Examining the interaction of density dependence and stochastic dispersal

over several life history scenarios

Heather Berkley

Bruce Kendall

David Siegel

Main Question

How does stochastic dispersal & demography interact to affect spatial & temporal variability in populations?

Future additions to the F3 model

Types of density dependence: Recruitment rate depends on adult density Production rate depends on adult density Adult mortality depends on adult density Production rate depends on adult density

Size & Age Structure Increasing time to maturity Increasing fecundity with age or size

Adult movement Variability in habitat quality (spatial & temporal)

Characterizing the existing model

Parameters that potentially impact variability in populations: Type of density dependence:

Recruitment rate depends on adult density Mortality Productivity Dispersal Distance Ndraw (number of draws from the kernel)

Equations

Model without harvest:

At equilibrium (Nt = K):

For stability analysis:

tcNt0t1t eNPSNN -

0PM

c1

K ln-

MP

M1NF 0

KNt t

ln-

Set Parameters

We chose the following values: Mortality:

0.5, based on lifespan of 2 years 0.05, based on lifespan of 20 years

Fixed kernel dispersal distance based on PLD: 70 km, based on PLD of 5 days 230 km, based on PLD of 50 days

Calculated Parameters

Productivity (P0) is calculated from value of M & by setting Eqn. for stability to monotonic (+0.5) or oscillating (-0.5) approach to stability

Density dependent term (c) is calculated by setting carrying capacity equation to 100 and given values of M and P0

100PM

c1

K0

ln-

MP

M1NF 0

KNt t

ln-

Parameter Combinations

M Po c DispD Stability

1 0.05 1101 0.099997 70 0.5 long lifespan, PLD ~ 5 days, monotonic

2 0.5 1.36 0.01000632 70 0.5 short lifespan, PLD ~ 5 days, monotonic

3 0.05 1101 0.099997 230 0.5 long lifespan, PLD ~ 50 days, monotonic

4 0.5 1.36 0.01000632 230 0.5 short lifespan, PLD ~ 50 days, monotonic

5 0.5 10.043 0.03000023 70 -0.5 short lifespan, PLD ~ 5 days, oscillating

6 0.5 10.043 0.03000023 230 -0.5 short lifespan, PLD ~ 50 days, oscillating

7 0.05 5.34E+11 0.29999956 70 -0.5 long lifespan, PLD ~ 5 days, oscillating

8 0.05 5.34E+11 0.29999956 230 -0.5 long lifespan, PLD ~ 50 days, oscillating

Model Settings & Calculations

Domain: Absorbing boundaries 3000 km, used only middle section Patches = 5km

Spatial variance calculated at last time step (100 yrs) over 300 patches

Temporal variance calculated for last 50 years Local: for each patch Total Population: for whole population (all 300 patches)

Autocorrelation (lag 1 only) Spatial Temporal

Local Total Population

Over a range of Ndraw values Values averaged over 50 simulations

Stochastic Dispersal

Ndraw For small values of Ndraw, each patch only sends out a

few groups of larvae to other locations At the receiving patch, the time between receiving larvae

groups can be very long For short-lived adults, natural adult mortality can drive the

population extinct until it receives a new group of larvae For large values of Ndraw, each patch is interacting with

almost all other patches Receiving patches should get larvae from many other

patches each year

Adult Population

Ndraw=10 Long-Lived

Short-lived

Short-lived,

oscillating

Distance (km)

1020 40 60 80 100 150 20060

80

100

120

140

160

180

200

220

Ndraw

Po

pu

lati

on

siz

e in

A 1 2 3 4 5 6

Short-lived, oscillating

Short-lived, monotonic

Long-lived

Short-lived, short PLDShort-lived, long PLDLong-lived, short PLDLong-lived, long PLDShort-lived, short PLD, oscillatingShort-lived, long PLD, oscillating

Parameter Combination #4

Ndraw=20 Short-lived

Population Size

1020 40 60 80 100 150 20060

80

100

120

140

160

180

200

220

Ndraw

Ad

ult

s

1020 40 60 80 100 150 2000

20

40

60

80

100

120

140

160

180

200

Ndraw

Rec

ruit

s

1020 40 60 80 100 150 20010

1

102

103

104

105

106

Ndraw

Set

tler

s

Short-livedLong-livedShort-lived, oscillating

Spatial Variance

1020 40 60 80 100 150 20010

0

101

102

103

104

105

106

Ndraw

Ad

ult

s

1020 40 60 80 100 150 20010

0

101

102

103

104

105

106

Ndraw

Rec

ruit

s

1020 40 60 80 100 150 20010

3

104

105

106

107

108

109

1010

1011

Ndraw

Set

tler

s

Short-livedLong-livedShort-lived, oscillating

Spatial Coefficient of Variation

1020 40 60 80 100 150 2000

0.5

1

1.5

2

2.5

Ndraw

Ad

ult

s

1020 40 60 80 100 150 2000

1

2

3

4

5

6

Ndraw

Rec

ruit

s

1020 40 60 80 100 150 2000.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Ndraw

Set

tler

s

Short-livedLong-livedShort-lived, oscillating

Temporal Variance(local)

1020 40 60 80 100 150 20010

0

101

102

103

104

105

106

Ndraw

Ad

ult

s

1020 40 60 80 100 150 20010

0

101

102

103

104

105

106

Ndraw

Rec

ruit

s

1020 40 60 80 100 150 20010

3

104

105

106

107

108

109

1010

1011

Ndraw

Set

tler

s

Short-livedLong-livedShort-lived, oscillating

Temporal Coefficient of Variation (local)

1020 40 60 80 100 150 2000

0.5

1

1.5

2

2.5

Ndraw

Ad

ult

s

1020 40 60 80 100 150 2000

1

2

3

4

5

6

Ndraw

Rec

ruit

s

1020 40 60 80 100 150 2000.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Ndraw

Set

tler

s

Short-livedLong-livedShort-lived, oscillating

Temporal Variance (population)

1020 40 60 80 100 150 20010

-2

10-1

100

101

102

103

Ndraw

Ad

ult

s

1020 40 60 80 100 150 20010

-2

10-1

100

101

102

103

Ndraw

Rec

ruit

s

1020 40 60 80 100 150 20010

0

101

102

103

104

105

106

107

108

Ndraw

Set

tler

s

Short-livedLong-livedShort-lived, oscillating

Temporal Coefficient of Variance (population)

1020 40 60 80 100 150 2000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Ndraw

Ad

ult

s

1020 40 60 80 100 150 2000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Ndraw

Rec

ruit

s

1020 40 60 80 100 150 2000.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

Ndraw

Set

tler

s

Short-livedLong-livedShort-lived, oscillating

Spatial Autocorrelation

1020 40 60 80 100 150 200-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

Ndraw

Ad

ult

s

1020 40 60 80 100 150 200-0.03

-0.02

-0.01

0

0.01

0.02

0.03

Ndraw

Rec

ruit

s

1020 40 60 80 100 150 200-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

Ndraw

Set

tler

s

Short-livedLong-livedShort-lived, oscillating

Temporal Autocorrelation (local)

1020 40 60 80 100 150 200-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

Ndraw

Rec

ruit

s

1020 40 60 80 100 150 200-13

-12

-11

-10

-9

-8

-7

-6

-5

-4

-3x 10

-3

Ndraw

Set

tler

s

1020 40 60 80 100 150 200-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Ndraw

Ad

ult

s

Short-livedLong-livedShort-lived, oscillating

TemporalAutocorrelation (population)

1020 40 60 80 100 150 200-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Ndraw

Rec

ruit

s

1020 40 60 80 100 150 200-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Ndraw

Set

tler

s

1020 40 60 80 100 150 200-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Ndraw

Ad

ult

s

Short-livedLong-livedShort-lived, oscillating

Next Steps

Add other forms of density dependenceAge/Size StructureAdult MovementSpatial/Temporal variability in habitat

quality