Population Connectivity and Management of an Emerging Commercial Fishery Crow White ESM 242 ProjectMay 31, 2007

Adult (15 cm) Recruits Kellet’s whelk Kelletia kelletii

Focus of developing fishery Sold to US domestic Asian market (mostly in LA) Mean price = $1.43/kg = ~$0.15/whelk Aseltine-Neilson et al. 2006

Caught as by-catch by commercial trap fishermen

Research questions:  What is the optimal harvest path that maximizes net present value of the Kellet’s whelk fishery?  Short-term.  Long-term.  How do they differ?

SBA NCI Focus on Santa Barbara area Two patches: SBA: Santa Barbara mainland NCI: Northern Channel Islands Patches differ with respect to: Habitat area, stock size & density Intra- and inter-patch dispersal dynamics Protection in reserves Santa Barbara

EQUATION OF MOTION (patch A): Adult stock [mt]Growth rate“Connectivity” = probability of dispersal Harvest [mt]Annual natural mortality rate Density dependent recruitment K = kelp [km2] Juvenile mortality t = time in years Tj = time until reproductively mature = age of legal size for fishery

CONSTRAINTS: Harvest in a patch must be equal or greater than zero, as well as equal or less than the current stock in that patch In Northern Channel Islands patch harvest may not reduce stock below 20% of its virgin size

12 reserves constituting ~20% of the NCI coastline

EQUATION OF MOTION (patch A): Adult stock [mt]Growth rate“Connectivity” = probability of dispersal Harvest [mt]Annual natural mortality rate Density dependent recruitment K = kelp [km2] Juvenile mortality t = time in years Tj = time until reproductively mature = age of legal size for fishery

SBA NCI Thanks Mike!

(N = 4) Pattern supported by lobster/Kellet’s whelk fisherman (John Wilson, per. comm. 16 May 2007)

Protected in reserves

EQUATION OF MOTION (patch A): Adult stock [mt]Growth rate“Connectivity” = probability of dispersal Harvest [mt]Annual natural mortality rate Density dependent recruitment K = kelp [km2] Juvenile mortality t = time in years Tj = time until reproductively mature = age of legal size for fishery

Mean size (n = 1000+) m = 1/mean age = Tj = ~6 years Annual natural mortality rate: Time until mature: Mature: (Growth data from D. Zacherl 2006 unpub. Res.)

EQUATION OF MOTION (patch A): Adult stock [mt]Growth rate“Connectivity” = probability of dispersal Harvest [mt]Annual natural mortality rate Density dependent recruitment K = kelp [km2] Juvenile mortality t = time in years Tj = time until reproductively mature = age of legal size for fishery

Kellet’s whelk, Kelletia kelletii larvae per egg capsule

Density dependence coefficient Given each patch is a closed system and T j = 1: N* = virgin carrying capacity.

EQUATION OF MOTION (patch A): Adult stock [mt]Growth rate“Connectivity” = probability of dispersal Harvest [mt]Annual natural mortality rate Density dependent recruitment K = kelp [km2] Juvenile mortality t = time in years Tj = time until reproductively mature = age of legal size for fishery

C source-destination: C SBA-SBA = 0.15 C SBA-NCI = 0.34 C NCI-NCI = 0.35 C NCI-SBA = 0.27 Gastropod larva K. kelletia settler (OIPL 2007) (Koch 2006) SBA NCI Thanks James!

Of the total number of settlers arriving at a patch: Santa Barbara AreaNorthern Channel Islands SBANCI Closed system:

Economics:  Revenue based on demand curve:  revenue(t) = choke price – (Harvest[t])(slope)  Cost based on stock effect:  cost(t) = θ / stock density  π(t) = (revenue[t] – cost[t])(1 – r)^-t  r = discount rate = 0.05 ∫

Choke price = max(Price [ ]) All whelks in system Profit calculated at end of each year’s harvest

mr = mc = θ / density, when density = 0.1*min(SBA* or NCI*)

mr, given supply = 1 mt Marginal profit calculated during harvest

Optimization procedure  Short-term: 40 years of harvest  Let un-harvested system equilibrate  Search for optimal harvest path: employ constrained nonlinear optimization function (derivative-based algorithm) in program Matlab.  Goal: find optimal H that maximizes NPV = ∑ π(t)  Long-term: Steady state (t → ∞)  Iterative exploration of all combinations of constant escapement (A – H ≈ 0 – 100%) in each patch.  run until system equilibrates  Goal: identify escapement combination that maximizes π at t = final.

Short-term (40-year) optimal harvest path

Harvest path is variable and different in the two patches

Higher Lower

Initial spike in harvest

Harvest limited by NCI reserve constraint

Harvest until mr = mc

Harvest path is semi-cyclic: due to delayed development?

NPV = ∑ π(t) = $1,279,900 ~$32,000/year

10,000 simulations: NPV H* H* - (v/2)(H*)H* + (v/2)(H*) H(t) = H* + U[-v/2, +v/2](H*)

10,000 simulations: 90% NPV H* H* - (v/2)(H*)H* + (v/2)(H*) NPV H(t) = H* + U[-v/2, +v/2](H*)

Long-term optimal harvest

Harvest everything

$68,067/year

Harvest everything $68,067/year 40-year horizon and r = 0.05: ~$31,000/year

Harvest everything $68,067/year 40-year horizon and r = 0.05: ~$31,000/year

Harvest everything $68,067/year 40-year horizon and r = 0.05: ~$31,000/year

Harvest everything 90%

Harvest everything Plenty Room for uncertainty: Little

90% NCI reserve constraint Harvest everything NCI used as a source, regardless of regulation!

Future research: 1.Improve accuracy in parameter estimates (e.g., λ, population density) and re-run analysis. 1.Relevance of NCI reserve constraint? 2.Incorporate known variability (e.g., connectivity across years) and uncertainty (e.g., in λ and form of density dependence function) into analysis. 3.Apply model to a variety of systems characterized by different levels of connectivity.

Currents Oceanographic boundaries (Gaylord & Gaines 2000)

Central CA Southern CA US Mexico Borders dividing fishery management jurisdictions Is cooperation in cross- border management part of the optimal solution?

Thank you!

10,000 simulations: 90% net present value H* H* - U[1-v,1+v](H*) H* + U[1-v,1+v](H*)

Southern hemisphere cetaceans (Hilborn et al. 2003)

“Rapid worldwide depletion of predatory fish” (Myers & Worm 2003)

Mean size (n = 1000+) m = 1/mean age = Annual natural mortality rate:

Excellent for lawn art (match gnomes beautifully!)

Thank you!