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A remark on the theory of risk-sensitive foraging
J. theor. Biol. (1984) 110, 217-222
A Remark on the Theory of Risk-sensitive Foraging KLAUS R E G E L M A N N
Arbeitsgruppe fiir Verhaltensforschung, Abteilung fiir Biologic, Ruhr-Universitiit Bochum, Postfach 102148, 4630 Bochum 1, West Germany (Received 28 January 1984, and in revised form 7 March 1984) Recent theoretical (Caraco, 1980; Real, 1980a, b; Stephens, 1981; Stephens & Charnov, 1982) and empirical (Caraco, Martindale & Whittam, 1980; Real, 1981; Caraco, 1981, 1983) studies have demonstrated that animal foraging decisions in a stochastic environment should be, and really are, attuned not only to expected net benefits but also to the associated variances. Two proxy attributes of fitness have been proposed in the literature to model and describe risk-sensitive foraging preferences. Caraco (1980) employed mathematical utility theory (Fishburn, 1970; Keeney & Raiffa, 1976) to develop a model where a linear combination of the mean and the variance of the reward distribution is used as a proxy attribute of fitness. The model has the form:
E ( X ) - K V ( X ) -~ Max where E ( X ) is the mean and V(X) is the variance of the reward distribution described by the random variable X, and K is a constant expressing the forager's risk-sensitivity. Similar models were proposed by Oster & Wilson (1978) and Real ( 1980 a, b). In an alternative approach Stephens ( 1981) and Stephens & Charnov (1982) suggested to use the probability of death due to starvation as a proxy attribute of fitness. Their model has the form:
P ( X < R) ~ Min where X is a random variable describing the reward distribution, and R is some minimum level of net energy requirement. Until now, however, a satisfying experimental test to decide which of the two approaches is fitted best to describe animal foraging decisions in a stochastic environment is lacking (Caraco & Lima, 1984). In this study I will analytically derive some general characteristics of the optimal solutions of the two optimality approaches. Numerical examples will be given to show that the two approaches are in general not equivalent, 217 0022-5193/84/180217 + 06 $03.00/0
O 1984 Academic Press Inc. (London) Ltd.
218
K. R E G E L M A N N
i.e. they may lead to different predictions regarding the most preferred foraging strategy. By reference to these numerical examples experiments are suggested that may help to decide between the two approaches. Like Caraco (1980),.I will consider an animal that forages for a fixed time T per day, where total feeding time T is constrained by some other vital requirements. The animal has to allocate its feeding time between n food patches. In line with Caraco's model, I will assume that on a given day foraging decisions are made before feeding commences (for a discussion of this assumption see Caraco, 1983). Let Xj (l-< i -< n) denote the net energetic benefit when exploiting patch i for time T (intake minus feeding and travel costs in energetic units). Xi is assumed to be a normally distributed random variable with mean/zi = E(Xi) and variance 0-~ = V(Xi). All Xi are assumed to be mutually independent. Furthermore, let p~ be the proportion of the total feeding time T spent in patch i, i.e. 0-< p~ ~ 1 for l ~ i ~ n and n ~=1P~ = 1. For convenience, let p = ( p l , . . . , p,) and let A be the set o f all feasible allocation vectors p, i.e.
A={p=(P','",P,)lO<--Pi<--lf°rl<--i<--nand~} ' , P'=I =, Finally, let R denote the minimum energy requirement o f the animal considered. It is assumed that the animal will die at the end of the day or in the night thereafter if the daily net energy intake is less than R units. The probability o f death due to starvation is then expressed by P(X < R) where X = X(p) = ~ ~=~ p~Xi.According to Caraco (1980) the optimal allocation vector p * = ( p * , . . . , p * ) is determined by solving the maximization problem: E (X(p)) - KV(X(p)) ->Max (Sl) subject to p c A. (Sl) may equivalently be written in the form:
~ p,tz _ K ~ i=1
p i202- 1 ~ M a x
(S1)
i=l n
subject to 0<-pi < - I for 1 -< i -< n and ~.~=~ p~ = 1. Here K is a constant expressing the forager's risk-sensitivity. Caraco (1980) suggested that risk-aversion (i.e. K > 0 ) should occur when the forager can expect a positive 24 hr energy budget and risk-prone behaviour (i.e. K < 0 ) should occur when the forager can expect a negative 2 4 h r energy budget ("24 hr energy budget rule"). Since the variables X~ are normally distributed and mutually independent, the variable X = X ( p ) = ~ . ~ = ~ p~X~ is normally distributed with mean /z = E(X) =~=1P;/~i and variance 0 - 2 = V(X) = E i = 2 2 The variable ( X " " l P~0-1-
RISK-SENSITIVE
FORAGING
219
/z)/o. is normally distributed with mean 0 and variance 1. The probability of starvation is then given by P(X
< R) = P((X
- ~)/o. < (R -
a)lo.) = @ ( ( R
- p,)/or).
As @(.) is the cumulative distribution function of the standard normal distribution, the probability of starvation increases in a strictly monotonic way with ( R - p~)/cr. Therefore, the probability of starvation is minimal if and only if ( R - p , ) / o . is minimal. Thus we arrive at the minimization problem: R - E(X(p))
~
Min
($2)
,/v(x(p)) subject to p ~ A which may equivalently be written in the form: R - ~ PiP-i i=l ~ Min
tl
($2)
subject to 0 -< Pi -< 1 for 1 - i -< n and Y.~=1Pi = 1. When deriving the optimality systems (S 1) and ($2) I have assumed that the random variables X~ are normally distributed. This assumption will only seldom hold in reality. The assumption of normality of X~ may however be relaxed, i.e. the derivation of the system (S1) and ($2) is valid for a wider class of random variables, if the central limit theorem for independent random variables is invoked (see Caraco, 1980; Stephens, 1981; Stephens & Charnov, 1982). Some general characteristics of the systems (S1) and ($2) can be given as follows. (1) If K > 0, (S 1) is a standard problem of quadratic optimization (Boot, 1964). (2) If K-< 0, there always exists a pure strategy, i.e. p = ( p ~ , . . . , p , ) ~ A and p~ = 1 for one i, which is an optimal solution of (S1). (3) The system ($2) does not belong to a category of standard optimizan tion problems. If ~ ~=~p~/~ -< R for all p = ( p ~ , . . . , p,) c A and o.2 > 0 for all i, there always exists a pure strategy which is an optimal solution of ($2). The following two examples will show that the optimal solutions of (SI) and ($2) will not generally be identical. Let n = 2 and XI, X2 be two normally distributed random variables with /~ = E ( X ~ ) = 9 0 , o.2= V ( X I ) = 2 5 and /~2 ----"E ( X 2 ) = 80, o .2 = V ( X 2 ) = 100 and finally let R = 100.
220
K. REGELMANN
A short calculation using (3) yields that both pure strategies, i.e. ( p*, p*) = (1,0) and (p*, p*) = (0, 1), are optimal solutions of ($2) but no mixed strategy can be optimal. If minimization of the probability of starvation is used as a proxy attribute of fitness one would consequently expect that individual foragers will exclusively visit either patch 1 or patch 2 and that preferences for either patch will vary between foragers. The optimal solution of (S1) has the following form: (p*, p * ) = (0, 1)
if K < - ~ ,
(Pl*, p*) = (0, l ) or (p*, p*) = ( 1, O)
ifK = - ~ ,
(p~,p2) = (1,0)
if-~
(2oK,, ,sK-I) ~-~_~
(P~*,P2*)=\ ~ / (
if K>½.
Obviously, the optimal solution of (S l) depends on the value of K. The predicted foraging strategy based on the optimal solution of (S1) will be identical to that based on the optimal solution of ($2) only for one single value of the parameter K, i.e. K = - ~ . In all other cases predictions will differ. The following second example shows that even alternative pure strategies may be predicted on the basis of the two systems (S1) and ($2), thus providing the possibility of a strong inference test to decide between the two approaches experimentally. Let n = 2 and Xt, X2 be two normally distributed random variables with 2 V(X2)= 100, tz~ = E ( X ~ ) = 9 5 , (r2~= V(X~)=20 and /z2 = E ( X 2 ) = 6 0 , (r2= and R = 100. It can easily be seen that (p*, p * ) = (1,0) is the only optimal solution of ($2). The optimal solution of (S l) has the following form: (p*, p ~ ) = (0, 1)
if K < - 7 ,
(PI*, P*) = (0, 1) or (p)*, P2*) = (1, 0)
ifK = _7,
(p~*,p2*) = (1,0)
if-7<
(P*'P*)
= ( 4 0 K +7 8 K - 7 ~ \ 48K ' 48K ]
K_< 7'
if K > 7.
Thus, for K < - 7 alternative pure strategies are optimal and would accordingly be predicted on the basis of the two systems (S1) and ($2). It has been demonstrated (Caraco e t a l . , 1980; Caraco, 1981, 1983) that the same individuals show either risk-averse or risk-prone foraging preferences depending on whether they can expect a positive or negative 24 hr
RISK-SENSITIVE
FORAGING
221
energy budget. Unfortunately, in all those experiments the simulated food patches had equal mean rewards. For such a situation it has been shown (Stephens, 198I) that the 24 hr energy budget rule follows from the minimization of the probability of starvation. Thus, applying the 24 hr energy budget rule, both approaches will yield qualitatively the same predictions regarding the optimal foraging strategy. It is therefore not possible to decide between the two approaches on the basis of Caraco's experiments. Caraco & Lima (1984) suggested a method to discriminate between the two approaches. Their method is based on establishing and comparing indifference curves of reward distributions with respect to the mean and the variance or with respect to the mean and the standard deviation (for details see Caraco & Lima, 1"984). However, they have to admit that the foragers' behaviour was not sufficiently precise to distinguish the two approaches by their method. I suggest a different method here. To interpret the results of Caraco et al. (1980) and Caraco (1981) within the framework of his model (Caraco, 1980) one has to assume that individuals alter their utility functions (i.e. the parameter K ) depending on their 24 hr energy budget. By manipulating the 24 hr energy budget it should thus be possible to generate foraging conditions as in the two examples I have constructed above. My second example shows that only a rough estimate of the parameter K ( K < - 7 , see example 2) is necessary to conduct a discriminating experiment. Methods for estimating the parameter K are provided and described by Real, Ott & Silverfine (1982) and Caraco & Lima (1984). Since in my second example alternative pure strategies are optimal for K < _ 7 , a discriminating experiment which is conducted according to this example provides the possibility of a strong inference test and may thus serve to decide which of the two approaches is fitted best to describe animal foraging decisions in a stochastic environment. I want to thank E. Curio, M. Milinski, M. Visser, and J. Matiak for reading and thoughtfully criticizing the manuscript. I am also grateful to T. Caraco for comments on a former version of the manuscript. The study was supported by a grant from the Deutsche Forschungsgemeinschaft Cu 4/25-3.
REFERENCES BOOT, J. C. G. (t964). Quadratic Programming. Amsterdam: North-Holland. CARACO, T. (1980). Ecology 61, 119. CARACO, T. (1981). Behav. Ecol. Sociobiol. 8, 213. CARACO, T. (1983). Behav. Ecol. Sociobiol. 12, 63. CARACO, T., MARTINDALE, S. ,g" WHI'VrAM, T. S. (1980). Anim. Behav. 28, 820.
222
K. R E ( 3 E L M A N N
CARACO, T. & LIMA, S. L. (1984). Sixth Harvard Symposium on Quantitative Analyses of Behavior. FISHBURN, P. C. (1970). Utility Theory for Decision Making. New York: Wiley. KEEN EY, R. L. & RAI FFA, H. (1976). Decisions with Multiple Objectives: Preferences and Value Trade-offs. New York: Wiley. OSTER, G. F. & WILSON, E. O. (1978). Caste and Ecology in the Social Insects. Princeton, New Jersey: Princeton University Press. REAL, L. A. (1980a). Am. Nat. 115, 623. REAL, L. A. (1980b). In: Limits to Action: the Allocation of Individual Behavior (Staddon, J. E. R. ed.) pp. 37-64. New York: Academic Press. REAL, L A. (1981). Ecology 62, 20. REAL, L. A., OTr, J. & SILVERFINE, E. (1982). Ecology 63, 1617. STEPHENS, D. W. (1981). Anita. Behav. 29, 628. STEPHENS, D. W. & CHARNOV, E. L. (1982). Behav. Ecol. Sociobiol. 10, 251.
A Remark on the Theory of Risk-sensitive Foraging KLAUS R E G E L M A N N
Arbeitsgruppe fiir Verhaltensforschung, Abteilung fiir Biologic, Ruhr-Universitiit Bochum, Postfach 102148, 4630 Bochum 1, West Germany (Received 28 January 1984, and in revised form 7 March 1984) Recent theoretical (Caraco, 1980; Real, 1980a, b; Stephens, 1981; Stephens & Charnov, 1982) and empirical (Caraco, Martindale & Whittam, 1980; Real, 1981; Caraco, 1981, 1983) studies have demonstrated that animal foraging decisions in a stochastic environment should be, and really are, attuned not only to expected net benefits but also to the associated variances. Two proxy attributes of fitness have been proposed in the literature to model and describe risk-sensitive foraging preferences. Caraco (1980) employed mathematical utility theory (Fishburn, 1970; Keeney & Raiffa, 1976) to develop a model where a linear combination of the mean and the variance of the reward distribution is used as a proxy attribute of fitness. The model has the form:
E ( X ) - K V ( X ) -~ Max where E ( X ) is the mean and V(X) is the variance of the reward distribution described by the random variable X, and K is a constant expressing the forager's risk-sensitivity. Similar models were proposed by Oster & Wilson (1978) and Real ( 1980 a, b). In an alternative approach Stephens ( 1981) and Stephens & Charnov (1982) suggested to use the probability of death due to starvation as a proxy attribute of fitness. Their model has the form:
P ( X < R) ~ Min where X is a random variable describing the reward distribution, and R is some minimum level of net energy requirement. Until now, however, a satisfying experimental test to decide which of the two approaches is fitted best to describe animal foraging decisions in a stochastic environment is lacking (Caraco & Lima, 1984). In this study I will analytically derive some general characteristics of the optimal solutions of the two optimality approaches. Numerical examples will be given to show that the two approaches are in general not equivalent, 217 0022-5193/84/180217 + 06 $03.00/0
O 1984 Academic Press Inc. (London) Ltd.
218
K. R E G E L M A N N
i.e. they may lead to different predictions regarding the most preferred foraging strategy. By reference to these numerical examples experiments are suggested that may help to decide between the two approaches. Like Caraco (1980),.I will consider an animal that forages for a fixed time T per day, where total feeding time T is constrained by some other vital requirements. The animal has to allocate its feeding time between n food patches. In line with Caraco's model, I will assume that on a given day foraging decisions are made before feeding commences (for a discussion of this assumption see Caraco, 1983). Let Xj (l-< i -< n) denote the net energetic benefit when exploiting patch i for time T (intake minus feeding and travel costs in energetic units). Xi is assumed to be a normally distributed random variable with mean/zi = E(Xi) and variance 0-~ = V(Xi). All Xi are assumed to be mutually independent. Furthermore, let p~ be the proportion of the total feeding time T spent in patch i, i.e. 0-< p~ ~ 1 for l ~ i ~ n and n ~=1P~ = 1. For convenience, let p = ( p l , . . . , p,) and let A be the set o f all feasible allocation vectors p, i.e.
A={p=(P','",P,)lO<--Pi<--lf°rl<--i<--nand~} ' , P'=I =, Finally, let R denote the minimum energy requirement o f the animal considered. It is assumed that the animal will die at the end of the day or in the night thereafter if the daily net energy intake is less than R units. The probability o f death due to starvation is then expressed by P(X < R) where X = X(p) = ~ ~=~ p~Xi.According to Caraco (1980) the optimal allocation vector p * = ( p * , . . . , p * ) is determined by solving the maximization problem: E (X(p)) - KV(X(p)) ->Max (Sl) subject to p c A. (Sl) may equivalently be written in the form:
~ p,tz _ K ~ i=1
p i202- 1 ~ M a x
(S1)
i=l n
subject to 0<-pi < - I for 1 -< i -< n and ~.~=~ p~ = 1. Here K is a constant expressing the forager's risk-sensitivity. Caraco (1980) suggested that risk-aversion (i.e. K > 0 ) should occur when the forager can expect a positive 24 hr energy budget and risk-prone behaviour (i.e. K < 0 ) should occur when the forager can expect a negative 2 4 h r energy budget ("24 hr energy budget rule"). Since the variables X~ are normally distributed and mutually independent, the variable X = X ( p ) = ~ . ~ = ~ p~X~ is normally distributed with mean /z = E(X) =~=1P;/~i and variance 0 - 2 = V(X) = E i = 2 2 The variable ( X " " l P~0-1-
RISK-SENSITIVE
FORAGING
219
/z)/o. is normally distributed with mean 0 and variance 1. The probability of starvation is then given by P(X
< R) = P((X
- ~)/o. < (R -
a)lo.) = @ ( ( R
- p,)/or).
As @(.) is the cumulative distribution function of the standard normal distribution, the probability of starvation increases in a strictly monotonic way with ( R - p~)/cr. Therefore, the probability of starvation is minimal if and only if ( R - p , ) / o . is minimal. Thus we arrive at the minimization problem: R - E(X(p))
~
Min
($2)
,/v(x(p)) subject to p ~ A which may equivalently be written in the form: R - ~ PiP-i i=l ~ Min
tl
($2)
subject to 0 -< Pi -< 1 for 1 - i -< n and Y.~=1Pi = 1. When deriving the optimality systems (S 1) and ($2) I have assumed that the random variables X~ are normally distributed. This assumption will only seldom hold in reality. The assumption of normality of X~ may however be relaxed, i.e. the derivation of the system (S1) and ($2) is valid for a wider class of random variables, if the central limit theorem for independent random variables is invoked (see Caraco, 1980; Stephens, 1981; Stephens & Charnov, 1982). Some general characteristics of the systems (S1) and ($2) can be given as follows. (1) If K > 0, (S 1) is a standard problem of quadratic optimization (Boot, 1964). (2) If K-< 0, there always exists a pure strategy, i.e. p = ( p ~ , . . . , p , ) ~ A and p~ = 1 for one i, which is an optimal solution of (S1). (3) The system ($2) does not belong to a category of standard optimizan tion problems. If ~ ~=~p~/~ -< R for all p = ( p ~ , . . . , p,) c A and o.2 > 0 for all i, there always exists a pure strategy which is an optimal solution of ($2). The following two examples will show that the optimal solutions of (SI) and ($2) will not generally be identical. Let n = 2 and XI, X2 be two normally distributed random variables with /~ = E ( X ~ ) = 9 0 , o.2= V ( X I ) = 2 5 and /~2 ----"E ( X 2 ) = 80, o .2 = V ( X 2 ) = 100 and finally let R = 100.
220
K. REGELMANN
A short calculation using (3) yields that both pure strategies, i.e. ( p*, p*) = (1,0) and (p*, p*) = (0, 1), are optimal solutions of ($2) but no mixed strategy can be optimal. If minimization of the probability of starvation is used as a proxy attribute of fitness one would consequently expect that individual foragers will exclusively visit either patch 1 or patch 2 and that preferences for either patch will vary between foragers. The optimal solution of (S1) has the following form: (p*, p * ) = (0, 1)
if K < - ~ ,
(Pl*, p*) = (0, l ) or (p*, p*) = ( 1, O)
ifK = - ~ ,
(p~,p2) = (1,0)
if-~
(2oK,, ,sK-I) ~-~_~
(P~*,P2*)=\ ~ / (
if K>½.
Obviously, the optimal solution of (S l) depends on the value of K. The predicted foraging strategy based on the optimal solution of (S1) will be identical to that based on the optimal solution of ($2) only for one single value of the parameter K, i.e. K = - ~ . In all other cases predictions will differ. The following second example shows that even alternative pure strategies may be predicted on the basis of the two systems (S1) and ($2), thus providing the possibility of a strong inference test to decide between the two approaches experimentally. Let n = 2 and Xt, X2 be two normally distributed random variables with 2 V(X2)= 100, tz~ = E ( X ~ ) = 9 5 , (r2~= V(X~)=20 and /z2 = E ( X 2 ) = 6 0 , (r2= and R = 100. It can easily be seen that (p*, p * ) = (1,0) is the only optimal solution of ($2). The optimal solution of (S l) has the following form: (p*, p ~ ) = (0, 1)
if K < - 7 ,
(PI*, P*) = (0, 1) or (p)*, P2*) = (1, 0)
ifK = _7,
(p~*,p2*) = (1,0)
if-7<
(P*'P*)
= ( 4 0 K +7 8 K - 7 ~ \ 48K ' 48K ]
K_< 7'
if K > 7.
Thus, for K < - 7 alternative pure strategies are optimal and would accordingly be predicted on the basis of the two systems (S1) and ($2). It has been demonstrated (Caraco e t a l . , 1980; Caraco, 1981, 1983) that the same individuals show either risk-averse or risk-prone foraging preferences depending on whether they can expect a positive or negative 24 hr
RISK-SENSITIVE
FORAGING
221
energy budget. Unfortunately, in all those experiments the simulated food patches had equal mean rewards. For such a situation it has been shown (Stephens, 198I) that the 24 hr energy budget rule follows from the minimization of the probability of starvation. Thus, applying the 24 hr energy budget rule, both approaches will yield qualitatively the same predictions regarding the optimal foraging strategy. It is therefore not possible to decide between the two approaches on the basis of Caraco's experiments. Caraco & Lima (1984) suggested a method to discriminate between the two approaches. Their method is based on establishing and comparing indifference curves of reward distributions with respect to the mean and the variance or with respect to the mean and the standard deviation (for details see Caraco & Lima, 1"984). However, they have to admit that the foragers' behaviour was not sufficiently precise to distinguish the two approaches by their method. I suggest a different method here. To interpret the results of Caraco et al. (1980) and Caraco (1981) within the framework of his model (Caraco, 1980) one has to assume that individuals alter their utility functions (i.e. the parameter K ) depending on their 24 hr energy budget. By manipulating the 24 hr energy budget it should thus be possible to generate foraging conditions as in the two examples I have constructed above. My second example shows that only a rough estimate of the parameter K ( K < - 7 , see example 2) is necessary to conduct a discriminating experiment. Methods for estimating the parameter K are provided and described by Real, Ott & Silverfine (1982) and Caraco & Lima (1984). Since in my second example alternative pure strategies are optimal for K < _ 7 , a discriminating experiment which is conducted according to this example provides the possibility of a strong inference test and may thus serve to decide which of the two approaches is fitted best to describe animal foraging decisions in a stochastic environment. I want to thank E. Curio, M. Milinski, M. Visser, and J. Matiak for reading and thoughtfully criticizing the manuscript. I am also grateful to T. Caraco for comments on a former version of the manuscript. The study was supported by a grant from the Deutsche Forschungsgemeinschaft Cu 4/25-3.
REFERENCES BOOT, J. C. G. (t964). Quadratic Programming. Amsterdam: North-Holland. CARACO, T. (1980). Ecology 61, 119. CARACO, T. (1981). Behav. Ecol. Sociobiol. 8, 213. CARACO, T. (1983). Behav. Ecol. Sociobiol. 12, 63. CARACO, T., MARTINDALE, S. ,g" WHI'VrAM, T. S. (1980). Anim. Behav. 28, 820.
222
K. R E ( 3 E L M A N N
CARACO, T. & LIMA, S. L. (1984). Sixth Harvard Symposium on Quantitative Analyses of Behavior. FISHBURN, P. C. (1970). Utility Theory for Decision Making. New York: Wiley. KEEN EY, R. L. & RAI FFA, H. (1976). Decisions with Multiple Objectives: Preferences and Value Trade-offs. New York: Wiley. OSTER, G. F. & WILSON, E. O. (1978). Caste and Ecology in the Social Insects. Princeton, New Jersey: Princeton University Press. REAL, L. A. (1980a). Am. Nat. 115, 623. REAL, L. A. (1980b). In: Limits to Action: the Allocation of Individual Behavior (Staddon, J. E. R. ed.) pp. 37-64. New York: Academic Press. REAL, L A. (1981). Ecology 62, 20. REAL, L. A., OTr, J. & SILVERFINE, E. (1982). Ecology 63, 1617. STEPHENS, D. W. (1981). Anita. Behav. 29, 628. STEPHENS, D. W. & CHARNOV, E. L. (1982). Behav. Ecol. Sociobiol. 10, 251.