Supplementary Material Resource competition induces heterogeneity and can increase cohort...

Supplementary Material Gosselin and Anderson

1

Resource competition induces heterogeneity and can increase cohort survivorship:

selection-event duration matters

Jennifer L. Gosselin1,* and James J. Anderson1,**

1 School of Aquatic and Fishery Sciences, University of Washington, Seattle, Washington, USA

* Email: [email protected]; ** Email: [email protected]

Fig. S1 Conceptual diagram of our treatment-challenge experiment. (1) In the treatment stage, guppies are grown with different levels of food availability and presence or absence of food competition. (2) Following the treatment stage, guppies are challenged under conditions of increased temperature and starvation. (3) The challenge stage vitality parameters are estimated from the observed times to mortality, and survivorship curves are fitted with the vitality model. The thick solid line depicts a treatment group with high food rations and the thin dashed line depicts a treatment group with low food rations

Observed survivorship data

4-day acclimation to increased temperature

21- to 24-day experienceof food availabilities and increased temperature

and starvation

Fitted survivorship curves

Su

rviv

orsh

ip(p

ropo

rtion

aliv

e)

Fish

con

ditio

n in

dex

(e.g

. mas

s)

Age (d)

Time scale and condi ons experienced in current study:

1. Treatment stage 2. Challenge stage 3. Vitality model

1- to 4- week challenge at

competition


2

Fig. S2 Depiction of Li and Anderson’s (2009) vitality model. (a) Individual random trajectories of vitality (blue jagged lines; Eq. 1 in Table S3) show how the vitality of each individual generally declines throughout its lifetime. Vitality densities (black curves; Eq. 2b in Table S3) are generated for the individual random trajectories of vitality. (b) The initial vitality (dotted purple curve) at t = 0 has a normal distribution with a CV value of u. The mean vitality (dashed red line) is generated with Eq. 7 in Anderson et al. (2008) and its slope represents r until fish begin to die. The grey shaded area depicts the vitality parameter s. The survivorship curve (large black curve; Eq. 3 in Table S3) is the integration of the probability density function of vitality (Eq. 2b in Table S3). The terms “relative density” in this study and “density” in Li and Anderson (2009) are equivalent

Vita

lity

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

density

Relative density

Time (d)

(a)

(b)Relative


3

Fig. S3 Frequency distribution of guppy initial mass in the challenge stage. Treatment groups are labeled in the top left corner of each graph. See Fig. 2 for definitions of treatment group names. Thick lines represent the mean mass of each treatment group. Different asterisks denote statistical differences with p < 0.05. CV stands for coefficient of variation

CV=0.26

*(a) FR10.C

CV=0.21

**(b) FR40.C

CV=0.15

***(c) FR50.NC

CV=0.15

****(d) FR100.NC

0

20

40

60

0

20

40

60

0

20

40

60

0

20

40

60

Mass (mg)

Freq

uenc

y

0 10 20 30 40 50


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Fig. S4 Sum of relative values of negative log-likelihood estimates (H) (Eq. 8 in Table S3) as a function of the fitting coefficient ϕ. The min H is circled

!

H

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.8 1.21.0 1.10.9

0.16


5

Fig. S5 Vitality densities (Eq. 2b in Table S3) of guppies at specific times in the challenge stage normalized to the mean initial vitality density of the FR100.NC group

(v0,4 ) . Vitality is a dimensionless measure of an individual’s capacity to survive.

Densities are depicted at various times (t) during the challenge: day 0 (thick, dark blue, solid line), day 2 (blue, dashed line), day 10 (thin, light blue, dotted line), and the day corresponding to mean time to mortality of the treatment group (T ; red, dotted-dashed line). Treatment groups are denoted in the top right corner of each graph. See Fig. 2 for definitions of treatment group names. The terms “relative density” in this study and “density” in Li and Anderson (2009) are equivalent

0

2

4

6

8 (a) FR10.C

T = 2.7

t = 0t = 2t = 10t = TT = 2.7T = 2.7T = 2.7

0

2

4

6

8 (b) FR40.C

T = 7.6T = 7.6T = 7.6T = 7.6

0

2

4

6

8 (c) FR50.NC

T = 7.8T = 7.8T = 7.8T = 7.8

0.0 0.5 1.0 1.50

2

4

6

8 (d) FR100.NC

T = 16T = 16T = 16T = 16

Vitality

Rel

ativ

e de

nsity

Challenge stage time (d)


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Table S1 Feeding schedules and rations during the treatment stage Treatment group

Number of feedings per day

Ration at each feeding (% of satiation amount)

Relative proportion of daily food rations

FR10.C 2 10% 0.10

FR40.C 2 40% 0.40

FR50.NC 1 100% 0.50

FR100.NC 2 100% 1.00


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Table S2 List of model parameters and their definitions. See Table S3 for equations

Symbol Eq. # Definition

α 5 Slope parameter of relationship between si and ri

β 6 Slope parameter of relationship between m0,i and v0,i/v0,4

CVmass 7 Coefficient of variation of mass at beginning of challenge stage

tε 1 White noise rate process

H 8 Group level fitting measure for model

i 4, 5, 6, 7, 8 Treatment groups 1, 2, 3 and 4 correspond to FR10.C, FR40.C, FR50.NC and FR100.NC respectively

l(t) 3 Survivorship at time t

λ 8 Constrained minimum negative log-likelihood estimate

L 8 Unconstrained minimum negative log-likelihood estimate

0m 6 Body mass in milligrams at beginning of challenge stage

( , )p v t′ 2a Probability distribution of vitality

( , )p v t 2b Probability distribution of normalized vitality

ϕ 7, 8 Slope parameter of relationship between ui and CVmass,i

Φ 3 Cumulative normal distribution function

r 2b, 3, 4, 5 Normalized mean rate of vitality loss, 0r vρ= (unit, t −1)

ρ 1, 2a, 4 Mean rate of vitality loss

σ 1, 2a, 4 Variability in the rate of vitality loss

s 2b, 3, 4, 5 Normalized variability in vitality loss rate, 0s vσ= (unit, t−1/2)

t 1, 2a, 2b, 3 Time in days

τ 2a Standard deviation of initial vitality distribution

T − Mean time to mortality

T * − Temporal threshold depicts differential effect of heterogeneity between two survivorship curves. Competition-induced initial heterogeneity increases survivorship for *t T> and decreases survivorship for *t T< .

cT − Duration of the challenge stage or selection event

u 2b, 3, 7 Coefficient of variation of initial vitality distribution, 0u vτ=

v0 2a, 4, 6 Mean initial vitality at beginning of the challenge stage

′v 1, 2a Vitality is an abstract measure of an individual’s capacity to survive at time t

v 2b Normalized vitality, 0v v v′=


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Table S3 Table of equations

Eq. # Equation

1 t

dvdt

ρ σε′= − +

2a ( )

( )( )( )

( )

2 2 200

22 2 2 2

2 2

2exp 1 exp2

( , )2

vv v t vt t

p v tt

ρτ σρστ σ τ σ

π τ σ

⎛ ⎞⎛ ⎞ ⎛ ⎞+′ − + ′⎜ ⎟⎜ ⎟ ⎜ ⎟− − −⎜ ⎟ ⎜ ⎟⎜ ⎟+ +⎝ ⎠ ⎝ ⎠⎝ ⎠′ =

+

2b ( )( )

( )( )

( )

2 2 2

2 2 2 2 2

2 2

21exp 1 exp

2( , )

2

v ru sv rtu s t s u s t

p v tu s tπ

⎛ ⎞⎛ ⎞ ⎛ ⎞+− + ⎜ ⎟⎜ ⎟ ⎜ ⎟− − −⎜ ⎟ ⎜ ⎟⎜ ⎟+ +⎝ ⎠ ⎝ ⎠⎝ ⎠=

+

3 ( ) ( )22 2

4 22 2 2 2

2 11 2 2( ) exp expr u s rtrt u r rl t kt

s su s t u s t

⎛ ⎞⎛ ⎞⎛ ⎞ + +⎛ ⎞−⎜ ⎟= Φ − + Φ − −⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟+ +⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠

4

if r4 =ρ

v0,4

and ri =ρ

v0,i

thenr4

ri

=v0,i

v0,4

if s4 =σv0,4

and si =σv0,i

thens4

si

=v0,i

v0,4

5 i is rα=

6

m0,i = β

v0,i

v0,4

7 mass,CVi iu ϕ=

8 4

1

( )( ) i

i i

HL

λ ϕϕ=

=∑

Note Eq. 2b corrects typo error in Eq. 15 of Li and Anderson (2009).


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Table S4 Two hypothetical examples of increased net reproductive rates (R0) and one of decreased R0 caused by increased heterogeneity among individuals, where T * = 7.5 Scenario 1: Late age reproduction (i.e. after T *) Simulation group x lx mx lxmx FR50.NC 0 1.00 0.0 0.00

1 1.00 0.0 0.00 2 1.00 0.0 0.00 3 1.00 0.0 0.00 4 0.97 0.0 0.00 5 0.91 0.0 0.00 6 0.78 0.0 0.00 7 0.61 0.0 0.00 8 0.42 0.5 0.21 9 0.28 1.0 0.28 10 0.17 1.5 0.26 11 0.09 1.0 0.09 12 0.05 0.8 0.04 13 0.02 0.5 0.01 14 0.01 0.2 0.00 15 0.01 0.0 0.00 16 0.00 0.0 0.00 R0 = 0.89

High CVmass 0 1.00 0.0 0.00 1 1.00 0.0 0.00 2 0.98 0.0 0.00 3 0.96 0.0 0.00 4 0.90 0.0 0.00 5 0.82 0.0 0.00 6 0.71 0.0 0.00 7 0.58 0.0 0.00 8 0.44 0.5 0.22 9 0.33 1.0 0.33 10 0.23 1.5 0.35 11 0.15 1.0 0.15 12 0.10 0.8 0.08 13 0.06 0.5 0.03 14 0.04 0.2 0.01 15 0.02 0.0 0.00 16 0.01 0.0 0.00 R0 = 1.17


10

Scenario 2: Late age reproduction (i.e. after T *) and high reproductive rates Simulation group x lx mx lxmx FR50.NC 0 1.00 0.0 0.00

1 1.00 0.0 0.00 2 1.00 0.0 0.00 3 1.00 0.0 0.00 4 0.97 0.0 0.00 5 0.91 0.0 0.00 6 0.78 0.0 0.00 7 0.61 0.0 0.00 8 0.42 0.0 0.00 9 0.28 0.0 0.00 10 0.17 0.0 0.00 11 0.09 0.0 0.00 12 0.05 10.0 0.50 13 0.02 20.0 0.40 14 0.01 10.0 0.10 15 0.01 0.0 0.00 16 0.00 0.0 0.00 R0 = 1.00

High CVmass 0 1.00 0.0 0.00 1 1.00 0.0 0.00 2 0.98 0.0 0.00 3 0.96 0.0 0.00 4 0.90 0.0 0.00 5 0.82 0.0 0.00 6 0.71 0.0 0.00 7 0.58 0.0 0.00 8 0.44 0.0 0.00 9 0.33 0.0 0.00 10 0.23 0.0 0.00 11 0.15 0.0 0.00 12 0.10 10.0 1.00 13 0.06 20.0 1.20 14 0.04 10.0 0.40 15 0.02 0.0 0.00 16 0.01 0.0 0.00 R0 = 2.60


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Scenario 3: Early age reproduction (i.e. before T *) Simulation group x lx mx lxmx FR50.NC 0 1.00 0.0 0.00

1 1.00 0.0 0.00 2 1.00 0.0 0.00 3 1.00 0.3 0.30 4 0.97 0.5 0.49 5 0.91 0.3 0.27 6 0.78 0.0 0.00 7 0.61 0.0 0.00 8 0.42 0.0 0.00 9 0.28 0.0 0.00 10 0.17 0.0 0.00 11 0.09 0.0 0.00 12 0.05 0.0 0.00 13 0.02 0.0 0.00 14 0.01 0.0 0.00 15 0.01 0.0 0.00 16 0.00 0.0 0.00 R0 = 1.06

High CVmass 0 1.00 0.0 0.00 1 1.00 0.0 0.00 2 0.98 0.0 0.00 3 0.96 0.3 0.29 4 0.90 0.5 0.45 5 0.82 0.3 0.25 6 0.71 0.0 0.00 7 0.58 0.0 0.00 8 0.44 0.0 0.00 9 0.33 0.0 0.00 10 0.23 0.0 0.00 11 0.15 0.0 0.00 12 0.10 0.0 0.00 13 0.06 0.0 0.00 14 0.04 0.0 0.00 15 0.02 0.0 0.00 16 0.01 0.0 0.00 R0 = 0.99

Supplementary Material Resource competition induces heterogeneity and can increase cohort...

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