Turbulent CoexistenceHeather Berkley, Satoshi Mitarai,

Bruce Kendall, David Siegel

Species Coexistence

• Consider two similar species A & B– Species A has a slightly better ability to utilize resources – Recruits compete for limited resources at settlement sites– Spawning timings are separated by weeks

• Compare cases with i) smooth dispersal kernel & ii) packet model for connectivity– Smooth dispersal kernel: spawning timing does not matter– Packet model: species A & B “catch” different eddies &

can settle at different sites

Diffusion Case

If they are put together, species B becomes extinct,species A thrives

On their own, both species can persist

Time (years)

IC’s: A = 100, B = 100

Packet Model

• Larval settlement as arrival of N packets

• L = domain size• l = eddy size (50 km)• T = Spawning time• t = eddy turnover rate (14 d)

t

T

l

LN

eddy size (l)

N larval packets

Spawning Window Overlap

• Specify how many days of overlap between spawning times for both species

• Makes some packets perfectly correlated for both species and others independent

Packets will have same settlement locations

Species A Spawning Window

Species B Spawning Window

TIME

Parameters

• Tsp (spawning time) = 30 days for both– Vary amount of overlap

• Fecundity of Sp.A = 0.5• Fecundity of Sp.B = 0.45• Adult Mortality = 0.09• Run time = 1000 yrs; • Patch size = 5 km; • Domain size = 500 km;• Averaged over 100 simulations • Larvae on larvae DD (total # of both sp)

BA

AA SSb

aSR

1

28 days

Theory

• Species A is stronger competitor (fA > fB)

• Want to know if the growth rate of Sp. B is positive when rare and Sp. A is at equilibrium

A

BB

A

AA bS

aSR

bS

aSR

+1=

+1= ;

mtN

tREm

tN

tRE

B

B

A

A )(

)]([ ;

)(

)]([

• Estimate E[RA] and E[RB] with Taylor expansion around mean number of settlers– need E[SA], E[SB], var(SA), & cov(SA,SB,)

( ) ( )

( ) ( )∑

∑

0=

=

yBBB

yAAAA

BftyxDtxS

KftyxDtxS

,,,

,,,

• Assume spatially homogenous Adult distribution:

• Settlers:

( ) ( )0= = BtxNKtxN BAA ,;,

1== ∑∑y

By

A DD

0=

=

BfSE

KfSE

BB

AAA

][

][ ( ) ( )( ) ( )BAABABA

AAAA

DDBKffSSDKfS

,cov,covvarvar

0

22

==

Coexistence Condition

• Coexistence when:

• Calculate var(DA) and cov(DA,DB) from Packet Model and Flow Simulations

BAAAA

AA

AA

AA

A

AA

AA

A

B

DDDKbfKbf

KbfKbf

DKbf

Kbf

f

f

,covvar11

1

var1

1 2

cov(DA,DB)

Coexistence line calculated with cov(DA,DB) & var(DA)

Equilibrium Pop B/A

Coexistence

Sp.B extinction

Equilibrium Pop B/A

Coexistence

Sp.B extinction

Recruitment Differences

• Key difference between species is density dependent per-capita recruitment rate, R/N

• For Species A:

A

AAA

bS

aS

KK

SR

1

1)(

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 10 20 30 40 50 60 70 80 90 100

Settler number (Sp A)

Pe

r-c

ap

ita

re

cru

itm

en

t

Species A

At equilibrium, average # Settlers is 50

KfS AA

Variability in Settlement

• If settlement varied between 20 & 80, then the per-capita recruitment rate would be less than if settlement was constant at 50

Recruitment Differences

• For Species B, recruitment is conditional on Sp. A settlers, too:

• So,

A

BBAB

bS

aS

BB

SSR

1

1),(

00

A

ABA

B

bS

SSaE

BS

B

RE

1

|1|

00

Recruitment Differences

A

A

A

BAB

A

BABA

B

S

aS

Kf

f

bS

afS

B

RE

1

1

11|

0

• Do some math, get:

• AB is the correlation in dispersal between species

• Low correlation, 1st part dominates• High correlation, 2nd part dominates

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 10 20 30 40 50 60 70 80 90 100

Settler number (Sp A)

Pe

r-c

ap

ita

re

cru

itm

en

t

Species A

Species B

Correlation in Settlement: = 0Expected per-capita R rate for Sp B

At equilibrium Settlement, R/N for Sp. A > Sp. B

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 10 20 30 40 50 60 70 80 90 100

Settler number (Sp A)

Pe

r-c

ap

ita

re

cru

itm

en

t

Species A

Species B

Nonlinear averaging leads to R/N Sp. B > Sp. A

Expected per-capita R rate for Sp B

Correlation in Settlement: = 0

Variable Settlement between 20 & 80

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 10 20 30 40 50 60 70 80 90 100

Settler number (Sp A)

Pe

r-c

ap

ita

re

cru

itm

en

t

Species A

Species B

Correlation in Settlement: = 0.5Expected per-capita R rate for Sp B

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 10 20 30 40 50 60 70 80 90 100

Settler number (Sp A)

Pe

r-c

ap

ita

re

cru

itm

en

t

Species A

Species B

Correlation in Settlement: = 1Expected per-capita R rate for Sp B

• Because of nonlinear averaging, variability in settlement will increase the average per-capita recruitment rate for Sp. B above that of Sp A, so Sp. B can invade when rare

• This advantage becomes weaker as the correlation in dispersal increases

Conclusions