1. “Fishing” Actually, coupled Flow, Fish and Fishing

2. Significance Interplay between spatial bioeconomics and environmental variability may be significant. Interplay between spatial bioeconomics and environmental variability may be significant. How should policy designed to manage these resources be informed by spatial and variable bioeconomics? How should policy designed to manage these resources be informed by spatial and variable bioeconomics? Approach – identify the optimal spatial harvest strategy, then interpret its characteristics. Approach – identify the optimal spatial harvest strategy, then interpret its characteristics.

3. Spatial economics Price-taking fishery Price-taking fishery Marginal harvest cost is stock-dependent Marginal harvest cost is stock-dependent Can vary across space (e.g. more costly to fish in deeper patch) Can vary across space (e.g. more costly to fish in deeper patch) MC increases over a season (as stock is drawn down), but not directly dependent on harvest. MC increases over a season (as stock is drawn down), but not directly dependent on harvest. Objective: Identify spatially-distinct optimal feedback control rule to maximize expected present value of harvest. Objective: Identify spatially-distinct optimal feedback control rule to maximize expected present value of harvest. This would allow us to answer the following questions… This would allow us to answer the following questions…

4. Key questions 1. What is the optimal (feedback) spatial harvest strategy for resources subject to spatial bioeconomic heterogeneity? 2. How does this strategy differ from the spatially- homogeneous case? What are the consequences of ignoring space? 3. How does the optimal policy depend on environmental variability and uncertainty? 4. Are permanent harvest closures ever optimal? Under what conditions? How should they be designed spatially? 5. Are temporary harvest closures ever optimal? When? 6. How does the implementation of a harvest closure (permanent or temporary) affect optimal management outside?

5. Approved Oct. 24, 2002 State waters are implemented Q1: Is this a “good” spatial reserve design? Q2: What happens outside the reserve?

6. Timing of production & harvest Adult population in a location Settlement and survival to adulthood Larval production Spawning population (Escapement) Harvest zfzf Random dispersal “D ij ” zSzS zmzm (Note here that harvest is location-specific) shock to adult survival shocks to settlement and larval survival shock to fecundity

7. Uncertainty and variability Adult population in a location Settlement and survival to adulthood Larval production Spawning population (Escapement) Harvest zfzf Random dispersal “D ij ” zSzS zmzm (Note here that harvest is location-specific) shock to adult survival shocks to settlement and larval survival shock to fecundity

8. Background MPAs and fishery yield (Hastings & Botsford) MPAs and fishery yield (Hastings & Botsford) MPAs and open access or regulated open access outside (Jim and Jim) MPAs and open access or regulated open access outside (Jim and Jim) Optimal spatial harvesting in deterministic environment (Neubert, Now Jim & Jim) Optimal spatial harvesting in deterministic environment (Neubert, Now Jim & Jim) Single-stock, no space – management under uncertainty (Reed) Single-stock, no space – management under uncertainty (Reed)

9. State and control vectors Objective: identify optimal feedback control rule to maximize expected present value from harvest in this fishery. Objective: identify optimal feedback control rule to maximize expected present value from harvest in this fishery. State vector: current stock in every patch (x 1, …, x I ) State vector: current stock in every patch (x 1, …, x I ) Control vector: harvest (alternatively, escapement) in every patch (e 1, …, e I ) Control vector: harvest (alternatively, escapement) in every patch (e 1, …, e I ) In time period t we have I state variables, and I control variables, t=1,….,T. In time period t we have I state variables, and I control variables, t=1,….,T. Typically e it * (x 1t, …, x It ) for all i=1,…,I Typically e it * (x 1t, …, x It ) for all i=1,…,I

10. Problem setup Maximize E{NPV} of profits from harvest. Find optimal harvest strategy: Maximize E{NPV} of profits from harvest. Find optimal harvest strategy: Equation of motion: Equation of motion: Dynamic Programming Equation: Dynamic Programming Equation:

11. Solution procedure Discrete-time stochastic dynamic programming. Discrete-time stochastic dynamic programming. Start at period T (end of planning horizon) Start at period T (end of planning horizon) Work backwards analytically Work backwards analytically Definition: An optimization problem is state separable if the necessary conditions are independent of the state vector. Definition: An optimization problem is state separable if the necessary conditions are independent of the state vector. If T=1, our problem becomes: If T=1, our problem becomes: The 1-period problem is state separable. The 1-period problem is state separable. Turns out that if an interior solution exists, the period-t problem is still state-separable Turns out that if an interior solution exists, the period-t problem is still state-separable

12. Results 1. In an interior solution, optimal policy varies across space, but is time and state independent. 2. If harvest cost and survival functions are linear, then solution is independent of dispersal. 3. Patch i should be closed to harvest if x(it)<e*(it). 4. If might close patch k next year, then reduce harvest in patch i for which D(ik)>0 5. If arbitrarily close patch k next year then increase harvest in patch I for which D(ik)>0.

13. Next Steps 1. Apply this model empirically to Channel Islands data 2. Model fleet dynamics and individual decision- making (John Lynham) 3. Does accounting for individual behavior qualitatively affect results (e.g. reserve/no reserve comparison)?