APS209 Animal Behaviour
Mathematical and theoretical insights
in animal behaviour
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 a mathematical approach in biology by being
able to follow some simple models.
This lecture on mathematical and theoretical insights will
introduce you to the use of maths in biology in the context of important
questions in animal behaviour. No mathematical background beyond GCSE is
necessary. The material will cover two topics. 1. Inclusive fitness theory; 2.
Game theory.
To maximize the number of copies of genes in the next
generation, an individual can:
1) Produce as many
offspring itself
2) Help to raise
siblings/ offspring of siblings
1. Inclusive fitness theory
The slides provide the information to understand this section.
We first consider Hamilton’s rule (Inclusive Fitness Theory) in the context of
cannibalism by tiger salamanders, following up on the research by David Pfennig
(see Alcock pp 95-96) showing that salamanders are less likely to be
cannibalistic when in a tank with siblings versus unrelated individuals. We can
consider cannibalism to be a social action (i.e., directed at members of the
same species in your environment) and determine how variation in relatedness
between individuals, and the cost and benefit of the action to actor and
recipient influence whether natural selection will favour carrying out that
action or not.
Hamilton’s
Rule states that a social action is favoured by natural selection if c <
b.r, where c is the cost to the actor (reduced fitness) and b the benefit to
the recipient (increased fitness), and r is their genetic relatedness. For
example, giving food to another animal will have a survival cost to the actor
and so will only be favoured by natural selection if the receiving individual is
kin (r positive). Note however, that costs and benefits can be negative.
Stealing food is also a social action. Who should actors steal from? In this
case the “cost” is negative, and the “benefit” is also negative. Hamilton’s
rule is more likely to be satisfied (i.e., stealing is favoured by natural
selection) if actors steal from non-kin (r = 0).
2. Game Theory: Introduction and sex
ratio
Human sports come in two main kinds. In
one type, it is you against the clock or some physical challenge. For example,
climbing a mountain or running 100m in the shortest possible time. In the other
type, you are playing directly against an opponent. For example, in tennis the
best place to put the ball is where your opponent isn’t. In tennis you have to
adjust your game according to the opponent’s behaviour. Similar things happen
with animals. Many of the challenges facing animals do not depend upon what
other animals are doing: a bird migrating alone across the sea, for example.
But many of the challenges faced by animals are more like a game of tennis. The
best place for a male to find a female may be where there are few other
competing males. The best place to find food may be where there are few other
foragers. Thus, the fitness benefits of behaviours often depend upon what
others are doing.
Game theory was originally developed in the 1940s by John
von Neumann as a tool to understand economic and military (“war games”)
behaviour. In the 1970’s John Maynard-Smith and George Price started to apply
game theory to animal behaviour. At more of less the same time other
biologists, particularly William Hamilton, also used game theory ideas to
investigate optimal sex ratios and group formation. Price died soon after, but
Maynard-Smith went on to develop the theory within a biological context. For
those of you who are interested in learning more, his book (Maynard Smith, J.
1982. Evolution and the theory of games.
Cambridge University Press) is well worth looking at, or see the references to
some of his shorter articles (Maynard Smith 1998, 2002).
On receiving the Kyoto Prize for evolutionary biology in
2001, Maynard Smith commented “I have
devoted most of my life to trying to apply mathematical reasoning, for example,
Game theory, to biology. Trivial as it may seem, mathematics is of great use in
biological studies. This is because verbal models can be interpreted in
different ways, depending on who the reader is. Any theory to explain the
complicated activities of organisms must always be simple. I felt the greatest
happiness in finding that my theory, which I had elaborated after the most
tenacious thinking, proved to be correct as some animal turned out to be doing
something odd that the theory predicted.” [www.inamori-f.or.jp]. Using game
theory in biology can be very mathematical. But it can also have zero or very
little mathematics. In these lectures on game theory we will stick to the
latter. As Maynard Smith (2002) notes, any theory investigating something as
complex as animal behaviour must be simple.
Sex
ratio
It
is perhaps surprising to think of a sex ratio as a behavioural strategy. But
consider a female who can choose the sex of her offspring. Should she choose
males or females or both? If most parents are producing daughters then it is
best to produce sons, because on average a son will produce more sets of
grand-offspring for his mother. If most parents are producing sons, then it is
better to have daughters. But when half of the offspring in the population are
sons and half are daughters, then offspring of either sex give a mother the
same number of grand-offspring. In other words an equal sex ratio is an ESS
(evolutionary stable strategy). It cannot be “invaded” by a better strategy. An
equal sex ratio is a “Nash equilibrium” (after the Princeton scientist who devised
it in his PhD research and for which he was awarded the Nobel prize for
Economics many years later. In a Nash equilibrium, the optimum strategy is its
own best reply.) That is, the best reply in a population where every female is
producing equal numbers of sons and daughters is to do the same yourself.
A little simple maths will make
things clear (see also the slides). Assume that every female has n surviving
offspring. Each daughter, therefore, results in n grand-offspring. How many
grand-offspring will a son provide? If there are equal numbers of males and
females in the population, then each son, on average, will have one mate. So
each son will also provide n grand-offspring. But imagine that there are twice
as many females as males in the population. Each son will now mate with two
females, on average. So each son provides 2n grand-offspring. This shows that
if the sex ratio in the population is female biased, then a female having sons
will have greater fitness (grand-offspring) that a female having daughters.
Likewise, if the sex ratio is male biased, a female having daughters will have
higher fitness than a female having sons. In this way, selection tends to cause
an even sex ratio.
Many animals, including humans, have
a genetic mechanism for determining sex (e.g., XY chromosomes) that can easily
give an even sex ratio. This is the proximate cause of an even sex ratio but it
is not the ultimate cause. Animals with chromosomal sex determination can
adjust offspring sex ratio. Seychelles warblers have chromosomal sex
determination (WZ; in birds the female is the heterogametic sex). Females
adjust the sex ratio of their brood according to territory quality. This shows
that, when there is an advantage for adjusting sex ratio, even animals with
chromosomal sex determination can do it. There are other forms of sex
determination some of which may make sex ratio adjustment a lot easier for the
mother. In Hymenoptera (sawflies, wasps, ants, bees) males are haploid and
females diploid. A female can control the sex of her offspring by choosing
whether or not to release sperm from the sperm storage organ as the egg is
laid. The sperm storage organ is connected to the oviduct via a duct controlled
by nerves and muscles. In honeybees
there is conflict between the mother queen and her daughter workers over the
sex ratio of young queens and males reared.
The workers may be able to cause a female-biased sex-allocation ratio by
selectively killing male larvae and the queen may be able to resist this by
laying few female eggs.
3. Game Theory: Hawk-Dove and
Hawk-Dove-Bourgeois games and cyclical dynamics.
In the sex ratio game, the
contestants (mothers choosing sex of offspring) were all playing each other,
which is known as playing the field,
as matings take place panmictically. However, many contests may take place
between paired opponents, (pairwise
contests). For example, if two individuals are contesting ownership of a
resource, which forms the next series of examples.
Animals often fight over resources such as food or females.
But not all do. Often, one animal quickly backs off. For example, in the
speckled wood butterfly, a resident male always wins an encounter with a
non-resident over the possession of a sun patch, a location where mating can
occur. Traditionally, it was considered that such non-aggressive contests were
for the good of the species. But game theory analyses have shown that
non-aggressive contests can be explained via their benefits to the individual.
Sometimes it is selfishly better to be less aggressive.
Consider a resource with a fitness value of v to whoever controls it. The resource
might be a feeding or mating location. We will consider that animals compete
for discrete resources in pairs and can have just two behavioural strategies:
Hawk and Dove.
|
Focal
individual
|
What happens in a pairwise contest over the resource
|
Opponent
|
Fitness
payoff to focal individual
|
|
Hawk
|
Always fights. Always wins if
opponent is Dove, and without paying the fighting cost. Wins half the time if
opponent is Hawk but pays fighting cost half the time if it loses.
|
Dove
Hawk
|
v
(v
- c)/2 on average
|
|
Dove
|
Never fights. Shares resource if
opponent is Dove, but always loses resource if opponent is Hawk. Never pays
the fighting cost.
|
Dove
Hawk
|
v/2
0
|
Hawk always wins when paired to a Dove. So it would seem
that the Dove strategy should be replaced by the Hawk strategy if both occur in
the population. But Hawk pays a fighting cost, c, when it pairs up with another Hawk. Dove does not pay this cost
because it never fights. Could this make Dove a winning strategy?
Consider a population of Doves with a lone Hawk. (We say
that the rare strategy is “invading”.) The Hawk can only pair with a Dove so
the rare Hawk’s payoff is always v.
The payoff of most Doves, who pair with other Doves, is v/2. This shows that Hawk can invade Dove because the rare Hawk has
a higher payoff than do Doves.
Now
consider a population of Hawks with a lone Dove. The Dove can only pair with a
Hawk so the rare Dove’s payoff is 0. The payoff of most Hawks is (v-c)/2. This is < 0 when c > v. So Dove can invade Hawk if c
> v. That is, when there is a high
fighting cost (greater than half the value of the resource being fought over).
In essence, the resource is not worth the fight because there can only be one winner
(hence the benefit is v/2 not v).
In
general, there are two conditions of cost and benefit that determine the best
strategy to use.
c >
v Dove
can invade Hawk. We already know that Hawk can invade Dove (because we assume
that v > 0, that is the resource has some positive value for survival or
reproduction), so this means that each strategy can invade a pure population of
the other. What we end up with is a “mixed ESS” (Evolutionarily Stable
Strategy) in which both strategies are present in the population. The exact
proportions depend on the values of c
and v. At the mixed ESS the fitness
of individuals playing each strategy will be equal. Hawks pass on as many
copies of their genes as Doves.
c <
v Dove
cannot invade Hawk. As we know that Hawk can invade Dove this means that Hawk
forms a “pure ESS”. The Dove strategy cannot persist. If it occurs in a
population, and a rare mutant Hawk occurs, then Hawk will replace it (referred
to as go to fixation).
The Hawk–Dove game is not meant to be a realistic model of
any real animal population. But it gives some important insights. It shows that
a non-fighting strategy can persist even when fighters occur, and that a
population can have animals playing two different strategies, each having equal
fitness. The two strategies do not have to be genetically determined, and in a
mixed ESS each individual could play both strategies. For example, if the mixed
ESS was 40% Doves, then this could come about by 40% always being Doves and 60%
Hawks, or via each individual playing both strategies at different times such
that, across all contests, 40% of the plays were Dove.
The payoffs to the different
strategies are often put into a payoff
matrix. In the matrix below (don’t be put off by the word matrix—it just
means a table), the payoffs relevant to a population with a rare Dove are in
column 1 (i.e., Dove can only pair with Hawk). The payoffs relevant to a
population with a rare Hawk are in column 2 (i.e., Hawk can only pair with
Dove). For example, cover up column 2 with your finger. This means that the
only possible opponent is a Hawk (i.e., everyone is a Hawk except for one Dove,
who cannot have itself as an opponent). The payoff to the rare Dove is 0 and to
the Hawks is (v-c)/2. This shows the
condition under which Dove can invade is 0 > v-c, which is the same as c
> v (i.e., cost is greater than
victory) Now cover up column 1. This means that the only possible opponent is a
Dove (i.e., everyone is a Dove except for one Hawk, who cannot have itself as
an opponent). The payoff to the rare Hawk is v and to the Doves is v/2.
As v > 0 (i.e., the resource is
worth something) this shows that the Hawk can invade a population of Doves.
Payoff Matrix for the Hawk-Dove Game. The payoff is the change in fitness of a focal individual
who is playing either Hawk or Dove when paired to an opponent who is playing
either Hawk or Dove.
Focal
Individual’s Strategy Opponent’s
Strategy
Hawk
(col. 1) Dove (col. 2)
Hawk (v-c)/2 v
Dove 0 v/2
Further strategies
Are
Hawk and Dove the only possible strategies? No, they are not. Maynard Smith
(2002) discusses some additional strategies, such as ‘Retaliator’ (plays Dove
initially but fights back if opponent fights), ‘Prober-Retaliator’
(occasionally fights but backs down if opponent fights), ‘Bully’ (always start
by fighting but back down if opponent fights) and ‘Bourgeois’ (see below).
Hawk-Dove-Bourgeois and the speckled wood butterfly (Alcock pages 278-279). Note: For the exam, you do not have to
understand the maths behind the H-D-B game but you do have to understand the
maths behind the H-D game.
Maynard-Smith considered various other
strategies and the one that seemed the most successful in many situations of
pairwise contests is Bourgeois, which is better than both Hawk and Dove and can
form a pure ESS. That is, everyone plays Bourgeois. Bourgeois also seems to be
quite common in nature. A good example of theory leading the way.
In
some animal contests, the strategy of an animal depends upon whether it is a resident
or not. Thus, male speckled wood butterflies can be either Hawks or Doves. They
play Hawk when they are the resident/owner of a sun patch, but play Dove if
they are the non-resident/intruder. Normally, when an intruder enters the patch
of another male butterfly, the resident flies up and chases him away. The
resident always wins after a short spiral flight. The intruder does not
contest. However, it is possible to have two residents. For example, by
experimentally moving one resident to the sun patch of another resident so that
both think that they are the resident. When they see each other, they both play
Hawk and this results in an “escalated” contest—a long spiral chase into the
tree tops. (The speckled wood is a common butterfly and it is easy to see these
behaviours.) So it seems that a butterfly plays Hawk when resident and Dove
when non-resident. This particular strategy has been called Bourgeois, in
reference to property owning people and their determination to hang on to their
house. It would perhaps have been more immediately obvious if the strategy name
Owner had been used instead of Bourgeois. How can we build upon our analysis of
the Hawk-Dove game to include Bourgeois? First, the basics…
|
Focal
individual
|
What happens in a pairwise contest over the resource
|
Opponent
|
Fitness
payoff to focal individual
|
|
Hawk
|
Always fights. Always wins if
opponent is Dove, and without paying the fighting cost. Wins half the time if
opponent is Hawk but pays fighting cost
|
Dove
Hawk
Bourgeois
|
v (as above)
(v-c)/2 (as above)
(3/4)v–c/4 (note 5)
|
|
Dove
|
Never fights. Shares resource if
opponent is Dove, but always loses resource if opponent is Hawk. Never pays
the fighting cost.
|
Dove
Hawk
Bourgeois
|
v//2 (as above)
0 (as above)
v/4 (note 4)
|
|
Bourgeois
|
Plays Hawk when resident and Dove
when non-resident
|
Dove
Hawk
Bourgeois
|
(3/4)v (note 3)
(v-c)/4 (note 2)
v/2 (note 1)
|
Note 1 Assume that when two Bourgeois meet,
one is the resident and one the non-resident so there is no fighting. On
average, individuals are resident half of the time so the average payoff is (v + 0)/2 = v/2.
Note 2 When Bourgeois is resident it plays
Hawk to an intruding Hawk and gets a payoff of (v-c)/2. When Bourgeois is non-resident it plays Dove when intruding
on a Hawk and gets a payoff of 0. On average individuals are resident half of
the time so the average payoff is ((v-c)/2
+ 0)/2 = (v-c)/4.
Note 3 When Bourgeois is resident it plays
Hawk to an intruding Dove and gets a payoff of v. When Bourgeois is non-resident it plays Dove when intruding on a
Dove and gets a payoff of v/2. On average individuals are resident half of the
time so the average payoff is (v + v/2)/2 = (3/4)v.
Note 4 When Dove plays Bourgeois it gets a payoff
of v/2 when it is resident and
Bourgeois is intruder, because both play Dove.
When Dove is non-resident it has a payoff of 0 when intruding on
Bourgeois, who plays Hawk. On average, Dove individuals are resident half of
the time so the average payoff is (0 + v/2)/2
= v/4.
Note 5 When Hawk plays Bourgeois it gets a
payoff of v when resident and
Bourgeois is intruder, because Bourgeois plays Dove. When Hawk is non-resident
it has a payoff of (v-c)/2 when
intruding on Bourgeois, who plays Hawk. On average, Bourgeois individuals are
resident half of the time so the average payoff is (v + (v-c)/2)/2 = (3/4)v – c/4.
Payoff Matrix for the
Hawk-Dove-Bourgeois Game. The Payoff is the
change in fitness of a focal individual who is playing either Hawk, Dove or
Bourgeois when paired to an opponent who is playing either Hawk, Dove or
Bourgeois.
Focal
Individual’s Strategy Opponent’s
Strategy
Hawk
(col 1) Dove (col 2) Bourgeois (col 3)
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
From
the matrix above we can determine whether the Bourgeois strategy is an ESS,
that is whether it gives a higher payoff than playing either Hawk or Dove. The
first question is, can Bourgeois invade a population of Doves? From Column 2 we
can see that the payoff to Bourgeois from playing against Dove, (3/4)v, is greater than the payoff to Dove
when playing Dove, v/2. Thus
Bourgeois can invade Dove. The second question is, can Bourgeois invade a
population of Hawks? From Column 1 we can see that the payoff to Bourgeois from
playing against Hawk is (v-c)/4
whereas the payoff to Hawk from playing against Hawk is (v-c)/2. The payoff to Bourgeois is greater than that to Hawk if v < c. Thus, Bourgeois can invade Hawk if v < c. Finally, if
everyone plays Bourgeois, can this be invaded by Hawk or Dove? Column 3 shows
that the payoff to Hawk, Dove and Bourgeois when playing Bourgeois are (3/4)v –
c/2, v/4, v/2, respectively. Clearly, Dove is always worse off than
Bourgeois, as v > 0, and Hawk is worse off than Bourgeois if v < c. The final condition is also the same as the condition that
allowed Bourgeois to invade Hawk. In summary, Bourgeois can always invade Dove
and is resistant to invasion by Dove, and can invade Hawk and is resistant to
invasion by Hawk if v < c.
This “Resident wins” situation is
called a conventional settlement. In other words, who wins is decided by
convention. It is logically possible to have the opposite convention,
non-resident wins, being the ESS. There are a few examples of this. It is also
possible that who wins is decided by some asymmetry in resource holding power,
such as “Bigger wins”. In the case of the speckled wood it seems that there is
also some asymmetry in resource holding power as the resident, who has been
sunning himself, may have higher body temperature.
Further insights from game theory
The
above examples will provide you with a start to understanding the use of
mathematics in the study of animal behaviour. Game theory has also made
numerous other insights. For example, some games may have cyclical dynamics,
with one strategy invading another, then being invaded by a third strategy
which in turn is invaded by the first strategy. This is similar to the
children’s game Rock, Scissors, Paper.
Rock beats scissors, paper beats rock, and scissors beats paper. Amazingly
enough, cyclical dynamics has been found in the reproductive behaviour of
lizards, Uta stansburiana (Sinervo,
B. Lively, C. M. 1996. Lizards play rock-scissors-paper. Nature 380: 240-243).
Maynard Smith (1998 p. 130-131) describes the situation in the following way. “Male...lizards have one of three mating
strategies. Orange-throated males establish large territories, within which
live several females. A population of such males can be invaded by males with
yellow throats: these “sneakers” do not defend a territory, but steal
copulations. The orange throats cannot defend all their females. However, a
population of yellow-striped males can be invaded by blue-throated males, which
maintain territories large enough to hold one female, which they can defend
against sneakers. Once sneakers become rare, it pays to defend a large
territory with several females. Orange males invade, and we are back where we
started. In the field, the frequencies of the three colour morphs cycled with a
period of about 6 years.”
Synopsis of the behaviours seen among
male morphs
The following descriptions are taken
from Barry Sinervo’s website. If you go to this website you can see video clips
of the behaviours of different male types on encountering each other.
http://bio.research.ucsc.edu/~barrylab/classes/animal_behavior/video_only.html
Yellow-Throated Sneaker Male Behaviour
Yellow
males are "sneakers" in that they mimic the throat colour of receptive
females, and yellows also mimic female behaviour. 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 behaviours 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 Behaviour
Blue
males 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, neighbouring males use a series of bobs to
communicate their identity, and the neighbours usually part without battle.
Orange-Throated Male Behaviour
Orange
males are ultra-dominant and very aggressive owing to high levels of testosterone,
and attack 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
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 symmetrical.
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