the south-facing plants under
observation tended to be large, reproducing plants when first surveyed and we had few
chances to observe the survival and growth trends of plants with fewer than 30 culms.
Across the four years of data collection, we did not record any mortality in south-

19
facing plants larger than 150 culms. However, the seedling data from planted seeds in
our open plots on south-facing slopes indicate extremely high mortality of seedlings
and small plants for several years until individuals reach sufficiently large size
thresholds. This high mortality was largely unaccounted for in the demography plots,
which may have resulted in an inaccurate picture of the survival rates of small plants
on south-facing slopes.
My models also indicate that retrogression plays an important role in the life
cycle of P. spicata plants. Though survival of plants with >150 culms approached
100% for both hillsides, the survival-growth kernels indicate that larger plants do not
tend to remain at large sizes indefinitely. The concentration of high elasticity values
around medium sized plants in the 35 culm to 100 culm range suggests that
retrogression may be an important step prior to P. spicata mortality. Prior research on
shrub-steppe and grassland ecosystems suggests that retrogression tends to occur when
individuals are experiencing environmental stress (Vega and Montaña 2014; Dalgliesh
et al. 2011; Busso and Richards 1995). As plants decrease in size in response to one
environmental variable, they are subject to the high mortality rates experienced by
smaller plants, possibly leading to an overall decline in population density at that
location (Busso and Richards 1995). Though the survival and growth patterns of the
two hillsides are not statistically significantly different, the projected size distributions
indicate that south-facing plants are expected to be larger than north-facing plants.
Since the survival of P. spicata appears to be closely tied to culm number, the large
proportion of small individuals in the predicted north-facing population may indicate
the eventual absence of this species at the study site. It is currently unclear why

20
formerly large plants that shrink to smaller sizes are subject to the same conditions as
plants that have remained a consistently smaller size, as shrinking plants presumably
maintain a large root system even as they shrink. The inability to investigate the root
systems of plants undergoing retrogression makes this question difficult to answer
without additional research.
In all likelihood, many of the differences seen between the north- and south-
facing hillsides are related to variables that were unaccounted for or obscured by the
simplified format of my projection model equations. For instance, competition from B.
tectorum, native perennials and other P. spicata individuals undoubtedly affected the
survival and growth of the monitored populations. On south-facing slopes, P. spicata
individuals are surrounded by and competing with a mixture of cheatgrass, other P.
spicata, and the occasional centaurea solstitialis (yellow starthistle). On north-facing
slopes, the majority of individuals can be found within close proximity to five or six
other native bunchgrasses and a variety of smaller grasses and annuals. Whereas south-
facing individuals may be experiencing greater stress from low moisture content due to
evaporate water loss, north-facing individuals experience a greater number of
competitors, which may create a situation where individuals on both hillsides are
experiencing low moisture or nutrient availability but for different reasons. Due to the
constraints of both the format of the model and my own mathematical capabilities, my
integral projection models did not account for differing species density and
composition on north- and south-facing hillsides. Moreover, I was parameterizing the
models in the hope of uncovering patterns that correlated with the moisture differences
between hillsides, but without actually including moisture or climate data in my

21
models. This indirect approach may have partially obscured the true impacts of
moisture differences between hillsides, as I was unable to include plant-specific
environmental conditions or variability in moisture between years. Previous studies
have suggested that water stress causes severe reduction in growth rates in grass
species (Busso and Richards 1995) and greatly reduces the rate of successful
recruitment (Wilson 2014). Though it is clear from both my models and previous
literature that moisture has a significant impact on P. spicata population dynamics,
addition research may be necessary to determine if other factors such as competition
are also strongly contributing to the differences between north- and south-facing
slopes.
The projected stable size distributions are another area of concern within my
model. Though the models predict that individuals on both north- and south-facing
hillsides will shrink to fewer than 50 culms, there are reasons to doubt this projection.
Based on the variability in P. spicata culm number between 2012 and 2015, there is no
reason to believe that neither the north- or south-facing populations are operating under
equilibrium conditions. Previous studies of projections calculated from population
eigenvectors have suggested that stable-size projection models perform poorly for
populations with non-equilibrium dynamics (Benton and Grant 1999). This critique
likely holds true for my own models, as the predicted-stable size distributions suggest
that plants will shrink to small sizes but somehow not experience the observed high
mortality of small plants. In all likelihood, a population with plants shrinking to fewer
than 50 culms will simply die off rather than reaching equilibrium. The non-
equilibrium dynamics of the studied populations undermines the validity of my

22
projections, leading to two improbable and likely inaccurate predictions of future size
distributions.
It is important to note that my model was only including 3 intervals of data,
including the very dry summer of 2015. When calculating the growth and survival
equations for the north- and south-facing P. spicata populations, it was immediately
obvious that the 2014 to 2015 transition was substantially different than the other two
time intervals. Individuals of all sizes experienced mortality and many of the largest
plants in the demography plots regressed to almost half their previous year’s size. With
only 3 intervals of data, it is difficult to know the long-term implications for the
population. For instance, while extreme temperature or soil moisture depletion may
increase mortality for multiple years, P. spicata may be sufficiently well adapted to
environmental variation to regrow once the environment returns to more favorable
conditions. Moreover, elasticity analysis in future years may reveal different trends
based on future moisture levels, which may shift our understanding of P. spicata’s
resilience to increased temperatures or moisture variability. Though my integral
projection model provides valuable insight into the population dynamics of the P.
spicata in the Wallula Gap, awareness of the model’s possible limitations indicates that
we should approach the model’s conclusions with caution.
The creation of this model was impeded by lack of climate information and
accurate seedling survival and germination rates. Though previous research on the
Wallula Gap P. spicata populations suggested a range of possible seedling survival
rates (Wilson 2014), my model was unable to completely account for differing seedling
survival and germination rates between hillsides. This incomplete information may

23
have led to slightly inaccurate kernel values in the integral projection model and
decreased the accuracy of elasticity calculations. Though my models indicate that
survival and growth of medium plants plays a more significant role than recruitment in
overall population growth, I suspect these results may not be entirely accurate. Based
on seedling survival rates for the few seedlings we did record in demography plots,
there appears to be a difference in recruitment success between north- and south-facing
hillsides. I have difficulty believing that these differences play no role in the two
population’s differing density and seed production. My inability to incorporate these
possible differences significantly undermines the capacity for my model to accurately
reflect the biology and population dynamics of the hillsides I was studying.
In order to assess the accuracy of my model’s results and expand our
understanding of P. spicata population dynamics, future data collection must include
both climate and seedling observations. Understanding the response of plant
communities to differing moisture availability or temperature conditions may help
scientists assess the effects of climate change to plant communities. Additional
competition research, especially with small and medium sized plants, may also assist in
accurate modeling techniques and help inform future conservation efforts.

24
Acknowledgements
Dr. Tim Parker for his guidance and patience throughout the thesis process; Whitman
College Perry Grant for funding; Dr. Doug Hundley for his assistance with calculus;
Chris Dailey, Erin Campbell, Molly Simonson, Alice Wilson, Kyle Moen, Emma
Oschrin, and Shelley Stephan for data collection.

25
Literature Cited
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Cheatgrass Competition on Seedling Development of Two Bunchgrasses.” Journal of
Range Management, 347–54.
Benton, Tim G., and Alastair Grant. 1999. “Elasticity Analysis as an Important Tool in
Evolutionary and Population Ecology.” Trends in Ecology & Evolution 14 (12): 467–
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Boyd, Chad S., and Jeremy J. James. 2013. “Variation in Timing of Planting Influences
Bluebunch Wheatgrass Demography in an Arid System.” Rangeland Ecology &
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Tenhumberg, and Daniel Riser-Espinoza. 2010. “Structured Population Dynamics: An
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0706.2012.00035.x.
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Adler. 2011. “Climate Influences the Demography of Three Dominant Sagebrush
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Easterling, M R. 1998. “Integral Projection Model: Theory, Analysis, and Application.”
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Carolina State University.
Easterling, Michael R., Stephen P. Ellner, and Philip M. Dixon. 2000. “Size-Specific
Sensitivity: Applying a New Structured Population Model.” Ecology 81 (3): 694–708.
Ellner, Stephen P., and Mark Rees. 2006. “Integral Projection Models for Species with
Complex Demography.” The American Naturalist 167 (3): 410–28.
Grant, A., and T. G. Benton. 2000. “Elasticity Analysis for Density-Dependent Populations
in Stochastic Environments.” Ecology 81 (3): 680–93.
Harris, Grant A. 1967. “Some Competitive Relationships between Agropyron Spicatum and
Bromus Tectorum.” Ecological Monographs 37 (2): 89–111. doi:10.2307/2937337.
Humphrey, L.David, and Eugene W Schupp. 2004. “Competition as a Barrier to
Establishment of a Native Perennial Grass (Elymus Elymoides) in Alien Annual Grass
(Bromus Tectorum) Communities.” Journal of Arid Environments 58 (4): 405–22.
doi:10.1016/j.jaridenv.2003.11.008.

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Melgoza, Graciela, Robert S. Nowak, and Robin J. Tausch. 1990. “Soil Water Exploitation
after Fire: Competition between Bromus Tectorum (cheatgrass) and Two Native
Species.” Oecologia 83 (1): 7–13.
O’Connor, Georganne. 2015. “Bunchgrass.” Pacific Northwest National Laboratory.
http://science-ed.pnnl.gov/pals/resource/cards/shrubsteppe.stm.
Oschrin, Emma. 2013. “Bluebunch Wheatgrass Success on North- and South-Facing
Slopes.” Washington: Whitman College.
Rees, Mark, Dylan Z. Childs, and Stephen P. Ellner. 2014. “Building Integral Projection
Models: A User’s Guide.” Edited by Tim Coulson. Journal of Animal Ecology 83 (3):
528–45. doi:10.1111/1365-2656.12178.
Simonson, Molly. 2014. “Conserving an Arid Species: The Effects of Aspect and
Competition From Cheatgrass on Bluebunch Wheatgrass Seedling Establishment.”
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Tilley, D, and L St. John. 2013. “Plant Fact Sheet for Bluebunch Wheatgrass
(Pseudoroegneria Spicata).” USDA Natural Resources Conservation Service.
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Vega, E., and C. Montaña. 2004. “Spatio-temporal variation in the demography of a bunch
grass in a patchy semiarid environment.” Plant Ecology. 175 (1): 107–120.
Wilson, Alice. 2014. “Identifying Key Factors Influencing Growth and Survival of
Pseudoroegneria Spicata (bluebunch Wheatgrass).” Washington: Whitman College.
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[Online]. U.S. Department of Agriculture, Forest Service, Rocky Mountain Research
Station, Fire Sciences Laboratory (Producer). Available:
http://www.fs.fed.us/database/feis/

27
Appendix A
Pre-Modeling Equations
Demography North Facing Equations
n = 465
South Facing Equations
n = 237
s(z) Survival .0723ln(z)+.06829 .0944ln(z)+.5109
G(z’,z) Growth Normal Distribution
in z’ with
σ 2
= 22.535
µ(z) = .646598z + 4.7886
Normal Distribution
in z’ with
σ 2
= 15.699
µ(z) = .879z + 5.682
Pb(z) Probability
of Producing
Seeds
0.0881ln(z) + 0.374 0.1485ln(z) + 0.0643
b(z) Density
Dependent
Seed
Survival
((1− d(z)(0.623755z - 0.518556))−.33 ((1− d(z))i (61.8411z − 855.048))−.33
d(z) tested at 0.010, 0.174, 0.338 and 0.500
C0 (x) Size
Distribution
for New
Seedlings
Normal Distribution
in x with
σ 2
= 4.921
µ(x) = 2.480
Normal Distribution
in x with
σ 2
= 7.731
µ(x) = 2.780
λ Calculated
Growth Rate 0.828 ± 0.083 0.644 ± 0.102
Table 1: Life History Functions for a P. spicata Population at the Wallula Gap
Biological Station near Wallula, Washington. Variables z and z’ represent the number
of culms at time t and time t+1 respectively.

28
Figure 1: Size-based Survival. A) North-facing survival from 2012 to 2015. r2
= 0.751
B) South-facing survival from 2012 to 2015. r2
= 0.676. There were no north-facing
plants larger than 250 culms within the studied plots. For both hillsides, individuals
smaller than 20 culms tended to experience low survival. Individual plant size was
measured as the number of visible culms at the time of data collection.

29
Figure 2: Growth Equations. (A) North-facing and (B) south-facing P. spicata
populations. The equations produced from multiple regression analysis are shown as a
solid line on each graph. There was no significant difference between the regression
slopes of the two hillsides, but south-facing slopes experienced higher rates of
retrogression between 2014 and 2015. Size was measured as in Fig. 1.

30
Figure 3: Probability of Spikes Based on Size. A) North-facing population. r2
= 0.921
B) South-facing population. r2
= 0.890. P. spicata on south-facing slopes had a higher
probability of having spikes for plants above 50 culms. Size was measured as in Fig. 1.

31
Figure 4: Seed Production. (A) North-facing and (B) south-facing populations. The
solid lines represent the overall seed production equations produced by multiple
regression analysis. Seed production was noticeably higher in south-facing population.
Size was measured as in Fig. 1.

32
Figure 5: Seedling Size Distributions. A) North-facing population. n = 52, µ = 2.481, s
= 2.218. B) South-facing population. n = 540, µ = 3.074, s = 2.780. Seedling data came
from uncovered and covered shade plots that simulated south-facing and north-facing
slope conditions respectively. Size was measured as in Fig. 1.

33
Figure 6: Average Population Size Distribution 2012 to 2015. A) North-facing
population. n = 465, = 11, s = 58.047 B) South-facing population. n = 237, = 17
s = 39.382. The south-facing population’s median was found to be significantly larger
than the median for north-facing P. spicata. Size was measured as in Fig. 1.

34
Integral Projection Model
Figure 7: North-facing Integral Projection Model. A) Survival and growth kernel can
be seen as the diagonal strip, with the reproductive kernel at the top. The dashed
diagonal line represents the kernel values for all transitions where plants remain the
same size as they were at time t. All values above the line indicate retrogression and all
values below represent growth. The model suggests that retrogression is currently very
common on north-facing slopes. B) Reproductive kernel in greater detail with seedling
survival = 0.338. Size was measured as in Fig. 1.
A
B

35
Figure 8: South-facing Integral Projection Model. A) Survival and growth kernel
components can be seen as the diagonal strip, with the reproductive kernel component
at the top. The dashed diagonal line is the same as seen in Fig. 7. The close proximity
of the survival and growth kernel components to the diagonal line indicates that
moderate retrogression is occurring but the plants are not currently shrinking
dramatically in size. B) Upper section of model in greater detail to show reproductive
kernel. Seedling survival = 0.338. Reproductive kernel values are smaller than those
seen on the north-facing integral projection model in Fig 7. Size was measured as in
Fig. 1.

36
d(z) value Description North-facing Value South-facing Value
0.010
Low Survival
Highest Kernel = 0.171
λ = 0.796
λ = 0.612
0.174
Medium
Survival
λ = 0.818
λ = 0.619
0.338 High Survival
λ = 0.828
λ = 0.644
0.500
Extremely High
Survival
λ = 0.840
Highest Kernel =0.081
λ = 0.688
Table 2: Effects of Different Seedling Survival Rates. As d(z) (the seedling survival
rate) increased, kernel values on both slopes increased. Though a seedling survival rate
of 50% (d(z) = 0.500) is unlikely to occur in the field, this value was chosen to see the
effects of extreme seedling survival on lambda and kernel values. No significant
impacts on data trends were observed when the model was run using different seedling
survival rates. These data indicate that significant differences in lambda could occur if
north and south-facing models were run with unequal seedling survival rates.

37
Stable State and Reproductive Value Predictions
Figure 9: Future Stable Population Distributions and Reproductive Outputs. A) North-
facing population. µ = 12.440, s =15.818. B) South-facing population. µ = 20.33, s =
66.136. The south-facing population is predicted to have both a higher reproductive
output and slightly larger plants in the population. Note that reproductive output refers
to the presence of spikes and seed production rather than reproductive success. The
north-facing population is expected to be primarily composed of plants with fewer than
45 culms. Size was measured as in Fig. 1.

38
Elasticity
Figure 10: North-facing Elasticity Kernel. A) Survival and growth kernel components
can be seen as the diagonal shape, with the reproductive kernel component as a thin
line across the top. B) Top section of model in greater detail. Seedling survival = 0.338.
Size was measured as in Fig. 1.

39
Figure 11: South-facing Elasticity Kernel. A) Survival and growth kernel components
can be seen as the diagonal shape, with the reproductive kernel component at the top.
The highest elasticity values can be found for plants with 30 to 60 culms at time t that
retain their size at time t+1. B) Top section of model in greater detail. Seedling survival
= 0.338. Transitions representing the production of seedlings by large plants had the
highest elasticity values for reproduction. Size was measured as in Fig. 1.

40
Appendix B
Integral Projection Model Matlab Code
Code shown uses north-facing model equations. Copy and paste into Matlab editor to
run.
%% Integral Kernel Build
% This will combine the growth, survival and fecundity equations and
return
% them as a matrix for graphing.
% The equations shown are for NF slope data
% This code requires the Matlab symbolic toolbox
%% Survive
% Builds the survival equation in the form y = a*ln(x)+b. Change exact
function to fit data.
syms x y
NFSurv = .0723*log(x)+.6829;
%% Grow
% G(z',z) represents the probability of that plant reaching size z' by
time t+1. The growth function is assumed to be a normally distributed
probability density function
% NFGrowrate is the linear regression for size data across time
% NFGrowstdv is the standard deviation within culm size
NFGrowrate = 0.646598*x+4.7886;
NFGrowstdv = 4.747;
NFGrow = (1/(NFGrowstdv*((2*pi).^(0.5))))*(2.7182818).^(-(0.5*(((y-
NFGrowrate)/NFGrowstdv).^(2))))
%% P(z',z)
% the survival-growth kernel
NFP= NFGrow*NFSurv;
%% Seedlings
%bz = (1-survival probability)*(number of seeds produced)^(-0.33)
%seed production will probably be a linear line based on culm number
NFbz = .0338*((1-0.0338)*(0.06238*y-0.51856)).^(-0.33);
% NFPbz=probability of producing seeds
NFPbz = 0.881*log(x)+0.374;
%NFC0 = size distribution for new seedlings
%newseed = the assumed size of future seedlings when first surveyed
newseed = 2;
% in this case 2.21 = standard dev and 2.480 is the mean size of
seedlings
NFC0 = (1/(2*pi*2.21).^(0.5))*(2.7182818).^(-0.5*(((newseed-
2.480)/2.21).^(2)));
%Fecundity kernel = F(z',z) = NFPbz*NFbz*NFC0
NFFec= NFPbz*NFbz*NFC0;
%% Overall Kernel
NFK = NFP+NFFec;
%% Conversion to "Real" Numbers
syms f(x,y)
f(x,y) = NFK;
[x,y] = ndgrid(linspace(1/10,500), linspace(1/10, 500))
g = matlabFunction(NFK);
NFKernel = real(g(x,y))

41
%% Graphing the IPM
figure
contourf(x,y,NFKernel) % contour map
clear title xlabel ylabel %Clears old runs index
xlabel ('Culms time t') %x-axis label
ylabel ('Culms time t+1') %y-axis label
axis ij %Flips y axis
title ({'North Facing Integral Projection Model', 'Seedling Survival =
0.0338'})
%% Finding Eiganvalues and Vectors
[NFV, NFD] = eig(NFKernel); % columns of V present eigenvectors of the
kernel. The diagonal matrix D contains eigenvalues of the kernel
NFlambda = max(NFD(:)) %pop growth rate
%% Repro
dNFV=NFV(:,1)
xxx = linspace(1/10,500)
figure
plot(xxx , abs(dNFV/sum(dNFV)),'--b') % plots with dotted blue line
xlabel ('Culms') %x-axis label
ylabel ('Stable Distribution') %y-axis label
title ({'NF Future Size Distribution and Reproductive Value',
'Seedling Survival = 0.0338'})
%% Stable Size Distribution
[NFVp, NFDp] = eig(NFKernel'); %Same as V and D but left
vectors/values instead
dNFVp=NFVp(:,1)
hold on %Plot on same figure as Repro
plot(xxx , abs(dNFVp/sum(dNFVp)), '-r') %plots with solid red line
legend('show')
legend ('Reproductive Output', 'Stable Size Distribution')

42
Elasticity Analysis Matlab Code
%% Elasticity and Sensitivity
[sens_kernel, elas_kernel] = sens(NFKernel,xxx)
% the function returns the sensitivity and elasticity of
% the kernel and then plots them. Change from mesh to contour if you
want
% a contour plot instead of a surface plot.
global wvpp %Somewhere around here the code graphs two extra figures
but I cannot figure out how to make that stop happening
[lambda, lammeth, eigfcns] = intlam(NFKernel);
wvpp=spline(xxx,eigfcns(:,2).*eigfcns(:,1));
q=quad('wvprod',min(xxx),max(xxx));
sens_kernel = eigfcns(:,2)*eigfcns(:,1)'/q;
sens_kernel=transpose(sens_kernel);
%% Graphing
% to suppress plotting, comment out all lines from here to the end
[xxx, yyy] = meshgrid(xx,xx);
figure
contourf(yyy,xxx,sens_kernel)
title ({'North Facing Sensitivity', 'Seedling Survival = 0.0338'})
elas_kernel = NFKernel.* sens_kernel/ lambda;
figure
contourf(yyy,xxx,elas_kernel)
title ({'North Facing Elasticity', 'Seedling Survival = 0.0338'})

43
Chapter 2: My Time With the Grass
https://www.youtube.com/watch?v=eNhyXhdVJKU

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