APS 209 Animal Behaviour

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APS 209 Animal Behaviour
Francis L. W. Ratnieks Laboratory of Apiculture
Social Insects
Department of Animal Plant Sciences University
of Sheffield
Lectures 6 7 Mathematical Theoretical
Insights Into Animal Behaviour
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Aims Objectives
Aims 1. To introduce the use of simple
mathematics in the study of animal behaviour. 2.
To show how to combine mathematics with
biology. Objectives 1. Understand the use of
figures and equations to express mathematical
ideas in biology. 2. Understand how to combine
mathematics with biology. 3. Lose fear of
mathematical approach in biology by being able to
follow some simple models.
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Hamiltons Rule Inclusive Fitness
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W. D. Hamilton
Hamilton, W. D. 1964. The genetical evolution of
social behaviour. Journal of Theoretical Biology
7 1-52. Inclusive Fitness Theory The social
behaviour of a species evolves in such a way that
in each distinct behaviour-evoking situation the
individual will seem to value his neighbours
fitness against his own according to the
coefficients of relationship appropriate to that
situation.
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Hamiltons Rule
Actor
Recipient
Hamiltons Rule c lt br (if true, then social
action is favoured) c cost to actor of social
action b benefit to recipient of social
action r genetic relatedness between actor and
recipient A social action is favoured by natural
selection if Hamiltons rule is true for the
appropriate values of c, b, and r. Note that the
cost and benefit can be negative (for example, if
actor eats recipient). Hamiltons rule is still
correct when this occurs. Some social actions
and their cost and benefit Give food to
recipient c is positive b is positive Defend
recipient c is positive b is positive Cannibalis
e recipient c is negative b is negative Steal
food from recipient c is negative b is
negative Swim past recipient c is zero b is zero
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Hamiltons Rule Cannibalism
Actor
Recipient
c cost to actor of social action b benefit to
recipient of social action r genetic relatedness
between actor and recipient Cannibalise
recipient c is negative b is negative -1
(meal) -10 (die) (In the above, the extra meal
for the actor is probably worth less than the
life of the recipient.) Hamiltons Rule c lt
br (if true, then social action is
favoured) -1 lt -10r More likely to favoured by
low relatedness. For example, if r 0.5
(siblings) then the above is not true as -1 gt -5.
(note -5 is less than -1).
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Hamiltons Rule Helping
Actor
Recipient
food
c cost to actor of social action b benefit to
recipient of social action r genetic relatedness
between actor and recipient Give food to
recipient c is positive b is positive 1
(less food) 3 (more food) (In the above, the
food may be worth more to the recipient than to
the actor. For example, an actor may less hungry
than a recipient.) Hamiltons Rule c lt br (if
true, then social action is favoured) 1 lt
3r More likely to favoured by high relatedness.
For example, if r 0.5 (siblings) then the above
is true as 1 lt 1.5
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Optimization Modelling Meets Inclusive Fitness
Theory
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Optimal Food Intake of Chicks
Survival of Chick
Food Eaten by Chick
The survival of a chick increases if it is given
more food by its parents. We dont know exactly
what this survival function looks like but it
almost certainly has an S-shape for birds with
altrichal young. Unless the chick receives a
certain amount of food it dies. Its survival then
increases to a maximum value beyond which more
food has no effect.
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Effect of Food on Chick Survival
Zero returns
Increasing additional returns
Decreasing additional returns
Zero additional returns
Survival of Chick
Food Eaten by Chick
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Optimal Feeding by Parents
Zero returns
Increasing additional returns
Decreasing additional returns
Zero additional returns
Survival of Chick
Food available Given to chick
Food Eaten by Chick
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Optimal Feeding by Parents Two Chicks
Zero returns
Increasing additional returns
Decreasing additional returns
Zero additional returns
Survival of Chick
Food available Given to chick
Food Eaten by Chick
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Optimal Feeding by Parents Two Chicks
Zero returns
Increasing additional returns
Decreasing additional returns
Zero additional returns
Survival of Chick
Food available Given to chick
Share between 2 chicks
Food Eaten by Chick
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Effect of Different Food/Survival Curves
Survival of Chick
Food Eaten by Chick
Left. Chick survival increases at the same rate
indefinitely with food. What is the optimal
feeding regime of the parents if there are two
(or more) chicks? It does not matter which chick
gets the food as an increment of food increases
the number of surviving chicks equally whichever
chick is fed. Right. The curve only has
decreasing returns. However much food the parents
have they will maximise the number of surviving
offspring by dividing the food equally among
them.
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Optimal Feeding Offspring Choice
Survival of Chick
Food Eaten by Chick
Total food available Chick As optimum Parents
optimum
to B
to A
A and B receive equal shares
Now consider the same situation but from the
perspective of one of the chicks. How should the
this focal chick want the food to be divided?
The focal chicks optimum is when the marginal
gains give equal inclusive fitness returns. For
example, if the chicks are related by 0.5 the
optimum occurs when the marginal gain (slope of
tangent) of the focal individual is half that of
the other chick. When this occurs, an increment
of food is as valuable to the focal chick whether
given to itself or its sib.
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Selfish Herd
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The Selfish Herd
A grass snake predator is about to prey on one of
several frogs sitting at the edge of a pond.
Where is the best place for a frog to sit to
minimise its risk? Assuming that the snake will
move with equal probability in all directions,
then the best place is next to another frog.
Then, if the snake attacks in your direction, you
are not the only target. This is a selfish
strategy. It benefits individuals but not the
group. By bunching prey, selfish herding may even
increase the total predation risk by making a
more tempting target. If you can understand the
above logic thencongratulationsyou have
understood an application of game theory to
animal behaviour. Hamilton WD 1971 J Theor Biol
31 295-311
Predation selects for bunching upselfish
herding
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The Selfish Herd
Is selfish herding good for the group? If the
bunched prey are a more tempting target it will
definitely not be good for the group. That is, it
will increase predation overall. However,
individuals who group will have lower predation
that those that do not. Everyone is worse off
than if they did not group. What would be the
likely fate of a rare mutant frog behaviour
which tried to reduce total predation risk on all
frogs by leaving the group and sitting on its
own? Would this spread out behaviour be
selected for or against?
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Game Theory
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Game Theory 1 Sex Ratio as a Game
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Sex Ratio as a Game Between Mothers
Determine Number Grandoffspring per Son or
Daughter Assume Sex ratio even 1F1M How many
mates does an average male get? What if each
female mates with two males?
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Sex Ratio as a Game Between Mothers
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per son
6
Number of grand- offspring
4
per daughter
2
0
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12
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Population sex ratio (FemalesMales)
When the population has more females than males
(left side), then a son will provide more
grandoffspring than a daughter (i.e., solid line
is above dashed line). When the population has
more males than females (right side), then a
daughter will provide more grandoffspring than a
son. The lines cross at an even sex ratio,
meaning that at an even sex ratio sons and
daughters are equally good at transmitting your
genes. The even sex ratio is an ESS.
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Game Theory 2 Pairwise Contests Hawk-Dove Game
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Game Theory 2
Pairwise Contests The Hawk-Dove Game The
Hawk-Dove-Bourgeois Game
John Maynard-Smith
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Pairwise Contest Hawk-Dove Game
Animals contest resources in pairs Strategies
considered Dove Prepared to share Never
fights Retreats if opponent fights Hawk Never
shares Always fights Payoffs Value of
resource v Cost of fighting to
loser c PAYOFF MATRIX Focal Individual
Opponent Hawk Dove Hawk v/2 -
c/2 v Dove 0 v/2
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Invasion of Rare Strategies
Focal Individual Opponent Hawk
Dove Hawk (v - c)/2 v Dove 0 v/2 DOVE
COMMON HAWK RARE. CAN HAWK INVADE? Hawk Dov
e Hawk v Dove v/2 Hawk invades if v
gt v/2 (i.e. always) HAWK COMMON DOVE RARE. CAN
DOVE INVADE? Hawk Dove Hawk (v -
c)/2 Dove 0 Dove invades if 0 gt (v - c)/2
(i.e. if c gt v)
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Invasion of Rare Strategies
1. Hawk can invade a population of Doves if v gt
v/2
All H
All D
2. Dove can invade a population of Hawks if c gt v
All H
All D
If 1 2 are both true then we get invasion from
both ends a mixed ESS
All H
All D
If only 1 is true then we get a pure ESS of Hawks
All H
All D
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Invasion of Rare Strategies
1. Hawk can invade a population of Dove if v gt v/2
All H
All D
You can think of the above line in the following
way.
1
Hawk Dove 1
Dove
0
0
1
Hawk
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Game Theory 2 Pairwise Contests Hawk-Dove-Bourgeo
is Game
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Hawk-Dove-Bourgeois Game
Animals contest resources in pairs STRATEGIES
CONSIDERED Dove Will share, never fights,
retreats if opponent fights Hawk Never shares,
always fights Bourgeois Plays Hawk when resident
and Dove when intruder PAYOFFS Value of resource
v Cost of fighting to loser c PAYOFF
MATRIX Focal Individual
Opponent Hawk Dove Bourgeois Hawk (v-c)/2
v (3/4)v - c/4 Dove 0 v/2 v/4 Bourgeois
(v-c)/4 (3/4)v v/2
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Is Bourgeois an ESS?
PAYOFF MATRIX Focal Individual
Opponent Hawk Dove Bourgeois Hawk (v-c)/2
v (3/4)v - c/4 Dove 0 v/2 v/4 Bourgeois
(v-c)/4 (3/4)v v/2 The notes show how the
payoffs for Bourgeois are calculated. Bourgeois
is the ESS if c gt v. See this from the final
column as follows. A rare Dove playing
Bourgeois always does worse that a Bourgeois
playing Bourgeois ( as v/4 lt v/2). A rare Hawk
playing Bourgeois gets (3/4)v - c/4 v/2
(v-c)/4. This is less than v/2 if c gt v.
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Game Theory 2 Bourgeois Butterflies Spiders
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Speckled Wood Butterfly
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Communal Spider Parawixia bistriata
Resident wins without fighting Non-resident
leaves Conventional Settlement to
Dispute Bourgeois if the ESS Evolutionry Stable
Strategy
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Game Theory 2 Pairwise Contests Rock-Scissors-Pap
er Game
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Rock-Scissors-Paper Game
Rock beats Scissors Paper beats Rock Scissors
beats Paper No single-strategy ESS is
possible Possible results 1. 1/3 of each
strategy 2. Cyclical dynamics
All Rock
All Rock
All Paper
All Scissors
All Paper
All Scissors
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Rock-Scissors-Paper Game
With three strategies, the proportions must add
up to one. As a result, you can plot the three
onto two dimensions. (In fact, there are only two
degrees of freedom. If Rock 0.2, and Paper
0.3, then Scissors must equal 0.5.) The 2
dimensional surface in dimensions x, y, x is such
that x y z 1. At the three apexes all
animals play one strategy only. The perimeter
line connecting two strategies is the case where
the population has animals playing only two
strategies. Locations on the surface away from
the edges represent populations where all three
strategies are played. The centre is where there
are equal proportions of all three strategies.
0
This way of representation is known as a De
Finetti diagram.
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Rock-Scissors-Paper Game Cyclical Dynamics
All Rock
All Rock
a)
b)
All Paper
All Scissors
All Paper
All Scissors
When will 1/3 each of Rock, Paper, and Scissors
not be an ESS? Imagine that winning is worth 2
units of fitness, losing 0 units, and drawing 1
unit. In this situation 1/3 of each is an ESS.
But if drawing is worth a little more than 1,
then 1/3 of each is not the ESS. For example, if
there are slightly more than 1/3 Rock, then Rock
will play Rock more often than Paper plays Paper,
or Scissors plays Scissors. Thus Rock will do
better than Scissors or Paper and increase still
further (a). However, if Rock is more common than
1/3, then Paper will benefit as Paper beats Rock
(a). So Paper becomes more frequent (b). But
Scissors beats Paper. So the things begin to
cycle (b). How far out from 1/3 each the cycles
are depends on how much greater the payoff of
playing against a partner of your own strategy is
then the average of winning and losing.
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Game Theory 2 Pairwise Contests Male Lizards
Play the Rock-Scissors-Paper Game
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Lizards Play Rock-Scissors-Paper
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Lizards Play Rock-Scissors-Paper
Males have three different mating strategies 1.
Large territory. Aggressive large territory with
several females (orange throat) 2. Sneakers.
Enter orange territories (yellow striped
throat) 3. Defenders. Detect sneakers defend
small territory with one female (blue throat)
Large Territory Holders
Sneakers Female Mimics
Defenders
Sinervo, B. Lively, C. M. 1996. The
rock-scissors-paper game and the evolution of
alternative male strategies. Nature 380 240-243.
http//bio.research.ucsc.edu/barrylab/
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Summary of Behaviours
Yellow-throated Sneaker Male Behavior Yellow
males are "sneakers" in that they mimic the
throat color of receptive females, and yellows
also mimic female behavior. When a yellow male
meets a dominant male, he pretends he is a female
-- a female that is not interested in the act.
The head bobs ("vibrations") involved in actual
female rejection behaviors are of very high
frequency compared to the low frequency
aggressive challenge displays shown by aggressive
males. In many cases, females will nip at the
male and drive him off. By co-opting the female
rejection display, yellow males use a dishonest
signal to fool some territory holding males. The
ruse of yellows works only on orange-throated
males. Blue-Throated Male Behavior Blues are not
fooled by yellows. Blue males root out yellow
males that enter the territory of the blue male.
Blue males are a little more circumspect when
they engage another blue male during territory
contests. Blue males spend a lot of time
challenging and displaying, presumably allowing
males to assess one another. Attack may or may
not follow as blue males very often back down
against other blue males. Indeed, neighboring
males use a series of bobs to communicate their
identity, and the neighbors usually part without
battle.
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Summary of Behaviours
Orange-Throated Male Behavior Orange males are
ultradominant and very aggressive owing to high
levels of testosterone, and attack other males
intruding blue males that typically have more
modest levels of testosterone. Attacks by orange
males on blue males do not involve the ritualized
head bobs that are seen when blue males engage
blue males -- orange males just attack with
little advanced warning. When an orange male
encounters a more equally matched orange male,
they are both a little more circumspect, and they
will not necessarily attack one another (as is
seen when blue meets blue). Summary Thus, each
strategy has a strength and a weakness and there
are strong asymmetries in contests between
morphs. Trespassing yellows, with their female
mimicry, can fool oranges. However, trespassing
yellows are hunted down by blue males and
attacked. While oranges with their high
testosterone and high stamina can handily defeat
blues, they are susceptible to the charms of
yellows. In contrast, contests between like
morphs (e.g., blue vs blue, orange vs orange or
yellow vs yellow) are usually more
symmetric. http//bio.research.ucsc.edu/barrylab
/classes/animal_behavior/video_only.html
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Yellow v Orange
Intruding Yellow (sneaker/female mimic) vs Orange
(large territory defender) "There's no action.
The most intriguing display that males make,
which is restricted to yellow-throated males, is
an imitation of the female rejection display.
This rejection display is characteristic of
post-receptive females and consists of a series
of rapid head vibrations which are so fast that I
used 1/2 times Slow Motion to capture it for you.
The male then extends his yellow throat, assumes
a humped back, and comes in and nips the dominant
male on the tail. The parallel between the yellow
male pictured here and an actual rejection by a
bonafide female below is extraordinary. Note how
the rapid buzz of the head is very distinct from
the usual male head bob which is displayed by the
male who was bitten. Near the end of the video,
the "rejected" male gives the usual pushup
pattern consisting of a small up, down, up, down,
followed by a large UP, DOWN that is typical for
males.
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Yellow (Female Mimic) v Orange
Y (female mimic)
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Blue (Defender) v Yellow (Female Mimic)
Vigilant Blue (defender) detects a Yellow
"Rooting a sneaker."When a blue spots a yellow,
it usually involves bobs by the blue and no
signalling by the yellow. Somehow the blue
recognizes the yellow and will attack if the
yellow is on the blue male's territory. In this
case, the blue roots out the sneaker, which is
hidden deep in the grass. There is a skirmish
(with sloMo replay at 1/4 time), the blue bobs
victoriously, the yellow slinks into a crack and
lives to sneak another day. Though the Yellow
sneakers are continually being run off by Blues,
the Yellows are also quite wary and always on the
look out for the dominant Blue. No doubt because
the dominant Blue can beat the snot out of them.
Intruding Yellow vs Blue "I'm Outa Here
Yellows are always watchful for the patrolling
blues. Here the Yellow sneaker spots the Blue and
slips into the rocks. The Blue, comes up to the
rocks, and sits there for about 1 minute (edited
out of the clip), waiting for the intruder to
show his face. When there is no action, the Blue
slips in after the Yellow, but comes up empty as
the Yellow jammed out the back door entrance.
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Blue (Defender) v Yellow (Female Mimic)
Y (female mimic)
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Orange (Aggressive) v Blue (Defender)
Intruding Orange vs Blue "I Am So BAAAD Really,
I Am"When you put an Orange on a Blue Males Turf,
the Blue is intimidated by just the color. Orange
is a "badge of status" the signifies to the blue
that the lizard is pretty tough in battle. Thus,
Blues take a long time to work up the "courage"
to come in a pick a fight. This male bobbed for
nearly 10 minutes before finally coming in to
pick a fight. The slow action of Blues is in
stark contrast to the rapid fire attacks of
oranges against blues or oranges against
orange. Intruding Blue vs Orange "Mutant Ninja
Lizards."Slow Motion Movie (1/4 time). The orange
male launches an attack against an "intruding
blue male". Oranges are very quick to attack and
in this case, a challenge display did not precede
the attack, a single bob was the only warning
that the hapless blue received. This is typical
behavior for oranges which attack intruding blue
males with little or no provocation. Such
preemptive attacks are usually won by the orange
owing their higher stamina. Note the back flip
and beautiful 4 point landing by the orange after
the first clash!
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Orange (Aggressive) v Blue (Defender)
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Game Theory 2 Bourgeois Brazilian Spiders
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Bourgeois Brazilian Spiders
Typical behaviours in Parawixia bistriata. a) The
spiders' daytime retreat. b) The leaving of the
retreat after sunset. c) Two spiders passing each
other while depositing silk on the main support
lines. d) A spider that has just completed her
web. e) A webless spider is being bounced at by a
web owner. f) A prey is shared between a web
owner, a spider from a neighbouring web and a
webless spider that was sitting at the edge of
the web.
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Communal webs of the orb-weaving spider Parawixia
bistriata have up to 1000 spiders. Webs are from
5-20m across and 2-5m high. The spiders build
communal main lines between trees and bushes but
each builds its own orb web. Webs are built new
each evening but the main lines are permanent and
get thicker and thicker with use. The spiders
cluster together in a bivouac in a bush in
daytime. At sunset they walking away from the
bivouac. Most spiders build their own individual
orb web. When a spider starts building a web it
repels other walking spiders, who have not yet
started building, by bouncing. The intruder
leaves without a fight, showing a "bourgeois
strategy. That is, the intruder backs off. Web
building takes about 20 minutes, starting with
the outer frame, then the radii, then the spiral.
Spiders with neigbouring orb webs sometimes
interfere, resulting in prolonged bouncing.
Neither leaves. Both play the owner, but soon
they habituate to each others bouncing and
stop. In crowded colonies some spiders dont
build a web but monitor an existing web. These do
not depart when bounced". If a large prey item
falls into a web up to 6 spiders eat it, but
small prey is eaten by the resident. So
non-building satellites get a chance of some
food. The whole thing is self-organized pattern
formation (orb webs well distributed over the
whole main web area) with simple rules. Spiders
without webs are repelled by spiders with webs
and walk further down the main lines. Spiders
with webs stay put. There is no difference in
weight between spiders who have webs and those
who do not. Males and females do not differ in
behaviour.
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Bourgeois Brazilian Spiders
Leaving the bivouac at sunset.
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Habitat
Main web along bushes beside road and to
neigbouring sugar cane field.
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Habitat
Main web in bushes in foreground.
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Bivouac Main Lines
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Bivouac
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Bivouac Close Up
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Leaving Bivouac at Sunset
Walking along main lines before starting to build
own orb web.
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Bourgeois Brazilian Spiders
Map of one study colony (2004-1) which was in
bushes and plants along the side of a road
bordering a sugar cane field at Fazenda
Aretuzina. Position of the daytime retreat or
"bivouac" and the main sections of thick, silken
support lines are shown. Numbers are the height
above ground of the various attachment points.
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Another Bivouac
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Another Bivouac
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Leaving Bivouac
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Passing Spiders on Main Line No Aggression
When spiders are walking they are not aggressive
or territorial.
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Resident Building Spiral of Own Web
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Adjacent Spiders Building Moving
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Resident in Centre of Own Web
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Resident in Centre of Own Web
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Turning Toward Intruder
A resident turns toward vibrations made by an
intruder.
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Bouncing Intruder
Resident (below) bouncing intruder walking along
bordering main line.
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Neighbouring Residents
Two residents. Resident on left is bouncing
resident on right..
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Neighbours Sharing a Large Prey Item
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Discarding Inedible Insect (Beetle)
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Eating Silk
Recycling. Some spiders eat loose web on the main
lines.
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Another Species Eating Silk in Daytime
Orb-weaver spider of another species eats bits of
silk from a Parawixia web main lines in day time.
Is is a parasite or just opportunistic?
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Number of Bouncing Behaviours
Building own webs
Dispersal from Bivouac
Web building complete
The total frequency of bouncing behaviour by
residents to intruders during dispersal from
bivouac (0-5 mins), web building (5-25 mins) and
when web building had been completed (gt30 mins).
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APS 209 Animal Behaviour
Francis L. W. Ratnieks Laboratory of Apiculture
Social Insects
Department of Animal Plant Sciences University
of Sheffield
Advice on Writing Exam Essays
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Essays are Worthwhile
Worthwhile intellectual challenge Preparation for
Level 3 projects and dissertation Career in
science Writing is part of most careers Some
students find essays easier than others Everyone
can write good essays if they make the effort.
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Read the Questions
Read over the essay questions Which do you have
the necessary knowledge to answer? Dont panic if
they seem very difficult On second reading they
will not seem so hard
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Consider What You Know
Which do you have enough knowledge to answer?
For which do you have the best knowledge?
Knowledge may come from the course lectures,
handouts, book etc. But you may also have
knowledge from other sources It does not matter
where you obtained your knowledge All that
matters is that it is relevant.
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ADVICE ON WRITING EXAM ESSAYS
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Essays are Worthwhile
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Read the Questions
84
Consider What You Know
85
What Is Relevant?
86
Formulate a Plan
87
Answer the Question
88
Its Not Just a List
89
Accuracy
90
Ending
91
Other Points
92
Practice
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Criteria For 1st Class