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Plant Reproduction and Environmental Noise: How do Plants Do It? Danielle Lyles, Todd S. Rosenstock, Alan Hastings
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Journal of Theoretical Biology
Cite this article as: Danielle Lyles, Todd S. Rosenstock, Alan Hastings, Plant Reproduction and Environmental Noise: How do Plants Do It?, Journal of Theoretical Biology, http://dx.doi.org/10.1016/j.jtbi.2015.02.009 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Plant Reproduction and Environmental Noise: How Do Plants Do It? Danielle Lyles a,1,∗, Todd S. Rosenstockb,2 , Alan Hastingsc a
Department of Environmental Science and Policy, One Shields Avenue, University of California, Davis, CA 95616 b Department of Plant Sciences, One Shields Avenue, University of California, Davis, CA 95616 c Department of Environmental Science and Policy, One Shields Avenue, University of California, Davis, CA 95616
Abstract Plant populations exhibit a wide continuum of reproductive behavior, ranging from nearly constant reproductive output on one end to the extreme of masting (synchronized, highly variable reproduction) on the other. Here, we show that including variability (noise) in density-dependent pollen limitation in current models for pollen-limited plant reproduction may produce any behavior on this continuum. We previously showed that (large) variability in pollination efficiency (a related phenomenon) may induce masting in non-pollen-limited plant populations. Other modeling studies have shown that including variability in accumulated resources (and/or the threshold for reproduction) may induce masting, but do account for masting in non-pollen-limited plant populations. Thus, our results suggest that the range of plant reproductive behavior may be explained with the simple resource budget model combined with the biological realism of variability ∗
Corresponding author Email address: [email protected] (Danielle Lyles ) 1 Current address: Department of Mathematics, The University of Texas at San Antonio, San Antonio, TX 78249 2 Current address: World Agroforestry Centre (ICRAF), PO Box 30677-00100, Nairobi, Kenya
Preprint submitted to Journal of Theoretical Biology
February 11, 2015
in density-dependent pollen limitation. This is a specific example of an important functional consequence of the interactions between stochasticity and nonlinearity, and highlights the importance of carefully considering both the biological basis and the mathematical effects of the noise term. Keywords: moran, masting, synchronized, hybrid, model
1
1. Introduction
2
Much of the development of the understanding of population dynamics in
3
ecology has come from models that are deterministic (Kingsland, 1995). How-
4
ever, as far back as the work of Moran (Moran, 1953) on synchrony, the impor-
5
tance of stochasticity, and in particular external environmental variability, in shap-
6
ing spatiotemporal dynamics has been recognized. More recently, in a variety of
7
areas there has been increasing attention paid to the importance of variability.
8
However, much of the stochastic theory has been based on linearizations, or on
9
assumptions that the level of noise is small (Lande et al., 2003). In contrast to this
10
theory it has been recognized that natural variability can be very large, due to envi-
11
ronmental cues such as highly variable rainfall patterns or timing of frosts, which
12
in turn produces highly variable population densities through time (Connell and
13
Sousa, 1983). Thus, it is important for ecologists to get an understanding of the
14
importance of large variability as an influence on spatiotemporal dynamics. Yet,
15
this understanding can only come from attention to specific systems, like masting
16
behavior of plants (Liebhold et al., 2004).
17
Masting is synchronized, highly variable reproduction in a population of plants.
18
Masting may be divided into three categories: strict (bimodal with no overlap be-
19
tween tails), normal bimodal (bimodal with overlap between tails, bimodality is
2
20
statistically significant), or normal switching (output varies greatly and the pres-
21
ence of switching is verified by negative autocorrelation of reproductive output)
22
(Kelly, 1994). Although masting has received much attention, there is no distinct
23
line that may be drawn between masting and non-masting species (Kelly, 1994).
24
Plants that truly mast should vary more than their environment (Kelly, 1994). The
25
predominant hypothesis for this, resource switching, suggests that reproduction
26
is carbon-limited and the individual plant alternates allocation of carbon between
27
reproductive and vegetative structures.
28
While resource switching and carbon allocation patterns might explain vari-
29
able reproduction,how might a population of plants synchronize their reproduc-
30
tion? Empirical studies have shown that weather cues may synchronize plants for
31
masting (Schauber et al., 2002; Kelly et al., 2000, 2008; Sel˙as, 2000; Kon et al.,
32
2005; Kon and Noda, 2007). Such studies also repeatedly find that both internal
33
resource dynamics (resource switching) and cues from appropriate weather vari-
34
ables (which are noisy) must be combined in order to explain masting data (Sel˙as,
35
2000; Kon et al., 2005; Rees et al., 2002; Lamontagne and Boutin, 2007). Further-
36
more, a meta-analysis of masting data on a global scale has shown that masting
37
species with higher variability generally are located in areas with higher variabil-
38
ity in rainfall (Kelly and Sork, 2002). Kelly et al. have also shown that, for a
39
given plant population, those at higher altitudes have lowered temperature thresh-
40
olds for a given flowering effort (Kelly et al., 2008), suggesting that the variability
41
of temperature is more important than absolute temperature.
42
Mathematical modeling studies support the idea that resource switching com-
43
bined with environmental noise is an important component of the mechanism of
44
masting in plant populations. Both Isagi (Isagi et al., 1997) and Satake & Iwasa
3
45
(Satake and Iwasa, 2000, 2002a) have shown that resource switching combined
46
with pollen limitation may cause masting in monoecious plants. Satake and Iwasa
47
have also shown that such a mechanism may be greatly enhanced by the addition
48
of environmental noise in the form of variability in accumulated resources and/or
49
reproductive thresholds of individual plants, but only in the presence of pollen lim-
50
itation (Satake and Iwasa, 2002b). Previously, we extended their model to show
51
that resource switching combined with (large) environmental noise in the form
52
of variability in pollination efficiency may cause masting in plant populations in
53
the absence of pollen limitation and over a wide range of conditions (Lyles et al.,
54
2009). These examples are similar to the Moran effect, whereby independent
55
(linear) populations are synchronized by spatially correlated environmental noise
56
over a large spatial scale.
57
Pollen limitation is common among plants (Obeso, 2002) and it’s statistical
58
significance varies among sites and years, so the pollination environment is not
59
constant (Burd, 1994). Here, we examine the effect of variability in density-
60
dependent pollen limitation on the reproductive output of pollen-limited plant
61
populations. This may also be viewed as variability in the effectiveness of abi-
62
otic pollination under different conditions. For example, suppose that pollination
63
is a process that has a certain probability of success p on each trial. The effective-
64
ness of pollination can be defined as the probability of successful pollination for
65
each trial. So if p is the probability of pollination in one trial, then the probability
66
of pollination after n trials is 1 − (1 − p)n . If the number of trials is proportional to
67
the number of flowering plants, then this is equivalent to density dependent pollen
68
limitation, and variation in p is analogous to variation in density-dependent pollen
69
limitation. Variability in density-dependent pollen limitation may arise in many
4
70
ways. For example, from a change in distance between partners (habitat frag-
71
mentation) (Aguilar et al., 2006) or asynchrony in flower development between
72
male and female flowers (Vankin et al. 2006). Synchronous flowering is neces-
73
sary to achieve high density-dependent pollen limitation and regional climate may
74
influence the timing of flowering among several species of plants (Post, 2003;
75
Schauber et al., 2002). Other aspects of the weather and its inherent variabil-
76
ity may also change density-dependent pollen limitation in some years especially
77
for wind-pollinated plants (Koenig and Ashley, 2003). For example, precipita-
78
tion during flowering may limit pollen dispersal and cause mechanical damage to
79
flowers (Knapp et al., 2001; Knops et al., 2007). Variability in density-dependent
80
pollen limitation may also reflect variability in the effectiveness of biotic pollina-
81
tion under different conditions, in the presence of pollinators that are not highly
82
specialized. For example, if the probability of transferring pollen among con-
83
specifics increases with the density of conspecifics. There is an active literature
84
in pollination ecology exploring this issue (Kunin, 1993; Crone, 2013). Thus,
85
density-dependent pollen limitation can vary, perhaps quite significantly, from
86
year-to-year. In this paper we investigate whether such noise in density-dependent
87
pollen limitation may contribute to masting behavior in the (pollen limited) plant
88
population model(s) of Satake and Iwasa (Satake and Iwasa, 2000, 2002a,b).
89
2. Modeling, Simulation, and Analysis
90
2.1. Model
91
The model we use extends our earlier work regarding plants that aren’t pollen-
92
limited (Lyles et al., 2009) by expanding upon the idea that large variability in
93
pollination efficiency (here represented by variability in density-dependent pollen 5
94
limitation) is an important element for determining the reproductive behavior of
95
plants. This work was based on the models by Satake and Iwasa (Satake and
96
Iwasa, 2000, 2002a,b; Iwasa and Satake, 2004) developed from a still earlier
97
model by Isagi et al. (Isagi et al., 1997). The models are developed in terms
98
of the resource budget for an individual plant which adds resources each year due
99
to photosynthesis.
100
Plants are assumed to flower if resources at the beginning of the year are above
101
a threshold level, and not to flower if resources are below the threshold. If plants
102
do flower, the amount of flowering depends on the difference between resources
103
available and the threshold level, and the amount of resources is reduced by an
104
amount depending on the fruit set. By nondimensionalization and rescaling, the
105
threshold level for flowering can be assumed to be zero, and in a deterministic
106
description the amount of resources added each year can be assumed to be one
107
(Satake and Iwasa, 2000). Letting y x (t) represent non-dimensionalized resource
108
stores of individual plant in a population (plant x) leads to the model:
if y x (t) < 0, y x (t + 1) = y x (t) + 1
(1)
if y x (t) > 0, y x (t + 1) = −k ∗ P(y x (t), β x (t)) ∗ y x (t) + 1
(2)
109
where k > 0 represents resource depletion due to reproduction (the cost of
110
reproduction), β x (t) is variable density-dependent pollen limitation, and the func-
111
112
tion P(y x (t), β x (t)) is a pollen limitation term that determines the contribution of pollen from nearby plants. In particular, P(y x (t), β x (t)) = ( 1n i∈U x [yi (t)]+ )βx (t)
113
where [y]+ = y if y ≥ 0 and [y]+ = 0 if y ≤ 0 and U x contains individuals
114
within the pollen dispersal distance D of plant x and n is the number of plants in
115
U x . The term P(y x (t), β x (t)) is the average pollen contribution of trees within the 6
116
pollen dispersal distance D of tree x (a number in [0,1]) raised to the power β x (t),
117
and so has values in the interval [0,1]. When β x (t) = 0, there is no pollen limi-
118
tation. When β x (t) is small, small amounts of pollen are magnified so the effects
119
of pollen limitation are minimal. When β x (t) is large, it takes larger amounts of
120
pollen to produce a similar pollen contribution, implying strong pollen limitation.
121
The key feature of β x (t) is that the ratio of seeds to flowers depends on the density
122
of flowering plants. This density dependence provides an endogenous feedback
123
that promotes synchrony. The noise term β x (t) is normally distributed with mean
124
βavg and coefficient of variation cv, so the standard deviation is cvβavg . The cor-
125
relation is described by a parameter R by writing β x (t) as a combination of two
126
variables as follows (Satake and Iwasa, 2002b): β x (t) = δ(t) + γ x (t)
(3)
129
where δ(t) is normally distributed with mean Rβavg and standard deviation √ cvβavg R and γ x (t) is normally distributed with mean (1 − R)βavg and standard √ deviation cvβavg 1 − R. Thus, when R = 1, β x (t) is the same value for each in-
130
dividual and when R = 0, β x (t) is independent among individuals. Intermediate
131
levels of correlation are obtain for values of R between 0 and 1. We allow β x (t) to
132
be normally distributed to compare to the case of noise in the threshold for repro-
133
duction and/or annual net production (Satake and Iwasa, 2002b). Since negative
134
β x (t) ignore biological realities, we set negative β x (t) values equal to zero in the
135
simulations.
136
2.2. Simulation
127
128
137
Individual plants were represented as positions in a 10 X 10 rectangular array,
138
which for the purposes of this paper, may be referred to as a ”forest”. Our results 7
139
with a 10 x 10 lattice match those of Satake and Iwasa with a 100 x 100 lattice
140
(Satake and Iwasa, 2002a), suggesting our lattice size is sufficient. Simulations
141
were started from a random initial condition, uniformly distributed throughout the
142
interval of possible values and run for 100 years unless otherwise stated.
143
2.3. Analysis
144
In order to study masting behavior, we must quantify ”synchronized, highly
145
variable reproduction”. To quantify synchrony, we use a measure of total syn-
146
chrony, S Y NC (as we did in (Lyles et al., 2009), following Satake and Iwasa
147
(Satake and Iwasa, 2002a)): S Y NC =
Vby , Vby + Vwy
(4)
148
where Vby represents average between-year variance of the population (repro-
149
ductive output of model population averaged for each year and then the variance
150
computed between years) and Vwy represents average within year variance of the
151
population (variance between individual reproductive output computed for each
152
year and then averaged). If the within year variance becomes arbitrarily small rel-
153
ative to between year variance, S Y NC → 1 and the population is considered to be
154
synchronized. Conversely, as within-year variance increases relative to between-
155
year variance S Y NC → 0 representing a population which is not synchronized.
156
Mean S Y NC values were calculated by averaging 100 S Y NC values that were
157
each calculated after running the simulation for 100 years (averaging 1,000 S Y NC
158
values provided similar results.) We verified whether moderate to high S Y NC
159
values corresponded to masting behavior by checking time series output of the
160
model. This was true for all cases except the globally coupled system when k < 1
8
161
and β x (t) is constant, which exhibits a stable equilibrium (highly synchronized but
162
not variable).
163
3. Results
164
We define masting to be synchronized, variable reproduction whose variabil-
165
ity is nearly double that of the environment and with a coefficient of variation
166
(CV) of at least one-half: environmental variability should be amplified and vari-
167
ability should be fairly large. Other studies have required CV to be larger than 1
168
(Liebhold et al., 2004; Lamontagne and Boutin, 2007) or larger than 1.6 (Kelly,
169
1994), however this is more applicable to plants whose average reproductive out-
170
put is near zero because CV increases without bound as the mean approaches
171
zero. Some plants, for example pistachio, may have average output much higher
172
than zero, decreasing the CV, though they exhibit a similar degree of variabil-
173
ity and other characteristics of masting behavior, such as delayed reproduction
174
(Stevenson and Shackel, 1998) and synchronous fruit production (Johnson and
175
Weinbaum, 1987; Lyles et al., 2009; Rosenstock et al., 2011). Here, we compare
176
both the synchrony S Y NC and variability CV of the model without noise (detailed
177
in (Satake and Iwasa, 2002a) and equivalent to our model with a constant β x (t)) to
178
that of the model with noise in density-dependent pollen limitation (β x (t)), and the
179
model with noise in accumulated resources/threshold for reproduction (detailed in
180
(Satake and Iwasa, 2002b) and equivalent to our model with constant β x (t) and a
181
noise term added to the second equation).
182
3.1. Synchrony
183
Pollen dispersal distance (D) has a dramatic effect on synchrony, hence mast-
184
ing,in the model. In the absence of noise, S Y NC increases with pollen dispersal 9
185
distance, for a given cost of reproduction k and average density-dependent pollen
186
limitation βavg (Figure 1). That is, as the pollen dispersal distance is increased,
187
masting is observed for larger ranges of k. 2
1 0.8 0.6
1
0.4
0.5
2
k
3
4
5
0.8 0.6
1
0.4
0.5
0.2 1
1
1.5 Average β
1.5 Average β
2
0
0.2 1
(a) D = 1 1
Average β
Average β
0.4
0.5
5
0
4
5
1 0.8 0.6
1
0.4
0.5
0.2 3
4
1.5
0.6 1
k
3
2
0.8
1.5
2
k
(b) D = 2
2
1
2
0
0.2 1
(c) D = 5
2
k
3
4
5
0
(d) D = 13 (Global pollen coupling)
Figure 1: No noise (β x (t) constant): S Y NC values of model versus k and βavg as pollen dispersal distance D is varied. S Y NC values increase with pollen dispersal distance D and densitydependent pollen limitation βavg , and decrease with increasing cost of reproduction k.
188
At a given set of parameter values (k, βavg ), variability in density-dependent
189
pollen limitation enhances synchrony, hence masting if both the spatial correlation
190
R and variability of the noise cv are large enough, as illustrated in Figure 2 for the
191
case of nearest neighbor coupling (D = 1). For example, when R = 0 increasing
192
cv causes S Y NC to decrease (compare Figures 2(a) & 2(b) to Figure 1(a)). In
193
contrast, when R = 0.5, increasing cv increases the S Y NC value at a given com-
194
bination of k and βavg (compare Figures 2(c) & 2(d) to Figure 1(a)). The same is 10
195
true for the model with noise in accumulated resources/threshold for reproduction
196
(Figure 3). The results for other pollen dispersal distances examined (D = 5 and
197
D = 13/Global) are analogous to those of the nearest neighbor case (unpublished
198
results). That is, both types of noise increase the parameter range of masting when
199
spatial correlation and variability of the noise are both large enough. 2
1 0.8 0.6
1
0.4
0.5
2
k
3
4
5
0.8 0.6
1
0.4
0.5
0.2 1
1
1.5 Average β
1.5 Average β
2
0
0.2 1
(a) R = 0, cv = 0.1 1
Average β
Average β
0.4
0.5
4
5
0
5
1 0.8 0.6
1
0.4
0.5
0.2 3
4
1.5
0.6 1
k
3
2
0.8
1.5
2
k
(b) R = 0, cv = 0.5
2
1
2
0
0.2 1
(c) R = 0.5, cv = 0.1
2
k
3
4
5
0
(d) R = 0.5, cv = 0.5
Figure 2: Noise in Pollination Efficiency: S Y NC values of model versus cost of reproduction k and average density-dependent pollen limitation βavg for various combinations of correlation R and coefficient of variation cv, under local pollen coupling (D = 1).
200
For the rest of this paper, we focus on a slice of k − β parameter space where
201
βavg = 1 and k varies. We do this to decrease the number of free parameters
202
and choose this particular slice because it seems to include a complete range of
203
possible behavior (see Figure 1). We also fix the spatial correlation for both types
204
of noise at r, R = 0.8 (highly correlated). We will consider both large (σ, cv = 11
2
1 0.8 0.6
1
0.4
0.5
2
k
3
4
5
0.8 0.6
1
0.4
0.5
0.2 1
1
1.5 Average β
1.5 Average β
2
0
0.2 1
(a) r = 0, σ = 0.1 1
Average β
Average β
0.4
0.5
4
5
0
5
1 0.8 0.6
1
0.4
0.5
0.2 3
4
1.5
0.6 1
k
3
2
0.8
1.5
2
k
(b) r = 0, σ = 0.5
2
1
2
0
0.2 1
(c) r = 0.5, σ = 0.1
2
k
3
4
5
0
(d) r = 0.5, σ = 0.5
Figure 3: Noise in accumulated resources/threshold for reproduction: S Y NC values of model versus cost of reproduction k and average density-dependent pollen limitation βavg for various combinations of correlation r and coefficient of variation σ, under local pollen coupling (D = 1).
12
205
0.5) and small (σ, cv = 0.1) noise and nearest-neighbor, intermediate, and global
206
pollen coupling distances (D = 1, 5, or D = 13/Global).
207
3.2. Variability
208
For the case with no noise, the coefficient of variation of reproductive output
209
(CV) is greater than one-half for k in [1, 2] under nearest neighbor coupling, and
210
the range of k for which CV is large enough increases as D is increased (Figure
211
4(a)), in parallel with the S Y NC results and confirming the presence of masting
212
under these conditions.
213
The inclusion of noise in density-dependent pollen limitation β x (t) may result
214
in near constant reproduction (CV less than that of the environment), resource
215
matching (variability/CV mirrors that of the environment), or masting (CV at least
216
twice as big as that of the environment), depending upon both the CV of the noise
217
and the extent of pollen coupling D. Reproductive behavior is nearly unchanged
218
from the deterministic (no noise) case for any cost of reproduction k, when both
219
the variability of β x (t) and the extent of spatial coupling are small (D = 1, Figure
220
4(b)). Resource matching is observed for k outside the interval [1,2] whenever
221
β x (t) is noisy enough and pollen coupling is limited to nearest neighbors (D = 1,
222
4(c)). Otherwise, when the spatial extent of pollen coupling is much larger (D = 5
223
or global pollen coupling), masting is observed for large noise in β x (t) (Figure
224
4(c)). In contrast to the case of noise in density-dependent pollen limitation, the
225
noise in accumulated resources/threshold for reproduction is always amplified by
226
the system under the parameter regime studied. This is illustrated in Figures 4(e)
227
and 4(d), where CV of the reproductive output is always larger than that of the
228
noise term in the model. Thus, there is never resource matching because environ-
229
mental variability is always amplified. Similarly, there is always masting if the 13
230
noise is large enough, so that the amplified environmental noise yields a CV of
231
reproductive output greater than one-half.
232
3.3. Types of Masting
233
As discussed in section 1, masting may be divided into three categories: strict,
234
normal bimodal, and normal switching (Kelly, 1994). In determining whether and
235
what type of masting is present, we will consider the long-term behavior of the
236
reproductive output of the model under nearest-neighbor coupling (D = 1). In the
237
absence of noise, masting only occurs when k is between 1 and 2 and it is strict
238
(Figure 5(a)). When large noise in density-dependent pollen limitation β x (t) is
239
included in the model, strict and possibly normal bimodal masting are observed
240
for k values in an interval around k = 1, and normal switching masting is observed
241
for k > 2 (Figure 5(b)). Similarly, when noise in accumulated resources/threshold
242
for reproduction is included in the model normal switching masting is observed
243
for most k values, with the exception of normal bimodal and/or strict masting for
244
a range of parameters near k = 1 and when the noise is small enough (Figures 5(c)
245
and 5(d)). The results for global coupling are analogous.
246
4. Discussion
247
In agreement with previous work (Satake and Iwasa, 2002b), we show that
248
including the biological realism of noise increases the parameter range for which
249
masting may occur. However, our results suggest that the type of environmental
250
noise a plant population experiences will have a profound effect on the dynamics
251
of reproductive output. This highlights the importance of choosing appropriate
252
noise sources in mathematical models and is described in more detail below.
14
1.5 1.2 1 CV
CV
1 0.8 0.6
0.5
0.4 0.2 0 0
1
2
k
3
4
0 0
5
1.2
k
3
4
5
1.2
1
1
0.8
0.8
CV
CV
2
(b) Noisy Pollination, cv = 0.1
(a) No noise
0.6 0.4
0.6 0.4
0.2 0 0
1
0.2 1
2
k
3
4
0 0
5
(c) Noisy Pollination, cv = 0.5
1
2
k
3
4
5
(d) Noisy Resources, σ = 0.1
CV
2
1.5
1 0
1
2
k
3
4
5
(e) Noisy Resources, σ = 0.5 Figure 4: Coefficient of variation of reproductive output (CV) under different types of noise. D = 1 (open circles), D = 5 (closed circles), and Global (stars). Simulations were run for 1,000 years and the coefficient of variation CV of reproductive output was calculated. Note the different scale in (d).
15
1
0.8
0.8
Reproductive Output
Reproductive Output
1
0.6 0.4 0.2 0 0
1
2
k
3
4
0.6 0.4 0.2 0 0
5
1
2
k
3
4
5
(b) Noisy Pollination, cv = 0.5
(a) No noise
4
Reproductive Output
Reproductive Output
5
3 2 1 0 0
1
2
k
3
4
1.2 1 0.8 0.6 0.4 0.2 0 0
5
(c) Noisy Resources, σ = 0.5
1
2
k
3
4
5
(d) Noisy Resources, σ = 0.1
Figure 5: Long term behavior for nearest neighbor coupling (D = 1). Note the different scale in (c). Simulations were run for 200 years and the last 100 years were plotted versus k. Results were similar when the simulations were run for 1,000 years, with the last 100 years shown.
16
253
Plant reproductive variability has been divided into five categories: nearly
254
constant, resource matching, and three types of masting behavior (strict, normal
255
bimodal, and normal switching). Since all plants are subject to environmental
256
variability, our results suggest that (pollen-limited) plant populations that exhibit
257
constant reproductive output, resource matching, and (to a lesser extent) strict
258
masting or normal bimodal masting may respond to noise in density-dependent
259
pollen limitation rather than noise in the threshold for reproduction and/or accu-
260
mulated resources. Those that exhibit the normal switching type of masting may
261
be sensitive to either type of noise. However, density-dependent pollen limitation
262
is likely to be highly variable and discrete because weather events may cause nu-
263
merous phenological responses. For example, partial failure of the pollination pro-
264
cess may be caused by poor male-female flower synchronization (Schauber et al.,
265
2002; Post, 2003) and precipitation during flowering (Knapp et al., 2001; Knops
266
et al., 2007). Thus, noise in density-dependent pollen limitation is a likelier candi-
267
date for the cause of normal switching masting in most pollen-limited plant pop-
268
ulations. Combined with the fact that noise in accumulated resources/threshold
269
for reproduction is largely ineffective for inducing masting in non-pollen-limited
270
plant populations (Satake and Iwasa, 2002b), while noise in pollination efficiency
271
induces masting over a wide parameter range in such plants (Lyles et al., 2009),
272
we suggest that noise in the pollination process is likely to be a proximate mech-
273
anism for variable reproduction in most plants, except for possibly the most vari-
274
able, pollen-limited ones. We hypothesize that the main mechanism by which
275
variability in accumulated resources/threshold for reproduction enhances mast-
276
ing behavior in the models is through adding variability in the pollen limitation
277
term. In contrast, including noise in density-dependent pollen limitation does not
17
278
change the range of y or the range of the pollen limitation term.
279
Our results also show that the presence of switching in the dynamics of a par-
280
ticular plant’s reproduction does not preclude the possibility that the reproductive
281
output of a population of such plants will exhibit resource matching or even near
282
constant reproduction. Moreover, the presence of switching does not confirm that
283
true masting is occurring in the absence of an idea of how variable the environment
284
is. Furthermore, given similar environmental variability, some plant populations
285
may exhibit masting, while others show resource matching or constant reproduc-
286
tion.
287
One may ask why our CV’s of reproductive output for the model with noisy
288
β x (t) are “small” compared to many published values for masting (Kelly, 1994;
289
Kelly et al., 2000; Kelly and Sork, 2002; Kelly et al., 2008). First, researchers
290
may focus on the most variable of species and/or those with zero reproduction
291
between mast years. As discussed in the first paragraph of the results section,
292
for a given amount of variability (standard deviation), the CV approaches infinity
293
as the mean approaches zero and is therefore sensitive to small changes in the
294
mean. Thus, plants with mean reproduction near zero may have CV larger than
295
one while a plant population with similar variability but mean farther from zero
296
has CV less than one. The historical origin of using a “null” CV of 1 for non-
297
masting species probably comes from comparing reproduction to a Poisson dis-
298
tribution (with mean=variance). But it’s not clear that plant reproduction should
299
be Poisson distributed, other than the fact that seeds are count data. Second, some
300
plant populations may be responding to noise in accumulated resources/threshold
301
for reproduction, which can cause higher CV’s in the models. And third, the
302
variability of density-dependent pollen limitation β x (t) may be more discrete than
18
303
represented by normally distributed noise. Although, the climate may vary ac-
304
cording to a normal distribution, its effect on the pollination process can be almost
305
all-or-none.
306
Given that the scale of pollen coupling appears to greatly influence whether or
307
not masting will occur, the persistence of masting behavior may be increasingly
308
threatened by large scale global change processes such as habitat fragmentation
309
and global climate change. Our results suggest that habitat fragmentation, which
310
is occurring at alarming rates in many parts of the world, will decrease pollen
311
exchange and inhibit pollen coupling between disparate individuals and popula-
312
tions. This is consistent with empirical studies that have shown negative effects
313
of pollen limitation on reproductive success in fragmented landscapes (Aguilar
314
et al., 2006). The consequences of global climate change on pollen coupling and
315
masting is less clear because the effect is likely to be species specific. Depend-
316
ing on the precise weather cue a plant population responds to, which can include
317
precipitation and temperatures at various parts of the year (Kelly and Sork, 2002),
318
and its particular values for k and average β x (t) (βavg ), climate change may either
319
favor or prohibit flowering synchrony. For example, species that respond to warm
320
temperatures to initiate bud induction (Kelly et al., 2000, 2008) may produce flow-
321
ers and seed more frequently while many species may suffer reproductive failure
322
due to more regular frost events (Inouye, 2000). Nevertheless the significance of
323
pollen coupling in our results suggests that climate change and habitat fragmenta-
324
tion, together, present considerable challenges to the longevity of pollen mediated
325
masting species.
19
326
5. Acknowledgements
327
Danielle Lyles was supported with an NSF Mathematical Sciences Postdoc-
328
toral Research Fellowship, award number DMS -0703700. Alan Hastings was
329
supported with NSF grants EF-0827460 and 1344187. Todd Rosenstock was sup-
330
ported by a grant from the California Pistachio Research Board and the Cali-
331
fornian Department of Food and Agriculture, Fertilizer Research and Education
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Program.
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(2000) 423–429.
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Periodic Reproduction out of Chaos, Journal of Theoretical Biology 203 (2000)
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D. Lyles, T. S. Rosenstock, A. Hastings, P. H. Brown, The role of large envi-
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ronmental noise in masting: General model and example from pistachio trees,
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tradeoff between growth and reproduction in California Oaks, PNAS 104 (43)
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M. T. Stevenson, K. A. Shackel, Alternate Bearing in Pistachio as a Masting Phe-
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Storage, Journal of the American Society for Horticultural Science 123 (6)
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(1998) 1069–1075.
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ternate bearing among kerman pistachio trees, Journal of the American Society
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24
Plant Reproduction and Environmental Noise: How do Plants Do It? Danielle Lyles, Todd S. Rosenstock, Alan Hastings
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Cite this article as: Danielle Lyles, Todd S. Rosenstock, Alan Hastings, Plant Reproduction and Environmental Noise: How do Plants Do It?, Journal of Theoretical Biology, http://dx.doi.org/10.1016/j.jtbi.2015.02.009 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Plant Reproduction and Environmental Noise: How Do Plants Do It? Danielle Lyles a,1,∗, Todd S. Rosenstockb,2 , Alan Hastingsc a
Department of Environmental Science and Policy, One Shields Avenue, University of California, Davis, CA 95616 b Department of Plant Sciences, One Shields Avenue, University of California, Davis, CA 95616 c Department of Environmental Science and Policy, One Shields Avenue, University of California, Davis, CA 95616
Abstract Plant populations exhibit a wide continuum of reproductive behavior, ranging from nearly constant reproductive output on one end to the extreme of masting (synchronized, highly variable reproduction) on the other. Here, we show that including variability (noise) in density-dependent pollen limitation in current models for pollen-limited plant reproduction may produce any behavior on this continuum. We previously showed that (large) variability in pollination efficiency (a related phenomenon) may induce masting in non-pollen-limited plant populations. Other modeling studies have shown that including variability in accumulated resources (and/or the threshold for reproduction) may induce masting, but do account for masting in non-pollen-limited plant populations. Thus, our results suggest that the range of plant reproductive behavior may be explained with the simple resource budget model combined with the biological realism of variability ∗
Corresponding author Email address: [email protected] (Danielle Lyles ) 1 Current address: Department of Mathematics, The University of Texas at San Antonio, San Antonio, TX 78249 2 Current address: World Agroforestry Centre (ICRAF), PO Box 30677-00100, Nairobi, Kenya
Preprint submitted to Journal of Theoretical Biology
February 11, 2015
in density-dependent pollen limitation. This is a specific example of an important functional consequence of the interactions between stochasticity and nonlinearity, and highlights the importance of carefully considering both the biological basis and the mathematical effects of the noise term. Keywords: moran, masting, synchronized, hybrid, model
1
1. Introduction
2
Much of the development of the understanding of population dynamics in
3
ecology has come from models that are deterministic (Kingsland, 1995). How-
4
ever, as far back as the work of Moran (Moran, 1953) on synchrony, the impor-
5
tance of stochasticity, and in particular external environmental variability, in shap-
6
ing spatiotemporal dynamics has been recognized. More recently, in a variety of
7
areas there has been increasing attention paid to the importance of variability.
8
However, much of the stochastic theory has been based on linearizations, or on
9
assumptions that the level of noise is small (Lande et al., 2003). In contrast to this
10
theory it has been recognized that natural variability can be very large, due to envi-
11
ronmental cues such as highly variable rainfall patterns or timing of frosts, which
12
in turn produces highly variable population densities through time (Connell and
13
Sousa, 1983). Thus, it is important for ecologists to get an understanding of the
14
importance of large variability as an influence on spatiotemporal dynamics. Yet,
15
this understanding can only come from attention to specific systems, like masting
16
behavior of plants (Liebhold et al., 2004).
17
Masting is synchronized, highly variable reproduction in a population of plants.
18
Masting may be divided into three categories: strict (bimodal with no overlap be-
19
tween tails), normal bimodal (bimodal with overlap between tails, bimodality is
2
20
statistically significant), or normal switching (output varies greatly and the pres-
21
ence of switching is verified by negative autocorrelation of reproductive output)
22
(Kelly, 1994). Although masting has received much attention, there is no distinct
23
line that may be drawn between masting and non-masting species (Kelly, 1994).
24
Plants that truly mast should vary more than their environment (Kelly, 1994). The
25
predominant hypothesis for this, resource switching, suggests that reproduction
26
is carbon-limited and the individual plant alternates allocation of carbon between
27
reproductive and vegetative structures.
28
While resource switching and carbon allocation patterns might explain vari-
29
able reproduction,how might a population of plants synchronize their reproduc-
30
tion? Empirical studies have shown that weather cues may synchronize plants for
31
masting (Schauber et al., 2002; Kelly et al., 2000, 2008; Sel˙as, 2000; Kon et al.,
32
2005; Kon and Noda, 2007). Such studies also repeatedly find that both internal
33
resource dynamics (resource switching) and cues from appropriate weather vari-
34
ables (which are noisy) must be combined in order to explain masting data (Sel˙as,
35
2000; Kon et al., 2005; Rees et al., 2002; Lamontagne and Boutin, 2007). Further-
36
more, a meta-analysis of masting data on a global scale has shown that masting
37
species with higher variability generally are located in areas with higher variabil-
38
ity in rainfall (Kelly and Sork, 2002). Kelly et al. have also shown that, for a
39
given plant population, those at higher altitudes have lowered temperature thresh-
40
olds for a given flowering effort (Kelly et al., 2008), suggesting that the variability
41
of temperature is more important than absolute temperature.
42
Mathematical modeling studies support the idea that resource switching com-
43
bined with environmental noise is an important component of the mechanism of
44
masting in plant populations. Both Isagi (Isagi et al., 1997) and Satake & Iwasa
3
45
(Satake and Iwasa, 2000, 2002a) have shown that resource switching combined
46
with pollen limitation may cause masting in monoecious plants. Satake and Iwasa
47
have also shown that such a mechanism may be greatly enhanced by the addition
48
of environmental noise in the form of variability in accumulated resources and/or
49
reproductive thresholds of individual plants, but only in the presence of pollen lim-
50
itation (Satake and Iwasa, 2002b). Previously, we extended their model to show
51
that resource switching combined with (large) environmental noise in the form
52
of variability in pollination efficiency may cause masting in plant populations in
53
the absence of pollen limitation and over a wide range of conditions (Lyles et al.,
54
2009). These examples are similar to the Moran effect, whereby independent
55
(linear) populations are synchronized by spatially correlated environmental noise
56
over a large spatial scale.
57
Pollen limitation is common among plants (Obeso, 2002) and it’s statistical
58
significance varies among sites and years, so the pollination environment is not
59
constant (Burd, 1994). Here, we examine the effect of variability in density-
60
dependent pollen limitation on the reproductive output of pollen-limited plant
61
populations. This may also be viewed as variability in the effectiveness of abi-
62
otic pollination under different conditions. For example, suppose that pollination
63
is a process that has a certain probability of success p on each trial. The effective-
64
ness of pollination can be defined as the probability of successful pollination for
65
each trial. So if p is the probability of pollination in one trial, then the probability
66
of pollination after n trials is 1 − (1 − p)n . If the number of trials is proportional to
67
the number of flowering plants, then this is equivalent to density dependent pollen
68
limitation, and variation in p is analogous to variation in density-dependent pollen
69
limitation. Variability in density-dependent pollen limitation may arise in many
4
70
ways. For example, from a change in distance between partners (habitat frag-
71
mentation) (Aguilar et al., 2006) or asynchrony in flower development between
72
male and female flowers (Vankin et al. 2006). Synchronous flowering is neces-
73
sary to achieve high density-dependent pollen limitation and regional climate may
74
influence the timing of flowering among several species of plants (Post, 2003;
75
Schauber et al., 2002). Other aspects of the weather and its inherent variabil-
76
ity may also change density-dependent pollen limitation in some years especially
77
for wind-pollinated plants (Koenig and Ashley, 2003). For example, precipita-
78
tion during flowering may limit pollen dispersal and cause mechanical damage to
79
flowers (Knapp et al., 2001; Knops et al., 2007). Variability in density-dependent
80
pollen limitation may also reflect variability in the effectiveness of biotic pollina-
81
tion under different conditions, in the presence of pollinators that are not highly
82
specialized. For example, if the probability of transferring pollen among con-
83
specifics increases with the density of conspecifics. There is an active literature
84
in pollination ecology exploring this issue (Kunin, 1993; Crone, 2013). Thus,
85
density-dependent pollen limitation can vary, perhaps quite significantly, from
86
year-to-year. In this paper we investigate whether such noise in density-dependent
87
pollen limitation may contribute to masting behavior in the (pollen limited) plant
88
population model(s) of Satake and Iwasa (Satake and Iwasa, 2000, 2002a,b).
89
2. Modeling, Simulation, and Analysis
90
2.1. Model
91
The model we use extends our earlier work regarding plants that aren’t pollen-
92
limited (Lyles et al., 2009) by expanding upon the idea that large variability in
93
pollination efficiency (here represented by variability in density-dependent pollen 5
94
limitation) is an important element for determining the reproductive behavior of
95
plants. This work was based on the models by Satake and Iwasa (Satake and
96
Iwasa, 2000, 2002a,b; Iwasa and Satake, 2004) developed from a still earlier
97
model by Isagi et al. (Isagi et al., 1997). The models are developed in terms
98
of the resource budget for an individual plant which adds resources each year due
99
to photosynthesis.
100
Plants are assumed to flower if resources at the beginning of the year are above
101
a threshold level, and not to flower if resources are below the threshold. If plants
102
do flower, the amount of flowering depends on the difference between resources
103
available and the threshold level, and the amount of resources is reduced by an
104
amount depending on the fruit set. By nondimensionalization and rescaling, the
105
threshold level for flowering can be assumed to be zero, and in a deterministic
106
description the amount of resources added each year can be assumed to be one
107
(Satake and Iwasa, 2000). Letting y x (t) represent non-dimensionalized resource
108
stores of individual plant in a population (plant x) leads to the model:
if y x (t) < 0, y x (t + 1) = y x (t) + 1
(1)
if y x (t) > 0, y x (t + 1) = −k ∗ P(y x (t), β x (t)) ∗ y x (t) + 1
(2)
109
where k > 0 represents resource depletion due to reproduction (the cost of
110
reproduction), β x (t) is variable density-dependent pollen limitation, and the func-
111
112
tion P(y x (t), β x (t)) is a pollen limitation term that determines the contribution of pollen from nearby plants. In particular, P(y x (t), β x (t)) = ( 1n i∈U x [yi (t)]+ )βx (t)
113
where [y]+ = y if y ≥ 0 and [y]+ = 0 if y ≤ 0 and U x contains individuals
114
within the pollen dispersal distance D of plant x and n is the number of plants in
115
U x . The term P(y x (t), β x (t)) is the average pollen contribution of trees within the 6
116
pollen dispersal distance D of tree x (a number in [0,1]) raised to the power β x (t),
117
and so has values in the interval [0,1]. When β x (t) = 0, there is no pollen limi-
118
tation. When β x (t) is small, small amounts of pollen are magnified so the effects
119
of pollen limitation are minimal. When β x (t) is large, it takes larger amounts of
120
pollen to produce a similar pollen contribution, implying strong pollen limitation.
121
The key feature of β x (t) is that the ratio of seeds to flowers depends on the density
122
of flowering plants. This density dependence provides an endogenous feedback
123
that promotes synchrony. The noise term β x (t) is normally distributed with mean
124
βavg and coefficient of variation cv, so the standard deviation is cvβavg . The cor-
125
relation is described by a parameter R by writing β x (t) as a combination of two
126
variables as follows (Satake and Iwasa, 2002b): β x (t) = δ(t) + γ x (t)
(3)
129
where δ(t) is normally distributed with mean Rβavg and standard deviation √ cvβavg R and γ x (t) is normally distributed with mean (1 − R)βavg and standard √ deviation cvβavg 1 − R. Thus, when R = 1, β x (t) is the same value for each in-
130
dividual and when R = 0, β x (t) is independent among individuals. Intermediate
131
levels of correlation are obtain for values of R between 0 and 1. We allow β x (t) to
132
be normally distributed to compare to the case of noise in the threshold for repro-
133
duction and/or annual net production (Satake and Iwasa, 2002b). Since negative
134
β x (t) ignore biological realities, we set negative β x (t) values equal to zero in the
135
simulations.
136
2.2. Simulation
127
128
137
Individual plants were represented as positions in a 10 X 10 rectangular array,
138
which for the purposes of this paper, may be referred to as a ”forest”. Our results 7
139
with a 10 x 10 lattice match those of Satake and Iwasa with a 100 x 100 lattice
140
(Satake and Iwasa, 2002a), suggesting our lattice size is sufficient. Simulations
141
were started from a random initial condition, uniformly distributed throughout the
142
interval of possible values and run for 100 years unless otherwise stated.
143
2.3. Analysis
144
In order to study masting behavior, we must quantify ”synchronized, highly
145
variable reproduction”. To quantify synchrony, we use a measure of total syn-
146
chrony, S Y NC (as we did in (Lyles et al., 2009), following Satake and Iwasa
147
(Satake and Iwasa, 2002a)): S Y NC =
Vby , Vby + Vwy
(4)
148
where Vby represents average between-year variance of the population (repro-
149
ductive output of model population averaged for each year and then the variance
150
computed between years) and Vwy represents average within year variance of the
151
population (variance between individual reproductive output computed for each
152
year and then averaged). If the within year variance becomes arbitrarily small rel-
153
ative to between year variance, S Y NC → 1 and the population is considered to be
154
synchronized. Conversely, as within-year variance increases relative to between-
155
year variance S Y NC → 0 representing a population which is not synchronized.
156
Mean S Y NC values were calculated by averaging 100 S Y NC values that were
157
each calculated after running the simulation for 100 years (averaging 1,000 S Y NC
158
values provided similar results.) We verified whether moderate to high S Y NC
159
values corresponded to masting behavior by checking time series output of the
160
model. This was true for all cases except the globally coupled system when k < 1
8
161
and β x (t) is constant, which exhibits a stable equilibrium (highly synchronized but
162
not variable).
163
3. Results
164
We define masting to be synchronized, variable reproduction whose variabil-
165
ity is nearly double that of the environment and with a coefficient of variation
166
(CV) of at least one-half: environmental variability should be amplified and vari-
167
ability should be fairly large. Other studies have required CV to be larger than 1
168
(Liebhold et al., 2004; Lamontagne and Boutin, 2007) or larger than 1.6 (Kelly,
169
1994), however this is more applicable to plants whose average reproductive out-
170
put is near zero because CV increases without bound as the mean approaches
171
zero. Some plants, for example pistachio, may have average output much higher
172
than zero, decreasing the CV, though they exhibit a similar degree of variabil-
173
ity and other characteristics of masting behavior, such as delayed reproduction
174
(Stevenson and Shackel, 1998) and synchronous fruit production (Johnson and
175
Weinbaum, 1987; Lyles et al., 2009; Rosenstock et al., 2011). Here, we compare
176
both the synchrony S Y NC and variability CV of the model without noise (detailed
177
in (Satake and Iwasa, 2002a) and equivalent to our model with a constant β x (t)) to
178
that of the model with noise in density-dependent pollen limitation (β x (t)), and the
179
model with noise in accumulated resources/threshold for reproduction (detailed in
180
(Satake and Iwasa, 2002b) and equivalent to our model with constant β x (t) and a
181
noise term added to the second equation).
182
3.1. Synchrony
183
Pollen dispersal distance (D) has a dramatic effect on synchrony, hence mast-
184
ing,in the model. In the absence of noise, S Y NC increases with pollen dispersal 9
185
distance, for a given cost of reproduction k and average density-dependent pollen
186
limitation βavg (Figure 1). That is, as the pollen dispersal distance is increased,
187
masting is observed for larger ranges of k. 2
1 0.8 0.6
1
0.4
0.5
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k
3
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1
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(b) D = 2
2
1
2
0
0.2 1
(c) D = 5
2
k
3
4
5
0
(d) D = 13 (Global pollen coupling)
Figure 1: No noise (β x (t) constant): S Y NC values of model versus k and βavg as pollen dispersal distance D is varied. S Y NC values increase with pollen dispersal distance D and densitydependent pollen limitation βavg , and decrease with increasing cost of reproduction k.
188
At a given set of parameter values (k, βavg ), variability in density-dependent
189
pollen limitation enhances synchrony, hence masting if both the spatial correlation
190
R and variability of the noise cv are large enough, as illustrated in Figure 2 for the
191
case of nearest neighbor coupling (D = 1). For example, when R = 0 increasing
192
cv causes S Y NC to decrease (compare Figures 2(a) & 2(b) to Figure 1(a)). In
193
contrast, when R = 0.5, increasing cv increases the S Y NC value at a given com-
194
bination of k and βavg (compare Figures 2(c) & 2(d) to Figure 1(a)). The same is 10
195
true for the model with noise in accumulated resources/threshold for reproduction
196
(Figure 3). The results for other pollen dispersal distances examined (D = 5 and
197
D = 13/Global) are analogous to those of the nearest neighbor case (unpublished
198
results). That is, both types of noise increase the parameter range of masting when
199
spatial correlation and variability of the noise are both large enough. 2
1 0.8 0.6
1
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k
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2
0
0.2 1
(c) R = 0.5, cv = 0.1
2
k
3
4
5
0
(d) R = 0.5, cv = 0.5
Figure 2: Noise in Pollination Efficiency: S Y NC values of model versus cost of reproduction k and average density-dependent pollen limitation βavg for various combinations of correlation R and coefficient of variation cv, under local pollen coupling (D = 1).
200
For the rest of this paper, we focus on a slice of k − β parameter space where
201
βavg = 1 and k varies. We do this to decrease the number of free parameters
202
and choose this particular slice because it seems to include a complete range of
203
possible behavior (see Figure 1). We also fix the spatial correlation for both types
204
of noise at r, R = 0.8 (highly correlated). We will consider both large (σ, cv = 11
2
1 0.8 0.6
1
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k
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0
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0
5
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(b) r = 0, σ = 0.5
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2
0
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(c) r = 0.5, σ = 0.1
2
k
3
4
5
0
(d) r = 0.5, σ = 0.5
Figure 3: Noise in accumulated resources/threshold for reproduction: S Y NC values of model versus cost of reproduction k and average density-dependent pollen limitation βavg for various combinations of correlation r and coefficient of variation σ, under local pollen coupling (D = 1).
12
205
0.5) and small (σ, cv = 0.1) noise and nearest-neighbor, intermediate, and global
206
pollen coupling distances (D = 1, 5, or D = 13/Global).
207
3.2. Variability
208
For the case with no noise, the coefficient of variation of reproductive output
209
(CV) is greater than one-half for k in [1, 2] under nearest neighbor coupling, and
210
the range of k for which CV is large enough increases as D is increased (Figure
211
4(a)), in parallel with the S Y NC results and confirming the presence of masting
212
under these conditions.
213
The inclusion of noise in density-dependent pollen limitation β x (t) may result
214
in near constant reproduction (CV less than that of the environment), resource
215
matching (variability/CV mirrors that of the environment), or masting (CV at least
216
twice as big as that of the environment), depending upon both the CV of the noise
217
and the extent of pollen coupling D. Reproductive behavior is nearly unchanged
218
from the deterministic (no noise) case for any cost of reproduction k, when both
219
the variability of β x (t) and the extent of spatial coupling are small (D = 1, Figure
220
4(b)). Resource matching is observed for k outside the interval [1,2] whenever
221
β x (t) is noisy enough and pollen coupling is limited to nearest neighbors (D = 1,
222
4(c)). Otherwise, when the spatial extent of pollen coupling is much larger (D = 5
223
or global pollen coupling), masting is observed for large noise in β x (t) (Figure
224
4(c)). In contrast to the case of noise in density-dependent pollen limitation, the
225
noise in accumulated resources/threshold for reproduction is always amplified by
226
the system under the parameter regime studied. This is illustrated in Figures 4(e)
227
and 4(d), where CV of the reproductive output is always larger than that of the
228
noise term in the model. Thus, there is never resource matching because environ-
229
mental variability is always amplified. Similarly, there is always masting if the 13
230
noise is large enough, so that the amplified environmental noise yields a CV of
231
reproductive output greater than one-half.
232
3.3. Types of Masting
233
As discussed in section 1, masting may be divided into three categories: strict,
234
normal bimodal, and normal switching (Kelly, 1994). In determining whether and
235
what type of masting is present, we will consider the long-term behavior of the
236
reproductive output of the model under nearest-neighbor coupling (D = 1). In the
237
absence of noise, masting only occurs when k is between 1 and 2 and it is strict
238
(Figure 5(a)). When large noise in density-dependent pollen limitation β x (t) is
239
included in the model, strict and possibly normal bimodal masting are observed
240
for k values in an interval around k = 1, and normal switching masting is observed
241
for k > 2 (Figure 5(b)). Similarly, when noise in accumulated resources/threshold
242
for reproduction is included in the model normal switching masting is observed
243
for most k values, with the exception of normal bimodal and/or strict masting for
244
a range of parameters near k = 1 and when the noise is small enough (Figures 5(c)
245
and 5(d)). The results for global coupling are analogous.
246
4. Discussion
247
In agreement with previous work (Satake and Iwasa, 2002b), we show that
248
including the biological realism of noise increases the parameter range for which
249
masting may occur. However, our results suggest that the type of environmental
250
noise a plant population experiences will have a profound effect on the dynamics
251
of reproductive output. This highlights the importance of choosing appropriate
252
noise sources in mathematical models and is described in more detail below.
14
1.5 1.2 1 CV
CV
1 0.8 0.6
0.5
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3
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CV
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(b) Noisy Pollination, cv = 0.1
(a) No noise
0.6 0.4
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k
3
4
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(c) Noisy Pollination, cv = 0.5
1
2
k
3
4
5
(d) Noisy Resources, σ = 0.1
CV
2
1.5
1 0
1
2
k
3
4
5
(e) Noisy Resources, σ = 0.5 Figure 4: Coefficient of variation of reproductive output (CV) under different types of noise. D = 1 (open circles), D = 5 (closed circles), and Global (stars). Simulations were run for 1,000 years and the coefficient of variation CV of reproductive output was calculated. Note the different scale in (d).
15
1
0.8
0.8
Reproductive Output
Reproductive Output
1
0.6 0.4 0.2 0 0
1
2
k
3
4
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1
2
k
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(b) Noisy Pollination, cv = 0.5
(a) No noise
4
Reproductive Output
Reproductive Output
5
3 2 1 0 0
1
2
k
3
4
1.2 1 0.8 0.6 0.4 0.2 0 0
5
(c) Noisy Resources, σ = 0.5
1
2
k
3
4
5
(d) Noisy Resources, σ = 0.1
Figure 5: Long term behavior for nearest neighbor coupling (D = 1). Note the different scale in (c). Simulations were run for 200 years and the last 100 years were plotted versus k. Results were similar when the simulations were run for 1,000 years, with the last 100 years shown.
16
253
Plant reproductive variability has been divided into five categories: nearly
254
constant, resource matching, and three types of masting behavior (strict, normal
255
bimodal, and normal switching). Since all plants are subject to environmental
256
variability, our results suggest that (pollen-limited) plant populations that exhibit
257
constant reproductive output, resource matching, and (to a lesser extent) strict
258
masting or normal bimodal masting may respond to noise in density-dependent
259
pollen limitation rather than noise in the threshold for reproduction and/or accu-
260
mulated resources. Those that exhibit the normal switching type of masting may
261
be sensitive to either type of noise. However, density-dependent pollen limitation
262
is likely to be highly variable and discrete because weather events may cause nu-
263
merous phenological responses. For example, partial failure of the pollination pro-
264
cess may be caused by poor male-female flower synchronization (Schauber et al.,
265
2002; Post, 2003) and precipitation during flowering (Knapp et al., 2001; Knops
266
et al., 2007). Thus, noise in density-dependent pollen limitation is a likelier candi-
267
date for the cause of normal switching masting in most pollen-limited plant pop-
268
ulations. Combined with the fact that noise in accumulated resources/threshold
269
for reproduction is largely ineffective for inducing masting in non-pollen-limited
270
plant populations (Satake and Iwasa, 2002b), while noise in pollination efficiency
271
induces masting over a wide parameter range in such plants (Lyles et al., 2009),
272
we suggest that noise in the pollination process is likely to be a proximate mech-
273
anism for variable reproduction in most plants, except for possibly the most vari-
274
able, pollen-limited ones. We hypothesize that the main mechanism by which
275
variability in accumulated resources/threshold for reproduction enhances mast-
276
ing behavior in the models is through adding variability in the pollen limitation
277
term. In contrast, including noise in density-dependent pollen limitation does not
17
278
change the range of y or the range of the pollen limitation term.
279
Our results also show that the presence of switching in the dynamics of a par-
280
ticular plant’s reproduction does not preclude the possibility that the reproductive
281
output of a population of such plants will exhibit resource matching or even near
282
constant reproduction. Moreover, the presence of switching does not confirm that
283
true masting is occurring in the absence of an idea of how variable the environment
284
is. Furthermore, given similar environmental variability, some plant populations
285
may exhibit masting, while others show resource matching or constant reproduc-
286
tion.
287
One may ask why our CV’s of reproductive output for the model with noisy
288
β x (t) are “small” compared to many published values for masting (Kelly, 1994;
289
Kelly et al., 2000; Kelly and Sork, 2002; Kelly et al., 2008). First, researchers
290
may focus on the most variable of species and/or those with zero reproduction
291
between mast years. As discussed in the first paragraph of the results section,
292
for a given amount of variability (standard deviation), the CV approaches infinity
293
as the mean approaches zero and is therefore sensitive to small changes in the
294
mean. Thus, plants with mean reproduction near zero may have CV larger than
295
one while a plant population with similar variability but mean farther from zero
296
has CV less than one. The historical origin of using a “null” CV of 1 for non-
297
masting species probably comes from comparing reproduction to a Poisson dis-
298
tribution (with mean=variance). But it’s not clear that plant reproduction should
299
be Poisson distributed, other than the fact that seeds are count data. Second, some
300
plant populations may be responding to noise in accumulated resources/threshold
301
for reproduction, which can cause higher CV’s in the models. And third, the
302
variability of density-dependent pollen limitation β x (t) may be more discrete than
18
303
represented by normally distributed noise. Although, the climate may vary ac-
304
cording to a normal distribution, its effect on the pollination process can be almost
305
all-or-none.
306
Given that the scale of pollen coupling appears to greatly influence whether or
307
not masting will occur, the persistence of masting behavior may be increasingly
308
threatened by large scale global change processes such as habitat fragmentation
309
and global climate change. Our results suggest that habitat fragmentation, which
310
is occurring at alarming rates in many parts of the world, will decrease pollen
311
exchange and inhibit pollen coupling between disparate individuals and popula-
312
tions. This is consistent with empirical studies that have shown negative effects
313
of pollen limitation on reproductive success in fragmented landscapes (Aguilar
314
et al., 2006). The consequences of global climate change on pollen coupling and
315
masting is less clear because the effect is likely to be species specific. Depend-
316
ing on the precise weather cue a plant population responds to, which can include
317
precipitation and temperatures at various parts of the year (Kelly and Sork, 2002),
318
and its particular values for k and average β x (t) (βavg ), climate change may either
319
favor or prohibit flowering synchrony. For example, species that respond to warm
320
temperatures to initiate bud induction (Kelly et al., 2000, 2008) may produce flow-
321
ers and seed more frequently while many species may suffer reproductive failure
322
due to more regular frost events (Inouye, 2000). Nevertheless the significance of
323
pollen coupling in our results suggests that climate change and habitat fragmenta-
324
tion, together, present considerable challenges to the longevity of pollen mediated
325
masting species.
19
326
5. Acknowledgements
327
Danielle Lyles was supported with an NSF Mathematical Sciences Postdoc-
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toral Research Fellowship, award number DMS -0703700. Alan Hastings was
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supported with NSF grants EF-0827460 and 1344187. Todd Rosenstock was sup-
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fornian Department of Food and Agriculture, Fertilizer Research and Education
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