1. Marine reserves and uncertain trade-offs Jon Pitchford York Centre for Complex Systems Analysis Departments of Biology and Mathematics University of York Edd Codling, Tanja Miethe, Chris West, Calvin Dytham, Dave Righton, Callum Roberts Nature-Investment Interaction Leicester, 21 st June 2010

2. World Cups 2002, 2006, Euro 2004 “... faced with a stochastic process, so we have to take expectations.” (Eric Naevdal, yesterday)‏

3. Next time, let’s agree to treat it as a temporally averaged process. No. I prefer to formulate the problem as the final state of continuous time stochastic process.

4. Plan 1. Uncertain investments – a (very) simple model 2. Marine reserves – simple ideas, complex realities. 3. Conclusions/ideas/Economics 101?

5. 1. Uncertain investments – a (very) simple model Q. How should I do a triathlon?

6. 1. Uncertain investments – a (very) simple model Q. How should I do a triathlon? “Invest” in training your swimming s, cycling c and running r. Constraint s + c + r = 1.

7. 1. Uncertain investments – a (very) simple model Q. How should I do a triathlon? “Invest” in training your swimming s, cycling c and running r. Constraint s + c + r = 1. A. Equal investment is optimal, s = c = r = 1/3.

8. 1. Uncertain investments – a (very) simple model Q. How should I do a triathlon? “Invest” in training your swimming s, cycling c and running r. Constraint s + c + r = 1. A. In a stochastic world, the best answer is “it depends”.

9. 2. Marine reserves - background How to manage a fishery? Free-for-all? Annual quotas? Long-term quotas? Ecosystem approach? Or a simple solution: just stop fishing?

10. The Facts: Scientists recommend that 20- 30% of our seas should be fully protected to ensure their survival. Over 60% of UK fisheries are unsustainable…. Marine Reserves can benefit divers, anglers, fishermen and biodiversity alike. New Zealand has 28 Marine Reserves, and 33% of the Great Barrier Reef is highly protected – many reserves have resulted in increased fish and shellfish populations, whilst biodiversity is protected from destruction. http://www.mcsuk.org/mcsaction/marinereserves/marine+reserves+now

11. Reserves will benefit to fish, tourism, and biodiversity. Question: Are reserves ever be beneficial to fishers? e.g. Overspill from reserve allows increased catch? Answer: (Deterministic) No, probably not. (Stochastic) Yes!

12. Simple discrete time single fishery model (e.g. Kot 2001). X(t+1) = (1 – m)X(t) + R(X(t)) – C X(t) : adult fish population m : natural mortality C : catch R(X(t)) = max { r X(t) (1 – X(t) / 2K), 0 } : recruitment MSY = K (r – m) 2 / 2 r

13. Extend to a reserve model …. X 1 (t+1) = (1 – m) X 1 (t) + R 1 (X 1 (t)) – f M(X 1 (t), X 2 (t))‏ X 2 (t+1) = (1 – m) X 2 (t) + R 2 (X 2 (t)) – C + f M(X 1 (t), X 2 (t))‏ Local recruitment (carrying capacity K), discrete time “diffusion”: R i (X i (t)) = max { r X i (t)(1 – X i / K), 0 } M(X 1, X 2 ) = max { (1 – m) X 1, 0 } – max {(1 – m) X 2 – C, 0 }

14. Do the maths…. standard phase plane analysis in variables ( X 1 + X 2 ) and ( X 1 - X 2 ). Result : At any stable equilibrium, C MSY > C RESERVE (or C MSY = C RESERVE when f = ½; perfect mixing between populations)‏ So the deterministic model says reserves are NOT BENEFICIAL (although they’re not very bad provided 0.2 < f < 0.5)‏ The end of the story? Add more REALISM, or ….

15. … add stochasticity. Recruitment : environmentally driven, notoriously variable Catch : inherent uncertainty, misreporting Simulate the noisy dynamical system under three “management regimes”: None: simple single fishery model, constant quota. MPA: marine reserve in half of area, same quota. HCR: set quota to target catch, unless population assessed as beneath threshold – quotas adjusted annually.

16. Example time series: fishing close to, but less than, deterministic MSY, f = 0.2 EXTINCT! Sustainable

17. Realised catch Quota

18. Catch variabilityP(fishery extinction)‏

19. No reserve : Poor and variable yield, poor sustainability, even when fishing well beneath MSY. Why? You’re exploiting the system close to a point of bifurcation – small errors are amplified. Total variance = interannual variance / (1 – |e|)‏ e.g. (Quota of 90% MSY) + (10% reporting variability) = disaster. Reserve : Good yield, usually sustainable when overexploited. Why? Uncertainty is buffered (MPA) or reacted to (HCR).

20. “I don’t believe you! You’re a liar!” Now add “reality”. Conclusions robust to: Type and source of stochasticity Reserve size Recruitment function and location Age structure Cheating But NOT (necessarily) robust to MOVEMENT and EVOLUTION.

21. Over exploitation could lead to earlier maturation in fish populations ( e.g. Olsen et al. Nature 2004 )? Suppose you are a fish with an (evolutionary, life-history based) option: mature after 1 year, have 10 babies every year; OR mature after 2 years, have 20 babies every year. What is your best option?

22. N 11 N 14 N 13 N 12 f2f2 f4f4 s4s4 s2s2 s5s5 h4h4h4h4 Small fish Large fish Gårdmark et al. (2003)‏  is the probability to mature small; fecundities f 2 << f 4. There is an evolutionary switch-point h 4 *.

23. N 11 N 14 N 13 N 12 f2f2 f4f4 s4s4 s2s2 s5s5 h4h4h4h4 Can reserves prevent this evolutionary switch? spill-over (d 4 )‏ Miethe et al. (2008, 2009, 2010)‏ N 21 N 24 N 23 N 22 YES! But...

24. Selfishly, some fish do not live in metapopulations…

25. Fish migrate between a spawning and feeding ground Fishing occurs on feeding ground but not in spawning ground

26. Analysis reveals possible bifurcations and bistabilities…are these important?

27. Drive with stochastic recruitment… regime changes and extinctions are possible.

28. Emergent spatial structure: aggregation/dispersal trade off

30. 3. Conclusions I : Basic model Economically pushing a nonlinear dynamical efficient = system close to a dangerous exploitation bifurcation (eigenvalue near 1)‏ Therefore deterministic models are likely to be misleading. Marine reserves do not generically benefit the fishery, on average (but they’re not too bad, either). Marine reserves can be very effective for non-migratory stocks.

31. Conclusions II: Complexity Real-world factors complicate the story – evolution, migration, bistability, emergent spatial structure. No simple panacea. No linear approximation. Rule of thumb: marine reserves will decrease yield (slightly) and decrease variability (significantly).

32. Conclusions III How should we manage in the face of uncertainty? “Optimum” depends on who is asking the question. Ongoing work?