Demographic projection of high-elevation white pines
infected with white pine blister rust: a nonlinear
disease model
Field S.G.∗†, Schoettle A.W.‡, Klutsch J.G.‡§, Tavener S.J.¶, and M.F. Antolin†
TimeStamp: 22–08–2011 @ 11:58
Running title: Disease modeling of wpbr in white pines
Manuscript type: Article (MS# 11–0470R)
Number of manuscript pages: 51
Number of words in Abstract: ∼300
Number of words in Main Body: ∼5500
Number of tables: 2
Number of figures: 11
Number of references: 52
e-mail:
1. Field: sgf@colostate.edu; +1 970 491 5744
2. Schoettle: aschoettle@fs.fed.us
3. Klutsch: jklutsch@gmail.com
4. Tavener: tavener@math.colostate.edu
5. Antolin: michael.antolin@colostate.edu
∗
Corresponding Author
Department of Biology, Colorado State University, Fort Collins, CO 80523-1878
‡
Rocky Mountain Research Station, USDA Forest Service, Fort Collins, CO 80526
§
Department of Agricultural & Resource Economics, Colorado State University, Fort Collins, CO 80523
¶
Department of Mathematics, Colorado State University, Fort Collins, CO 80523
†
1
Disease modeling of wpbr in white pines – Field et al.
2
1
Abstract
2
Matrix population models have long been used to examine and predict the fate of
3
threatened populations. However, the majority of these efforts concentrate on long-term
4
equilibrium dynamics of linear systems – and their underlying assumptions – and
5
therefore omit the analysis of transience. Since management decisions are typically
6
concerned with the short-term (< 100 years), asymptotic analyses could lead to
7
inaccurate conclusions or worse yet, critical parameters or processes of ecological concern
8
may go undetected altogether.
9
We present a stage-structured, deterministic, nonlinear, disease model which is
10
parameterized for the population dynamics of high-elevation white pines in the face of
11
infection with white pine blister rust (wpbr). We evaluate the model using
12
newly-developed software to calculate sensitivity and elasticity for nonlinear population
13
models at any projected time step. We concentrate on two points in time: i) during
14
transience and ii) at equilibrium, and under two scenarios: i) a regenerating pine stand
15
following environmental disturbance and ii) a stand perturbed by the introduction of
16
wpbr.
17
The model includes strong density-dependent effects on population dynamics,
18
particularly on seedling recruitment, and results in a structure favoring large trees.
19
However, the introduction of wpbr and its associated disease-induced mortality alters
20
stand structure in favor of smaller stages. Populations with infection probability (β) & 0.1
21
do not reach a stable coexisting equilibrium and deterministically approach extinction.
22
The model enables field observations of low infection prevalence among pine seedlings
23
to be reinterpreted as resulting from disease-induced mortality and short residence time
24
in the seedling stage.
25
Sensitivities and elasticities, combined with model output, suggest that future efforts
26
should focus on improving estimates of within-stand competition, infection probability,
27
and infection cost to survivorship. Mitigating these effects where intervention is possible
28
is expected to produce the greatest effect on population dynamics over a typical
29
management timeframe.
Disease modeling of wpbr in white pines – Field et al.
30
Keywords: Cronartium ribicola, disease prevalence, elasticity, five-needle pine,
31
nonlinear disease model, Pinus albicaulis, Pinus flexilis, sensitivity, stage-structured
32
model.
3
Disease modeling of wpbr in white pines – Field et al.
4
33
1
34
Infectious diseases, particularly introduced non-native species, play an important role in
35
the stability and ultimate fate of host species populations (Anderson and May, 1986;
36
Crowl et al., 2008). Mathematical models that include infection status and pathogen
37
transmission aim to predict the trajectory of populations over time, highlight the most
38
important parameters influencing dynamics of populations in the face of emerging
39
pathogens, and constitute an important preliminary step for eventual management and/or
40
policy decisions (e.g. Keane et al., 1996; Keeling et al., 2003). Models of infectious
41
disease are often in the form of the three-class sir (Susceptible–Infected–Recovered)
42
series of continuous-time differential equations (e.g. Anderson and May, 1979, see
43
Keeling and Rohani (2008)), but discrete time models in the form of iterated maps (Oli
44
et al., 2006) are also well suited to examine disease systems. More complex stage- or age
45
structured matrix models allow for inclusion of additional heterogenieties, such as
46
differences in survival among reproductive and non-reproductive classes (Caswell, 2001),
47
and stage-specific nonlinearities like density-dependence (Caswell, 2008).
48
Introduction
Efficient analysis of nonlinear multi-stage disease models can be achieved by
49
calculating transient sensitivities and elasticities by use of the chain rule (Tavener et al.,
50
2011). Sensitivities, as well as sensitivity rescaled to reflect relative change (i.e.
51
elasticities), provide insight into which model parameters must be estimated most
52
accurately and which parameters can be approximated without losing model accuracy.
53
Methods to analyze nonlinear matrix models (and generally, iterated maps) have been
54
extended to allow the evaluation of sensitivity with respect to any parameter and at all
55
time steps during population transience (Caswell, 2008; Tavener et al., 2011). Sensitivity
56
during transience, defined as the trajectory i) from arbitrary initial conditions toward a
57
stable equilibrium solution, ii) following a disturbance, or iii) following a perturbation
58
that changes the equilibrium solution, is of increasing interest to ecologists (Fox and
59
Gurevitch, 2000; Caswell, 2007; Haridas and Tuljapurkar, 2007). Transient analysis is
60
especially important in applying models to management of infectious diseases because the
61
time-scale of management/conservation projects usually coincides with the short-term
Disease modeling of wpbr in white pines – Field et al.
62
transient phase, rather than an equilibrium eventually reached in the long term (Ezard
63
et al., 2010). Management activities are by definition short-term perturbations that
64
require analysis of transient behavior, particularly in long-lived species.
5
65
Here we develop and apply a stage-structured population model to examine the effects
66
of an introduced disease on the dynamics and stand structure of long-lived high-elevation,
67
five-needle white pines (Pinus albicaulis and P. flexilis) in western North America. These
68
species grow at or near the alpine treeline and play an important role in high-elevation
69
ecosystems. They provide seed for a variety of sub-alpine wildlife species such as the
70
Clark’s nutcracker, Nucifraga columbiana (Tomback, 1982), squirrels and other rodents,
71
and grizzly bears (Pease and Mattson, 1999). These critical tree species, like all
72
five-needle pines, are highly susceptible to white pine blister rust (Hoff et al., 1980). In
73
North America, white pine blister rust (wpbr) is caused by the non-native fungal
74
pathogen Cronartium ribicola and is lethal to high-elevation white pines as infected trees
75
typically do not recover. Since its introduction to the western coast of British Columbia
76
in ∼ 1910 (Hoff and Hagle, 1990), wpbr has gradually expanded southeast, inland and
77
into the high-elevation Rocky Mountain ranges.
78
The emergence of the ecosystem management perspective in the 1990s shifted from
79
managing forests for relatively few tree species that provide wood commodities to broader
80
management of forests for ecosystem services. As a result, the importance of traditionally
81
unmanaged high elevation white pine forests increased when it was recognized that they
82
provide watershed protection and wildlife habitat (Schoettle, 2004; Tomback and Achuff,
83
2010). Management of wpbr in the high-elevation white pines now concentrates on both
84
multigenerational population sustainability and understanding the effects of wpbr on
85
short- and long-term population dynamics (Schoettle and Sniezko, 2007). Previous
86
modeling efforts in this pathosystem include both linear matrix models (Ettl and
87
Cottone, 2004) and nonlinear process models of fire succession that incorporate disease
88
(Keane et al., 1996).
89
Considering the importance of transient sensitivity analysis to management decisions
90
(see above), we construct a nonlinear, stage-structured, sir-type infection model and
91
examine both transient and equilibrium sensitivity/elasticity. These results are useful in
Disease modeling of wpbr in white pines – Field et al.
92
an applied conservation context to manage wpbr, identify critical parameters, and
93
suggest future avenues of research.
6
Disease modeling of wpbr in white pines – Field et al.
7
94
2
Methods
95
2.1
96
Because high-elevation white pines are long-lived species, predicting the effects of wpbr
97
on forest dynamics depends on the development of disease models that incorporate
98
nonlinear processes like density-dependent survival, seed dispersal, and seedling
99
recruitment. We model a one hectare, closed system, high-elevation white pine
Model construction
100
population using a six-stage, nonlinear matrix projection model. Infection is included by
101
allowing each stage to be either susceptible or infected with wpbr, resulting in a total of
102
12-stages. See Appendix 1 for details regarding parameter estimation and Appendix 2 for
103
an explicit summary of modeling assumptions related to the system. Both can be found
104
in the Ecological Archives.
105
2.2
106
We model a stage-structured population in the form of a nonlinear map. Vectors are
107
indicated using the ~v notation, thus ~x is the state vector of the population and
108
xi , i = 1, . . . , N, is the number of individuals in the ith stage. Matrices are indicated
109
using capital bold fonts (e.g. A), and parameters of the model are represented by
110
p~k , k = 1, . . . , K.
111
112
Generalized model framework
The elements of ~x{n} (t) ∈ Rn contain the population in each stage of a basic n-stage
structured model at time t. The basic iterated map takes the form:
~
~x{n} (t + 1, p~) = h{n} ~x{n} (t, p~), p~
~x{n} (0, p~) = ~x{n},0 ,
(1)
(2)
113
where ~h{n} : Rn 7→ Rn . In § 2.2.1 we construct a general disease model from equation (1).
114
We then separate the vector-valued function ~h{n} into fecundity and survivorship &
115
transition. For ease of exposition, we drop explicit dependence on p~ and let
~x{n} (t + 1) = ~g{n} ~x{n} (t) + f~{n} ~g{n} ~x{n} (t)
(3)
Disease modeling of wpbr in white pines – Field et al.
8
116
where f~{n} and ~g{n} are vector-valued functions Rn 7→ Rn that control fecundity and
117
survivorship & transition respectively.
118
2.2.1
119
A typical sir model with n base individual stages has a total of 3n classes. Our model is
120
atypical in that infection is neither density-dependent (SI), nor frequency-dependent
121
( SI
). This is because the life cycle of wpbr does not involve direct tree-to-tree infection,
N
122
but rather the infection process occurs through an alternate host. Further, high-elevation
123
white pines do not recover from infection and thus the model includes only susceptible
124
and infected classes (i.e. no recovered class). Thus, we assume a constant background of
125
infective spores and that the probability of infection is independent of the number of
126
susceptible and infected individuals. The state vector becomes
Incorporating disease
~x{2n}


S
~x{n} 
=
.
~xI{n}
(4)
127
We assume that the infection status affects survivorship and fecundity independently and
128
ν
let the cost of infection to survivorship and transition be C{n}
, where
ν
C{2n}


0 
I{n}
=

ν
0 C{n}
and Cnν = diag(ci ),
i = 1, . . . , n,
129
where I{n} is the n × n identity matrix. We define an intermediate, post-survival,
130
population that has not yet undergone reproduction or infection as
ν
~y{2n} = C{2n}
∗ ~g{2n} ~x{2n} .
(5)
(6)
131
The effect of infection on fecundity is modeled as a weighting of the infected individuals
132
(i.e. infection cost). We define
ϕ
~
~
f{2n} = f{2n} C{2n} ∗ ~y{2n} ,
(7)
Disease modeling of wpbr in white pines – Field et al.
133
9
ϕ
and let the cost of infection to fecundity of infected individuals be C{n}
, where


0 
I
ϕ
C{2n}
=
 .
ϕ
0 C{n}
134
(8)
Lastly, infection is modeled by the linear operation
B{2n} ∗ ~x{2n}
135
(9)
where


I{n} − B{n} 0 
B{2n} = 

B{n}
I{n}
and B{n} = diag(βi ) ,
i = 1, . . . , n .
(10)
136
Once again I{n} is the n × n identity matrix. Finally, combining equations (6), (7), and
137
(9) the complete general nonlinear map is given by
ϕ
~x{2n} (t + 1) = B{2n} ∗ ~y{2n} + f~{2n} C{2n} ∗ ~y{2n}
.
(11)
138
For the generalized case above, all three operations – the cost of infection to survival &
139
transition, the cost of infection to fecundity, and infection probability – could be modeled
140
as nonlinear processes.
141
2.3
142
The pine population was subdivided into six stages: 1) seeds, 2) primary seedlings,
143
3) secondary seedlings, 4) saplings, 5) young adults, and 6) mature adults. We initially
144
define the seed stage as 0 – 1 years. Other stages were identified from age and size
145
dependent factors related to survival, reproductive capability and infection cost
146
(Tomback et al., 1993; Smith and Hoffman, 2000, 2001; Conklin, 2004; Kegley and
147
Sniezko, 2004; Burns, 2006; Smith et al., 2008), where age and size relationships were
148
estimated from tree ring analysis (J. Coop and A. Schoettle, unpublished data). Primary
The six-stage population model
Disease modeling of wpbr in white pines – Field et al.
10
149
seedlings (sd1 ) are defined as 1 – 4 year olds, a period of low survivorship for most forest
150
trees (Shepperd et al., 2006; Woodward, 1987). By age 5, seedling survivorship increases
151
(Maher and Germino, 2006), and we define secondary seedlings (sd2 ) as seedlings 5 years
152
old until they reach a height of definable diameter at breast height (dbh; 1.37 m). Based
153
on age/height relationships for P. flexilis and P. aristata (J. Coop and A. Schoettle,
154
unpublished data) this corresponds to ∼20 years old. We define saplings (sa) as trees of
155
21 years (i.e. >1.37 m) until reproductive age, which we set at 40 years, since
156
high-elevation white pines have first reproductive output between ages 30 – 50
157
(McCaughey and Schmidt, 1990). We accordingly define young adults (ya) as
158
reproductive trees ages 41 – 90 years and mature adults (ma) as greater than 90 years old
159
with full reproductive capacity (Table 1). Delineation of ya and ma was estimated from
160
field observations of reproductive capacity, and age/size measurements from P. flexilis
161
(Burns et al., 201X, J. Coop and A. Schoettle, unpublished data). Based on this stage
162
structure, the mean dbh for saplings, young adults, and mature adults was estimated to
163
be 2.05, 12.5, and 37.0 cm respectively (Table 2).
For each iteration of the map we assume the following sequence of biological events
164
165
(changing this order implicitly changes model assumptions and thus alters the model).
Calculation of lai(~x), equation (25)
↓
Seedling Recruitment, equation (30)
↓
Survival & Transition, equation (17)
↓
Calculation of lai(~y ), equation (25)
↓
Fecundity, equation (33)
↓
Infection, equation (43)
166
2.3.1
Mathematical description
167
Let xi denote the number of individuals in stage i where i = 1, . . . , 6. Let the elements of
168
~x{6} (t) ∈ R6 contain the populations in each stage of a six-stage structured model at time
169
t,
Disease modeling of wpbr in white pines – Field et al.

~x{6}
170
  
seeds
x1
 sd1  x2 

  
 sd2  x3 

 =  .
=
 x4 
sa

  
 ya  x5 
ma
x6
11
(12)
We define the nonlinear map
~x{6} (t + 1) = ~g{6} (~x{6} (t)) + f~{6} (~g{6} (~x{6} (t)))
(13)
171
where f~ and ~g represent vector-valued functions that control i) fecundity and
172
ii) survivorship & transition functions respectively.
173
2.3.2
174
We model fecundity and seedling recruitment as nonlinear processes, while survivorships
175
and transitions are assumed to be linear. Correspondingly, we split ~g{6} (~x{6} ) into linear
176
and nonlinear components. We discuss nonlinear components below in § 2.3.3. Let
Linear model components: survivorship & transition
~g{6} (~x{6} ) = S{6} ∗ ~x{6} + ~γ{6} (~x{6} ) ,
(14)
177
where S{6} is the linear survivorship matrix (defined below), and ~γ{6} (~x{6} ) represents a
178
vector-valued nonlinear function acting upon ~x{6} , which defines the transition from seed
179
to sd1 recruitment process described in § 2.3.3 and defined in equation (30). For
180
notational convenience we let
181
182
~x
b{6} = ~x{6} + ~γ{6} (~x{6} ),
(15)
~g{6} (~x{6} ) = S{6} ∗ ~x
b{6} = ~y{6} ,
(16)
to define
where ~y{6} represents an intermediate population. The projection matrix S{6} is
Disease modeling of wpbr in white pines – Field et al.
12

S{6}
183

0 0 0 0 0 0
0 s 2 0 0 0 0 


0 t2 s3 0 0 0 

 ,
=

0
0
t
s
0
0
3
4


0 0 0 t4 s5 0 
0 0 0 0 t5 s6
with the following coefficients along the diagonal,
s1 = 0, s2 = 0.636, s3 = 0.8391, s4 = 0.9310, s5 = 0.9653, s6 = 0.995 ,
184
(17)
(18)
and the following coefficients along the sub-diagonal,
t1 = 0, t2 = 0.212, t3 = 0.0559, t4 = 0.0490, t5 = 0.0197.
(19)
185
The S21 (t1 ) and S11 (s1 ) entries are both 0 since the germination and production of seeds
186
are modeled as nonlinear processes and are included later when nonlinearities are
187
calculated in § 2.3.3. We assume seeds either germinate to primary seedlings or become
188
nonviable within one year (i.e. no seed bank). Thus, s1 = 0; see equation (17).
189
190
Survivorship and transition (i.e. viability) probabilities for primary seedlings through
mature adults were calculated as
si =
ti =
1
1−
Ri
∗ (1 − mi ),
1
∗ (1 − mi ),
Ri
i = 2, . . . , 6 ,
i = 2, . . . , 5 ,
(20)
(21)
191
where si is the proportion of individuals surviving and remaining in the same stage i, ti is
192
the proportion of individuals within stage i that grow into the next stage, Ri is the
193
residence time for stage i, and mi is the mortality of stage i individuals. Note that the
194
entries ti reflect the combination of transition and survivorship from stage i → i + 1, such
195
that survivorship is contained within the transition probability. Table 1 shows the
196
calculated values for si and ti . See Table 2 for mi and Ri values.
Disease modeling of wpbr in white pines – Field et al.
197
2.3.3
198
Modeling density-dependence is inherently nonlinear because the previous population
199
(~x(t, p~)) vector affects current parameters, and because the population depends
200
implicitly upon time. There are two forms of density-dependence:
ii) Fecundity (i.e. female production of seed/cones).
203
204
Nonlinear model components: density-dependence
i) Seedling recruitment (i.e. germination)
201
202
13
Leaf area index (lai), the amount of projected leaf area over a given ground area
m2
, is commonly used to reflect vegetation density (Steltzer and Welker, 2006) and
2
m
205
relates to ecosystem parameters like carbon, water and energy flux, and competitive
206
interactions between and within species. Here, density-dependent processes are mediated
207
via lai (see below). Lower values of lai represent sparsely populated stands whereas
208
larger lai values represent more dense, shaded ones. Typical lai estimates, depending on
209
succession stage, environmental conditions, and species of interest, plateau at
210
approximately 9 for numerous ecosystems (Gower et al., 1999). Pine forests, however,
211
rarely exceed an lai of 8 (Brown, 2001) and Law et al. (2001) estimated the range for P.
212
ponderosa as 0.59 – 2.77. Our equilibrium values of lai without and with infection were
213
4.71 and 2.49 respectively, which is in accordance with these estimates. Perhaps most
214
importantly, at no point during the path to equilibrium does lai exceed 5.0, well within
215
the reported upper threshold for pine ecosystems.
216
The relationship between leaf area (la) and diameter at 1.37 m (dbh) was estimated
217
by maximum likelihood estimation (mle) assuming the general form y = axb fit to data
218
from (Callaway et al., 2000) supplemented with unpublished data for P. albicaulis (A.
219
Sala, unpublished data). Leaf area was calculated using equations (23) and (24). mle and
220
95% confidence intervals for α2 and α3 are shown in Table 2. Primary seedlings have
221
negligible la and thus do not contribute to lai. For secondary seedlings, a la estimate of
222
0.456 m2 was estimated from data in Schoettle and Rochelle (2000) and Schoettle (1994)
223
because this stage is below 1.37 m in height (and therefore dbh = 0).
Disease modeling of wpbr in white pines – Field et al.
14
224
To determine lai at a given time step, la by stage was calculated and multiplied by
225
the number of individuals present in each stage, then summed to obtain the total la of
226
the population. This value was divided by land area (10,000 m2 ) to obtain lai. Finally,
227
because natural populations are seldom monocultures, laib was added to represent
228
background interspecific competition contributed by leaf cover of other tree species.
229
Specifically, the la for each stage is calculated as a function of the diameter at breast
230
height (1.37m) according to equation (23). The contributions of each stage to the overall
231
leaf area index are weighted by the populations within each stage and then scaled by the
232
area in equation (25). Secondary seedlings have d3 = 0 but we assume they contribute to
233
stand leaf area, thus l3 is defined as the constant α1 . Using the mean dbh measurements
234
for the six stages,
d1 = 0, d2 = 0, d3 = 0, d4 = 2.05, d5 = 12.5, d6 = 37.0 ,
235
(22)
and
l1 = 0, l2 = 0, l3 = α1
li = α2 (di )α3 ,
236
i = 4, . . . , 6 ,
where
α1 = 0.456, α2 = 0.0736, α3 = 2.070 ,
237
(23)
we calculate lai as
lai(~x6 ) =
(~l, ~x{6} )
+ laib .
10000
(24)
(25)
238
Nonlinear seedling recruitment
239
We model the transition of the seed → sd1 (t1 ) as a density-dependent process. Seed
240
germination depends upon a number of factors including seed production, predation,
241
dispersal, environmental conditions, and seedbed conditions (Woodward, 1987; Keane
242
et al., 1990; Ettl and Cottone, 2004). We modified a germination equation from Keane
Disease modeling of wpbr in white pines – Field et al.
243
15
et al. (1990) to obtain the germination probability in equation (29). First, we define
SpB(~x{6} ) =
x1
,
nBirds
0.73
+ 0.27 ,
1 + exp (31000 − SpB(~x{6} ))/3000
1
rALs (~x{6} ) =
,
1 + exp(2 (lai(~x{6} ) − 3))
rcache (~x{6} ) =
(26)
(27)
(28)
244
where x1 is the number of seeds at time t, nBirds is the number of Clark’s nutcrackers
245
per hectare, SpB is the number of seeds per bird, rcache is a reduction factor for the
246
propensity to cache seeds, and rALs is a reduction factor for the available light and is
247
directly dependent upon lai. See Appendix 1 and 2 for further description and derivation
248
of these two density-dependent reduction factors.
249
Finally, the probability of seeds germinating in a given year is defined by
(1 − Pf ind ) (1 − Pcons )
r2 (~x{6} ) =
∗ rcache (~x{6} ) ∗ rALs (~x{6} ) .
SpC
(29)
250
The following parameters are independent of population size: Pcons , the proportion of
251
seeds consumed by nutcrackers during a caching season (Keane, pers. obs. in Cottone,
252
2001; vander Wall, 1988), Pf ind , the proportion of seeds reclaimed from caches by the
253
nutcrackers, and SpC, the average seeds per cache (range = 2.6 – 5.0; vander Wall and
254
Balda, 1977; Tomback, 1982; vander Wall, 1988; Tomback et al., 2005, see Table 2). The
255
number of new seedlings is then given by
~γ{6} (~x{6} ) = r2 (~x{6} ) ∗ x1 ∗ ~e2 ,
(30)
256
where ~e2 ∈ R6 is the unit vector with a single nonzero entry in the second position.
257
Nonlinear fecundity
258
We assume that fecundity differences arise entirely through cone production and that
259
pollen is non-limiting (see Appendix 2; #8). We assume only ya and ma stages (> 40 yr)
260
produce cones. The yearly maximum number of cones per ma individual was defined as
Disease modeling of wpbr in white pines – Field et al.
16
261
Cmax = 7.5 (Schwartz et al., 2006) and is consistent with data from McKinney and
262
Tomback (2007). We further assume ya individuals produce a maximum of 10% (ρ = 0.1)
263
of a ma individual (J. Coop and A. Schoettle, unpublished data; calculated from primary
264
data from (Burns, 2006)).
265
The number of cones is a function of the number of young and mature adults and their
266
total fecundity. Seed production is determined by the number of seeds per cone, Scone ,
267
and the number of cones per tree, Ctree (~y{6} ). We define
Ctree (~y{6} ) =
0.5
+ 0.5 ∗ Cmax ,
1 + exp(5 (lai(~y{6} ) − 2.25))
(31)
= rcones ∗ Cmax
r1 (~y{6} ) = Scone ∗ Ctree (~y{6} )
(32)
268
where Cmax and Scone are fecundity parameters (Table 2). The number of seeds produced
269
is given by
f~{6} (~y{6} , ρ) = r1 (~y{6} ) ∗ (ρ ∗ y5 + y6 ) ∗ ~e1 ,
(33)
270
where ~e1 ∈ R6 is a unit vector with a single nonzero entry in the first position, ρ is the
271
reduction in seed production of ya relative to ma, and ~y{6} , the intermediate population
272
vector, is defined in equation (16).
273
2.3.4
274
Combining equations (15), (16), (30), and (33) obtains the six-stage, nonlinear map of
275
the population without infection dynamics,
The six-stage nonlinear map
~x6 (t + 1) = ~y{6} + f~{6} ~y{6} , ρ .
(34)
Disease modeling of wpbr in white pines – Field et al.
17
276
2.4
The twelve-stage disease model
277
When incorporating disease we assume all stages except seeds are susceptible to infection
278
(i.e. x7 = 0). Including disease doubles the number of stages, as described in § 2.2.1, and
279
we define

~x{12}
280
2.4.1
281
We define
seeds

 susceptible sd1

 susceptible sd2

 susceptible sa


 susceptible ya
S 
 susceptible ma
~x{6}

=
=

I
~x{6}

0

 infected sd1


 infected sd2

 infected sa


 infected ya
infected ma



x1
  
  x2 
  
  x3 
  
  x
  4
  
  x5 
  
  x
  6
 =  .
  x7 
  
  x8 
  
  
  x9 
  
  x10 
  
  
  x11 
x12
(35)
Survivorship, transition and disease


S
~x{6} 
~g{12} (~x{12} ) = ~g{12} 

~xI{6}
(36)
282
as follows. The nonlinear transition function for seedling recruitment ~γ{6} is described in
283
§ 2.3.3. Certain components of this function (e.g. lai and SpB ) are independent of
284
disease status and must be extended to be functions of ~x{12} . Similarly, the fecundity
285
function r1 (·) as described in § 2.3.3 must be extended to be a function of ~y{12} . Let
~x
b{12}

 

S
SI
S
I
~x
b{6}  ~γ{6} ~x{6} , ~x{6} 
= I +

~x
b{6}
0
(37)
Disease modeling of wpbr in white pines – Field et al.
18
286
SI
where the nonlinear function ~γ{6}
is the modification of ~γ{6} . Since there are no infected
287
seeds, infected sd1 individuals cannot arise via seedling recruitment. We define

 
S{6} 0  I{6}
~g{12} = 
∗
0 S{6}
0
288
 

S
~
0  x
b{6} 
ν
b{12} = ~y{12} .
 ∗  I  = S{12} ∗ C{12} ∗ ~x
ν
~
C{6}
x
b{6}
(38)
The viability cost of infection is

ν
C{12}
I{6}
0
ν
=
, where C{6}
ν
0 C{6}
0
0

0
=
0

0
0

0 0 0 0 0
c2 0 0 0 0 

0 c3 0 0 0 
,
0 0 c4 0 0 

0 0 0 c5 0 
0 0 0 0 c6
(39)
289
where the cost of infection is either a constant (for stages without a definable dbh) or a
290
function of dbh (sa, ya and ma). Specifically, the viability cost reduction coefficients are
c1 = 0, c2 = 0.01, c3 = 0.13,
ci = 1 − exp(−δ · di ),
(40)
i = 4, . . . , 6 ,
(41)
291
where the parameter δ is a coefficient that influences the cost of infection for trees taller
292
ν
than 1.37 m. The matrix product S{12} ∗ C{12}
in equation (38) is post-multiplied because
293
we assume transition from stage i → i + 1 occurs after survivorship (see Appendix 2 for
294
additional details).
295
2.4.2
296
White pine blister rust is not vertically transmitted from adults to seeds, therefore both
297
susceptible and infected adults produce susceptible seeds. Infection with wpbr reduces
298
cone production only since there is no evidence for reduced pollen production with wpbr
299
infection. We again assume pollen is non-limiting and that fecundity of infected ya and
300
ma are reduced by the same proportion (Cf ; Table 2). Fecundity for the twelve-stage
Fecundity and disease
Disease modeling of wpbr in white pines – Field et al.
301
model is defined by
f~{12} (~y{12} , ρ, Cf ) = r1 (~y{12} ) ∗
302
19
(ρy5 + y6 ) + Cf (ρy11 + y12 ) ∗ ~e1
(42)
where, as with the six-stage model, ρ is a measure of the effect of stage-class on
303
fecundity and Cf is the effect of infection on fecundity (compare equation (33)). Now
304
~e1 ∈ R12 is a unit vector with a single nonzero entry in the first position.
305
2.4.3
306
Simplifying assumptions about wpbr infection of high-elevation white pines were
307
implemented. We model disease prevalence assuming the probability of infection is
308
constant and is independent of stage, time, and the solution. We define
Infection

B{12} =
309
I{6} − B{6} 0
, where B{6}
B{6}
I{6}

0 0 0 0 0 0
0 β2 0 0 0 0 


0 0 β3 0 0 0 

=
0 0 0 β4 0 0  ,


0 0 0 0 β5 0 
0 0 0 0 0 β6
(43)
where the infection probability is
β1 = 0, β2 = β3 = β4 = β5 = β6 = 0.044 .
(44)
310
Note that both c1 = 0 and β1 = 0 to emphasize that seeds cannot become infected.
311
2.4.4
312
ν
The matrices B{12} , C{12}
, and S{12} are independent of both the solution and time. We
313
also emphasize that the order of biological events reflect that infection occurs after
314
i) seedling recruitment, ii) survivorship and transition, iii) and fecundity. Thus, infection
315
is the final process and combining equations (37), (38), (42), and (43), we obtain the
316
twelve-stage, nonlinear map,
The twelve-stage nonlinear map
~x{12} (t + 1) = B{12} ∗ ~y{12} + f~{12} (~y{12} , ρ, Cf ) .
(45)
Disease modeling of wpbr in white pines – Field et al.
20
317
2.5
Sensitivity analysis
318
Sensitivity and elasticity analyses were performed using the software package sensai
319
(Tavener et al., 2011) which analyzes deterministic, multi-stage, multi-parameter
320
nonlinear population models. This program efficiently calculates sensitivity at all time
321
points during transience, rather than focusing on long-term asymptotic dynamics, which,
322
as noted earlier, may result in misleading conclusions that affect management decisions
323
(Fox and Gurevitch, 2000; Ezard et al., 2010). The basic nonlinear iterative process is
~x(t + 1, p~) = ~h ~x(t, p~), p~ ,
t>0
(46)
~x(0) = ~z.
324
Using index notation, we rewrite equation (46) as
xi = hi (~x, p~),
325
i = 1, . . . , N.
(47)
Differentiating equation (47) with respect to parameters, pk , gives

N
∂xi (t + 1) X ∂hi ∂xm (t)
∂hi 


=
∗
+

∂pk
∂x
∂p
∂p
m
k
k
m=1


∂xi (0)


=0
∂pk
i = 1, . . . , N, k = 1, . . . , K.
(48)
326
Observe that to evolve equation (48) to determine the stability of xi with respect to pk at
327
any t > 0, we need to evaluate ∂hi /∂xm and ∂hi /∂pk .
328
329
To determine stability with respect to the initial conditions, we differentiate
equation (47) with respect to the initial conditions to give

N

∂xi (t + 1) X ∂hi ∂xm (t) ∂hi 


=
∗
+


∂zk
∂x
∂z
∂z

m
k
k

m=1

∂xi (0)
=1


∂zi




∂xi (0)


= 0, k 6= i,

∂zk
i = 1, . . . , N,
k = 1, . . . , N ,
(49)
Disease modeling of wpbr in white pines – Field et al.
330
21
or alternatively using the Kronecker delta,
∂xi (0)
= δij
∂zj
where δij = 1,
if i = j,
0 otherwise.
(50)
331
To solve this for the population and its stability with respect to parameters and initial
332
conditions we evolve equations (47), (48) and (49) simultaneously.
333
Elasticities are defined in terms of relative sensitivities. Let
∆ξ =
334
∆x
x
and ∆κ =
∆p
,
p
then the elasticity of x with respect to p
∂ξ
∆ξ
p
∆x
p ∂x
= lim
=
lim
=
.
∂κ ∆κ→0 ∆κ
x ∆p→0 ∆p
x ∂p
335
We define the elasticity of the ith variable with respect to the k th parameter, Ei,k as
Ei,k =
pk (t) ∂xi
(t).
xi (t) ∂pk
(51)
336
2.6
Modeling software
337
The model was constructed primarily in R (R Development Core Team, 2010) and all
338
figures were produced using its default pdf graphics device. Sensitivity analyses were
339
carried out in MATLAB (MathWorks: R2010a) via the front end software sensai (Tavener
340
et al., 2011) which combines the MATLAB platform and Maple (Maplesoft: v14.0) to
341
calculate derivatives. sensai is freely available from:
342
http://www.fescue.colostate.edu/SENSAI
Disease modeling of wpbr in white pines – Field et al.
343
3
344
3.1
345
Following a period of transience during regeneration, a disease-free population starting
346
with 1000 sd1 individuals reaches an equilibrium stable stage distribution after
347
approximately 600 years (Fig. 3). The equilibrium stage distribution without disease is
348
349
22
Results
Disease-free solutions and sensitivity analysis
~x>
{12} = (62580, 38, 79, 65, 91, 353, 0, 0, 0, 0, 0, 0), with the total tree population
6
P
xi = 626. This equilibrium solution is used in analyses that include rust infection,
i=2
350
351
where we perturb the population from this disease-free equilibrium (§ 3.2).
The mature adult stage (x6 ) quickly dominates the landscape as a result of low
352
mortality and shading effects on younger tree stages and germination rate. The effect of
353
shading through lai is particularly apparent during the transient phase of regeneration
354
when ma trees decline after ∼ 200 years. The remaining stages respond and increase in
355
size between 300–400 years as a result of increased seed germination (Fig. 3). This
356
equilibrium pine stand closely matches field observations of age structure in
357
high-elevation white pines (Burns, 2006; Burns et al., 201X).
358
Sensitivity analyses with respect to model parameters focused on two quantities i) the
359
total population (excluding seeds), and ii) the mature adult population. Analysis of the
360
disease-free population (i.e. regenerating scenario; β = 0) in Fig. 3 reveals that three sets
361
of parameters have large effects (Fig. 4, top-left). Mortality (p1 , . . . , p5 ), infection
362
(p10 , . . . , p14 ), and the lai parameters (p22 , p23 ) all reduce the ma population, especially
363
in the transient phase during regeneration when there are relatively few ma individuals.
364
At this time period the ma population depends on younger stages to transition into the
365
ma stage (Fig. 4). At equilibrium, however, the most sensitive parameters for the ma
366
stage are only ma mortality and ma infection probability. Increasing β would have a
367
strong, negative effect on the susceptible ma population as susceptible ma individuals
368
become infected and eventually die.
369
The total susceptible population (x2 , . . . , x6 ) is also sensitive to these same three
370
groups of parameters, both early in the population projection and at equilibrium. The
Disease modeling of wpbr in white pines – Field et al.
371
exception is ma mortality (m6 ), which has a positive effect on the total population
372
(Fig. 4d) as the suppressing effect of density-dependence by ma is released.
373
23
Elasticity is a rescaling of sensitivity to obtain the relative effect of a parameter on the
374
quantity of interest. Model elasticity for ma (Fig. 4, top-right) revealed that seed and
375
cone parameters have a positive effect on the ma population. In addition, increasing Pf ind
376
(p29 ), the proportion of seeds found and consumed by Clark’s nutcrackers, has a negative
377
influence on the ma population. These effects, however, are only important during the
378
transient phase of a regenerating population (i.e. < 200 years). Once again, as with the
379
sensitivity, elasticity revealed a consistent pattern of leaf area parameters (α2 , α3 ) with a
380
strong negative effect on both the ma and total populations.
381
3.2
382
We repeated the analysis with βi = 0.044, i = 2, . . . , 6, introduced to the disease-free
Introducing
WPBR
384
equilibrium population (Fig. 5) and once again examined i) the mature adult population
12
P
(x6 + x12 ) and, ii) the total population ( xi )with respect to all model parameters.
385
During the transient phase following infection, elasticities of the parameters revealed a
386
similar pattern to elasticities calculated without infection. Leaf area parameters α2 (p22 )
387
and α3 (p23 ), seed parameters Cmax (p25 ) and Scone (p26 ), and germination parameters
388
Pf ind (p29 ) and SpC (p31 ) again had the largest influence. However, the cost of infection
389
to survivorship (δ) becomes a critical parameter. Decreasing infection cost (i.e. increasing
390
δ (p20 ), see Fig. 2d), positively affects the ma population whereas it negatively affects the
391
overall population (Fig. 5a–b).
383
i=2
392
Lastly, the total tree population becomes sensitive to changes in the mean dbh of
393
mature adults (p17 ). This parameter influences leaf area and thus highlights both the
394
suppressive effects of density-dependence and the co-dependence of model parameters.
395
Interestingly, The ma population also becomes sensitive to changes in dbh of adult classes
396
at equilibrium (not shown), because the ma population ultimately depends on a supply of
397
seedlings transitioning through the initial stages to become adults. A similar line of
398
reasoning explains the change in magnitude of the seed germination parameters Pf ind
Disease modeling of wpbr in white pines – Field et al.
24
399
(p29 ), Pcons (p30 ), and SpC (p31 ). Mature adults depend only indirectly on these
400
parameters as seedlings eventually transition to become adults, whereas the total
401
population includes many early stages that depend directly on germination.
402
3.2.1
403
As predicted by the sensitivity analysis, introducing disease to the equilibrium population
404
dramatically changes the trajectory (Fig. 6) and stage-structure (Fig. 7) of the
405
population. Low values of β actually have a positive effect on the total population as
406
infected ma individuals suffer increased, infection-induced mortality, allowing other
407
classes to increase in number which again highlights the effect of density-dependence
408
mediated by lai (Fig. 6a). With low β, there are more trees overall, but the population
409
has a vastly different stage structure. This can be seen in Fig. 6b where β has a
410
consistent, negative effect on the ma population. Higher values of β (> 0.10) have a
411
consistent, negative effect on the entire population and results in eventual extinction.
Infection probability (β)
412
Fig. 7 depicts the effect of β on the population trajectory and structure during the
413
transient phase following infection for low (β = 0.016), medium (the mle; β = 0.044),
414
and high (β = 0.20) transmission probabilities. Following rust introduction a rapid
415
deviation from the initial equilibrium stage structure occurs as the population shifts
416
towards younger stages. At lower β (< 0.10), the density-dependent effects of lai
417
predominate, and increased mortality in adults allows younger stages to increase.
418
However, when β = 0.20, the population declines deterministically to extinction because
419
infection overcomes the positive effects of removing density-dependent mechanisms on
420
smaller tree stages.
421
This effect is even greater with additional interspecific competition via laib
422
(Fig. 7e–h). This background shading prevents the population from reaching a viable
423
stable equilibrium even at low transmission probabilities (Fig. 7b–c vs. f–g) as
424
competition from other tree species inhibits seed germination and the supply of younger
425
individuals to the higher stages.
Disease modeling of wpbr in white pines – Field et al.
25
426
3.2.2
Infection cost (δ)
427
The strength of the infection cost is mediated by the coefficient δ (Fig. 2d). As predicted
428
by the elasticity analysis (Fig. 5), lowering the cost of infection (δ ↑) has a positive effect
429
on the ma population. Conversely, increasing the cost of infection (δ ↓) has a positive
430
effect on the total population (compare Fig. 8a vs. 8b). This is reflected by the change in
431
sign of the elasticities of δ (p20 ) in Fig. 5a and 5b. Simultaneously considering both δ and
432
β further supports this relationship (Fig. 9). With low β and low δ (high cost), the total
433
population increases dramatically in the first 100 years. However when both the infection
434
probability and the cost of infection are high (front corner of Fig. 9), the total population
435
rapidly goes extinct. The suppressive density-dependent effect of mature adults on the
436
rest of the population is eventually overwhelmed by wpbr infection. A similar pattern is
437
observed at equilibrium for both the total and mature adult populations, but extinction
438
occurs in a much larger region of parameter space. This indicates that 100 years is too
439
early in the trajectory to encapsulate population extinction (compare right corner of
440
Fig. 9b,d).
441
3.2.3
442
We define the stage-specific and total prevalence as the proportion infected individuals
443
defined by
Rust prevalence (κ)
κi =
444
x6+i
,
xi + x6+i
and
i = 2, . . . , 6
12
P
κT =
i=8
12
P
(52)
xi
(53)
xi
i=2
445
respectively. When introduced into a fully susceptible equilibrium population, the total
446
prevalence (κT ) rapidly increased for all three values of β (Fig. 10). High infection
447
prevalence is maintained in the ma stage (κ6 ) because they i) have lower cost of infection
448
and, ii) are longer-lived than other tree stages and more opportunities to become
449
infected. For β = 0.20, ma infection prevalence quickly reaches 100% as all trees become
Disease modeling of wpbr in white pines – Field et al.
26
450
infected in the years leading up to adulthood (90 years; Fig. 10c). Because all stages are
451
infected with the same probability (equation (44)), smaller tree stages also become
452
infected, but suffer such a high cost that they are quickly removed from the population.
453
Fig. 11 shows the cumulative sum of dead seedling (sd1 ) individuals during the transient
454
phase following rust introduction into an equilibrium population. For high β, the
455
majority of sd1 individuals either die or transition to the sd2 stage. Thus the combined
456
effect of infection-induced mortality and short residence time maintains a low infected
457
sd1 (x8 ) population (i.e. low κ2 ).
Disease modeling of wpbr in white pines – Field et al.
27
458
4
Discussion
459
We analyzed sensitivity and elasticity of a stage-structured, nonlinear disease model of a
460
high-elevation white pine stand in the face of infection with wpbr. In the absence of
461
genetic resistance, our model shows that sustainability of high-elevation white pine stands
462
infected with wpbr depends on two dominant effects: i) infection probability and,
463
ii) regeneration mediated via competition (e.g. lai). Parameters controlling these effects
464
disproportionately remove smaller stages via infection induced mortality and by limiting
465
seedling establishment. More generally, parameters and factors that reduce the seedling
466
population impede long-term population viability. Sensitivity analysis further highlights
467
the co-dependence of model parameters, as some parameters indirectly influence the
468
seedling population through other parameters and/or factors, particularly those involved
469
in density-dependence. For example, mean dbh of mature adults (p17 ), suppresses
470
population growth because it is the largest contributor to the leaf area calculation
471
equation (23), and contributes to lai and ultimately the suppression of the regeneration
472
cycle. The potential for parameter co-dependence in complex, nonlinear models highlights
473
an advantage of the sensitivity analysis performed here, that unexpectedly influential
474
parameters can be readily identified.
475
We considered two time steps at which to perform sensitivity analyses, i) during
476
transience at 100 years and, ii) at equilibrium (> 1000 years), because sensitivities at
477
these time steps tell us different things about the dynamics of the system. Early transient
478
dynamics are important for analysis of a stand in a state of flux following disturbance
479
(e.g. fire or rust introduction), whereas sensitivities at equilibrium relate to processes in
480
stable, subalpine stands. Further, transient dynamics are likely more informative for
481
management strategies on a realistic timescale. For example, at 100 years, ma individuals
482
are sensitive to both the mortalities and infection probabilities of all stages beneath them
483
because adult stages in a regenerating population arise from the supply of smaller stages
484
transitioning into the ma stage (Fig. 4 - left). At equilibrium, however, the ma stage is
485
only sensitive to changes in the mortality and infection probability of its own stage (i.e.
486
p5 and p14 ). This pattern is even more apparent when one considers the total susceptible
Disease modeling of wpbr in white pines – Field et al.
28
487
population (bottom two of Fig. 4 - left). Initially the population is sensitive to numerous
488
parameters related to mortality and infection, but at equilibrium the population as a
489
whole is most sensitive to the mortality and infection of ma only, the stage that
490
dominates at equilibrium.
491
Starting with the disease-free equilibrium and default parameters, an infected
492
high-elevation white pine population reaches a new diseased equilibrium in less than 500
493
years, however with a stage distribution that is much less dominated by mature adults.
494
Increasing infection probability (up to β ≈ 0.07) causes a shift in age structure towards
495
younger age classes. Fig. 7 suggests that high-elevation white pine populations are indeed
496
capable of tolerating moderate levels of wpbr infection as long as seedling recruitment is
497
maintained and stands are not simultaneously suppressed by i) other competing tree
498
species, or ii) other agents of mortality (e.g. mountain pine beetle, Dendroctonus
499
ponderosae).
500
Traditional stability analysis of sir models includes the index R0 , the basic
501
reproductive ratio, which is a measure of the linear stability (or instability) of the
502
disease-free equilibrium solution (Keeling and Rohani, 2008). In contrast to density- and
503
frequency-dependent disease models that include terms like
504
that infection is from an evenly distributed cloud of spores from alternate hosts (i.e. Ribes
505
spp.). The life cycle of wpbr does not involve direct tree-to-tree transmission, so the
506
assumption of a constant β, independent of the population, seems reasonable. However, a
507
non-trivial disease-free equilibrium solution only exists for the special case when β = 0
508
(i.e. the absence of Ribes). In this case, the equilibrium solution is always stable with
509
respect to perturbation with infected individuals (R0 < 1), since there is no transmission
510
pathway when β = 0. In this context, the traditional notion of R0 is defined, but
511
uninformative because any infected individuals introduced to the population simply die
512
out. In our model, the appropriate analogue is not equilibrium stability with respect to
513
adding infected individuals, but rather stability with respect to the addition of infected
514
Ribes which complete the transmission pathway. For this class of model, an alternative
515
measure of the population’s susceptibility to disease could be the sensitivity of the
SI
N
or SI, our model assumes
Disease modeling of wpbr in white pines – Field et al.
516
diseased population
12
P
29
xi with respect to transmission probability (β) when β = 0.
i=8
517
Rust prevalence in the sd1 (κ2 ) is low when infection probability β = 0.044, yet
518
primary seedlings become infected (4.4%/year). Rust prevalence remains low because of
519
the combination of high infection cost, high natural mortality, and low residence time
520
(Fig. 11). Therefore, in natural populations, the sd1 population may appear uninfected
521
(or escaping infection), but our model suggests infected sd1 simply do not remain long
522
enough, either as dead trees on the landscape or as transitioned maturing seedlings, to be
523
reliably sampled. This may account for the low seedling rust prevalence found in field
524
surveys (Burns, 2006; Kearns, 2005). The converse is also true. The model predicts that
525
the stage with i) the largest residence time and, ii) the lowest mortality will accumulate
526
the highest rust prevalence in the population, namely the mature adults (Fig. 10). High
527
rust prevalence in larger size classes has been observed by Conklin (2004) in P.
528
strobiformis, Burns et al. (201X) in P. flexilis, and Smith and Hoffman (2000) in P.
529
albicaulis.
530
This model lays the framework for studying wpbr infection in a stage-structured,
531
deterministic, nonlinear map, but could be extended to include broader ecosystem
532
interactions or disturbances via external forces (e.g. the effect of climate on seedling
533
establishment). Further, alternative infection dynamics could be incorporated (e.g.
534
density-dependent infection) as well as age of infection, multiple infections, and location
535
of infections on trees. Finally, heritable resistance to wpbr has been described at low
536
frequency in high-elevation white pines (Hoff et al., 1980), and may include a mechanism
537
that is controlled by a single dominant gene (Kinloch and Dupper, 2002). Using the
538
mathematical framework developed in Tavener et al. (2011) this model can be readily
539
extended to include single-locus genetics as an additional nonlinearity. The effect of
540
genetic resistance in this host-pathogen system is the basis of forthcoming papers.
541
Conclusions
542
Our model clearly demonstrates a strong effect of wpbr on population structure. The
543
sensitivity and elasticity analyses indicate that future research should focus on improving
Disease modeling of wpbr in white pines – Field et al.
30
544
estimates of both infection probability and infection cost. They also suggests the
545
exploration of the effects of competition (i.e. density-dependence) on population
546
dynamics, especially seedling recruitment, is warranted. These efforts should be used to
547
develop management strategies to mitigate these effects. For example, stimulating
548
natural regeneration or planting genetically resistant individuals, which have been
549
suggested as potentially critical management solutions (Schoettle and Sniezko, 2007),
550
would likely lower the effects of both infection probability and cost, especially if infection
551
is found to be density-dependent as suggested by Hatala et al. (2011).
We further propose an alternative interpretation to field observations of both high
552
553
prevalence in larger sized trees and particularly low prevalence in young (e.g. sd1 ) size
554
classes. Prevalence in younger stages may be low not because of a low infection
555
probability, as previously assumed, but caused by a combination of high infection cost
556
and short residence time (and vice versa for larger trees). If so, a more careful evaluation
557
of seedling mortality could reveal additional management strategies.
Our model provides an example of how sensitivity analysis can be used to determine
558
559
critical parameters in complex, nonlinear models under transient and/or equilibrium
560
conditions in an applied ecological context.
561
5
562
Supplemental online information regarding model parameter estimation and explicit
563
modeling assumptions can be found in the Ecological Archives supplement to this article
564
(see Appendix 1 and 2).
Ecological Archives
Disease modeling of wpbr in white pines – Field et al.
31
565
6
Acknowledgements
566
We thank the following for primary or unpublished data: K. Burns, D. Conkin, J. Coop,
567
M. Germino, A. Sala, and D. Tomback. We thank B. Keane, S. McKinney, and R.
568
Sniezko for insightful discussions. Funding was provided by USDA Forest Service Rocky
569
Mountain Research Station (# 07-RJVA-11221616-252) to M.F.A. and USDA Economic
570
Research Service Program of Research on the Economics of Invasive Species Management
571
(PREISM: # 58-7000-8-0096) to A.W.S. We thank members of the “Flexible and
572
Extendible Scientific Undergraduate Experience” program (fescue) for valuable
573
discussions and model development. Lastly, the final version of the manuscript was
574
greatly improved by comments from two anonymous reviewers.
Disease modeling of wpbr in white pines – Field et al.
32
575
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Disease modeling of wpbr in white pines – Field et al.
717
7
38
Tables
Table 1: Survivorship (si ) and transition (ti ) probabilities used in equation (17) and calculated using equations (20) and (21). See Table 2 for estimates of residence time (Ri ) and
mortality (mi ). We assume no seed bank, thus s1 = 0 and t1 in equation (17) is calculated
via the nonlinear (nl) function in equation (29). Note that s6 = 1 − m6
stage class
age
si (ci95)
ti (ci95)
0
nl
seeds
1
0–1
sd1
2
1–4
0.6360 (0.5888 − 0.6742) 0.2120 (0.1962 − 0.2248)
sd2
3
5 – 20
0.8391 (0.7238 − 0.9056) 0.0559 (0.0482 − 0.0604)
sa
4
21 – 40
ya
5
41 – 90 0.9653 (0.9575 − 0.9712) 0.0197 (0.0195 − 0.0198)
ma
6
> 90
0.9310 (not available)
0.9950 (0.9840 − 1.000)
0.0490 (not available)
−
Disease modeling of wpbr in white pines – Field et al.
39
Table 2: Parameters used in the model. The parameter numbers (pk , k = 1, . . . , 31)
correspond to the sensitivity and elasticity analysis. ci95 are from mle estimates
Parameter
pk
Symbol
Mortality seeds
−
m1
Mortality sd1
1
m2
Mortality sd2
2
m3
Mortality sa
3
m4
Mortality ya
4
m5
Mortality ma
5
m6
Residence time seeds
−
R1
Residence time sd1
6
R2
Residence time sd2
7
R3
Residence time sa
8
R4
Residence time ya
9
R5
Residence time ma
−
R6
Transmission probability 10 − 14 β2 , . . . , β6
Mean dbh sa
15
d4
Mean dbh ya
16
d5
Mean dbh ma
17
d6
Infection cost (sd1 )
18
c2
Infection cost (sd2 )
19
c3
Infection cost (viability)
20
δ
la sd2
21
α1
la coeff 1
22
α2
la coeff 2
23
α3
Background lai
24
laib
Max. cones per tree
25
Cmax
No. seeds per cone
26
Scone
Infection cost (fecundity)
27
Cf
No. Clark’s nutcrackers
28
nBirds
Prop. seeds found
29
Pf ind
Prop. seeds consumed
30
Pcons
No. seeds per cache
31
SpC
Fecundity ratio (ya:ma)
−
ρ
Default value
ci95
1
0.152
0.105
0.020
0.015
0.005
1
4
16
20
50
∞
0.044
2.05
12.5
37.0
0.01
0.13
0.15
0.456
0.0736
2.070
0
7.5
46
0.125
3
0.8
0.3
3.7
0.1
−
0.101 − 0.215
0.034 − 0.228
not available
0.009 − 0.023
0.000 − 0.016
0.037 − 0.052
0 − 0.03
0.10 − 0.16
0.011 − 2.701
1.932 − 5.220
Disease modeling of wpbr in white pines – Field et al.
40
718
8
Figure Legends
719
Figure 1. Life cycle graph of the high-elevation white pine disease model. The cycle
720
begins with seed and moves counter-clockwise to mature adults (ma). White nodes
721
represent susceptible stages; black nodes represent infected stages. Black arrows represent
722
either survivorship or transitions, grey arrows represent infection processes, and white
723
arrows represent the fecundity process. The transition arrow from seed → sd1 is dotted
724
to emphasize the fact that germination is a density-dependent process and therefore
725
differs from the other black arrows (which represent linear processes).
726
Figure 2. Reduction factors for seedling recruitment (a, b), fecundity (c), and infection
727
cost (d). The cost of infection, ci , is a function of tree size (c4→6 ). Lines represent
728
δ = 0.05, 0.10, 0.15, 0.20, 0.25. Default value δ = 0.15 (solid line). Higher δ values shift
729
curves closer to 1.0 and thus exhibit a lower cost to survivorship.
730
Figure 3. Population projection of a pine stand (#/ha) to equilibrium with parameter
731
defaults (see Table 2) and without disease (βi = 0), regenerated from 1000 sd1
732
individuals (seed population (x1 ) is not shown).
733
Figure 4. Sensitivity (left column) and elasticity (right column) plots for the
734
regenerating, disease-free population (Fig. 3). Parameter numbers on the x-axis are
736
defined in Table 2. Quantities of interest are both the number of mature adult trees (x6 )
6
P
and the total number of trees ( xi ) with respect to each model parameter, at two time
737
steps i) during the transient phase of regeneration – 100 years and, ii) at the stable
738
equilibrium – >1000 years. Bar shading from black → light-gray groups associated
739
parameters into clusters to facilitate the identification of related parameters.
740
Figure 5. Elasticity analysis with disease (βi = 0.044) during the transience (at 100
741
years) following infection of a fully susceptible, disease-free equilibrium population
735
i=2
743
(Fig. 3). Quantities of interest are (a) the total mature adult population (x6 + x12 ) and
12
P
(b) the total population ( xi ). Parameter numbers on the x-axis are defined in Table 2.
744
Sensitivities were qualitatively similar (not shown). Note the different scales of the y-axes.
742
i=2
Disease modeling of wpbr in white pines – Field et al.
745
Figure 6. (a) Surface plot of the total population
12
P
41
xi as a function of infection
i=2
746
probability (β) and time and, (b) surface plot of the total mature adults ma (x6 + x12 ) as
747
a function of β and time. Initial conditions were the disease-free equilibrium solution
748
(Fig. 3).
749
Figure 7. Population projections and stand structure without additional shading from
750
other tree species, laib = 0 (top row; a – d) and with shading by other trees, laib = 2
751
(bottom row; e – h). Initial conditions were the disease-free equilibrium (Fig. 3). From
752
left to right β = 0, 0.016, 0.044, 0.20, corresponding to zero, low, medium, and high
753
infection probability. All other parameters set to default values.
754
Figure 8. (a) Surface plot of the total population
755
of infection (δ) and time and, (b) surface plot of the total mature adults ma (x6 + x12 ) as
756
a function of δ and time. The default δ = 0.15. A low value of delta corresponds to a
757
high cost of infection (Fig. 2d). Initial conditions were the disease-free equilibrium
758
(Fig. 3), all other parameters set to default values.
12
P
xi as a function of both the cost
i=2
759
760
Figure 9. Surface plot of the relationship between probability of infection (β), the cost
12
P
of infection coefficient (δ), and (a) the total population
xi and, (b) mature adults
i=2
761
(x6 + x12 ) 100 and > 1000 (c, d) years after infection was introduced. The front corner of
762
the graphs represents high probability of infection and high cost of infection while the
763
back is low probability of infection and low cost of infection.
764
Figure 10. Stage-specific and total prevalence (κi , i = 2, . . . , 6 and κT ) of white pine
765
blister rust for low (β = 0.016), medium (mle; β = 0.044), and high (β = 0.20)
766
probability of infection over time. The bold solid line represents the overall (total)
767
prevalence of wpbr in the population.
768
Figure 11. Population projections for the sd1 class only (x2 and x8 ) introducing wpbr
769
to an equilibrium structured population. The cumulative sum of dead individuals and the
770
wpbr prevalence of sd1 (κ2 ) is also shown for (a) β = 0 and (b) high infection
771
probability, β = 0.20.
Disease modeling of wpbr in white pines – Field et al.
772
9
42
Figures
SP
SD2
YA
SD1
MA
SEED
iSD1
iMA
iSD2
Arrows:
iSP
Survival & transition
Infection
Fecundity
iYA
Nodes:
Susceptible
Infected
Figure 1:
Disease modeling of wpbr in white pines – Field et al.
43
0.8
0.6
0.4
0.2
0.0
0.0
0.2
0.4
rALs
r cach e
0.6
0.8
1.0
(b)
1.0
(a)
0
1
2
3
4
5
2
6
7
0
10
20
30
40
50
Seeds per bird (x1000)
)
1.0
1.0
Leaf Area Index (m m
2
0.8
0.9
0.6
δ = 0.05
0.2
0.4
c i = 1 − e (−δ • dbh)
0.8
0.7
0.6
0.0
0.5
0.4
0.5
r cones =
1 + exp (5(LAI − 2.25))
+ 0.5
δ = 0.25
0
1
2
3
4
2
Leaf Area Index (m m
2
5
0
10
20
dbh (cm)
)
(c)
(d)
Figure 2:
30
40
44
1400
Disease modeling of wpbr in white pines – Field et al.
800
600
400
200
0
No. Individuals
1000
1200
SD1
SD2
SA
YA
MA
0
200
400
time
Figure 3:
600
800
Sensitivity
Sensitivity
Sensitivity
Sensitivity
-5000
-20000
-10000
-50000
-5000
-25000
20000
-30000
3
1
3
3
1
1
3
1
5
5
5
5
9
11
13
15
17
19
21
23
25
7
7
7
13
15
17
19
21
11
13
15
17
19
21
Total Tree Population - 100 yrs
11
23
23
9
13
15
17
19
21
Parameter (p)
11
23
Total Tree Population - Equilibrium
9
9
25
25
25
Mature Adult Tree Population - Equilibrium
7
27
27
27
27
29
29
29
29
Figure 4:
31
31
31
31
Elasticity
Elasticity
Elasticity
Elasticity
1
-5 -3 -1
0
-4
-8
0
-4
-8
0
-4
-8
Mature Adult Tree Population - 100 yrs
3
1
3
3
1
1
3
1
5
5
5
5
9
11
13
15
17
19
21
23
25
7
7
7
13
15
17
19
21
11
13
15
17
19
21
Total Tree Population - 100 yrs
11
23
23
9
13
15
17
19
21
Parameter (p)
11
23
Total Tree Population - Equilibrium
9
9
25
25
25
Mature Adult Tree Population - Equilibrium
7
Mature Adult Tree Population - 100 yrs
27
27
27
27
29
29
29
29
31
31
31
31
Disease modeling of wpbr in white pines – Field et al.
45
Disease modeling of wpbr in white pines – Field et al.
46
(a)
-2
-5
-4
-3
Elasticity
-1
0
1
Mature Adult Tree Population - 100 yrs
1
3
5
7
9
11
13
15
17
19
21
23
25
27
29
31
Parameter (p)
0
Total Tree Population - 100 yrs
δ
-5
P f i nd
MA dbh
-10
Elasticity
α2
α3
1
3
5
7
9
11
13
15
17
19
Parameter (p)
(b)
Figure 5:
21
23
25
27
29
31
600
800
Trees
Total
0
50
e
tim
100
150
200
(a)
0.20
0.15
ta
Be
0.10
0.05
0.00
Figure 6:
250
100
150
200
u
re Ad
200
Matu
400
300
350
50
e
tim
100
150
(b)
200
0.20
0.15
ta
Be
0.10
0.05
0.00
Disease modeling of wpbr in white pines – Field et al.
47
lts
50
0
100
No. Individuals
150
200
250
300
350
0
50
100
No. Individuals
150
200
250
300
350
SD1
SD2
SA
YA
MA
50
50
100
100
time
time
50
0
0
50
0
0
50
0
(e)
(a)
150
150
β=0
β=0
200
200
SD1
SD2
SA
YA
MA
No. Individuals
100
150
200
No. Individuals
250
300
350
0
50
100
150
200
250
300
350
0
0
50
50
(f)
100
time
100
time
(b)
150
150
200
200
Figure 7:
β = 0.016
β = 0.016
No. Individuals
100
150
200
No. Individuals
250
300
350
0
50
100
150
200
250
300
350
0
0
50
50
(g)
(c)
100
time
100
time
150
150
β = 0.044
β = 0.044
200
200
No. Individuals
100
150
200
No. Individuals
250
300
350
0
50
100
150
200
250
300
350
0
0
(h)
50
50
(d)
100
time
100
time
150
150
β = 0.2
β = 0.2
200
200
Disease modeling of wpbr in white pines – Field et al.
48
2000
2500
Trees
Total
500
50
e
tim
100
150
200
(a)
0.00
0.05
0.10
de
0.15
lta
0.20
0.25
0.30
Figure 8:
250
100
150
200
u
re Ad
1000
Matu
1500
300
350
50
e
tim
100
150
(b)
200
0.00
0.05
0.10
de
0.15
lta
0.20
0.25
0.30
Disease modeling of wpbr in white pines – Field et al.
49
lts
Disease modeling of wpbr in white pines – Field et al.
50
(a)
(b)
Trees
Total
Matu
2500
300
2000
re Ad
200
1500
ults -
- 100
1000
100
100 y
yrs
500
rs
0.00
0.30
0
0.00
0.30
0.25
0.25
0.05
0.05
0.20
Be 0.10
ta
0.15
0.10
de
0.20
Be 0.10
ta
lta
0.15
0.15
0.10
de
lta
0.15
0.05
0.20
0.05
0.00
0.20
Trees
Total
Matu
2500
0.00
300
re Ad
2000
1500
200
ults >
> 100
1000
100
1000
0
0.00
yrs
0 yrs
500
0.30
0
0.00
0.30
0.25
0.25
0.05
0.05
0.20
Be 0.10
ta
0.20
Be 0.10
ta
0.15
0.10
d
ta
el
0.15
0.15
0.10
0.15
0.05
0.20
0.05
0.00
0.20
(c)
0.00
(d)
Figure 9:
d
ta
el
0
Total
SD1
SD2
SA
YA
MA
50
100
150
200
β = 0.016
0
50
100
150
200
β = 0.044
Rust Prevalence
Rust Prevalence
1.0
0.8
0.6
0.4
(a)
Figure 10:
(b)
time
0.2
0.0
Rust Prevalence
1.0
0.8
0.6
0.2
0.0
0.4
1.0
0.8
0.6
0.4
0.2
0.0
time
0
50
(c)
time
100
150
β = 0.2
200
Disease modeling of wpbr in white pines – Field et al.
51
0
10
20
x2
x8
cumulative dead x 2 + x 8
cumulative dead x 8
κ2
Primary seedlings (SD1)
300
250
200
150
100
50
0
(a)
time
30
40
x2
dead x 2 + x 8
κ2
50
1.0
0.8
0.6
0.4
0.2
0.0
Figure 11:
300
250
200
150
100
50
0
0
10
time
(b)
20
30
40
x2
dead x 8
dead x 2 + x 8
κ2
β = 0.2
50
x8
1.0
0.8
0.6
0.4
0.2
0.0
β=0
Disease modeling of wpbr in white pines – Field et al.
52
Rust Prevalence (κ2)