Search problems: ecological and
evolutionary perspectives?
Lorentz Center, Leiden, May 2012
Jon Pitchford
York Centre for Complex Systems Analysis
Departments of Biology and Mathematics
University of York
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.
1. What would you do if you were a bone?
Bones need to be STIFF and TOUGH.
Crane compact
bone in tension
Strain
Stress
Total work vs Mineral
Log scales
E vs Mineral
35
20
30
-3
Work (MJ m )
Young's modulus (GPa)
10
25
20
15
5
2
1
10
0.5
5
0.2
0
200
220
240
260
-1
Mineral (mg g )
“Stiffness”
(pre-yield)
280
300
200
220
240
260
Mineral (mg g-1)
“Toughness”
(post-yield)
280
300
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
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.
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
fitness
This is the ESS
(Evolutionarily Stable
Strategy, Maynard Smith
1982).
0
1
m
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
m(1  m   ) f ( )d
where the random variable ξ, with probability density function f(ξ ),
represents the variability in T, and x is defined by


x
f ( )d  p
thereby ensuring one considers only the fittest p offspring.
And it turns out that
1 
m*  md 
f ( )d

2p x
m* exceeds md, by an amount which increases with intraspecific competition for survival in the next generation (1/p).
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.
1 : Simple models of foraging fish larvae
BBC
• We need more fish
• To get more fish (“stock”) we need baby fish to grow
and survive (“recruitment”)
• It all looks very random
FAO
ESA
www.richard-seaman.com
How should I move if my world is patchy, and dynamic,
uncertain, and turbulent?
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)
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
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 = 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?
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
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 s2.
“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 –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
?
m = 3.8
m = 2.2
http://chaos.utexas.edu/research/annulus/rwalk.html
Theory: Lévy walks are “optimal” when m = 2.
?
?
But the devil is in the detail….
Algorithm for success!
1. Choose animal
2. Analyse movement data
3. Fit power law, m = 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 random
walks might be just as
good (Benhamou, Ecology 2007,
Codling, Plank, Benhamou Ecology
2008)
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.
Let
.
Then x has PDF
Hence
, a line with slope m – 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 Lévylike 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
Fixed-step
Ballistic
1.0
2.0
3.0
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?
Analytical model: Stochastic saltatory foraging, “Magic Frogs”
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?
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:
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
Mean
Speed
Alpha
Male
0.435m/s 1.46
Female
0.215m/s 1.23
“Females make males go ballistic. Rapidly.”
How about the times spent flying versus sitting?
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?
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.
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.