Search problems: ecological and evolutionary perspectives?

1. Lorentz Center, Leiden, May 2012 Search problems: ecological and evolutionary perspectives? Jon Pitchford York Centre for Complex Systems Analysis Departments of Biology and Mathematics University of York

2. The Plan 1 : Irrelevant introduction Data, imagination, Darwin. 2 : Stochastic models for fish and fisheries Scaling from individuals to populations – physics and uncertainty. 3 : Optimal Lévy foraging in biology? “Seminal” and “correct” are different. Both are good. 4 : Are butterflies princesses or monsters? Shaky speculation on a real conservation problem.

3. What would you do if you were a bone? Bones need to be STIFF and TOUGH. Crane compact bone in tension Strain Stress

4. “Toughness” (post-yield) “Stiffness” (pre-yield)

6. Messages from data: average properties are governed by mineral content . Pre-yield properties (“stiffness”) are tightly determined by mineral content, post-yield properties are much less tightly determined. Does this matter??? J. D. Currey, J. W. Pitchford, P. D. Baxter, J. Roy. Soc. Interface, 2007

7. What is evolution by natural selection? “The survival of the fittest”

8. What is evolution by natural selection? “The survival of the fittest” Herbert Spencer

9. CHAPTER IV NATURAL SELECTION; OR THE SURVIVAL OF THE FITTEST Summary of Chapter. … owing to their geometrical rate of increase, a severe struggle for life at some age, season, or year, and this certainly cannot be disputed; … … if variations useful to any organic being ever do occur, assuredly individuals thus characterised will have the best chance of being preserved in the struggle for life; …and from the strong principle of inheritance, these will tend to produce offspring similarly characterised.

10. Bones need to be both stiff (more mineral) and tough (less mineral). Suppose stiffness proportional to mineral, toughness to (1-mineral): S = m, T = 1-m Deterministic model: If “fitness” = S * T then we can easily solve the optimisation problem. Optimal choice: md =1/2 This is the ESS (Evolutionarily Stable Strategy, Maynard Smith 1982). fitness 1 0 m

11. Add stochasticity: S = m ; T is a random variable with E(T) = (1-m) Add population dynamics: Only the fittest fraction p of each generation survive to reproduce next time. The best choice, m*, is that which maximizes where the random variable ξ, with probability density function f(ξ ), represents the variability in T, and x is defined by thereby ensuring one considers only the fittest p offspring.

12. And it turns out that m* exceeds md, by an amount which increases with intra-specific competition for survival in the next generation (1/p).

13. Some useful general messages? • Invest in your more tightly determined trait, then hope for the best. • “Bio-inspired” is not necessarily biological reality. • Local deterministic optimisation can be very misleading – but we do it a lot! • This is NOT “reliability” – evolution doesn’t do this.

14. 1 : Simple models of foraging fish larvae BBC FAO • We need more fish • To get more fish (“stock”) we need baby fish to grow and survive (“recruitment”) • It all looks very random

15. ESA www.richard-seaman.com How should I move if my world is patchy, and dynamic, uncertain, and turbulent?

16. Foraging and growth: some observations Laboratory experiments Coupled ODE models for zooplankton and fish (unless food unrealistically abundant) Data from fish in “identical” real environments (Dower, Pepin, Leggett, Fish. Oceanogr. 2002)

17. What’s going wrong? Fish, especially larvae, are SMALL : relative to turbulence, predators STUPID : behaviour mainly visual, no evidence for memory or complex behaviours (?) DEAD : massive mortality } We can build models } But not ODEs

18. Model 1: Stochastic cruise foraging, Poisson process

19. Do the maths: entering and leaving patches becomes an alternating renewal process (Cox 1962). Mean encounters per patch visit = g / b Mean time per foraging cycle (find patch, eat, leave) = 1 / a + g t / b + 1 / b Mass balance: (1 – V) a = V b where V = proportion volume of patches Put these together: encounter rate depends only on average prey conc. This generalises quite easily. So patchiness is pointless?

20. What does this mean for little fish in turbulent oceans? Use turbulent encounter theory and Stokes drag to ask: What swimming speed maximises (encounter rate) – (swimming cost)? Pitchford, James and Brindley, MEPS, 2003

21. Growth models: Consider the simplest deterministic growth model: fish of mass M grows at constant rate r, up to maturity at Mmat. A surviving fish reaches maturity at time so its probability of surviving to recruitment is simply

22. Now take the SAME model, but add noise: where W(t) is a white noise process. M(t) then becomes a simple diffusion process (Brownian motion with drift): and maturity time becomes a random variable:

23. Recruitment probability is then It’s ugly. It’s different to deterministic. But is it useful? i.e. stochasticity is ALWAYS BENEFICIAL, especially in a high mortality (or low growth rate environment).

24. Probability of reaching maturity in both stochastic and deterministic environments. Pitchford, James and Brindley, Fish. Oceanogr. (2005)

25. Open problem? (But an easy one!) IS THIS WRONG? Swimming speed influences mean r and variance s2. “Optimise” (stochastic gain) – (deterministic cost)?

27. 3: Lévy walks: optimal searching in biology? Lévy walk: search for prey by moving a random distance, l, between random reorientations, with density function f (l) ~ l–m with m “typically” between 1 and 3. N.B. m < 3 has infinite variance m < 2 has infinite mean and variance m > 3 is essentially diffusive movement ? ?

28. m = 2.2 m = 3.8 http://chaos.utexas.edu/research/annulus/rwalk.html

29. Theory: Lévy walks are “optimal” when m = 2. ? ? But the devil is in the detail….

30. Algorithm for success! 1. Choose animal 2. Analyse movement data 3. Fit power law, m = 2, optimal! 4. Publish paper

31. SIMULATIONS: A power law exponent of 2 is “optimal” for mean resource acquisition rate. TRUE… but not universal – depends on the details e.g. prey and patch regeneration time, how to simulate from distributions with infinite moments. James and Plank J. Roy. Soc Interface (2008); James, Pitchford, Plank (2010) And superimposed random walks might be just as good (Benhamou, Ecology 2007, Codling, Plank, Benhamou Ecology 2008)

32. How to analyse movement data? Ask a physicist. We suspect f (l) ~ l –m , so that ln (f(l)) is a straight line with slope –m. Easy! Plot a histogram of your logarithmically binned data, log the vertical axis, find the slope. Simulated data, m = 2… hang on… Sims, Righton, Pitchford, “Minimizing errors in identifying Levy flight behaviour of organisms” Journal of Animal Ecology 76, 2007, 222-229.

33. Let . Then x has PDF Hence , a line with slope m – 1 i.e. (diffusive movements) + (wrong analysis) = “optimal” Levy?

36. “Seminal” and “correct” are different Can we salvage something… 1) numerically, and 2) analytically? FASHION: Lots of animal movements appear to follow Lévy-like distributions (power law tail). SIMULATIONS / THEORY: If this power law has an exponent of 2, then mean resource acquisition rate is “optimal”. FACT:Planktonic prey is patchy (e.g. Lough and Broughton, 2007) FACT: Most fish are dead, only the tails of distributions are important (e.g. Pitchford et al. (2005))

37. Careful IBM simulations (boundaries, finite movements, …) … asking biologically relevant questions: “How quickly can you find 50 items of food?”

38. Recipe: • Simulate forager performing fixed-step or Levy-like random walks. • Generate distributions of hitting times. • Add mortality, as a simple Poisson Process. (Less time foraging = less likely to be eaten.) • Examine results in evolutionary context: • What strategy is best on average? • What strategy gives the best recruitment probability?

39. Interesting thing 0 2 4 6 1.0 2.0 3.0 Fixed-step Ballistic Levy walk Diffusive walk

41. So Levy walks might give an advantage, on average, but only slightly, and only in patchy environments? Comparisons: Expected hitting time Diffusive approximation for hitting time distribution Full hitting time distribution (Blue is uniform prey, red is patchy prey)

43. In fact, there’s no generic “best” strategy unless you consider the underlying (and unmodelled) selection pressure acting on the populations... ... bimodality in strategies? ... small selection pressure on foraging strategy?

44. Space Space Space Imagine a 1-D patchy prey distribution, and a blind frog who has to decide how to jump (saltatory foraging e.g. cod larvae). Analytical model: Stochastic saltatory foraging, “Magic Frogs” What jump strategy is optimal?

45. With a Levy jump distribution, the number of foraging locations N visited in time t is NegBin… … and with a Levy prey distribution, the number of prey Y encountered at each foraging location is NegBin:

46. Independent of spatial pattern, no strategy is “optimal” Variance grows faster than t So no strategy looks optimal for a small stupid forager in a patchy environment?

47. NO! Variance matters... ... when you’re almost certain to die, stochasticity works in your favour. Another (tractable?) open problem?

48. 4. Princesses or monsters? Melissa blue (Lycaeides melissa) High resolution movements of around 100 individuals in native and exotic host plants; 1s time step, 24000 observations. Matt Forister (Nevada), Paul Armsworth (UTK), Mark Preston (LSHTM)

49. characterize differences in speed of movement and diffusivity among males and females; • develop a model which quantifies interactions between males, females, and host plants; • investigate variation among male search strategies as a driver of variation in encounter rates and time spent with females.

50. Data analysis: • Males move faster than females • Males move in a more direct manner • Location is important, but the differences are consistent