Search problems: ecological and evolutionary perspectives?

Jon PitchfordYork Centre for Complex Systems AnalysisDepartments of Biology and Mathematics

University of York

Lorentz Center, Leiden, May 2012

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.

The Plan

1. What would you do if you were a bone? Bones need to be STIFF and TOUGH.

Crane compactbone in tensionStrain

Stress

E vs Mineral

Mineral (mg g-1)

200 220 240 260 280 300

You

ng's

mod

ulus

(GP

a)

0

5

10

15

20

25

30

35

“Stiffness”(pre-yield)

Total work vs MineralLog scales

Mineral (mg g-1)

200 220 240 260 280 300W

ork

(MJ

m-3

)0.2

0.5

1

2

5

10

20

“Toughness”(post-yield)

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

What is evolution by natural selection?

“The survival of the fittest”

What is evolution by natural selection?

“The survival of the fittest”

Herbert Spencer

… 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.

CHAPTER IVNATURAL SELECTION; OR THE SURVIVAL OF THE FITTEST

Summary of Chapter.

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

m10

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

x

dfmm )()1(

where the random variable ξ, with probability density function f(ξ ), represents the variability in T, and x is defined by

pdfx

)(

thereby ensuring one considers only the fittest p offspring.

x

d dfp

mm )(21*

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).

• 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.

Some useful general messages?

• We need more fish

• To get more fish (“stock”) we need baby fish to grow and survive (“recruitment”)

• It all looks very random

FAOBBC

1 : Simple models of foraging fish larvae

How should I move if my world is patchy, and dynamic, uncertain, and turbulent?

www.richard-seaman.comESA

Laboratory experiments

Coupled ODE models for zooplankton and fish

Data from fish in “identical” real environments (Dower, Pepin, Leggett, Fish. Oceanogr. 2002)

(unless food unrealistically abundant)

Foraging and growth: some observations

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 canbuild

models

But notODEs

Model 1: Stochastic cruise foraging, Poisson process

Do the maths: entering and leaving patches becomes an alternating renewal process (Cox 1962).

Mean encounters per patch visit = /

Mean time per foraging cycle (find patch, eat, leave) = 1 / + / + 1 /

Mass balance: (1 – V) = V 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?

What does this mean for little fish in turbulent oceans?

Pitchford, James and Brindley, MEPS, 2003

Use turbulent encounter theory and Stokes drag to ask:

What swimming speed maximises (encounter rate) – (swimming cost)?

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

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:

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).

Probability of reaching maturity in both stochastic and deterministic environments.

Pitchford, James and Brindley, Fish. Oceanogr. (2005)

Open problem? (But an easy one!)

IS THIS WRONG? Swimming speed influences mean r and variance 2.

“Optimise” (stochastic gain) – (deterministic cost)?

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 –

with “typically” between 1 and 3.

N.B. < 3 has infinite variance < 2 has infinite mean and variance > 3 is essentially diffusive movement

??

= 3.8 = 2.2

http://chaos.utexas.edu/research/annulus/rwalk.html

Theory: Lévy walks are “optimal” when = 2.

??

But the devil is in the detail….

Algorithm for success!

1. Choose animal

2. Analyse movement data

3. Fit power law, = 2, optimal!

4. Publish paper

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 randomwalks might be just asgood (Benhamou, Ecology 2007, Codling, Plank, Benhamou Ecology 2008)

How to analyse movement data? Ask a physicist.

We suspect f (l) ~ l –, so that ln (f(l)) is a straight line with slope –.

Easy! Plot a histogram of your logarithmically binned data, log the vertical axis, find the slope.

Simulated data, = 2… hang on…

Sims, Righton, Pitchford, “Minimizing errors in identifying Levy flight behaviour of organisms” Journal of Animal Ecology 76, 2007, 222-229.

Let .

Then x has PDF

Hence , a line with slope – 1

i.e. (diffusive movements) + (wrong analysis) = “optimal” Levy?

“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))

Careful IBM simulations (boundaries, finite movements, …)

… asking biologically relevant questions:

“How quickly can you find 50 items of food?”

Recipe:

1.Simulate forager performing fixed-step or Levy-like random walks.

2.Generate distributions of hitting times.

3.Add mortality, as a simple Poisson Process. (Less time foraging = less likely to be eaten.)

4.Examine results in evolutionary context:

What strategy is best on average?

What strategy gives the best recruitment probability?

Interesting thing

0 2 4 6 1.0 2.0 3.0 Fixed-step Ballistic Levy walk Diffusive walk

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)

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?

Imagine a 1-D patchy prey distribution, and a blind frog who has to decide how to jump (saltatory foraging e.g. cod larvae).

Space

Space

Space

What jump strategy is optimal?

Analytical model: Stochastic saltatory foraging, “Magic Frogs”

With a Levy jump distribution, the number of foraging locationsN visited in time t is NegBin…

… and with a Levy prey distribution, the number of prey Y encountered at each foraging location is NegBin:

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?

NO! Variance matters...

... when you’re almost certain to die, stochasticity works in your favour.

Another (tractable?) open problem?

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)

(1) characterize differences in speed of movement and diffusivity among males and females;

(2) develop a model which quantifies interactions between males, females, and host plants;

(3) investigate variation among male search strategies as a driver of variation in encounter rates and time spent with females.

Data analysis:

•Males move faster than females

•Males move in a more direct manner

•Location is important, but the differences are consistent

Mean Speed Alpha

Male 0.435m/s 1.46

Female 0.215m/s 1.23

Data analysis:

•Males move faster than females

•Males move in a more direct manner than females

•Location is important, but the differences are consistent

“Females make males go ballistic. Rapidly.”

How about the times spent flying versus sitting?

Mean Speed Alpha

Male 0.435m/s 1.46

Female 0.215m/s 1.23

Similar data for flying, and same data exist for males.

Data-driven simulation algorithm

•Create an environment of host plants, “uniform” or “clumped”, at some specified density.

Then, for each individual butterfly,

•choose flight duration of τ seconds from empirical distribution•each plant within a distance τv is assigned a weighted probability according to its distance, τ, v and α•the next host plant is chosen via these probabilities; if there are no plants within τv then the original host plant is selected •the individual moves to the chosen host plant after a flight of τ seconds. Repeat for all individuals, for 10000s.

Results

Run simulations for various male speeds, at different plant densities, for different courtship times.

Which speed gets a male the most new mates?

No real answers, just questions:

• Can this tell us about what drives successful courtship?

•Do monsters always move faster, and more directly, than princesses?

•If you were a princess, how many monsters would you want to meet?

•What will happen in new (or fragmented) habitats?

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.

The Plan

Conclusions:

Ecology provides “noise”, evolution needs “noise”; biological models which don’t allow for this might miss the point?

Biological stochasticity is IMPORTANT and TRACTABLE:

• nonlinear systems (everywhere)

• rare events (recruitment, evolution)

• systems close to bifurcation (managed and exploited systems)

• buffering uncertainty (group movement, reserves)

Thanks: Paul Armsworth, Paul Baxter, John Brindley, Edd Codling, John Currey, Matt Forister, Alex James, Qiming Lv, Mike Plank, Mark Preston, Dave Righton, David Sims.