A true story...

as told by Steven Hamblin

IntroductionInformation use

Foraging in spaceConclusion

Optimal foraging

Social foraging

Producer-scrounger game

producer

scrounger

Evolutionarily stable strategy (ESS)

mechanism regulating the frequency of alternatives in the population.An initial population of pure producer, for instance, could be invaded byalleles for scrounger because the initially rare scroungers would do muchbetter than any of the individuals bearing producer alleles (Fig. 1A).Scrounger alleles could not go to fixation because a population made upof pure scroungers would be invaded by producers. So, neither producernor scrounger alone can be evolutionarily stable. The ESS in this case ismixed, allowing the frequency of scrounger alleles to increase until thefitness of scroungers drops to the fitness of producers.

2. The Behaviorally Stable Strategy

In most behavioral instances, however, animals reach game solutions byadjusting their use of strategies according to the conditions in which theyare playing the game. The mechanism of adjustment in this case is not

A

B

C

Pay

offs

Proportion scrounger

FIG. 1. The payoff functions of the producer–scrounger game. The three panels givedifferent possible effects of scroungers on the producers’ (thin line) and the scroungers’ payoffs(thick line). Panel A gives the classic producer–scrounger payoffs: producer and scroungerpayoffs are depressed by increased proportion of playing scrounger but scroungers are affectedmore strongly. Panel B shows a case where the producers are unaffected by scroungers,whereas in panel C, producer payoffs seem to benefit from increased frequencies of scrounger.

66 LUC‐ALAIN GIRALDEAU AND FREDERIQUE DUBOIS

Payo

ffs

Proportion scrounger

adapted from Giraldeau & Dubois, 2008

Scrounger

Producer

ESS assumes !a genetic model.

housed in flocks of six in common cages (59!32 and46 cm high) made of galvanized wire mesh and kept on a12:12 h light:dark cycle at 27"C (#2"). They were fed adlibitum on a mixture of white and red millet seeds andoffered ad libitum water. Each bird was marked with aunique combination of two coloured leg bands. Inaddition, the tail and neck feathers of each individualwere coloured with acrylic paint to allow individualidentification from a distance.

ApparatusThe purpose of the experimental apparatus was to

constrain subjects to act as either producers or scroungersin order to manipulate the frequency of each tactic

within a flock. The apparatus consisted of an indoor cage(273!102 cm and 104 cm high) with a producer and ascrounger compartment divided by a series of 22 patches,of which every second one contained seeds (Fig. 2a). Anopaque barrier placed length-wise from ceiling to floorprevented birds from moving between the producer andscrounger compartments (Fig. 2a).

Each patch consisted of a seed container and a stringthat prevented the seeds from falling out. Pulling thestring caused the seeds to fall into a 2!2 cm collectingdish located directly below the seed container. Oncein the collecting dish the seeds were available to theindividual that pulled the string from the producercompartment and all individuals within the scrounger

BarrierScrounger side

Producer side

Seed container

Division

Collecting dish

String

Perch

Scrounger sideProducer side

(b)

(a)

Figure 2. Top view of the experimental apparatus (a) and foraging patch (b). Individuals could search for seed-containing patches by pullingthe string associated with each patch. Strings were available only in the producer compartment. Birds in the scrounger compartment searchedfor individuals feeding from produced patches. When the top portion of an opaque barrier was in place, the birds in one compartment couldnot move into the other compartment. A close-up view of the patch (b) shows that producers had to sit on a perch directly in front of a patchto pull the string associated with that patch, and if seeds were present, they were released into the collecting dish. From the perch, a producercould reach the collecting dish by stretching its neck through a small hole in the division placed between compartments. The arrow indicatesthe direction in which the string had to be pulled to release the seeds.

343M O TTLEY & G IRAL D EA U : CONVERGING ON PS EQUILIBRIA

But individuals sample and learn.

Mottley & Giraldeau, 2000

Information use

Personal information

Social informationSocial

Foraging in space...

Video of zebra finches in the lab?

some words onmethodology

Individual-based models

Cellular automata

Genetic algorithms

selection

reproduction

termination

initialization

IntroductionInformation use

Foraging in spaceConclusion

Personal information use:

Learning rules

Learning rule: “A learning rule is defined as a rule which assigns for every possible behaviour the probability of displaying that behaviour at each trial of a game as a function of previous payoffs.” (Harley, 1981)

where 0 < x < 1 is a memory factor,

ri > 0 is the residual value associated with alternative i,

Pi(t) is the payo� to alternative i at time t, and

Si(t) is the value that the animal places on the behavioural alternative i at

time t.

Si(t) = xSi(t� 1) + (1� x)ri + Pi(t)

Relative payoff sum

where 0 < x < 1 is a memory factor,

ri > 0 is the residual value associated with alternative i,

Pi(t) is the payo� to alternative i at time t, and

Si(t) is the value that the animal places on the behavioural alternative i at

time t.

Si(t) = xSi(t� 1) + (1� x)ri + Pi(t)

Relative payoff sum

where 0 < x < 1 is a memory factor,

ri > 0 is the residual value associated with alternative i,

Pi(t) is the payo� to alternative i at time t, and

Si(t) is the value that the animal places on the behavioural alternative i at

time t.

Si(t) = xSi(t� 1) + (1� x)ri + Pi(t)

Relative payoff sum

where 0 < x < 1 is a memory factor,

ri > 0 is the residual value associated with alternative i,

Pi(t) is the payo� to alternative i at time t, and

Si(t) is the value that the animal places on the behavioural alternative i at

time t.

Si(t) = xSi(t� 1) + (1� x)ri + Pi(t)

Relative payoff sum

where 0 < x < 1 is a memory factor,

ri > 0 is the residual value associated with alternative i,

Pi(t) is the payo� to alternative i at time t, and

Si(t) is the value that the animal places on the behavioural alternative i at

time t.

Si(t) = xSi(t� 1) + (1� x)ri + Pi(t)

Relative payoff sum

Relative payoff sum

Linear operator

Perfect memory

Hamblin & Giraldeau, 2009

selection

reproduction

termination

initialization

+

Time

Popu

latio

n The RPS rule dominates LOP and PM.

01

23

45

Group size

Para

met

er v

alue

s

●

●

● ●

●

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10 40 90 160 360 1000

Producer residual

Scrounger residual

Memory factor

Why are the producer residuals so high?

Social learning

Social learning heuristic:A “rule of thumb” in which individuals!observe their neighbours and adopt the!strategy which led to the highest payoff!in their neighbourhood.

possible updating rules, including stochastic rules that allow for a more realistic

(but no longer replicable) updating, and which may have an effect on the results

(e.g. Moyano and Sánchez 2009), but we do not deal with these here.

1.500.75

2.00

1.50

2.00 2.00

1.50 2.25

1.50

S S S S

S

S

S

SSSSS

S

S

S

S

SS

P

S

P

S S

S S

1.500.75

2.75

1.50

2.75 2.75

1.50

1.50

S S S S

S

S

S

SSSSS

S

S

S

S

SS

P

S

P

S

S

2.75

P

P

Figure 5.2: Updating a cell. As before, red is scrounger and blue is producer. Cellscalculate their payoffs against each of their neighbours, in this case using GE (1) andan� of 0.75. Here, we look at the cell in the centre of the grid section pictured. On theleft hand side, the scrounger in the bottom-right of the focal cell’s neighbourhoodhas a higher payoff than the focal cell, so the focal cell will become a scrounger inthe next time step (though this is not depicted, so will the other two producers). Onthe right hand side, the addition of one more producer in the neighbourhood drivesthe focal cell’s fitness high enough that it will no longer change to scrounger in thenext time step.

112

Producer

ScroungerFormer producer

Former scrounger

Producer

ScroungerFormer producer

Former scrounger

020

40

60

80

100

1-5

6-10

11-15

16-20

Figure 5.7: A chaotic outcome. Shown is the first twenty steps of a run that did notachieve an fixed outcome after 10000 time steps, with the population proportion ofscrounger and producer over the twenty steps graphed below. Red is a scrounger,blue is a producer, green is a scrounger that was a producer in the previous timestep, and yellow is a producer that was a scrounger in the previous time step; thegraph lines include cells that switched, such that the red scrounger line is the totalof the red and green cells in that step, just as the blue producer line includes the blueand yellow cells.

122

Coevolving information: !Predator-prey dynamics

Predator search efficiency

Dispersed prey favours producers...

... while clumped prey induces increased scrounging.

Thus: will prey evolve to manipulate !predator information use?

selection

reproduction

termination

initialization

selection

reproduction

termination

initialization

+ +

Prey: !clumpiness

Predators:!NI!SI!

●

●

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●

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●0.

00.

20.

40.

60.

8Fr

eque

ncy

of sc

roun

ger A

●

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1 5 10 15 20 25 30 35 40 45 50

300

500

700

Prey clump size

Prey

surv

ival

B

~60-65%

Roughly !constant

No social !information

Social !information

010

2030

4050 A

0.0

0.2

0.4

0.6

0.8

1.0

010

2030

4050

Prey

clu

mp

size

B

0.0

0.2

0.4

0.6

0.8

1.0

Freq

uenc

y of

scro

unge

r

0 100 200 300 400 500

010

2030

4050

Generation

C0.

00.

20.

40.

60.

81.

0solid line: prey clump size!dashed line: predator scrounging

Prey clump in a way that induces maximum!social information use among predators.

●

●

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250

300

350

400

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Prey

surv

ival

100 200 300 400 500 100 200 300 400 500 100 200 300 400 500

No social information Social information Public information

IntroductionInformation use

Foraging in spaceConclusion

Where is as important as what.

Foraging with personality.

Boldness

Bold

Shy

Bold

Shy

~ producer

~ scrounger

Can we get such a dimorphism to evolve?

selection

reproduction

termination

initialization

+

No ... not within a population.

Genome by Patch Richness and Density, Population Size 50

Boldness

Scrounging

0.2

0.4

0.6

0.8

0.2 0.4 0.6 0.8

: Patch Richness { 5 } : Patch Richness { 10 }

0.2 0.4 0.6 0.8

: Patch Richness { 20 } : Patch Richness { 30 }

: Patch Richness { 40 } : Patch Richness { 50 } : Patch Richness { 60 }

0.2

0.4

0.6

0.8

: Patch Richness { 70 }

0.2

0.4

0.6

0.8

: Patch Richness { 80 }

0.2 0.4 0.6 0.8

: Patch Richness { 90 } : Patch Richness { 100 } Patch Density5102030405060

Patch density ▸ Boldness Patch richness ▸ Scrounging

Shy scroungers

Bold producers

Genome by Patch Richness and Density, Population Size 50

Boldness

Scrounging

0.2

0.4

0.6

0.8

0.2 0.4 0.6 0.8

: Patch Richness { 5 } : Patch Richness { 10 }

0.2 0.4 0.6 0.8

: Patch Richness { 20 } : Patch Richness { 30 }

: Patch Richness { 40 } : Patch Richness { 50 } : Patch Richness { 60 }

0.2

0.4

0.6

0.8

: Patch Richness { 70 }

0.2

0.4

0.6

0.8

: Patch Richness { 80 }

0.2 0.4 0.6 0.8

: Patch Richness { 90 } : Patch Richness { 100 } Patch Density5102030405060

Patch density ▸ Space Patch richness ▸ Information

Shy scroungers

Bold producers

Conclusion

IntroductionInformation use

Foraging in space

selection

reproduction

termination

initialization

Down the road...

Landscapes project

landscape had a mean of exactly six neighbours, though therewas variation about this value within individual landscapeinstances. The CGD virtual landscapes all resembled pixe-lated versions of the Dirichlet landscape. However, both thevisual and mathematical approximation improved as theresolution of the underlying raster was increased, asdemonstrated by both the mean and standard deviation ofthe number of neighbours (Table 1). Cells in the aggregatemap had approximately six neighbours, and were a range ofshapes because the sequential building rules meant thatgrowing cells were often geometrically constrained byneighbours. Of the geometries tested in this study, the meannumber of neighbours of a cell was six, or its approximation,with the exception of the rasters. There was variation in thedistribution of cell sizes within the irregular virtual land-scapes (Table 1).We measured and compared all possible unique cell-to-cell

step lengths (measured between centre-of-mass centroids) infive landscapes: the three regular landscapes, and singleinstances of the Dirichlet and the CGD4 landscapes (Figure 3).In the von Neumann and hexagonal landscapes, only one steplength was ever possible, with lengths 1 km and 1.074 km,respectively. In the Moore landscape, two steps were equallyprobable, with lengths 1 km and 1.41 km producing a meanstep of 1.21 km per landscape. Step lengths in the Dirichletlandscape were gamma distributed (Figure 3) with a mean of1.095 km, which is close to that found in the hexagonallandscape; the step lengths of each cell in the CDG4landscape were similarly distributed with a mean of 1.18km, though the distribution was less smooth as a result of thefinite distribution of cell shapes and hence step lengths(Figure 3).

Moving across Model LandscapesAccessibility. We used three methods to investigate move-

ment (of individuals or information) across our virtuallandscapes; these were accessibility, random movement, anddirected random movement. Accessibility (sensu [43]) meas-ured the shortest possible sequence of cell-to-cell stepsbetween two points in the virtual landscape. We implementedthis as the maximum geographical distance accessible from acommon origin in a fixed number of steps (Figure 2). Therewere striking differences between the accessibility of theregular virtual landscapes and those with an irregularstructure (Figure 2). The mean minimum steps required toaccess a fixed distance (effectively the inverse of Figure 2)varied considerably between the regular models (to travel 100km took a mean of 125.9, 90.7, and 102.6 steps for the vonNeumann, Moore, and hexagonal virtual landscapes, respec-tively) and were large compared to the mean minimum stepsrequired in the irregular landscapes (approximately 73 stepsin all five irregular landscapes).There was considerable directional bias shown in the

accessibility of the three regular virtual landscapes. Themaximum distance accessible in a fixed number of steps inthe von Neumann, Moore, and hexagonal landscapes pro-duced a distinctive shape dependent on their neighbourhoodrules: a diamond, a square, and a hexagon, respectively. Incontrast, accessibility in the irregular virtual landscapes wasalways circular. The angular variation in maximum distanceaccessible is demonstrated numerically by the standarddeviation of the minimum number of steps required to travel

Figure 1. Example Instances of Eight Virtual Landcapes

Example virtual landscape geometries (7 km 3 7 km section). (A) vonNeumann and (B) Moore neighbourhoods in a raster grid; (C) hexagonal;(D) Dirichlet tessellation; CGD tessellation with a mean of (E) four, (F)nine, and (G) 16 raster cells per km2; (H) land cover aggregate map. Theneighbourhood (grey) of a focal cell (black) is highlighted in each virtuallandscape.doi:10.1371/journal.pcbi.0030200.g001

PLoS Computational Biology | www.ploscompbiol.org October 2007 | Volume 3 | Issue 10 | e2001981

Geometry of Virtual Landscapes

Genetic programming.

Learning in butterflies

Emilie Snell-Rood!University of Minnesota

Thanks to this guy, and his lab...

... and these groups ...

... and a big thank you !to Brandy, for

!

(among many other things) !!

visual support

Incompatibility

Patch discovery

Scroungers converge