Posts Tagged ‘ ESS ’

Evolution of Animal Fighting Behaviour

It is recommended that you read ‘An Introduction to the Evolution of Animal Fighting Behaviour‘ before you read this, as there are some concepts explained in the earlier article which are used without explanation in this article.


Success in fighting behaviour is frequency dependent i.e. as the population size increases fighting success decreases. Fighting behaviour amongst a species is often explained in terms of roles, for example we have previously looked at the fairly simplistic Hawk/Dove model which gives animals the role of either hawk or dove.

When determining which strategy will be dominant amongst a population we check it against the ‘standard conditions’ of Maynard Smith. If a strategy is adopted amongst the majority of a population we call it an evolutionarily stable strategy (ESS). An ESS is a strategy which, if adopted by most members of a population, cannot be invaded by a mutant strategy which is initially rare. The standard conditions to determine if a strategy (e.g. strategy I) is an ESS are:


1 – E (I, I) > E (J, I)


2 – If E (I, I) = E (J, I) then E (I, J) > E (J, J)

E (I, I) is the payoff of strategy I against strategy I. Therefore by the conditions of 1, strategy I is an ESS if the payoff of I vs. I is greater than the payoff J receives from fighting I i.e. E (J, I).

The Hawk/Dove/Retaliator Model

The Hawk/Dove model makes certain assumptions:

  • Animals fight with equal ability
  • Animals fight in pairs
  • Animals can only display, escalate or retreat
  • Encounters are random

Due to the limitations of the Hawk/Dove model an additional strategy was added – retaliator

  • Hawks (H)– Escalate immediately, retreat if injured
  • Doves (D)– Display immediately, retreat if opponent escalates
  • Retaliators (R)– Immediately display, escalate if opponent escalates and retreat if injured

The payoff matrix looks like this:

Vs. > H D R
H (V-C)/2 V (V-C)/2
D 0 V/2 (V/2)x 0.9
R (V-C)/2 (V/2)x 1.1 V/2
  • V = Value of resource.
  • C = Cost paid attempting to gain resource.
  • The reason for the x1.1 is that retaliators do slightly better than doves (10%) as they sometime escalate so their payoff is increased.
  • The reason for the x0.9 is the same as above, but because the doves sometimes lose, their payoff is reduced by 10%.

When V>C: Hawk is not an ESS, doves are unable to invade and therefore retaliator is an ESS.

When V<C: Hawk is not an ESS, doves are unable to invade and therefore retaliator is an ESS.

The War of Attrition

When 2 animals meet in a contest for a resource, the amount of energy they are willing to invest to win that resource are predetermined. The animal will therefore display until this time/energy is up. This value is not modified during the display. In the contest, the animal which wins is therefore the one who invested the most predetermined energy. We can model this:

  • Rate of cost accumulation: c
  • Contest length: T
  • Cost of contest: cT
  • Resource value: V
  • Animal A persistence time: TA
  • Animal B persistence time: TB

In this example TA > TB

Payoff to animal A = V – cTB (Animal A wins the resource V but still has to pay the cost, c for the length of the contest, the length of the contest would therefore be TB as the contest ended when animal B gave in.)

Payoff to animal B = – cTB (Animal B does not win any resource, but must still pay the energy that was used during the contest)

A population does not evolve a constant persistence time, however:

  • If cT < V it is worth persisting longer for a resource as the payoff is greater than the cost.
  • If cT > V then a persistence time of 0 spreads amongst the population as to engage in contest for the resource will mean a loss of energy (even if winning). It is therefore better to not engage and lose nothing.

The length animals choose as their persistence time follows a negative exponential distribution, i.e. many choose short times and a very limited number choose long times. The length of contests will therefore also follow this distribution. The log of the number of contests plotted against the length of the contests will give a straight line.


  • Damselfly larvae compete for perching space. Intruders encroaching on perching space will be warned by a ritualised display of the abdomen. The intruder may either leave or contest against the perch space owner. The contests are slow, but their duration follows the negative exponential predicted earlier, however 70-80% of the contests are won by the original occupant and not the 50% you would expect.
  • The fighting of male dungfly over female dungfly (which can be considered a resource) follows the same negative exponential pattern.

The assumption that all contests are fought symmetrically (equal chances of winning) is false, we can assume asymmetry because:

  • Resource Value – Resources are worth more or less to different animals, e.g. a piece of food may be worth more to a hungry animal than a recently fed animal.
  • Resource Holding Power (RHP) – The fighting ability of the animal, this will vary amongst the population, those with a higher RHP will be more likely to win a contest.
  • Uncorrelated Asymmetry – This is any asymmetry which is not correlated to the value of the resource.

Resource Holding Power (RHP):

  • RHP is the fighting ability of an animal, therefore the animal with the greatest RHP is going to keep the resource and the animal with the lower RHP will retreat.
  • Animals must therefore find a way to assess the RHP of others.
  • If RHPs are of similar value, this is when a fight will escalate.
  • For example:
    • The roar contest in red deer helps to determine the RHP. The deer with the lower roaring rate retreats as it is very likely to have a lower RHP. This type of contest is a true signal of RHP (unlike size for example).
    • Croaking in toads when trying to find/compete for a female allows the toads to determine RHP and thus whether or not to attack. Larger males produce deeper croaks and are determined to have a higher RHP.

Bourgeois Strategy

The Bourgeois strategy is a method used to determine the payoff values for competing for resources. It is substituted into the Hawk/Dove model. The Bourgeois strategy is:

  • If it is the owner of a resource, it plays hawk
  • If it is the intruder, it plays dove
  • The assumption is 50% of the time; the Bourgeois is the owner of the resource.

When we put this into a payoff matrix we get the results:

  • If V>C – The hawk strategy is an ESS
  • If V<C – The Bourgeois strategy is an ESS

e.g. The speckled wood butterfly protect areas of sunlight as they are looking for a mate. When an intruder approaches a short spiral fight occurs. The owner of the path of sunlight always wins the spiral fight. If there is confusion over the ownership of the sunlight patch then the spiral fight lasts much longer.

There also exists an Anti-Bourgeois strategy where the intruder always wins the resource for example:

  • In certain spider species, intruders always displace the owner of a web funnel.
  • Seagulls on a flag post always give up the space immediately to invaders.

Introduction to the Evolution of Animal Fighting Behaviour


Animal fighting behaviour can be introduced using the simple models discussed here; one of these is the ‘Hawk/Dove’ model by Maynard Smith. From this model, we can construct payoff matrixes which can then be used to determine evolutionarily stable strategies (defined below).

Evolutionarily stable strategy – An evolutionary stable strategy or ESS is a strategy which, if adopted by most members of the population, cannot be invaded by a mutant strategy which is initially rare.

Maynard Smith’s Model

The evolution of ritualised behaviour has evolved for the benefit of individuals. Simply, we can imagine that a ritualised behaviour has evolved to allow an animal to avoid participating in conflict, when it is aware that the opposition is more capable. By avoiding conflict, the animal does not have to pay a ‘cost’ of injury. This is looked at in the Maynard Smith Hawk/Dove model.

There are many examples, such as the domestic dog will roll on to its back making itself vulnerable to the opposition, signifying that it does not want to participate in the conflict and the opposition may take the resource for which the conflict arose

Another example is the roaring behaviour of the Red Deer. Typically a roaring contest (another type of ritualised behaviour) will mark the start of a conflict; this allows each Red Deer to gauge the prowess of one another, from which they can decide whether or not to elevate the aggression.

The payoff of conflict is frequency dependent i.e. if there are very few dominant aggressive males, they will obtain a large majority of the resources as the other males are highly likely to show submissive behaviour and flee.

The payoff of conflict can be broken down into two simple values:

  • V – The value of the resource (This could be food, females etc.)
  • C – The cost of the conflict (Injury), however if the animal retreats, no cost is therefore paid

From this we can build the Hawk/Dove Model.

Building the Hawk/Dove Model

The hawk/dove model is a simplistic model concerning the possibilities and outcomes of conflict. By simplifying the animal’s behaviour, we can break down their responses to conflict into 3 choices, either:

  • The animal displays ritualised behaviour (E.g. the roaring contest of the Red Deer)
  • The animal retreats
  • Or the animal elevates the conflict and engages in a fight

Using these three possible outcomes, 2 strategies are built, the Hawk strategy and the Dove strategy. Animals can be related to either the dove strategy or the hawk strategy.

The Hawk strategy consists of the following behaviour:

  • The hawk will engage in conflict immediately and only retreat if it becomes injured

The Dove strategy consists of the following behaviour:

  • The dove will display immediately and only retreat should its opponent escalate the conflict

By winning in a conflict, the animal will gain the resource V, by losing the animal will have to pay the cost C. Some possible scenarios are listed below (Hawk – H, Dove – D)

  • H vs. H – Both will escalate the conflict meaning one gets injured and retreats. This means there is a 50% chance of winning resource V and a 50% chance of paying cost C. (This is assuming both individuals are equally matched and of the same fitness – see assumptions section below). In mathematical terms, E (the energy gain from the conflict, or payoff) of the conflict between two hawks is E(H,H). We can equate this to the value and cost, so:
  • E(H,H) = (V-C)/2
  • H vs. D – The hawk immediately escalates the fight and so the dove retreats. This means the hawk always gets the resource V and the dove always gets nothing – but does not pay a cost as it retreats. So:
  • E(H,D) = V, E(D,H) = 0
  • D vs. D – Both immediately display but as both are of equal fitness (see assumptions below) they must either share the resource or one randomly wins the resource, either way they receive the equivalent of half the resource. So:
  • E(D,D) = V/2


The Hawk/Dove model retains its simplicity due to some assumptions, these are:

  • All individuals are of the same Darwinian fitness, making them evenly matched.
  • The V gained and C paid are the same for all individuals e.g. the cost of an injury costs the same amount of energy in all ‘dove’ individuals.
  • All interactions are completely at random.

Payoff Matrixes

Payoff matrixes are grids which determine whether an ESS is in place, by inputting the values of V and C we can see whether or not hawks, doves or a mixture of both give an ESS.

From previous knowledge (above) we know the following information:

Vs. > H D
H (V-C)/2 V
D 0 V/2

To determine which strategy is an ESS depends on whether or not V<C or V>C, which would depend on the situation. Each has been equated below:


If V>C e.g. V=4, C=2, we would get the following information (by substituting the values into the table above):

Vs. > H D
H 1 4
D 0 2

What this shows us is that, because in column 1, H vs. H = 1 and D vs. H = 0, Doves are unable to invade a hawk population – This means that the Hawk Strategy when V>C is an ESS. We back this up by looking at column 2 and seeing D vs. D = 2 and H vs. D = 4. This means Hawks are able to invade doves, so doves therefore cannot be an ESS.


If V<C e.g. V=2, C=4, is there a difference when compared to V>C? Again by substituting the values into the table above, the following information is obtained:

Vs. > H D
H -1 2
D 0 1

Column 1 – D vs. H = 0, H vs. H = -1. This means that Doves are able to invade hawks, does this mean hawk is not an ESS?

Column 2 – D vs. D = 1, H vs. D = 2. This means that Hawks can invade doves.

As both strategies are able to invade one another, when V<C, a mixed ESS arises.

Determining Proportions in a Mixes ESS

In a mixed ESS, we are able to determine the proportion of each strategy by using a simple equation, p=V/C. The equation is derived initially from a more complex equation however:

W=Fitness, W(H) = Fitness of hawks, W(D) = Fitness of doves, W0 = basic fitness p = proportion

We assume that W(H) = W(D)

  • W(H) = W0 + p[(V-C)/2] + pV
  • Where p[(V-C)/2] is the proportion of occasions that we see H vs. H
  • Where pV is the proportion of occasions we see H vs. D
  • W(D) = W0 + p0 + p[V/2]
  • Where p0 is the proportion of occasions we see D vs. H
  • Where p[V/2] is the proportion of occasions we see D vs. D

Because we assume that W(H)=W(D) we can equate these equations to one another, therefore:

  • W0 + p[(V-C)/2] + pV =  W0 + p0 + p[V/2]
  • p[(V-C)/2] + pV = p[V/2]
  • p=V/C

So when simplified we get p = V/C which means the proportion of a strategy in the mixed ESS depends entirely on the value of the resource and cost of injury. Using the values we saw in the V<C example above (V=2, C=4) we get p(H)=2/4. This equates to 0.5 or 50%, therefore the proportion of hawks in this mixed ESS is 50%.

What we can conclude from this is that behavioural variation in a population is suited to evolve that way, especially when V<C. Also that it is frequency dependent. Also as the cost of injury increases, more contests for resources within the species will be settled by ritualised displays.