J Ethol (2013) 31:23–28 DOI 10.1007/s10164-012-0344-y
ARTICLE
Foraging behavior in stochastic environments Hiromu Ito • Takashi Uehara • Satoru Morita Kei-ichi Tainaka • Jin Yoshimura
•
Received: 25 April 2012 / Accepted: 19 August 2012 / Published online: 1 September 2012 Japan Ethological Society and Springer 2012
Abstract How do temporally stochastic environments affect risk sensitivity in foraging behavior? We build a simple model of foraging under predation risks in stochastic environments, where the environments change over generations. We analyze the effects of stochastic environments on risk sensitivity of foraging animals by means of the difference between the geometric mean fitness and the arithmetic mean fitness. We assume that foraging is associated with predation risks whereas resting in the nest is safe because it is free of predators. In each generation, two different environments with given food amounts and predation risks occur with a certain probability. The geometric mean optimum is independent of food amounts. In most cases of stochastic environments, risk-averse tendency is increased, but in some limited conditions, more risk-prone behavior is favored. Specifically, risk-prone tendency is increased when the variation in food amount increases. Our results imply that the optimal behavior depends on the
Electronic supplementary material The online version of this article (doi:10.1007/s10164-012-0344-y) contains supplementary material, which is available to authorized users. H. Ito T. Uehara S. Morita K. Tainaka J. Yoshimura (&) Department of Systems Engineering, Shizuoka University, 3-5-1 Johoku, Naka-ku, Hamamatsu 432-8561, Japan e-mail:
[email protected] J. Yoshimura Department of Environmental and Forest Biology, State University of New York College of Environmental Science and Forestry, Syracuse, NY 13210, USA J. Yoshimura Marine Biosystems Research Center, Chiba University, Uchiura, Kamogawa, Chiba 299-5502, Japan
probability distribution of environmental effects under all selection regimes. Keywords Stochastic environment Foraging theory Risk sensitivity Geometric mean fitness
Introduction It has long been recognized that adaptation in stochastic environments is often drastically different from that in stable environments (Cohen 1966; Lewontin and Cohen 1969; Schaffer 1974; Stearns 1976; Yoshimura and Clark 1991). Geometric mean fitness is used as a more precise measure of adaptation, instead of the usual ‘arithmetic’ mean fitness. The effects of environmental stochasticity have classically been thought to be limited just to certain life-history traits, e.g., risk-spreading dispersal, dormancy, and overlapping generations (Schaffer 1974; Stearns 1976). Recently, however, stochastic effects have been found to affect all phenotypic traits and these effects depend solely on the probability distribution of environmental effects, but not on the type of the trait itself (Yoshimura and Clark 1991; Yoshimura and Jansen 1996; Yoshimura et al. 2009). Therefore, any phenotypic trait can change dramatically when an environment varies stochastically across generations, e.g., clutch size, the size and number trade-off of seeds or eggs, and foraging behavior under predation risks (Yoshimura and Clark 1991). Foraging behavior under varying predation risks is a compelling topic, since it considers the essential stochasticity in an animal’s environment: uncertainty in both food and predators (Real and Caraco 1986; Stephens and Krebs 1986; Stephens et al. 2007; Houston et al. 2011). In classical analyses of foraging, animals are shown to respond to
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risk and variation in food quantity on a daily basis (Caraco 1980; Stephens and Charnov 1982). Thus, the optimality of foraging behavior is considered within the lifetime of an individual, i.e., during one generation. Here, risk sensitivity is demonstrated as avoidance of variance, skew, etc. (Caraco and Chasin 1984; Yoshimura and Shields 1987). In principle, the optimality criterion of foraging behavior in stochastic environments over multiple generations is known to be different from that within a single generation (Yoshimura and Clark 1991). However, differences in the predicted optimal behavior have remained unknown both on a qualitative and quantitative basis. We here investigate the effects of long-term environmental stochasticity on optimal foraging strategies using a simple foraging model. We evaluate both qualitative and quantitative differences between the geometric mean fitness and the traditional arithmetic mean.
in a stable or constant environment from the aspect of multiple generations. Average food amounts and predation risks of a generation do not change over generations. Risk sensitivity is thus evaluated in a ‘stable’ environment. However, the environments may also vary drastically over generations. Therefore, we compare the risk sensitivity in the environments varying over generations (geometric mean fitness) with those varying within a generation (arithmetic mean fitness).
Results The arithmetic mean fitness A and the geometric mean fitness G are given by X A ¼ p1 u1 þ p2 u2 ¼ pu ð2Þ i¼1;2 i i and
Model
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Here, we build a simple model of foraging time allocation under varying predation risks and food quantity. We assume that foraging is associated with risks, while remaining in the nest is safe because it is free of predators (Lima et al. 1985; Lima 1985; Stephens and Krebs 1986; Yoshimura and Clark 1991). We set parameters ai and bi as the amount of food and the abundance of predators, respectively, in the i-th environment. Then, we assume predation events occur randomly over foraging time rate x (0 B x B 1). From this assumption, the probability of no predation (0 times of predation) for the duration of x follows a Poisson distribution, ebi x (Lima et al. 1985; Lima 1985). In addition, we can express food gain such as ai 9 x. The fitness function is defined as the product of the food gain and the probability of survival escaping from predation (survival rate). This is the simplest model of foraging with a basic trade-off between food gains and predation risks. We evaluate the long-term optimality of foraging behavior in animals with discrete generations (Yoshimura and Clark 1991). Animals in one generation are assumed to experience one of two distinctive environments (i = 1, 2), with a given probability (pi, where p1 ? p2 = 1). Here, the fitness ui(x) at one generation is expressed as ui ðxÞ ¼ ai x ebi x
ð1Þ
In order to evaluate the effects of stochastic environments on the risk sensitivity of a foraging animal, we compare the geometric mean fitness of Eq. (1) with its equivalent arithmetic mean fitness. In the traditional studies of foraging behavior, risk sensitivity is analyzed
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Note that A C G for any x. The maximum A (= A* = A(xA )) cannot be solved analytically, but determined numerically by calculating its derivative dA dx ¼ 0. In contrast, the optimal G (= G* = G(xG )) is solved directly as xG ¼
1 1 ¼P b1 p1 þ b2 p2 i¼1;2 bi pi
ð4Þ
Surprisingly, the optimal foraging time xG is independent of food amounts ai, but the value of the optimal geometric mean fitness G(xG ) depends on ai. We here evaluate the effects of stochastic environments in terms of the two parameters: food amount ai and predator abundance bi (Fig. 1). When b1 = b2, there is no stochastic environment effect and in stochastic environments, such that xG = xA . Therefore, we restrict our analysis to the conditions under which one environment (E1) will have fewer predators than the other environment (E2), such that b1 \ b2, i.e., environment E1 is advantageous with respect to predation pressure (1/b1 [ 1/b2). Figure 1 shows examples of fitness functions against foraging time duration x. We plot the fitness functions in environments E1 and E2 together with their arithmetic mean fitness A and geometric mean fitness G, assuming the two environments occur with equal probability, i.e., p1 = p2 = 0.5 (Fig. 1a, c). The arithmetic and geometric means are generally unimodal with a single peak, or monotonic increasing or decreasing functions. The relationship between the arithmetic means and the geometric means are plotted in A–G phase trajectories indicating the
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Fig. 1 Fitness functions for foraging time allocation under predation risks in stochastic environments. a, b Environment E1 has fewer predators and more food than E2, such that b1 \ b2 and a1 [ a2. Parameters are b1 = 3, b2 = 24, a1 = 5, and a2 = 2.5. c, d E1 has fewer predators but less food than E2, such that b1 \ b2 but a1 \ a2. Parameters are b1 = 3, b2 = 24, a1 = 5, and a2 = 60. a, c Potential fitness functions of E1 and E2, u1 and u2, and their corresponding
arithmetic and geometric means, A and G, are plotted against foraging time ratio x (x = 0,…,1). b, d The trajectory of foraging time is plotted in the phase plain of arithmetic mean and geometric mean fitnesses. Thin arrows indicate the trajectory of foraging time starting at the origin (x = 0) to the end (x = 1). Max {G(x)} and max {A(x)} are indicated by down arrows and left arrows, respectively. The probabilities of E1 and E2 are set equal, such that p1 = p2 = 0.5
differences in two optima (Fig. 1b, d). Note that the inequality in the two means remains constant, such that G(x) B A(x). The A–G trajectory always starts at the origin, i.e., x = 0 and increases towards x = 1 (arrows in Fig. 1b, d). We define behavior as ‘more risk-averse’ when the geometric mean optimum xG (down arrow) is reached before the arithmetic mean counterpart xA (left arrow), i.e., xG \ xA (Fig. 1b), ‘more risk-prone’ when the order is opposite, i.e., xG [ xA (Fig. 1d), and ‘equivalently risksensitive’ when the two optima are equal, i.e., xG = xA . We now analyze the difference in the optimal behavior D = xA - xG (Fig. 2). By definition, foraging behavior is more risk-averse if D [ 0, equivalent if D = 0 and more
risk-prone if D \ 0.Assuming D = 0, we get (see Supporting Information for derivations): b2 b1 ð5Þ a2 ¼ ca1 where c ¼ exp b1 p1 þ b2 p2 Therefore, we can partition ai-parameter space into the following three cases (Fig. 2a): Case 1: More risk-averse (D [ 0), if a2 \ ca1 Case 2: Equivalently risk-sensitive (D = 0), if a2 = ca1 and Case 3: More risk-prone (D \ 0), if a2 [ ca1 We also plot D, when the probabilities of the two environments are varied (p1 = 0–1 and p2 = 1 - p1).
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Fig. 2 The risk sensitivity analyses of foraging time allocation in stochastic environments. a The phase plane of food abundance a1 and a2. Equivalently risk-sensitive boundary is indicated by the solid line, where its right side is more risk-averse (xG \ xA ) and its left side is more risk-prone (xG [ xA ). Note that darker colors indicate large differences between the two optimal behaviors xG and xA . The probability of E1 and E2 are set equal, i.e., p1 = p2 = 0.5. The predator abundances are set: b1 = 2 and b2 = 10. b The effects of probability p1 (=1 - p2) on the difference in the optimal behaviors
D = xA - xG . The food amount in E2 is varied, such that a2 = 1 (filled circle), 6 (filled triangle), 20 (filled square), 40 (cross), and 200 (open circle). The other parameters are kept constant: (a1, b1, b2) = (4, 2, 10). c The effects of food amount a1 = 0–10 on the optimal strategies xA (dots) and xG (solid line). The food amount in E2 is varied, such that a2 = 5 (filled circle), 10 (filled triangle), 20 (open circle). The other parameters are kept constant: (b1, b2) = (2, 10). The probabilities of E1 and E2 are set equal, such that p1 = p2 = 0.5
When food amount a2 is increased in E2 (note b1 \ b2), more risk-prone behavior is predicted (Fig. 2b). We also show each optimal strategy xA (various dots) and xG (solid line), when the food amount in E1 is varied (a1 = 0–10). The other parameters are kept constant (Fig. 2c). The geometric mean optimal behavior xG is constant because it is independent of the amount of food available. The arithmetic mean optimal behavior xA depends on food amounts a1 and a2. More risk-prone behavior (xG [ xA ) appears when environment E1 has a small food amount (small a1), as well as when environment E2 has a large food amount (large a2). Typical examples of A–G trajectories are shown in Fig. 3. If a1 [ a2, the optimal behavior is always more risk-averse (D [ 0), since E1 is better than E2 with respect to both ai and bi (since 1/b1 [ 1/b2) (Fig. 3a–c). In contrast, for a1 \ a2, the optimal behavior is more risk-prone when the difference in predator abundance is small between the two environments (Fig. 3d, e), but becomes more risk-averse when the difference becomes large (Fig. 3f). Note that the difference in bi increases with the absolute value of D (|D|).
Discussion
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We have here refined and extended traditional mathematical foraging models under stochastic environments. It is very surprising that the food amounts (ai) have no effect on xG (the optimal foraging time), even though food quantity (ai) affects optimal geometric mean fitness G(xG ). This is because this model assumes that fitness is proportional to the collected food amount (aix). By introducing thresholds for starvation and/or full stomach, the geometric mean optimum may become dependent on the food amount, i.e., ui ðxÞ ¼ f ðai Þx ebi x , where f(ai) is a nonlinear function. Such effects may be easily shown by using dynamic modeling approaches (Mangel and Clark 1986, 1988), but the analytic solution may not be tractable. In previous studies of foraging, risk-averse behavior has been frequently predicted (see Stephens and Krebs 1986; Stephens et al. 2007). Our results suggest that stochastic environments generally promote more risk-averse tendencies (Figs. 1a, b, 2, 3a–c, f). However, we show more riskprone behavior may be promoted under a wide range of conditions (Figs. 1c, d, 2, 3d, e). Note that more risk-averse
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Fig. 3 Various trajectories of foraging time in the phase plain of arithmetic mean and geometric mean fitnesses. The optimal behavior is more risk-averse in a–c and f, while more risk-prone in d and
e. Parameters are set: a–c (a1, a2) = (5, 2.5), d–f (a1, a2) = (5, 60), a, d (b1, b2) = (12,24), b, e (b1, b2) = (3,24), and c, f (b1, b2) = (1,24). See Fig. 1 for notations
tendency can be very strong, while more risk-prone is relatively weak (Fig. 2). Thus, the prediction by geometric mean fitness may be either more risk-averse or more riskprone, depending not only on the values of the two parameters (ai and bi) but also on the probability distributions (pi) of each environment. Interestingly, the equivalently risk-sensitive boundary (D = 0) is solved analytically (Eq. 5), though the arithmetic mean optimum xA is not analytically solvable (see Supporting Information). Thus, our analyses suggest that more risk-averse behavior should be much more common in foraging behavior, but more risk-prone behavior may occur under rather limited conditions. Traditional studies evaluate risk sensitivity in the context of environmental variation and risks in daily life (Caraco 1980; Stephens and Charnov 1982; Caraco and Chasin 1984; Real and Caraco 1986; Stephens and Krebs 1986; Houston and McNamara 1999; Stephens et al. 2007). In stochastic environments, animals, including humans, are expected to have evolved risk sensitivity as a response favoring an
increase of geometric mean fitness (Yoshimura and Clark 1991). Our results show that generally more risk-averse behavior is further promoted by stochastic environments. But when food is abundant, the results may be reversed to more risk-prone behavior (Fig. 3). Such adaptation may broaden variation in behavioral strategies, such as variation in niche breadth and nutrient value (Houston et al. 2011; Mayntz et al. 2005; Dussutour et al. 2010), reducing digestive rates (Armstrong and Schindler 2011). In view of arithmetic mean fitness, this would appear to be a maladaptive strategy. Our analyses may be applicable to the cases of apparent maladaptive strategy that cannot be explained by the traditional approaches (McNamara and Houston 1987; Henly et al. 2008). Our results may imply that the effects of inter-generationally stochastic environments can be not only quantitatively different but also qualitatively different from stable environments. Even though more risk-averse behavior is likely to be the prevailing strategy, more risk-prone behavior may become adaptive when the probability distributions of the environments (pi) are altered (Fig. 2b).
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28 Acknowledgments We thank Derek A. Roff and Donald G. Miller for the comments on the manuscript. This work was supported by grants-in-aid from the Ministry of Education, Culture, Sports, Science and Technology of Japan to J.Y. (No. 22370010) and K.T. (No. 20500204).
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