1. Spatial models (meta-population models)

2. Readings Hilborn R et al. (2004) When can marine reserves improve fisheries management? Ocean and Coastal Management 47:197-205 Hilborn R et al. (2006) Integrating marine protected areas with catch regulation. CJFAS 63:642-649

3. Overview Why worry about space? General metapopulation model One-dimensional model of marine protected areas

4. Why worry about space?

5. Rijnsdorp AD et al. (1998) Micro-scale distribution of beam trawl effort in the southern North Sea between 1993 and 1996 in relation to the trawling frequency of the sea bed and the impact on benthic organisms. ICES Journal of Marine Science 55:403-419 England Netherlands Dutch beam trawl fleet

6. Jennings S & Lee J (2012) Defining fishing grounds with vessel monitoring system data. ICES Journal of Marine Science 69:51-63 Beam trawlers Dredgers Netters Otter trawlers Potters All combined UK fisheries

7. Swain DP & EJ Wade (2003) Spatial distribution of catch and effort in a fishery for snow crab (Chionoectes opilio): tests of predictions of the ideal free distribution. CJFAS 60:897-909 CPUE (constant) Effort (variable) Abundance (variable) Ideal free distribution

8. Murawski SA et al. (2005) Effort distribution and catch patterns adjacent to temperate MPAs. ICES Journal of Marine Science 62:1150-1167 Marine protected area (MPA) MPA

9. Reason for “fishing the line” Hours trawled Dollars per hour CV of dollars Murawski SA et al. (2005) Effort distribution and catch patterns adjacent to temperate MPAs. ICES Journal of Marine Science 62:1150-1167

10. Metapopulation models A metapopulation is a series of discrete populations that are isolated but have limited exchange.

11. Generalized metapopulation model Immigration: movement from area j into area i Emigration: movement out of area i into area j Environmental conditions in area i in time t Numbers in area i in time t

12. Important details Population dynamics within an area Models of dispersal Environmental models, process error, catastrophic events

13. Boundaries? Absorbing boundary – Disappear/die on hitting the boundary – E.g. another country, advection into open ocean, natural range bounds Reflective boundary – Remain in the cell next to the boundary – E.g. mountain range, river barrier Pac-man (circular coastline) – Reappear on the opposite side – E.g. circumpolar, islands http://www.google.com/pacman/

14. One-dimensional logistic-growth model with harvesting Only survivors of harvest will move Movement rate the same in all cells Numbers in area i in time t Exploitation rate Cell on the left Cell on the right Harvest, then movement Density dependence Harvest Immigration Emigration Logistic model

15. 16 spatial models animation.r

16. Diffusion scenario 21 areas, 50 time steps, migration rate 0.2, r = 0.2, K = 1000, reflective boundary Diffusion scenario: starting population = K in center cell, 0 in all other cells, exploitation rate 0 16 spatial models animation.r

17. Diffusion, no harvesting, N 11 = K (black = first year, light gray = last year) 16 spatial models animation.r Abundance Cell number

18. Marine protected area scenario As before: 21 areas, 50 time steps, migration rate 0.2, r = 0.2, K = 1000, reflective boundary MPA scenario: starting population = K in all cells, exploitation rate 0 in center 5 cells (numbers 9–13), exploitation rate 0.2 in all other cells 16 spatial models animation.r

19. MPA: no harvest in center cells (black = first year, light gray = last year) Abundance 16 spatial models animation.r Cell number

20. Do protected areas increase yields? 51 areas, migration rate m = 0.2, r = 0.2, K = 1000, start population = K in all areas Run with u = 0, 0.01, 0.02, …, 0.9 After a large number of time steps (1000) the model is at equilibrium, and yield in the final time step is the equilibrium yield for each value of u No MPA: all areas have harvest rate = u MPA: 5 middle areas have zero harvest rate, other have harvest rate = u 16 MPA yield.r

22. Time series of catches Year Yield in each year No MPA MPA u = 0.1 u = 0.25 Almost at equilibrium yield 16 MPA yield.r Initially the no MPA scenario has higher catches; after year 14 with high u = 0.25 the MPA has higher catches; with no MPA, the population can go extinct

23. Do protected areas increase yield? MPA No MPA Higher yield without MPA Yield maximized at higher harvest rate with MPA At u > r, MPA halts collapse (insurance policy) u = r = 0.2 u MSY = r/2 = 0.1 16 MPA yield.r Harvest rate (u) Equilibrium yield

24. Zoomed in MPA No MPA u MSY without MPAu MSY with MPABreak-even point 16 MPA yield.r Harvest rate (u) Equilibrium yield

25. Bigger MPA (close 25 cells) (previously, close 5 of 51 cells) MPA No MPA Closing more areas reduces the maximum yield more Maximum yield reduced from 2550 to 1377 in this simulation 16 MPA yield.r Harvest rate (u) Equilibrium yield

26. MPA No MPA MPA: u = 0.10 produces MSY No MPA: two values of u result in equivalent yield: u = 0.032 and u = 0.168 For the same yield what is u? 16 MPA yield.r Harvest rate (u) Equilibrium yield

27. For the same yield, what is biomass? MPA u = 0.1 No MPA u = 0.032 No MPA u = 0.168 16 MPA yield.r Cell number Abundance

28. For the same yield, what is CPUE? (assume effort is proportional to harvest rate u) MPA u = 0.1, yield = 1370, CPUE = 13700 No MPA u = 0.032, yield = 1370, CPUE = 1370/0.032 = 42800 No MPA u = 0.168, yield = 1370, CPUE = 8200 16 MPA yield.r

29. Lessons Closing areas reduces the maximum yield The more areas closed, the lower the maximum yield Closed areas provide insurance against high fishing pressure (bad management, lack of enforcement) For every level of yield with an MPA there is an equivalent yield without an MPA which has: – higher biomass outside the MPA – lower biomass inside the MPA – lower harvest rate where fishing occurs – higher CPUE and hence greater profits – no completely protected areas

30. Old target New target Tradeoffs of fishing Worm B et al. (2009) Rebuilding global fisheries. Science 325:578-585 Exploitation rate Percent of maximum

31. Lessons Spatial scale and pattern matters Simple movement models Marine protected areas: insurance vs. yield Trade-offs between catch, profit, and biodiversity