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