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.

Introduction

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:

Either:

1 – E (I, I) > E (J, I)

Or

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: T_A
• Animal B persistence time: T_B

In this example T_A > T_B

Payoff to animal A = V – cT_B (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 = – cT_B (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.

Examples:

• 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.