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?

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)

Adult abundance at location x during time-step n+1 # of adults harvested Natural mortality of un-harvested adults Fecundity Larval survival Larval dispersal Larval recruitment at x Number of larvae that successfully recruit to location x An integro-difference model describing coastal fish population dynamics:

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  Productivity (P 0 ) is calculated to give either monotonic or oscillating approach to stability  Density dependent term (c) is calculated to set carrying capacity to 100

Parameter Combinations MP0P0 cDispDStability long lifespan, PLD ~ 5 days, monotonic short lifespan, PLD ~ 5 days, monotonic long lifespan, PLD ~ 50 days, monotonic short lifespan, PLD ~ 50 days, monotonic X long lifespan, PLD ~ 5 days, oscillating short lifespan, PLD ~ 5 days, oscillating X long lifespan, PLD ~ 50 days, oscillating short 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 200 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

Parameter Combination #4 Ndraw=20 Short-lived Long PLD distance (km)

Adult Population Ndraw=10  Long-Lived, Long PLD  Short-lived, Long PLD  Short-lived, Long PLD, oscillating distance (km)

Monotonic Approach to Equilibrium: Long lived, short PLD Short lived, short PLD Long lived, long PLD Short lived, long PLD Population Size

Oscillating Approach to Equilibrium: Long lived, short PLD Short lived, short PLD Long lived, long PLD Short lived, long PLD

Spatial Coefficient of Variation Monotonic Approach to Equilibrium: Long lived, short PLD Short lived, short PLD Long lived, long PLD Short lived, long PLD

Spatial Coefficient of Variation Oscillating Approach to Equilibrium: Long lived, short PLD Short lived, short PLD Long lived, long PLD Short lived, long PLD

Temporal Coefficient of Variation (local) Monotonic Approach to Equilibrium: Long lived, short PLD Short lived, short PLD Long lived, long PLD Short lived, long PLD

Temporal Coefficient of Variation (population) Monotonic Approach to Equilibrium: Long lived, short PLD Short lived, short PLD Long lived, long PLD Short lived, long PLD

Temporal Coefficient of Variation (local) Oscillating Approach to Equilibrium: Long lived, short PLD Short lived, short PLD Long lived, long PLD Short lived, long PLD

Temporal Coefficient of Variation (population) Oscillating Approach to Equilibrium: Long lived, short PLD Short lived, short PLD Long lived, long PLD Short lived, long PLD

Spatial Autocorrelation Monotonic Approach to Equilibrium: Long lived, short PLD Short lived, short PLD Long lived, long PLD Short lived, long PLD

Spatial Autocorrelation Oscillating Approach to Equilibrium: Long lived, short PLD Short lived, short PLD Long lived, long PLD Short lived, long PLD

Temporal Autocorrelation (local) Monotonic Approach to Equilibrium: Long lived, short PLD Short lived, short PLD Long lived, long PLD Short lived, long PLD

Temporal Autocorrelation (population) Monotonic Approach to Equilibrium: Long lived, short PLD Short lived, short PLD Long lived, long PLD Short lived, long PLD

Temporal Autocorrelation (local) Oscillating Approach to Equilibrium: Long lived, short PLD Short lived, short PLD Long lived, long PLD Short lived, long PLD

Temporal Autocorrelation (population) Oscillating Approach to Equilibrium: Long lived, short PLD Short lived, short PLD Long lived, long PLD Short lived, long PLD

Future Structure of the F 3 model Types of density dependence:  Recruitment rate depends on adult density  Production rate depends on adult density  Adult mortality depends on adult density  Recruitment rate depends on larval density Size & Age Structure  Increasing time to maturity  Increasing fecundity with age or size Adult movement Variability in habitat quality (spatial & temporal)

Next Steps Add other forms of density dependence Age/Size Structure Adult Movement Spatial/Temporal variability in habitat quality

Equations used to calculate parameters Non-Spatial model without harvest: P 0 = Productivity = Fecundity X Larval Survival At equilibrium (N t = K): For stability analysis:

Calculated Parameters Productivity (P 0 ) 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 P 0

Spatial Variance Monotonic Approach to Equilibrium: Long lived, short PLD Short lived, short PLD Long lived, long PLD Short lived, long PLD

Spatial Variance Oscillating Approach to Equilibrium: Long lived, short PLD Short lived, short PLD Long lived, long PLD Short lived, long PLD

Temporal Variance (local) Monotonic Approach to Equilibrium: Long lived, short PLD Short lived, short PLD Long lived, long PLD Short lived, long PLD

Temporal Variance (local) Oscillating Approach to Equilibrium: Long lived, short PLD Short lived, short PLD Long lived, long PLD Short lived, long PLD

Temporal Variance (population) Monotonic Approach to Equilibrium: Long lived, short PLD Short lived, short PLD Long lived, long PLD Short lived, long PLD

Temporal Variance (population) Oscillating Approach to Equilibrium: Long lived, short PLD Short lived, short PLD Long lived, long PLD Short lived, long PLD

Spatial Autocorrelation (run 2) Monotonic Approach to Equilibrium: Long lived, short PLD Short lived, short PLD Long lived, long PLD Short lived, long PLD

Spatial Autocorrelation (run 2) Oscillating Approach to Equilibrium: Long lived, short PLD Short lived, short PLD Long lived, long PLD Short lived, long PLD