1
Trait selection during food web assembly:
2
The roles of interactions and temperature
3
Online Appendices
4
5
Isabelle Gounand*1,2,3,4,5, Sonia Kéfi1, Nicolas Mouquet1 and Dominique Gravel2,3
6
7
Affiliations
8
1. Institut des Sciences de l’Evolution, Université de Montpellier II, CNRS, IRD, CC 065,
9
Place Eugène Bataillon, 34095 Montpellier Cedex 05, France
10
2. Université du Québec à Rimouski, Département de Biologie, Chimie et Géographie,
11
300 Allée des Ursulines, Québec G5L 3A1 Canada.
12
3. Quebec Center for Biodiversity Science.
13
4. Department of Evolutionary Biology and Environmental Studies, University of Zurich,
14
Winterthurerstrasse 190, CH-8057 Zürich, Switzerland.
15
5. Eawag, Swiss Federal Institute of Aquatic Science and Technology, Department of
16
Aquatic Ecology, Überlandstrasse 133, CH-8600 Dübendorf, Switzerland.
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18
19
*Corresponding author: [email protected]
20
List of Supplementary Material
21
Online Appendix A: Equilibria of small modules (Table 3)
22
Online Appendix B: Model comparison (Table 4)
23
Online Appendix C: Interaction domain (Fig. 6)
24
Online Appendix D: Invasion analyses, and the effect of temperature
25
1) Invasion criteria (Table 5 and Figs 7 and 8)
26
2) Effect of temperature (Table 6 and Fig. 9)
27
2
28
29
30
31
Appendix A
32
Equilibria of small modules
33
Table 3
34
Equilibria for models 1 to 4 when one producer P (3.1), or one herbivore H feeding on
35
one producer P (3.2), are present (small modules). For clarity we used the following
𝐸(𝑇−𝑇0 )
𝑎𝑦 𝑖 𝑀𝑖−0.25 𝑒 𝑘𝑇𝑇0
36
symbols: 𝑌𝑖 =
37
the metabolic rate, and 𝑐𝑖 =
38
intractable for model 4 when herbivore is present (see Table 1 for symbols).
3.1 Without
𝑌𝑖
𝑎𝑥
= 𝑎 𝑖 , the ratio of the biological rates. Equilibria were
𝑦𝑖
Model
R*
P*
Model 1
𝑚𝑃
𝑎𝑃 𝜀𝑃
𝐼𝜀𝑃 𝑜𝑢𝑡𝑅
−
𝑚𝑃
𝑎𝑃
Model 2
𝑐𝑃
𝜀𝑃
1 𝐼𝜀𝑃
(
− 𝑜𝑢𝑡𝑅 )
𝑌𝑃 𝑐𝑃
Model 3
𝑐𝑃
𝜀𝑃 𝐼
( − 𝑜𝑢𝑡𝑅 )
𝑌𝑃 𝑐𝑃
Model 4
𝑐𝑃
𝑜𝑢𝑡𝑃
(1 +
)
𝜀𝑃
𝑋𝑃
1
𝐼𝜀𝑃
(
− 𝑜𝑢𝑡𝑅 )
𝑌𝑃 𝑐 (1 + 𝑜𝑢𝑡𝑃 )
𝑃
𝑋𝑃
𝐼
𝑚𝐻
𝑎𝐻 𝜀𝐻
1
(𝜀 𝑎 𝑅 ∗ − 𝑚𝑃 )
𝑎𝐻 𝑃 𝑃
𝑐𝐻
𝜀𝐻
1
(𝜀 𝑌 𝑅 ∗ − 𝑋𝑃 )
𝑌𝐻 𝑃 𝑃
𝑐𝐻
𝜀𝐻
(𝑌 𝑅∗ − 𝑋𝑃 )
𝑌𝐻 𝑃
Herbivore
(Fig. 1a)
𝑋𝑖
the maximum consumption rate, 𝑋𝑖 =
𝐸(𝑇−𝑇0 )
𝑎𝑥 𝑖 𝑀𝑖−0.25 𝑒 𝑘𝑇𝑇0
Model 1
𝑜𝑢𝑡𝑅 +
3.2 With
Herbivore
𝑎𝑃 𝑚𝐻
𝑎𝐻 𝜀𝐻
𝐼
Model 2
(Fig. 1b and 1c)
𝑌𝑐
𝑜𝑢𝑡𝑅 + 𝑃 𝐻
𝜀𝐻
𝐼
Model 3
𝑌𝑐
𝑜𝑢𝑡𝑅 + 𝑃 𝐻
𝜀𝑃
H*
39
3
40
Appendix B
41
Model comparisons
42
Table 4
43
Model comparisons for a compartment i fed upon by a compartment j, with 𝐵𝑖 the density
44
of compartment 𝑖. For clarity we used the following symbols: 𝑌𝑖 = 𝑎𝑦 𝑖 𝑀𝑖−0.25 𝑒
45
maximum per capita uptake rate, and 𝑋𝑖 = 𝑎𝑥 𝑖 𝑀𝑖−0.25 𝑒
46
allometric constants 𝑎𝑥 𝑖 and 𝑎𝑦 𝑖 (for the biological rates measured at temperature T0)
47
are different between producers and herbivores but they do not vary within a trophic
48
level in the multispecies simulations. Models 3 and 4 are variants of model 2 and the
49
differences from model 2 are highlighted with grey background. The supply and loss rate
50
of the inorganic resources are the same for all models: 𝐼 − 𝑜𝑢𝑡𝑅 𝑅𝑖 . (see Table 1 for
51
symbols)
𝐸(𝑇−𝑇0 )
𝑘𝑇𝑇0
𝐸(𝑇−𝑇0 )
𝑘𝑇𝑇0
the
the metabolic rate. Note that
Description
Uptake
Production
Catabolism
Exportation
Model 1
Classical ConsumerResource model
−𝑎𝑖 𝐵𝑖 𝐵𝑗
+𝑎𝑖 𝜀𝑖 𝐵𝑖 𝐵𝑗
−𝑚𝑖 𝐵𝑖
-
−𝑌𝑖 𝐵𝑖 𝐵𝑗
+𝑌𝑖 𝜀𝐵 𝐵𝑖 𝐵𝑗
−𝑋𝑖 𝐵𝑖
-
−𝑌𝑖 𝐵𝑖 𝐵𝑗
𝜀𝑖
+𝑌𝑖 𝐵𝑖 𝐵𝑗
−𝑋𝑖 𝐵𝑖
-
−𝑌𝑖 𝐵𝑖 𝐵𝑗
+𝑌𝑖 𝜀𝐵 𝐵𝑖 𝐵𝑗
−𝑋𝑖 𝐵𝑖
−𝑜𝑢𝑡𝑖 𝐵𝑖
Bioenergetics
Model
Model
2
Efficiency in production
Model
3
Efficiency in uptake
Model
4
Supplementary losses
52
53
54
55
4
56
57
APPENDIX C
58
Interaction domain
59
60
Fig. 6
61
Interactions between herbivores and producers are based on their body mass 𝑀. We
62
defined the domain of interaction (grey area) using the 10th and 90th quantile regressions
63
on an empirical relationship between the logarithm of body mass of preys and predators
64
(Brose et al. 2006a, Gravel et al. 2013). Given its own body mass 𝑀𝐻 , an herbivore 𝐻
65
feed on all the producers whose log(𝑀) fall into the domain of interactions (dotted lines).
66
Given our initial distributions of body mass species (log⁡(𝑀) ∈ 𝒰[10−8 , 108 ]), the largest
67
producers escape herbivory
68
5
69
APPENDIX D
70
Invasion analysis, and the effect of temperature
71
We consider the situations where a producer or an herbivore tries to invade an
72
ecosystem as described in the Fig. 1a, 1b and 1c.
73
1) To invade the ecosystem, a species must grow when initially rare despite the
74
competition with the residents.
75
2) Changes in temperature may affect this outcome through variation in resource
76
availability or in species performance..
77
78
1) Invasion criteria
79
When a competitor is already present in the ecosystem, the invader must win the
80
competition to persist. It means that its growth rate must be positive when it tries to
81
invade the ecosystem at equilibrium. For instance in the case described in Fig. 2b, we
82
ask whether the producer P2, initially rare (𝑃2 → 0), and sharing the same resource than
83
P1 and the same herbivore H, can invade the system and exclude P1. The criteria to be
84
satisfied is:
85
86
87
𝑑𝑃2
|
> 0⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡[13]
𝑑𝑡 𝑅∗,𝑃1∗,𝐻 ∗,𝑃2≪𝑃1∗
⁡
Taking model 1 as an example, the system can be written:
𝑑𝑅
= 𝐼 − 𝑜𝑢𝑡𝑅 𝑅 − (𝑎𝑃1 𝑃1 + 𝒂𝟐 𝑷𝟐 )𝑅
𝑑𝑡
𝑑𝑃1
= 𝑎𝑃1 𝜀𝑃1 𝑃1 𝑅 − 𝑚𝑃1 𝑃1 − 𝑎𝐻 𝑃1 𝐻
𝑑𝑡
⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡[14]
𝑑𝑃2
= 𝑎𝑃2 𝜀𝑃2 𝑃2 𝑅 − 𝑚𝑃2 𝑃2 − 𝑎𝐻 𝑃2 𝐻
𝑑𝑡
𝑑𝐻
{ 𝑑𝑡 = (𝑎𝑃1 𝑃1 + 𝒂𝟐 𝑷𝟐 )𝑎𝐻 𝜀𝐻 𝐻 − 𝑚𝐻 𝐻
6
88
Since 𝑃2 tends to 0 at invasion, we can approximate the system by deleting the terms
89
containing 𝑃2 in the equation of R and H in the system [14] (in bold and red). Then the
90
invasion criteria becomes:
𝑑𝑃2
|
= 𝑎𝑃2 𝜀𝑃2 𝑃2 𝑅 ∗ − 𝑚𝑃2 𝑃2 − 𝑎𝐻 𝑃2 𝐻 ∗ > 0⁡⁡⁡⁡⁡⁡⁡⁡⁡[15]
𝑑𝑡 𝑅∗,𝑃1∗,𝐻 ∗,𝑃2 ≪𝑃1∗
91
𝐼
1
92
with: 𝑅 ∗ =
93
The condition [15] can be written as in Table 5. At the upper trophic level, this condition
94
implies the minimization of the resource requirement (R* rule). For the producers under
95
herbivory, the apparent competition changes this condition: the winner will be the one
96
that sustains the highest densities of herbivores at equilibrium. Table 5 shows the
97
invasion condition for the competition in the four models derived from the same
98
procedure.
99
Our interpretation of trait selection is as follows:
𝑎𝑃 𝑚𝐻
𝑜𝑢𝑡𝑅 + 1
𝑎𝐻 𝜀𝐻
and 𝐻 ∗ = 𝑎 (𝜀𝑃1 𝑎𝑃1 𝑅∗ − 𝑚𝑃1 ) (see Appendix A for the equilibria)
𝐻
100
For the models 1 and 2, the invasion criteria involves combinations of several traits in
101
which traits of P1 cannot be expressed relative to traits of P2. We derived the direction of
102
trait selection from the form of the invasion condition and from supplementary numerical
103
analyses (Figs. 7 and 8):
104
For model 1, since the R∗ is positive by definition, the invasion condition (Table 5)
105
is more easily satisfied if 𝜀𝑃2 𝑎𝑃2 is large relative to 𝜀𝑃1 𝑎𝑃1 (Fig. 7) and if 𝑚𝑃2 is small
106
compared to 𝑚𝑃1 . We conclude that competition should select species with higher
107
conversion efficiency and attack rate, and lower mortality rate.
108
109
For model 2, the invasion criteria (Table 5) is more easily satisfied if 𝜀𝑃2 is large
relative to 𝜀𝑃1 and 𝑀𝑃2 small compared to 𝑀𝑃1 . The invader P2 is more likely to
7
110
outcompete the resident P1 if it has the smallest body mass possible and the highest
111
conversion efficiency (Fig. 8a compared to Fig. 8b). We conclude that competition
112
should select species with smaller body mass and higher conversion efficiency.
113
For models 3 and 4, the species with the smallest body mass wins the competition
114
(Table 5).
115
8
116
117
Table 5
118
Invasion conditions for competition in small modules (see Figs 1a, 1b, 1c). For clarity,
119
we used the following simplifications: 𝑠𝑎 =
120
the temperature-dependent term: 𝑓(𝑇) = 𝑒
𝑜𝑢𝑡𝑃
𝑎𝑥𝑃
𝑚𝐻
, 𝑠𝑏 = 𝑎
𝐻 𝜀𝐻
𝐸(𝑇−𝑇0 )
𝑘𝑇𝑇0
, 𝑠𝑐 =
𝑜𝑢𝑡𝑅
𝑎 𝑦𝑃
, 𝑠𝑑 = 𝜀
𝑎𝑥𝐻
𝐻 𝑎𝑦𝑃 𝑎𝑦𝐻
, and of
.
Herbivore
Model
Producer invasion condition
invasion
(P2 the invader)
condition
(H2 the invader)
General
𝑅2∗ < 𝑅1∗
condition
Model 1
𝑚𝑃2
𝑚𝑃1
<
𝑎𝑃2 𝜀𝑃2 𝑎𝑃1 𝜀𝑃1
Herbivore
Model 2
𝜀𝑃2 > 𝜀𝑃1
(Fig. 1a)
Model 3
ø
Model 4
⁄𝑓(𝑇) 1 + 𝑠𝑎 𝑀𝑃0.25
⁄𝑓(𝑇)
1 + 𝑠𝑎 𝑀𝑃0.25
2
1
<
𝜀𝑃2
𝜀𝑃1
5.1
Without
General
𝐻1∗ < 𝐻2∗
condition
5.2 With
𝜀𝑃2 𝑎𝑃2 − 𝜀𝑃1 𝑎𝑃1 >
Model 1
Herbivore
𝑃2∗ < 𝑃1∗
𝑚𝑃2 − 𝑚𝑃1
𝑚𝐻2
𝑚𝐻1
<
𝑎𝐻2 𝜀𝐻2 𝑎𝐻1 𝜀𝐻1
𝐼
𝑜𝑢𝑡𝑅 + 𝑎𝑃1 𝑠𝑏
(Figs 1b
and1c)
Model 2
1
𝑀𝑃0.25
2
(
Model 3
𝜀𝑃2 𝐼
− 𝑎 𝑥𝑃
𝑠 𝑓(𝑇)
𝑠𝑐 + 𝑑 0.25
𝑀𝑃1
>
1
𝑀𝑃0.25
1
)
𝑀𝑃2 < 𝑀𝑃1
(
𝜀𝑃1 𝐼
− 𝑎 𝑥𝑃
𝑠 𝑓(𝑇)
𝑠𝑐 + 𝑑 0.25
𝑀𝑃1
𝜀𝐻2 > 𝜀𝐻1
)
ø
121
122
9
123
124
Fig. 7
125
Outcomes of invasions with model 1 (Classical consumer-resource model) for the case
126
shown in Fig. 2b (a producer P2 try to invade an ecosystem at equilibrium where a
127
producer P1 consume a resource R and is fed upon by an Herbivore H), according to the
128
attack rates and conversion efficiencies of P1 and P2. Parameter values: 𝜀𝐻 = 0.8,
129
𝑚𝑃1 = 𝑚𝑃2 = 0.2; panel a: 𝜀𝑃1 = 0.6 and 𝜀𝑃2 = 0.8; panel b: 𝜀𝑃1 = 𝜀𝑃2 = 0.8; panel c
130
𝜀𝑃1 = 0.8 and 𝜀𝑃2 = 0.6. See Table 1 for other parameters
131
10
132
133
134
Fig. 8
135
Outcomes of invasions with model 2 (bioenergetic model with efficiency at the
136
production), for the case shown in Fig. 2b (a producer P2 try to invade an ecosystem at
137
equilibrium where a producer P1 consume a resource R and is fed upon by an Herbivore
138
H), according to log10 of body masses and conversion efficiencies of P1 and P2.
139
Parameter values: 𝜀𝐻 = 0.8; panel a 𝜀𝑃1 = 0.2 and 𝜀𝑃2 = 0.8; panel b: 𝜀𝑃1 = 𝜀𝑃2 = 0.8;
140
panel c 𝜀𝑃1 = 0.8 and 𝜀𝑃2 = 0.2. See Table 1 for the other parameter
141
11
142
2) Effect of temperature
143
An increase in temperature has two effects on invasion success. First, it reduces some
144
equilibrium densities. Table 6 shows these changes in equilibrium densities for models 2
145
to 4. In model 2 the main consequence is that the reduction of 𝑃∗ in absence of
146
herbivores makes harder the success of invasion by herbivores, because of reduced
147
resource availability (P* without the herbivore diminishes).
148
Second, it may affect the outcome of competition. For model 2, the temperature has no
149
effect on the selection at the upper trophic level, but it affects the invasion criteria for the
150
producer under herbivory (see expressions enlightened in red in Table 5). Below we
151
give the extensive analysis of temperature effect on this condition. The results are
152
summarized at the end in italic.
153
In case of herbivory, the condition for the invader P2 (case of Fig. 2b) to successfully
154
invade the ecosystem is:
155
156
𝐼(𝜀𝑃2 𝑀𝑃−0.25
− 𝜀𝑃1 𝑀𝑃−0.25
)
2
1
𝑠𝑐 +
𝑠𝑑 𝑀𝑃−0.25
𝑓(𝑇)
1
with the following simplifications: 𝑠𝑐 =
− 𝑎𝑥𝑃 (𝑀𝑃−0.25
− 𝑀𝑃−0.25
) > 0⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡[16]
2
1
𝑜𝑢𝑡𝑅
𝑎𝑦𝑃
, 𝑠𝑑 = 𝜀
𝑎𝑥𝐻
𝐻 𝑎𝑦𝐻
, and 𝑓(𝑇) = 𝑒
𝐸(𝑇−𝑇0 )
𝑘𝑇𝑇0
157
158
This condition is valid only if the herbivore has enough resource to invade. This means
159
that the P* without the herbivore must be higher than the expected P* with the herbivore,
160
which can be expressed as follows:
161
1
𝑎𝑦𝑃 ⁡𝑀𝑃−0.25 𝑒
162
𝐼𝜀𝑃
𝑐𝐻
(
−
𝑜𝑢𝑡
)
>
⁡
> 0⁡⁡⁡⁡⁡⁡[17]
𝑅
𝐸(𝑇−𝑇0 ) 𝑐
𝜀𝐻
𝑃
𝑘𝑇𝑇0
𝑎 𝑥𝑃
with the following simplifications: 𝑐𝑃 = 𝑎
𝑦𝑃
𝑎𝑥𝐻
and 𝑐𝐻 = 𝑎
𝑦𝐻
12
163
We calculated the partial derivative of the left term of condition [16] with respect to
164
temperature, yielding:
−𝑎𝑐𝑒
165
(𝑏 +
𝐸(𝑇−𝑇0 )
𝑘𝑇𝑇0
𝐸
𝑘𝑇 2 ⁡⁡⁡⁡⁡⁡[18]
𝐸(𝑇−𝑇0 ) 2
𝑐𝑒 𝑘𝑇𝑇0 )
𝑜𝑢𝑡𝑅
𝑎𝑥𝐻
166
with the following simplifications: 𝑎 = 𝐼(𝜀𝑃2 𝑀𝑃−0.25
− 𝜀𝑃1 𝑀𝑃−0.25
), 𝑏 =
2
1
167
𝑑 = 𝑎𝑥𝑃 (𝑀𝑃−0.25
− 𝑀𝑃−0.25
)
2
1
168
Since the sign of the partial derivative [18] does not depend on temperature but only on
169
𝑎, the left term of inequality [16] for the successful invasion of P2 is monotonous. The
170
outcome of the competition change with temperature only if the sign of this left term
171
changes with temperature and the condition of herbivore maintenance [17] is still valid
172
for the parameters of P1 and P2. Actually, if the herbivore dies, then the condition for
173
invasion becomes the R* rule (Table 5.1). We derived analytically the critical
174
temperatures for which these three condition changes, 𝑇𝑖𝑛𝑣𝑃2 , 𝑇𝐻𝑒𝑟𝑏𝑃1 and 𝑇𝐻𝑒𝑟𝑏𝑃2 ,
175
respectively:
176
177
𝑇𝑖𝑛𝑣𝑃2 =
𝑎 𝑦𝑃
,𝑐=𝜀
1
𝐻 𝑎𝑦𝐻
𝑀𝑃−0.25
,
1
⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡[19]
𝐼(𝜀𝑃2 𝑀𝑃−0.25
− 𝜀𝑃1 𝑀𝑃−0.25
) 𝑜𝑢𝑡𝑅
𝜀𝐻 𝑎𝑦𝐻
1 𝑘
2
1
−
𝑙𝑜𝑔
((
−
)
𝑇0 𝐸
𝑎𝑦𝑃 𝑎𝑥𝐻 𝑀𝑃−0.25 )
𝑎𝑥𝑃 (𝑀𝑃−0.25
− 𝑀𝑃−0.25
)
1
2
1
𝑇𝐻𝑒𝑟𝑏𝑃1 =
1
𝐼𝜀𝑃1 𝑜𝑢𝑡𝑅
𝜀𝐻 𝑎𝑦𝐻
1 𝑘
−
𝑙𝑜𝑔
((
−
)
𝑇0 𝐸
𝑎 𝑥𝑃
𝑎𝑦𝑃 𝑎𝑥𝐻 𝑀𝑃−0.25 )
⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡[20]
1
178
𝑇𝐻𝑒𝑟𝑏𝑃2 =
1
𝐼𝜀𝑃2 𝑜𝑢𝑡𝑅
𝜀𝐻 𝑎𝑦𝐻
1 𝑘
−
𝑙𝑜𝑔
((
−
)
𝑇0 𝐸
𝑎 𝑥𝑃
𝑎𝑦𝑃 𝑎𝑥𝐻 𝑀𝑃−0.25 )
⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡[21]
2
13
179
Given that the left terms of conditions [16] and [17] are monotonous with temperature,
180
the outcome of competition between P1 and P2 changes with temperature only if:
(𝑇𝑖𝑛𝑣𝑃2 < 𝑇𝐻𝑒𝑟𝑏𝑃1 ) ∩ (𝑇𝑖𝑛𝑣𝑃2 < 𝑇𝐻𝑒𝑟𝑏𝑃2 )⁡⁡⁡⁡⁡[22]
181
182
183
184
185
186
From [19], [20] and [21], the conditions [22] can be re-written:
(0 <
𝜀𝑃2 𝑀𝑃−0.25
− 𝜀𝑃1 𝑀𝑃−0.25
2
1
𝑀𝑃−0.25
− 𝑀𝑃−0.25
2
1
< 𝜀𝑃1 ⁡) ∩ (0 <
𝜀𝑃2 𝑀𝑃−0.25
− 𝜀𝑃1 𝑀𝑃−0.25
2
1
𝑀𝑃−0.25
− 𝑀𝑃−0.25
2
1
< 𝜀𝑃2 )⁡⁡⁡⁡[23]
Two cases have to be considered:
(a) the derivative [18] is negative, that is:
𝜀𝑃2 𝑀𝑃−0.25
− 𝜀𝑃1 𝑀𝑃−0.25
> 0⁡⁡⁡⁡⁡⁡⁡[24]
2
1
187
It follows that the left term of inequality [16] is positive at low temperature and
188
decreases when temperature increases. This means that P2 is more competitive than
189
P1 and invades the ecosystem at low temperature, but its relative competitiveness
190
decreases with temperature (Fig. 9a). It might even fail to invade at high temperature
191
if the following conditions, derived from [23] and [24] are met (Fig. 9b):
192
(𝜀𝑃2 𝑀𝑃−0.25
− 𝜀𝑃1 𝑀𝑃−0.25
> 0⁡) ∩ (𝑀𝑃2 < 𝑀𝑃1 ) ∩ (𝜀𝑃2 < 𝜀𝑃1 )⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡[25]
2
1
193
194
195
(b) the derivative [18] is positive, that is:
𝜀𝑃2 𝑀𝑃−0.25
− 𝜀𝑃1 𝑀𝑃−0.25
< 0⁡⁡⁡⁡⁡⁡⁡[26]
2
1
196
It follows that the left term of inequality [16] is negative at low temperature and
197
increases when temperature increases. This means that P2 is less competitive than
198
P1 and fail to invade at low temperature, but its relative competitiveness increases
199
with temperature (Fig. 9c). It might even succeed to invade at high temperature if the
200
following conditions, derived from [23] and [26] are met (Fig. 9d):
14
(𝜀𝑃2 𝑀𝑃−0.25
− 𝜀𝑃1 𝑀𝑃−0.25
< 0⁡) ∩ (𝑀𝑃2 > 𝑀𝑃1 ) ∩ (𝜀𝑃2 > 𝜀𝑃1 )⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡[27]
2
1
201
202
203
In summary, in general, the producer with the smallest body mass wins the competition.
204
However, under herbivory, the competitiveness of the smallest producer tends to
205
decrease with temperature (Fig. 9). In the case it has lower conversion efficiency than its
206
competitor, it can even lose the competition if its body mass is not sufficiently smaller
207
than the one of its competitor. In terms of trait selection, this means that the smallest
208
body masses and the highest conversion efficiencies should still be selected but
209
temperature increase may temper the strength of this selection when species traits are
210
very similar.
211
15
212
Table 6
213
Arrows show the direction of changes in equilibrium densities with increasing
214
temperature derived from the examination of equilibria shown in Table 3. Equilibria were
215
intractable for model 4 with the herbivore.
Structure
Models
𝑹∗
𝑷∗
6.1 Without
Model 2
∅
𝑃∗ ↓
Model 3
∅
𝑃∗ ↓
Model 4
𝑅∗ ↓
non-linear
Model 2
𝑅∗ ↓
∅
𝐻∗ ↓
Model 3
𝑅∗ ↓
∅
𝐻∗ ↓
Herbivore
𝑯∗
(Fig. 1a)
6.2 With
Herbivore
(Fig. 1b and 1c)
216
217
218
219
16
Competition outcome changes
0.15
b
0.10
8
no H
MP2 < MP1
0
0.00
2
0.05
4
6
P2 invades
10
20
30
T2
40
0
0
−4
−2
T1
P2 excluded
c
0
10
20
30
40
0
10
20
−0
.
1
5 −0
.
1
0 −0
.
0
5 0.00
−2
T2
−6
Invasion condition (if positive, P2 invades)
10
a
0.20
No change
30
40
T1
MP2 > MP1
d
0
10
20
30
40
Temperature (°C)
220
221
Fig. 9
222
Outcomes of competition between P1 and P2 for the case shown in Fig. 2b (with
223
herbivory) with model 2, according to temperature and traits of P1 and P2. The line
224
shows the left part of inequation [16]. In the top panels conversion efficiencies are 𝜀𝑃2 =
225
0.90 < 𝜀𝑃1 = 0.91, and P2 has a smaller body mass than P1: (a) 𝑀𝑃2 = 0.0001 < 𝑀𝑃1 =
226
0.001; (b) 𝑀𝑃2 = 0.0001 < 𝑀𝑃1 = 0.00011. In the bottom panels 𝜀𝑃2 = 0.91 > 𝜀𝑃1 = 0.90,
227
and P2 has a larger body mass than P1: (c) 𝑀𝑃2 = 0.001 > 𝑀𝑃1 = 0.0001; (d) 𝑀𝑃2 =
228
0.00011 > 𝑀𝑃1 = 0.0001. If the difference between 𝑀𝑃1 and 𝑀𝑃2 is small enough,
229
temperature may change the outcome of competition (right panels). In all panels: 𝜀𝐻 =
230
0.5 and 𝑀𝐻 = 0.1; see Table 1 for the other parameters
17