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ONLINE SUPPORTING INFORMATION:
Individual phenotypic variation reduces interaction strengths in a consumerresource system
Jean P. Gibert and Chad E. Brassil
INDEX
Appendix 1: Parameter values ..………………………………………………………..
2-4
Appendix 2: Attack rate and Handling time ...…………………………………………
5
Appendix 3: Elasticity for attack rates and handling times ……………………………..
6-7
Appendix 4: Asymmetric trait distribution …………………………………………….. 8 - 10
Appendix 5: Asymmetric functional forms for attack rate and handling time ………… 11 - 14
Appendix 6: Asymmetric trait distributions and asymmetric attack rate and handling
time………………………………………………………………………………………15 - 16
Appendix 7: Species persistence ………………………………………………………
17
Appendix 8: Stability ……………………………………………………………………18 - 19
References: ………………………………………………………………………………… 20
1
51
APPENDIX 1: Exploring different parameter values
52
In this section we assess how robust our results are to a change in parameter values.
53
We did so by exploring other possible values for da , dh , t and n . For changes in da and
54
dh our qualitative results hold, but an increase in dh seems to have a less pronounced effect
55
than one in da (Fig. S1-1; also see Appendix 3). As t and n increase, the effect of individual
56
variation decreases (Fig. S1-2). This occurs because the attack rate and the handling time
57
become constant, and largely independent of the value of the controlling trait. Small t or n
58
leads to a large dependency of the attack rate and the handling time upon the underlying trait
59
value, and hence, to an increased effect of individual trait variation (Fig. S1-3).
60
61
Fig S1-1: Plots of interaction strength against increasing individual variation (gray: resource,
62
black: consumer). (a) a = 1, hmax = 2, hmin = 1, t =3, n = 1, da = 2, dh = 0. (b) same as (a) but
63
for da = 0, dh = 2.
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2
65
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Figure S1-2: Plots of interaction strength against individual variation measured as s 2 .
67
Parameter values: (a) a = 1, hmax = 2, hmin = 1, t =3, n = 1, da = 0, dh = 0. (b) same as (a) but
68
for t =1, n = 3. (c) same as (a) but for t =3, n = 3.
3
69
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Figure S1-3: Plots of interaction strength against individual variation measured as s 2 .
71
Parameter values: a = 1, hmax = 2, hmin = 1, t =0.1, n = 0.1, da = 0, dh = 0.
72
73
74
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76
77
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79
80
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4
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APPENDIX 2: Mean attack rate and mean handling time
In what follows we show how the mean attack rate and the mean handling time change
89
with increasing levels of individual variation. While attack rate decreases with individual
90
variation whenever phenotypic mismatch is small, handling time increases (Fig. S2-1a).
91
When phenotypic mismatch is large, however, attack rate increases at first with variation and
92
then decreases, and the opposite is true for handling time (Fig. S2-1b).
93
94
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Figure S2-1: Plots of how mean attack rate (black) and mean handling time (grey) change
96
with individual variation under small phenotypic mismatch (a) and larger phenotypic
97
mismatch (b). Parameter values: (a) a max = 2, hmax = 2, hmin = 1, e = 0.5, t =1, n = 1,
98
da = dh = 0 ; (b) same as in (a) but for da = dh = 2 .
5
99
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APPENDIX 3: Elasticity
The elasticity is a measure of model sensitivity defined as the absolute value of
101
¶log( f ) ¶log(a) , where f is the function of interest (interaction strength in this case), and
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a is the parameter of interest (attack rate or handling time in this case). The larger the
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elasticity, the more sensitive the function is to a change in the parameter.
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The effects of individual variation upon consumer-resource dynamics seem to be
105
mainly driven by variation in the attack rate, as its elasticity is generally larger than that of the
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of handling time regardless of phenotypic mismatch or individual variation (Fig. S3-1).
107
Although Jensen’s inequality predicts opposite effects of variation in attack rate and handling
108
time when considered independently (Fig. 1a, 1b), interaction strengths incorporating
109
individual variation in both attack rate and handling time simultaneously seem to mainly be
110
affected by variation in attack rate.
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Fig S3-1: Plot of the elasticity of the interaction strengths for with respect to the attack rate
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(black) and the handling time (gray). (a) a = 1, hmax = 2, hmin = 1, t =1, n = 1, da = 0, dh = 0.
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(b) same as (a) but for da = 2. (c) same as (a) but for dh = 2.
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127
7
128
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APPENDIX 4: asymmetric trait distribution
In the main text we assumed the trait that controls the ecological interaction through
130
its effect on attack rate and handling time to be normally distributed. However, the
131
distribution of some traits is highly asymmetric and skewed (Gouws et al. 2011). In this
132
section, we break this assumption by incorporating an asymmetric distribution (log-normal
133
distribution, Fig. S4-1). We show that the effect of individual variation is not largely affected
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by the choice of the underlying trait distribution but the range of scenarios at which
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interaction strength decreases with individual variation becomes larger when asymmetry is
136
taken into account.
137
Here, we assumed both attack rate and handling time to depend on the value of a log-
138
normally distributed trait with location parameter x and scale parameter s 2 . Then its density
139
in the population is:
é ( log(x) - x ) 2 ù
ú.
Lp(x, x ) =
exp ê2s 2
êë
úû
x 2ps 2
1
140
(1)
141
Note that as both the location and scale parameter control the shape of the distribution, the
142
variance of the distribution, and hence, individual variation, now depends on both parameters.
143
For simplicity, we focus on the case where only s 2 varies. We have numerically integrated
144
I R,L (a, h ) and IC,L (a, h ) to find the interaction strength with varying levels of individual
145
variation s 2 as:
146
I R,L (a, h ) = -R
¥
a (x)
ò 1+ a (x) h(x) R Lp(x, x ) dx
(2)
a (x)
Lp(x, x ) dx
2
-¥ (1+ a (x) h (x) R)
(3)
-¥
147
148
149
I C, L (a, h ) = e C
¥
ò
We found that the interaction strength has a qualitatively similar behavior with respect
to individual variation than in the case with a symmetric distribution. This is, there is a range
8
150
of scenarios at which the interaction strength decreases monotonically with individual
151
variation, and a range of scenarios at which the interaction strength is maximized by
152
intermediate values of individual variation (see main text). Indeed, there is an optimal amount
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of individual variation that maximizes interaction strength when trait mismatch is large, if the
154
average trait value in the population is smaller than the selective optimum ( da << 0 or dh << 0
155
, Fig S4-2a), and this behavior is also quantitatively comparable to the one obtained with a
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symmetric trait distribution. The interaction strength still decreases with individual variation
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whenever trait mismatch is small ( da ~ 0 and dh ~ 0 , Fig S4-2b), but this is also true for
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cases where the average trait value in the population is larger than the selective optimum (
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da >> 0 or dh >> 0 , Fig S4-2c). Thus, asymmetric trait distributions can increase the range of
160
scenarios in which interaction strengths decreases with individual variation.
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Figure S4-1: Plot of a symmetric distribution (e.g. normal) and an asymmetric distribution
164
(e.g. log-normal). The log-normal distribution used in the supplementary material mainly
165
differs from the normal distribution used in the main text in that it the former is more skewed
166
than the latter.
9
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Figure S4-2: Plots of interaction strength against individual variation measured as s 2 .
169
Phenotypic mismatch is large (a) and (c), and small in (b). Parameter values: (a) a = 1, hmax =
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2, hmin = 1, t =1, n = 1, da = -2, dh = 0. (b) same as (a) but for da = 0. (c) same as (a) but for
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da = 2.
10
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APPENDIX 5: asymmetric functional forms for attack rate and handling time
In the main text, we assumed the attack rate and handling time to be non-linear, yet
174
symmetric functional forms of the underlying controlling quantitative phenotypic trait.
175
However, these ecological attributes could be asymmetric, as found in most thermal response
176
curves (Vasseur et al. 2014). The asymmetry of these functional forms generally arise from
177
important physiological or biomechanical constrains (Vucic-Pestic et al. 2010), which need to
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be taken into account to accurately describe the non-linear relationship between underlying
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phenotypic traits and the ecological attributes they influence. In this section, we break the
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assumption of symmetry for the attack rate and the handling time, by incorporating
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asymmetric functional forms (Fig. S5-1). We found that the asymmetry in attack and handling
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times can have a quantitative effect in the way individual variation affects interaction
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strengths, mostly by reducing the range of possible scenarios in which interaction strength
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decreases monotonically with increasing individual variation.
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The now asymmetric predator’s attack rate, a asymm (x) , can be assumed to be maximal
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at a given optimal trait value x = qa , and to decrease away from that maximum at a different
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rate depending on the direction. Such a scenario can be modeled by:
é
( log(x) - log(qa ) )
2
ù
ú,
úû
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a asymm (x) = a max - a max exp ê-
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where a max is the maximal attack rate (Fig. S5-1a) and the rest of the parameters are as
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described in the main text. Similarly, the predator’s handling time, hasymm (x) , is minimal at
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the given optimal value x = qh , and increases away from that minimum at a different rate
192
depending on the direction like:
193
êë
2t 2
é log(x) - log(q ) 2 ù
(
h ) ú
,
hasymm (x) = (hmax - hmin ) expê2
ê
ú
2
n
ë
û
(4)
(5)
11
194
where hmax and hmin are maximal and minimal handling times respectively (Fig. S5-1b) and
195
the rest of the parameters are as described in the main text. Because of the asymmetry, it is
196
now impossible to derive analytic expressions for the mean (asymmetric) attack rate and
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handling times, so we have numerically integrated I R, asymm (a, h ) and IC, asymm (a, h ) to find the
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interaction strength with varying individual variation s 2 as:
I R, asymm (a, h ) = -R
199
¥
ò 1+ a
-¥
I C, asymm (a, h ) = e C
200
¥
ò
-¥
201
a asymm (x)
p(x, x ) dx
asymm (x) hasymm (x) R
(1+ a
a asymm (x)
asymm
(x) hasymm (x) R)
2
p(x, x ) dx
(6)
(7)
Overall, we found that the asymmetry in attack rate and handling time seems to
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preclude a monotonically decreasing relation of interaction strengths with individual
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variation. If phenotypic mismatch is large enough and the average trait value in the population
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is smaller than the selective optimum ( da << 0 or dh << 0 ), both the symmetric and the
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asymmetric case predict a hump shaped relationship between interaction strengths and
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individual variation. If phenotypic mismatch is small ( da ~ 0 and dh ~ 0 ), interaction seems
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to only increase with individual variation when asymmetric attack and handling rates are
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considered, rather than showing a monotonic decrease as with symmetric attack rates and
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handling times (Fig. S5-2b). Finally, if the average trait value in the population is larger than
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the selective optimum ( da >> 0 or dh >> 0 ), both the symmetric and the asymmetric case are
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congruent.
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Figure S5-1: Plots of attack and handling time against a given quantitative phenotypic trait,
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where qa and qh are the optimal trait values for attack rate and handling time respectively.
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Note that the ecological attributes are now asymmetric with respect to the trait of interest in
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contrast to what was assumed in the main text (Fig. 2, main text).
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Figure S5-2: Plots of interaction strength against individual variation measured as s 2 .
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Phenotypic mismatch is large in (a) and (c), and small in (b). Parameter values: (a) a = 1,
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hmax = 2, hmin = 1, t =1, n = 1, da = -3, dh = 0. (b) same as (a) but for da = 0. (c) same as (a)
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but for da = 3.
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APPENDIX 6: Asymmetric trait distributions, and asymmetric attack rate and handling time
In this section, we incorporate asymmetric trait distributions as well as asymmetric
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attack rate and handling times by means of equations (1), (4) and (5) of the supporting
235
information. Because of the asymmetry, it is now impossible to derive analytic expressions
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for the (asymmetric) attack rate and handling time, so we have numerically integrated
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I R, Lasymm (a, h ) and IC, Lasymm (a, h ) to find the interaction strength with varying individual
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variation s 2 as:
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I R, Lasymm (a, h ) = -R
¥
ò 1+ a
-¥
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I C, Lasymm (a, h ) = e C
¥
ò
a asymm (x)
Lp(x, x) dx
asymm (x) hasymm (x) R
a asymm (x)
-¥ (1+ a asymm (x) hasymm (x) R )
2
Lp(x, x ) dx
(8)
(9)
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The results for asymmetric distribution and asymmetric attack rate and handling time
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are comparable to those found in Appendix S5. Specifically, whenever phenotypic mismatch
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is large enough and the average trait value in the population is smaller than the selective
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optimum ( da << 0 or dh << 0 ), the symmetric and the asymmetric cases yield comparable
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predictions (Fig. S6-1a). Conversely, the interaction strength seems to be maximized by
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intermediate levels of individual variation whenever phenotypic mismatch is small ( da ~ 0
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and dh ~ 0 ), but this differs from what is predicted by the symmetric case (Fig. S6-1b).
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Finally, whenever the average trait value in the population is larger than the selective
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optimum ( da >> 0 or dh >> 0 , Fig. S6-1c), both symmetric and asymmetric cases are
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congruent. Overall, it seems that asymmetric relationships between the attack rate and the
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handling time with the underlying controlling quantitative trait precludes interaction strengths
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to decrease with individual variation, but the opposite is truth whenever only asymmetric
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distributions are considered.
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Figure S6-1: Plots of interaction strength against individual variation measured as s 2 .
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Phenotypic mismatch is large in (a) and (c), and small in (b). Parameter values: (a) a = 1,
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hmax = 2, hmin = 1, t =1, n = 1, da = -2, dh = 0. (b) same as (a) but for da = 0. (c) same as (a)
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but for da = 5.
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APPENDIX S7: Consumer persistence
Large values of individual variation can lead to consumer extinction (Fig S7-1), as
suggested by eqn 14 and eqn 15 of the main text.
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Figure S7-1: Outcome of the consumer-resource interaction as a function of individual
267
variation ( s 2 ) and phenotypic mismatch between preys and predators ( d 2 ). In the black
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region, consumers go extinct but the resource survives, while in white and grey regions both
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consumers and resources coexist. Parameter values: a max = 2, hmax = 2, hmin = 1, e = 0.5, t =1,
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n = 1, da = dh , K=1, b = 0.1.
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APPENDIX S8:
Here we show that for those values of s 2 for which coexistence is ensured, the larger
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s 2 is, the more stable the system becomes. To do so, we observe that, if s 2 is very small,
276
then the following equality holds,
¥
a (x ) R
RC a (x)
=ò
p(x, x ) dx ,
1+ a (x ) h (x ) R -¥ 1+ a (x) h(x) R
277
278
(10)
where:
¥
a (x ) = ò a (x)p(x, x ) dx
279
-¥
é
ù
a max t
da2
,
exp ê2
2 ú
ë 2 (s + t ) û
s 2 +t 2
=
280
(11)
¥
h (x ) = ò h(x)p(x, x ) dx
281
-¥
= hmax -
282
n (hmax - hmin )
s 2 +n 2
é
ù
dh2
exp êú,
2
2
ë 2 (s + n ) û
(12)
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and da = x - qa and dh = x - qh , are the distance between the mean trait in the population and
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the adaptive optimum (phenotypic mismatch).
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Hence, assuming that individual variation is small enough, we can assess local
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stability of the dynamic system by replacing the functional response defined in the main text
287
(in eqn 13 of the main text, or right side of eq. 10 in appendix) by the functional response
288
evaluated at a (x ) and h (x ) , and by then calculating the Jacobian of the system at its
289
equilibrium:
290
æ
ö
é
ù
ç - r d ëe - Kea (x )h (x ) + d h (x ) (1+ K a (x )h (x ))û - d ÷
ç
K a (x )h (x )e (e - d h (x ))
e ÷
÷.
J R*,C* = ç
æ d
ö
ç
÷
rç
+ e - d h (x )÷
0 ÷
çç
÷
è K a (x )
ø
è
ø
(13)
18
291
The system is stable, if and only if the determinant of J R*,C* is positive but its trace is
292
negative. The latter is true whenever:
293
d<
e + d h (x )
e
and a (x) <
. We can now use (11) of the appendix to obtain:
K h (x) (e + d h (x ))
h (x )
294
é
ù
amax t
da2
e + d h (x )
.
exp ê<
ú
2
2
ë 2 (s + t ) û K h (x ) (e + d h (x ))
s 2 +t 2
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If phenotypic mismatch is small ( da2 ~ 0 ), we can rearrange the eq. 14 to obtain:
296
s2 >
a max t K h (x ) (e - d h (x )) 2
-t .
e + d h (x )
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Finally, if we further assume that variation in attack rate has a larger effect than that in
298
handling time, as observed in appendix 3, we get eq. 3.3 of the main text:
299
s2 >
amax t K hmax (e - d hmax ) 2
-t .
e + d hmax
(14)
(15)
(16)
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Eq. 16 implies that for the system to be stable, individual variation needs to be larger than a
301
certain amount. This is supported by our simulations (Fig 3, main text), as increasing
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variation forces the system through a Hopf bifurcation, from an attractive limit cycle to an
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attractor node. Although the limit cycle is orbitally stable, the population fluctuations
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underwent by both interacting species makes the system more likely to lose species due to
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demographic or environmental variability.
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Vasseur DA, Delong JP, Gilbert B, Greig HS, Harley CDG, McCann KS, Savage V, Tunney
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TD, O’Connor MI. 2014 Increased temperature variation poses a greater risk to species
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than climate warming Increased temperature variation poses a greater risk to species
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Vucic-Pestic O, Rall BC, Kalinkat G, Brose U. 2010 Allometric functional response model:
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