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Prior fitness and the evolutionarily stable strategy for intraspecific conflict
i. theor. Biol. (1980) 84, 567-573
Prior Fitness and the Evolutionarily Stable Strategy for lntraspecific Conflict MICHEL TREISMAN AND JOHN 8. COLLINS
Department of Experimental Psychology, University of Oxford, South Parks Road, Oxford, OXI 3 UD, England (Received 12 July 1979 and in revised form 3 January 1980) Maynard Smith (1974) employed the game "Hawks and Doves" as a model for one type of intraspecific conflict and showed that the evolutionarily stable probability that a contestant will choose the more aggressive of two options is a function of V, the value of the prize, and D, the damage inflicted in a successful attack. Here it is demonstrated that in addition the parameter 17o, the animal's fitness prior to a contest, may affect the evolutionarily stable strategy for that contest. When F0 is low the expected probability of attack may be higher, for D large, than would otherwise be predicted. Computer simulations of selection over 1000 generations indicate that the effect may be less marked ~vhen multiple contests may ensue than for a single contest.
Ritualization of intraspecific conflicts, which limits the damage contestants inflict on one another, is ubiquitous, which raises the question why animals do not fight as savagely as they are able. In an attempt to answer this, Maynard Smith and his colleagues d e v e l o p e d the concept of an evolutionarily stable strategy (ESS) (Maynard Smith & Price, 1973; Maynard Smith, 1974, 1978; Maynard Smith & Parker, 1976). Such'conflicts were analysed in game theory terms, and the ESS was defined as a strategy such that the fitness of members playing it would be greater than the fitness of any behavioural mutant which might invade the population. A useful model for some types of intraspecific conflict is provided by the game "Hawks and Doves" (Maynard Smith & Parker, 1976; Maynard Smith 1978; Treisman, 1977) in which each contestant must make a choice between two strategies, " A t t a c k " or "Display", with values or costs attaching to each alternative as shown in the payoff matrix below. In each cell the figure on the left is the return or expected return to contestant I, that on the right the return to contestant J. 567 0022-5193180/110567 + 07 $02.00/0
O 1980 Academic Press Inc. (London) Ltd.
568
M. T R E I S M A N
AND
J. S. C O L L I N S
J (pj) ATTACK
V-D V-D 2 ' 2
(P0 Attack (1 - P l ) Display
o,
v
(1 - pj) DISPLAY
v,
0
V/2,
V/2
(Ml)
This game provides a simple model of a contest between randomly chosen rivals. V is the value of the prize, measured as a contribution to the fitness of the winner. If each opponent adopts the more violent strategy, "Attack", one will inflict damage - D on the other, and will itself gain V. Which contestant suffers and which gains is determined bychance. "Display" is a strategy in which the contestant retreats if the opponent Attacks: the latter then gains V, the former nothing. If both Display no damage is inflicted and one or Qther gains the prize, with probability 0.5. The strategy I is represented by (p~, 1 -p~), the probabilities of choosing the gambits Attack and Display respectively. Then type ! is that section of the population genetically committed to strategy I, while type J are those that play J = ( p j , 1 - p . l ) . The better strategy is that which assures greater fitness for the type playing it, and this will be positively selected. Maynard Smith (1974) has defined the conditions for an evolutionarily stable strategy. To discover what strategy will be stable we may examine the expected returns for the alternative gambits. Thus in Hawks and Doves the expected returns for an Attack or Display by I against J are given by E (Attack; I, J ) = p j ( V - D ) / 2 + ( 1 - p j ) V
(1)
E (Display;/, J) = (1 - p , ) V/2
(2)
and
and the overall return for strategy I played against J is
E(LJ)=pjE(Attack;/, J)+(1 -pj)E(Display;/, J).
(3)
If V > D then E(Attack; I, J) is always greater than E ( D i s p l a y ; / , J) and thus the equilibrium strategy, which is also evolutionarily stable, is I = (1, 0). For V = D, the return for an Attack is greater than that for a Display unless pj = 1, when they are the same. Again I = (1, 0) is a stable strategy and evolutionarily stable. For V < D , we may equate the two expected returns and obtain Pl = V/D. Then I = (V/D, 1 - V/D) is evolutionarily stable.
FITNESS
AND THE
ESS
569
These results are summarized by the expression px = min. (V/D, 1).
(4)
~/here pl ~s the stable probability of Attack and "min. (a, b)" is read " a or b, ~vhichever is the less". The object of the present note is to extend this analysis by examining the effects of a further parameter which may affect the ESS and cause departures from the predictions given by equation (4). This is the average fitness of members of the population arising from causes other than the contest under consideration, which we shall refer to as the prior fitness, Fo. This parameter has been noted before (Maynard Smith, 1978) but it has implicitly been assumed that it is sufficiently large relative to the fitness changes resulting from a contest not to influence the derivation above. But prior fitness need not always be large, and when it is not so big that net fitness is always positive, this may affect the derivation of the stable attack probability. Suppose that an animal is involved in a contest or series of contests in which it attacks and loses. Then its final fitness, i.e. its eventual reproductive success, will be correspondingly curtailed. If it attacks and loses sufficiently often for the final fitness to be reduced to zero, it will not reproduce at all. Nor will it reproduce if its final fitness is less than zero, an outcome which is the more likely to occur the smaller the value of its fitness prior to these contests. Thus, from the point of view of the individual animal, an eventual fitness of zero is exactly equivalent in its effect to a fitness having any negative value. But, in the derivation of the evolutionarily stable attack probability [equation (4)], increments and decrements to fitness are treated linearly. That is, the argument is based on expected values obtained by giving the same weight to one animal's gain (V) as to another animal's loss ( - D ) if the animals fall in the same category. But this is valid only if no loss takes an animal's fitness below zero. To put it another way, animals reproduce as individuals, and if instead of being available for reproduction an animal is dead (negative or zero fitness) it does not matter how dead he is. This problem is the more likely to arise the smaller Fo and the larger D. Thus if at any time a contestant's net fitness is 5 units, D is 10, and there is only one more contest to go, he is in the same position as if D were 5. Having little to lose he should attack more recklessly than if Fo were greater. That is, the frequency with which he chooses to attack should be higher than the value given by equation (4). If an individual animal engaged in no more than one contest, the effect of Fo could be allowed for by rewriting equation (1) as: E (Attack; I, J ) = p j ( V - k D ) / 2 + (1 - p j) V
(5)
570
M. T R E I S M A N
A N D J. S. C O L L I N S
where k = min. (Fo/D, 1)
(6)
and represents the average effectiveness of a defeat. The ESS would then become: pi = min. (V/kD, 1).
(7)
This shows that there will be a sharp transition in the effect of prior fitness as it falls below D. Above this point it has no effect on p~. But below this threshold, px increases inversely with Fo (for pz < 1, of course), at least when only one contest may occur. When more than one battle is possible, the definition of k becomes more complex, since the probabilities of different numbers of contests, some of which m a y a n d some of which may not take Fo below D, must be considered. To examine the effects that might be expected with multiple contests, computer simulations of selection directed by the Hawks and Doves game were run. Each simulation commenced with a population of 1010 animals rectangUlarly distributed over attack probability. That is, 10 animals were characterized by a probability of attack, p, of 0; 10 had p =0.01; 10 had p = 0 . 0 2 ; . . . 10 had p = 1.00. Each animal had the same initial fitness, F0. 1010 contests then took place, the contestants in each being randomly selected from the whole population with replacement. Thus an animal might expect to be involved in two contests, but the number might be greater or less than this. In each contest, each animal attacked with a probability determined by his personal value of p, and each received 0, V, or - D as prescribed by Matrix (M1). At the end of the 1010 contests, each animal whose final fitness was greater than zero was considered a survivor, and was allowed to reproduce himself in proportion to his fitness. That is, if survivor X, with attack probability pi, had final fitness Fx, he contributed a number (F,,/F) 1010 animals (rounded to an integer) to category pi in the next generation, where F represents fitness summed over all survivors. (As an animal was not automatically removed if his fitness fell to or below zero before the end o| the series of contests, he had the chance of being drawn for further contests in that generation and retrieving his fortunes.) This describes one generation of selection. Each new generation commenced with 1010 animals (or slightly more or less due to rounding-off error), each with the same prior fitness, Fo. But the number in each attack category depended on the outcomes of the selection process in previom generations. For each simulation of 1000 generations of selection, V=2 and Fo and D were given fixed values.
571
F I T N E S S A N D T H E ESS
~ o-oo o4
0,5
P~ 0-2
t 'll),i I1\\ )(
-tlf
1~
t
,,',
.
0-9
08
'.,,
II
l \ " , \l ~~ .........
0
0
1 o-1 _
.
0
200
1
400
= I,~: ]
............
~
600
/.,
800
0
I000
:
"~" . . . . . . . . . . . . . . . . . . . . . . ". . .""-,.-" . . . . . . . . . . . . . ""
.
4
0.7
6
. . . 200 400
600
800
I000
05
Generofions
FIo. 1. Simulations of 1000 generations of selection. In each case V = 2. Fo = 10 or 100. D = 1, 10, 15, 20 or 30 as indicated by the parameters on the curves. The value of p~ predicted by equation (4) is 1 for D = 1; for the other values of D the values of p~ predicted by this equation are indicated by horizontal straight lines. The scale of p, for D = 1 is shown on the tight; for the other values of D it is on the left.
The results of simulations with Fo = 10 or 100, and D = 1, 10, 15, 20, or 30 are illustrated in Fig. 1: the average value of pi is plotted for every 50th generation. These simulations illustrate that when the contribution of the contest to total fitness is relatively small (Fo= 100) the value of p approached by selection approximates the value predicted by equation (4). As the contribution of the contest to total fitness is small, these limits are approached slowly. (For D = 1, the curve reaches 0.95 at 2000 generations). When the contest bulks large in determining total fitness (Fo = 10) selection is rapid. But when D approaches or exceeds Fo the probability of attack does not fall to the levels predicted by the standard theory [equation (4)]; for D = 15, 20 and 30, p~ remains instead at about 0.15-0.17. It is interesting to examine the value of pi reached after 1000 generations for different levels of Fo. Table I shows the values of p, predicted by equations (4) and (7) (i.e., for the standard theory, and for a single contest), and those obtained by simulation of multiple contests, for the levels of Fo and D given. We see that the effect of Fo is somewhat less for multiple contests than for a single contest, but clearly evident. When Fo is greater than or equal to D,
M. TREISMAN AND J. S. COLLINS
572
TABLE 1 Values o[ p~ after 1000 generations
Predicted pt = V/kD Fo: ~
D 15 20 30
0-133 0.100 0.067
Simulated Pt = V/D
30
20
10
30
0-133 0.100 0.067
0-133 0-100 0-100
0.200 0.200 0.200
0.140 0.098 0-068t
20
10
0.139 0.168~; 0-093t 0.173:i: 0.086:[: 0.151,
tFo=D *Fo
the end-point obtained by simulation is not distinguishable from that predicted by the standard theory. Only when Fo is less than D do we see a rise in p,, and the asymptotic value is intermediate between those given by equations (4) and (7). It appears that the threshold is the same for single and multiple contests. For Fo < D multiple contests give a lower final value than do single contests [equation (7)], presumably because of cases in which a series of early successes raise Fo above D. Thus the effect of prior fitness may be seen most sharply in single make-or-break encounters. Figure 1 also suggests that, at least on the present assumptions, once Fo is less than D the actual value of D has little if any effect on the final stable level of pi. It is likely that in a real population both Fo and D will have some variance and thus the threshold transition may be more blurred than the simple theory suggests. It is a familiar observation and may seem a natural one that when animals have suffered a sequence of adversities, such as famine and harsh weather, they may fight more fiercely over whatever goods come their way. As well as seeming "natural," i.e. familiar, it should now be evident that this is a predictable consequence of evolutionary pressures, which will tend to make Pl an inverse function of Fo, rather than of D, when the former is sufficiently small. Thus we do not need to attribute this behaviour to the "desperation" of the animals concerned (unless we use the term to describe a mediating mechanism); we may note instead that it is an instance of the good game sense of Evolution. An animal may face a contest which will occur once only but which is not the sole determinant of its fitness; for example, it may have the chance of taking over its sire's harem, or of being expelled and having to obtain mates by recruitment or to rely on insemination as a hanger-on. But if all its chances of future reproduction depend on a contest which will occur once
F I T N E S S A N D T H E ESS
573
0nlY then necessarily Fo = 0 and by equations (6) and (7) it must always attack. Something close to this situation may arise when the established naale, in sole possession of a harem (and who, if displaced, will find no other) faces a younger challenger. Certainly the older male should always attack as fiercely as he can. A possibly analogous example of the importance o f t h e parameter Fo in determining the ESS may be the familiar observation that a tree whose ~4talityhas been impaired by damage may set more blossom, i.e. it devotes proportionately more of its resources to the risky adventure of reproduction, and less to the more prudent course of somatic maintenance, than would otherwise have been the case. (Its opponent, in this case, is the weather). The value of the prize in Hawks and Doves may also sometimes be subject to an analogous non-linearity: at some level of repletion the increment in fitness accruing from the acquisition of additional food or additional territory may become less than that which was at stake in earlier battles, and so the likelihood of an attack may decrease. Thus pt may tend to fall when an animal has already had a number of successful fights. The importance of this effect will depend on the population and type of conflict modelled. This work was done while in receipt of support from the Science Research Council. REFERENCES MAYNARD SMITI:t, J. (1974). Z theor. BioL 47, 209. MAYNARD SMITH, J. (1978). ScL Am. 239, 136. MAYNARD SMITH, J. & PARKER, (3. A. (1976). Anita. Behav. 24, 159. MAYNARD SMITtl, J. & PRICE, G. R. (1973). Nature 246, 15. TREISMAN, i . (1977). J. l~fath. Psychol. 16, 167.
Prior Fitness and the Evolutionarily Stable Strategy for lntraspecific Conflict MICHEL TREISMAN AND JOHN 8. COLLINS
Department of Experimental Psychology, University of Oxford, South Parks Road, Oxford, OXI 3 UD, England (Received 12 July 1979 and in revised form 3 January 1980) Maynard Smith (1974) employed the game "Hawks and Doves" as a model for one type of intraspecific conflict and showed that the evolutionarily stable probability that a contestant will choose the more aggressive of two options is a function of V, the value of the prize, and D, the damage inflicted in a successful attack. Here it is demonstrated that in addition the parameter 17o, the animal's fitness prior to a contest, may affect the evolutionarily stable strategy for that contest. When F0 is low the expected probability of attack may be higher, for D large, than would otherwise be predicted. Computer simulations of selection over 1000 generations indicate that the effect may be less marked ~vhen multiple contests may ensue than for a single contest.
Ritualization of intraspecific conflicts, which limits the damage contestants inflict on one another, is ubiquitous, which raises the question why animals do not fight as savagely as they are able. In an attempt to answer this, Maynard Smith and his colleagues d e v e l o p e d the concept of an evolutionarily stable strategy (ESS) (Maynard Smith & Price, 1973; Maynard Smith, 1974, 1978; Maynard Smith & Parker, 1976). Such'conflicts were analysed in game theory terms, and the ESS was defined as a strategy such that the fitness of members playing it would be greater than the fitness of any behavioural mutant which might invade the population. A useful model for some types of intraspecific conflict is provided by the game "Hawks and Doves" (Maynard Smith & Parker, 1976; Maynard Smith 1978; Treisman, 1977) in which each contestant must make a choice between two strategies, " A t t a c k " or "Display", with values or costs attaching to each alternative as shown in the payoff matrix below. In each cell the figure on the left is the return or expected return to contestant I, that on the right the return to contestant J. 567 0022-5193180/110567 + 07 $02.00/0
O 1980 Academic Press Inc. (London) Ltd.
568
M. T R E I S M A N
AND
J. S. C O L L I N S
J (pj) ATTACK
V-D V-D 2 ' 2
(P0 Attack (1 - P l ) Display
o,
v
(1 - pj) DISPLAY
v,
0
V/2,
V/2
(Ml)
This game provides a simple model of a contest between randomly chosen rivals. V is the value of the prize, measured as a contribution to the fitness of the winner. If each opponent adopts the more violent strategy, "Attack", one will inflict damage - D on the other, and will itself gain V. Which contestant suffers and which gains is determined bychance. "Display" is a strategy in which the contestant retreats if the opponent Attacks: the latter then gains V, the former nothing. If both Display no damage is inflicted and one or Qther gains the prize, with probability 0.5. The strategy I is represented by (p~, 1 -p~), the probabilities of choosing the gambits Attack and Display respectively. Then type ! is that section of the population genetically committed to strategy I, while type J are those that play J = ( p j , 1 - p . l ) . The better strategy is that which assures greater fitness for the type playing it, and this will be positively selected. Maynard Smith (1974) has defined the conditions for an evolutionarily stable strategy. To discover what strategy will be stable we may examine the expected returns for the alternative gambits. Thus in Hawks and Doves the expected returns for an Attack or Display by I against J are given by E (Attack; I, J ) = p j ( V - D ) / 2 + ( 1 - p j ) V
(1)
E (Display;/, J) = (1 - p , ) V/2
(2)
and
and the overall return for strategy I played against J is
E(LJ)=pjE(Attack;/, J)+(1 -pj)E(Display;/, J).
(3)
If V > D then E(Attack; I, J) is always greater than E ( D i s p l a y ; / , J) and thus the equilibrium strategy, which is also evolutionarily stable, is I = (1, 0). For V = D, the return for an Attack is greater than that for a Display unless pj = 1, when they are the same. Again I = (1, 0) is a stable strategy and evolutionarily stable. For V < D , we may equate the two expected returns and obtain Pl = V/D. Then I = (V/D, 1 - V/D) is evolutionarily stable.
FITNESS
AND THE
ESS
569
These results are summarized by the expression px = min. (V/D, 1).
(4)
~/here pl ~s the stable probability of Attack and "min. (a, b)" is read " a or b, ~vhichever is the less". The object of the present note is to extend this analysis by examining the effects of a further parameter which may affect the ESS and cause departures from the predictions given by equation (4). This is the average fitness of members of the population arising from causes other than the contest under consideration, which we shall refer to as the prior fitness, Fo. This parameter has been noted before (Maynard Smith, 1978) but it has implicitly been assumed that it is sufficiently large relative to the fitness changes resulting from a contest not to influence the derivation above. But prior fitness need not always be large, and when it is not so big that net fitness is always positive, this may affect the derivation of the stable attack probability. Suppose that an animal is involved in a contest or series of contests in which it attacks and loses. Then its final fitness, i.e. its eventual reproductive success, will be correspondingly curtailed. If it attacks and loses sufficiently often for the final fitness to be reduced to zero, it will not reproduce at all. Nor will it reproduce if its final fitness is less than zero, an outcome which is the more likely to occur the smaller the value of its fitness prior to these contests. Thus, from the point of view of the individual animal, an eventual fitness of zero is exactly equivalent in its effect to a fitness having any negative value. But, in the derivation of the evolutionarily stable attack probability [equation (4)], increments and decrements to fitness are treated linearly. That is, the argument is based on expected values obtained by giving the same weight to one animal's gain (V) as to another animal's loss ( - D ) if the animals fall in the same category. But this is valid only if no loss takes an animal's fitness below zero. To put it another way, animals reproduce as individuals, and if instead of being available for reproduction an animal is dead (negative or zero fitness) it does not matter how dead he is. This problem is the more likely to arise the smaller Fo and the larger D. Thus if at any time a contestant's net fitness is 5 units, D is 10, and there is only one more contest to go, he is in the same position as if D were 5. Having little to lose he should attack more recklessly than if Fo were greater. That is, the frequency with which he chooses to attack should be higher than the value given by equation (4). If an individual animal engaged in no more than one contest, the effect of Fo could be allowed for by rewriting equation (1) as: E (Attack; I, J ) = p j ( V - k D ) / 2 + (1 - p j) V
(5)
570
M. T R E I S M A N
A N D J. S. C O L L I N S
where k = min. (Fo/D, 1)
(6)
and represents the average effectiveness of a defeat. The ESS would then become: pi = min. (V/kD, 1).
(7)
This shows that there will be a sharp transition in the effect of prior fitness as it falls below D. Above this point it has no effect on p~. But below this threshold, px increases inversely with Fo (for pz < 1, of course), at least when only one contest may occur. When more than one battle is possible, the definition of k becomes more complex, since the probabilities of different numbers of contests, some of which m a y a n d some of which may not take Fo below D, must be considered. To examine the effects that might be expected with multiple contests, computer simulations of selection directed by the Hawks and Doves game were run. Each simulation commenced with a population of 1010 animals rectangUlarly distributed over attack probability. That is, 10 animals were characterized by a probability of attack, p, of 0; 10 had p =0.01; 10 had p = 0 . 0 2 ; . . . 10 had p = 1.00. Each animal had the same initial fitness, F0. 1010 contests then took place, the contestants in each being randomly selected from the whole population with replacement. Thus an animal might expect to be involved in two contests, but the number might be greater or less than this. In each contest, each animal attacked with a probability determined by his personal value of p, and each received 0, V, or - D as prescribed by Matrix (M1). At the end of the 1010 contests, each animal whose final fitness was greater than zero was considered a survivor, and was allowed to reproduce himself in proportion to his fitness. That is, if survivor X, with attack probability pi, had final fitness Fx, he contributed a number (F,,/F) 1010 animals (rounded to an integer) to category pi in the next generation, where F represents fitness summed over all survivors. (As an animal was not automatically removed if his fitness fell to or below zero before the end o| the series of contests, he had the chance of being drawn for further contests in that generation and retrieving his fortunes.) This describes one generation of selection. Each new generation commenced with 1010 animals (or slightly more or less due to rounding-off error), each with the same prior fitness, Fo. But the number in each attack category depended on the outcomes of the selection process in previom generations. For each simulation of 1000 generations of selection, V=2 and Fo and D were given fixed values.
571
F I T N E S S A N D T H E ESS
~ o-oo o4
0,5
P~ 0-2
t 'll),i I1\\ )(
-tlf
1~
t
,,',
.
0-9
08
'.,,
II
l \ " , \l ~~ .........
0
0
1 o-1 _
.
0
200
1
400
= I,~: ]
............
~
600
/.,
800
0
I000
:
"~" . . . . . . . . . . . . . . . . . . . . . . ". . .""-,.-" . . . . . . . . . . . . . ""
.
4
0.7
6
. . . 200 400
600
800
I000
05
Generofions
FIo. 1. Simulations of 1000 generations of selection. In each case V = 2. Fo = 10 or 100. D = 1, 10, 15, 20 or 30 as indicated by the parameters on the curves. The value of p~ predicted by equation (4) is 1 for D = 1; for the other values of D the values of p~ predicted by this equation are indicated by horizontal straight lines. The scale of p, for D = 1 is shown on the tight; for the other values of D it is on the left.
The results of simulations with Fo = 10 or 100, and D = 1, 10, 15, 20, or 30 are illustrated in Fig. 1: the average value of pi is plotted for every 50th generation. These simulations illustrate that when the contribution of the contest to total fitness is relatively small (Fo= 100) the value of p approached by selection approximates the value predicted by equation (4). As the contribution of the contest to total fitness is small, these limits are approached slowly. (For D = 1, the curve reaches 0.95 at 2000 generations). When the contest bulks large in determining total fitness (Fo = 10) selection is rapid. But when D approaches or exceeds Fo the probability of attack does not fall to the levels predicted by the standard theory [equation (4)]; for D = 15, 20 and 30, p~ remains instead at about 0.15-0.17. It is interesting to examine the value of pi reached after 1000 generations for different levels of Fo. Table I shows the values of p, predicted by equations (4) and (7) (i.e., for the standard theory, and for a single contest), and those obtained by simulation of multiple contests, for the levels of Fo and D given. We see that the effect of Fo is somewhat less for multiple contests than for a single contest, but clearly evident. When Fo is greater than or equal to D,
M. TREISMAN AND J. S. COLLINS
572
TABLE 1 Values o[ p~ after 1000 generations
Predicted pt = V/kD Fo: ~
D 15 20 30
0-133 0.100 0.067
Simulated Pt = V/D
30
20
10
30
0-133 0.100 0.067
0-133 0-100 0-100
0.200 0.200 0.200
0.140 0.098 0-068t
20
10
0.139 0.168~; 0-093t 0.173:i: 0.086:[: 0.151,
tFo=D *Fo
the end-point obtained by simulation is not distinguishable from that predicted by the standard theory. Only when Fo is less than D do we see a rise in p,, and the asymptotic value is intermediate between those given by equations (4) and (7). It appears that the threshold is the same for single and multiple contests. For Fo < D multiple contests give a lower final value than do single contests [equation (7)], presumably because of cases in which a series of early successes raise Fo above D. Thus the effect of prior fitness may be seen most sharply in single make-or-break encounters. Figure 1 also suggests that, at least on the present assumptions, once Fo is less than D the actual value of D has little if any effect on the final stable level of pi. It is likely that in a real population both Fo and D will have some variance and thus the threshold transition may be more blurred than the simple theory suggests. It is a familiar observation and may seem a natural one that when animals have suffered a sequence of adversities, such as famine and harsh weather, they may fight more fiercely over whatever goods come their way. As well as seeming "natural," i.e. familiar, it should now be evident that this is a predictable consequence of evolutionary pressures, which will tend to make Pl an inverse function of Fo, rather than of D, when the former is sufficiently small. Thus we do not need to attribute this behaviour to the "desperation" of the animals concerned (unless we use the term to describe a mediating mechanism); we may note instead that it is an instance of the good game sense of Evolution. An animal may face a contest which will occur once only but which is not the sole determinant of its fitness; for example, it may have the chance of taking over its sire's harem, or of being expelled and having to obtain mates by recruitment or to rely on insemination as a hanger-on. But if all its chances of future reproduction depend on a contest which will occur once
F I T N E S S A N D T H E ESS
573
0nlY then necessarily Fo = 0 and by equations (6) and (7) it must always attack. Something close to this situation may arise when the established naale, in sole possession of a harem (and who, if displaced, will find no other) faces a younger challenger. Certainly the older male should always attack as fiercely as he can. A possibly analogous example of the importance o f t h e parameter Fo in determining the ESS may be the familiar observation that a tree whose ~4talityhas been impaired by damage may set more blossom, i.e. it devotes proportionately more of its resources to the risky adventure of reproduction, and less to the more prudent course of somatic maintenance, than would otherwise have been the case. (Its opponent, in this case, is the weather). The value of the prize in Hawks and Doves may also sometimes be subject to an analogous non-linearity: at some level of repletion the increment in fitness accruing from the acquisition of additional food or additional territory may become less than that which was at stake in earlier battles, and so the likelihood of an attack may decrease. Thus pt may tend to fall when an animal has already had a number of successful fights. The importance of this effect will depend on the population and type of conflict modelled. This work was done while in receipt of support from the Science Research Council. REFERENCES MAYNARD SMITI:t, J. (1974). Z theor. BioL 47, 209. MAYNARD SMITH, J. (1978). ScL Am. 239, 136. MAYNARD SMITH, J. & PARKER, (3. A. (1976). Anita. Behav. 24, 159. MAYNARD SMITtl, J. & PRICE, G. R. (1973). Nature 246, 15. TREISMAN, i . (1977). J. l~fath. Psychol. 16, 167.