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Significance of memory properties in prey choice decisions
Ecological Modelling 115 (1999) 177 – 189
Significance of memory properties in prey choice decisions Heikki Hirvonen *, Esa Ranta, Hannu Rita, Nina Peuhkuri Integrati6e Ecology Unit, Di6ision of Population Biology, Department of Ecology and Systematics, PO Box 17, FIN-00014, Uni6ersity of Helsinki, Helsinki, Finland
Abstract To forage efficiently in a spatially and temporally heterogeneous environment requires that an individual’s information from the immediate past is combined with information from the more distant past to track environmental change. We made use of a model involving exponentially devaluating weights for past events to emulate behaviour of the individual’s memory. As the devaluation rate increases, more weight is given to the most recent events. First, performance of individuals with different memory properties was tested in simulations in which two prey types with different profitabilities were available in different proportions. In a structurally stable prey environment a low memory devaluation rate gave better estimation of prey proportions than a high memory devaluation rate. In a highly variable environment, on the contrary, individuals with high devaluation rate could more quickly correct their estimates as prey availability changed, although this was achieved with the cost of high error rate of the estimate. Second, the ability to reliably assess relative abundances of the prey types proved to increase an individual’s success in prey choice (according to the decision rules by the optimal prey choice model). Third, in further simulations individuals were allowed to adjust their memory devaluation rate according to experience from their success in prey choice decisions in previous patches. We found that there was no need to adjust a high devaluation memory in a highly variable environment, but foragers starting with low devaluation value rather rapidly shifted to high devaluation rates. In a relatively stable environment the situation was reversed and finally all foragers used low devaluation rates. These results imply that the variation in estimation efficacy of prey availability may be critical in terms of optimal prey choice and thus memory properties should be included in examinations of prey choice. Including individual variation in foraging performance in individual-based models could increase our understanding of the consequences of these differences at the population and community levels. © 1999 Elsevier Science B.V. All rights reserved. Keywords: Foraging theory; Learning; Memory; Optimal foraging; Prey selection
1. Introduction * Corresponding author. Tel.: + 358-9-1917374; fax: + 3589-1917492; e-mail: [email protected].
‘‘It’s a poor sort of memory that only works backwards’’ White Queen to Alice in ‘Through the
0304-3800/99/$ - see front matter © 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 8 0 0 ( 9 8 ) 0 0 1 9 1 - 4
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Looking-Glass’ by Lewis Carroll, MacMillan, London, 1896. Understanding the patterns and ways in which animals utilize food resources is fundamental to the study of any animal population and community. Resource intake rate for individual foragers may be very different, even when the resource is plentiful (Sutherland, 1996). This is due to the fact that foraging efficiency may differ considerably between classes of animals within the population and between individuals within a class. Class differences have been related to, for example, size or age (Marchetti and Price, 1989; Hirvonen and Ranta, 1996a). However, even between individuals within a class there may be big differences in rates of energy intake (Ranta and Nuutinen, 1985; Ehlinger, 1989). This might be because of differences in individual abilities in utilizing the prey available, but also because individuals utilize different prey species or sizes of those available (Holbrook and Schmitt, 1992). These sort of differences can have a genetic base (Lemon, 1993), or maybe due to learning (Hughes, 1997).
1.1. Optimal foraging and indi6idual-based modelling When having access to the same prey environment, different predator individuals may adopt different strategies to utilize the food resource. Optimal foraging theory (Stephens and Krebs, 1986) explains how foragers maximizing their net energy intake may do this in different types of prey environments. How individuals make their prey choice decisions has profound consequences on their own energetic input and thereby on the output, i.e. growth, development, and body reserves, which in turn influence age of maturation and reproductive success (Calow, 1994), but also on the dynamics of the prey populations and structure of the prey community (Persson et al., 1997). For example, size-selective predation has been shown to regulate the structure of zooplankton communities (Zaret, 1980; Lazzaro, 1987) and may have a major impact on prey population dynamics (Fryxell and Lundberg, 1997). Fryxell and Lundberg (1997) constructed a model using individuals with two different prey
choice strategies, those taking the prey indiscriminately when encountered and those adopting the optimal prey choice strategy (Charnov, 1976a; Stephens and Krebs, 1986). When prey abundances are low, there are no differences between the impacts of the two strategies, but as prey abundances increase, differences in prey selection may cause profound differences on dynamics and stability (Fryxell and Lundberg, 1997). This example shows that an individual-based approach has the potential to increase our understanding of fundamental processes in population ecology. However, in the above model individuals adopting a certain foraging strategy were considered similar to each other in terms of foraging efficiency and prey choice. This is certainly not the case in nature, recent empirical and theoretical work has indicated that many properties of an individual influence its feeding rate and diet (Blanckenhorn, 1991; Krebs and Inman, 1992) and these in turn affect population interactions (Werner, 1992; Persson et al., 1997). Real individual-based models (Huston et al., 1988; DeAngelis and Matsinos, 1996; Uchman´ski and Grimm, 1996) take the advantage of individual differences to better capture and understand population and community dynamics. Foraging theories, on the other hand, have emphasized individuals as the units of selection (Ce´zilly et al., 1991) and hence the significance of optimal behaviour on individual fitness, but have paid little attention to the consequences of individual foraging decisions on dynamics of populations and communities. However, combining individualbased modelling and foraging theory would prove a productive approach to proceed. Individualbased modelling has recently been used to simulate, for example, fish predation and population growth (DeAngelis et al. 1991; Petersen and DeAngelis, 1992; Rice et al., 1993; Scheffer et al., 1995). Still, Breck’s study (Breck, 1993) on first-year growth of bluegill sunfish is probably the only one involving optimal foraging theory, at least to some extent, in individual-based modelling. One of the major properties of a foraging individual is its ability to estimate prey abundance, which is dependent on its cognitive abilities (Krebs and Inman, 1992). Therefore any differences in these
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abilities might also affect prey selectivity and consequently the rate of net energy intake, which is often used as a measure of individual fitness, but can also influence prey population dynamics. Thus far, differences in individual cognitive abilities have mostly been neglected in studies of prey choice. When aiming at building individual-based models including foraging behaviour, the effect of individual properties on prey choice and proportional prey mortality should be integrated into model construction. As a first step to this direction, we first present a model of forager memory, and then show how memory quality affects prey choice decisions of individuals adopting the same foraging strategy, optimal prey choice. Finally we discuss the implications of differences in individual memory properties for population and community consequences.
2. Foraging and memory Functional models of foraging are concerned with behaviours that represent goals, but usually they do not consider the adaptive processes involved in attaining these goals (Provenza and Cincotta, 1993). For example, conventional foraging models assume that individual foragers maximise the net rate of energy gain while foraging and regard this as a correlate of individual fitness (Stephens and Krebs, 1986). These models often assume environments as static and that foraging individuals have complete information of the global properties of the environment, and consequently, of the model parameters. Indeed, foraging success of individuals has been shown to increase individual fitness (Blanckenhorn, 1991). However, to forage efficiently in a spatially and temporally heterogeneous environment requires that individuals are able to acquire and integrate different sources of information from their environment. Therefore a fundamental question about behavioural ecology of foraging is how such information from the immediate past is combined with information from the more distant past to track environmental change. As foraging individuals are not omniscient, the way this information is acquired is still unclear (Shettleworth et al.,
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1993) and is one of the most challenging questions in the study of foraging behaviour (Krebs et al., 1983; Krebs and Inman, 1992; Stephens, 1993). To take advantage of the heterogeneous environment an individual forager should be able to assess differences in resource distribution among patches (Milinski and Regelmann, 1985; Valone, 1992a) and the resource abundance of specific patches (McNamara, 1982; McNamara et al., 1993). Depending on an individual’s cognitive properties, two main outcomes are possible. First, if we assume that the individual forager is not capable to acquire patch-specific information, all patches should be treated identically (McNair, 1983; Valone, 1992b). On the other hand, if the individual can assess resource density instantaneously and accurately, foraging decisions should —in terms of energy maximisation— generally follow the predictions set by theories on prey selection and patch usage (Charnov, 1976a,b; Stephens and Krebs, 1986). Foraging individuals in nature probably lie somewhere between the two extremes, which means that they should be, at least to some extent, capable of assessing patch quality and resource availability within patches. The accuracy needed to meet the rate maximising principle, however, depends on the decision process used (Valone and Brown, 1989). Natural selection would thus be expected to favour individuals with memory update systems that permit efficient exploitation of available food resources. Empirical evidence suggests that foraging individuals may be capable to assess prey density, e.g. by using encounter rates (Hughes, 1979; Jaeger et al., 1982; Shettleworth and Plowright, 1992; Samu, 1993). Foragers may also be able to track changes in the prey environment in the course of foraging (Shettleworth et al., 1988; Cuthill et al., 1990, 1994; Hirvonen and Ranta, 1996b). Such tracking skills require some type of memory, time perception and learning. To track changes in a useful way, individuals must distinguish between local stochastic fluctuations in parameter values from more significant directional movement. Two approaches have been used to model declining reliability of information over time. Discarding is used in memory window models to estimate environmental parameters over a range of experiences,
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dropping the oldest events as new ones are added to the estimate (Cowie, 1977; Valone, 1992a). Another approach involves devaluating outdated information with relative weighting of past and present experience (Kacelnik et al., 1987; Devenport and Devenport, 1994). The way to track environmental change with minimal memory load is to form the current estimate of the environmental parameter by using an exponentially weighted moving average, EWMA (Killeen, 1981; Dow and Lea, 1987; McNamara and Houston, 1987a). Here we make use of a model involving exponentially devaluating weights for past events to emulate behaviour of an individual’s memory. We then investigate how memory devaluation influences (i) foragers’ estimation of prey availability, and (ii) their subsequent performance in prey choice decisions in terms of maximising net rate of energy gain in different types of environments. As there should be an optimal memory window length or devaluation factor that depends on the predictability of the environment, we tested (iii) how memory devaluation rate of individual foragers changes in the course of foraging through a series of patches in different types of environments. 3. A model for memory The treatment here is based on the optimal prey model (Stephens and Krebs, 1986; Crawley and Krebs, 1992), except that we relax the complete information assumption. We assume that a forager discovers a food patch and commences encountering the prey items, which are of two types, 1 (the more profitable) and 2 (the less profitable), in a random sequence with given average encounter rates. The decision rule is that the better prey type should be always taken upon encounter. The decision to take the poorer prey depends on the encounter rate, l1, of the better type only. If it is high enough to promote specialisation in the better type (in terms of net rate of energy gain), the poorer prey should be ignored, regardless of its encounter rate, l2. Thus the decision to specialise or generalise is based on a threshold encounter rate with the better prey type, which is set by the model (Stephens and Krebs, 1986).
Let Ik be the indicator for the type of the kth prey item, if it is of the more profitable type 1, Ik = 1, otherwise Ik = 0. The index k runs from 0 to . The indicator I0 is for the current prey item and Ik for the prey k items earlier. Throughout the current treatment the occurrences of the prey types are assumed to be independent. The proportion p= P(Ik = 1) of the more profitable prey in the patch is an indicator of availability of that prey type and it is estimated by:
pˆ = (1− e − a) % e − akIk
(1a)
k=0
The parameter a\ 0 and describes the devaluation behaviour of the memory by determining the relative weight of recent and past experience (note that the weights wk = (1− e − a)× e − ak sum up to unity). Small values of a indicate that somewhat equal weight is given to the memory of past experience, while with large a only the most recent encounters are weighted. Basic properties of the memory model are treated in Appendix. The Eq. (1a) assumes an indefinitely long foraging history. Obviously, this is not always the case. Therefore the Eq. (1a) has to be modified to acknowledge that the forager already has encountered —in addition to the current prey— a limited number, m, of prey items. The estimate of the more profitable prey items in the patch, based on the m+1 prey encountered, is: pˆm =
m 1− e − a % e − akIk − (m + 1)a 1− e k=0
(1b)
Note, that in the summation the index m affects the memory window length. This, in turn, is dependent on the value of the parameter a (however, practice shows that values of m\ 100 are of little importance for Eq. (1b)). In our model the estimate of p is updated upon every encounter.
3.1. Memory in estimating prey a6ailability The model Eq. (1b) was used to explore how individual foragers with low (a=0.01) and high (a= 1.0) devaluation rates of the past events perform in estimating availability (the proportion, p= P(Ik = 1)) of the more profitable prey type in a stable and a variable environment with two
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types of prey. The appearance of the two prey types in our simulations is random and sequential, as is assumed by the optimal prey model (Stephens and Krebs, 1986). The first assessment was run in a prey environment where the proportion of the better prey type was p =0.65, with a total of 100 encounters. For both the low and high memory devaluation rates the procedure was repeated for 100 times. As a measure of performance in estimation of the availability of the better prey type we scored mean of 100 foragers with 95% confidence limits of the mean. The second measure of the forager’s performance, called the error rate, is the proportion of the estimates falling outside 910% margin of p. The second evaluation of memory performance of foragers with either a low or a high memory devaluation rate was run in an environment where two patch types alternated as follows: the first patch included 50 encounters with p= 0.7; the second 50 encounters, p= 0.2; the third 20 encounters, p = 0.7; and the fourth 100 encounters with p = 0.2. Simulations in this environment were also run for 100 times. In the structurally stable environment foragers with low memory devaluation rate performed better in estimating prey proportions (Fig. 1A). Error rate of their estimates also decreased with a decelerating rate with increasing number of encounters with prey (Fig. 1C). In contrast, estimates of p by foragers with high memory devaluation rate tended to fluctuate around the real value with rather broad 95% confidence limits (Fig. 1B) and error rate of their estimate remained high through the whole sequence of encounters (Fig. 1C). In the fluctuating environment, however, it appeared that foragers with high memory devaluation rate performed better in estimation of p. High devaluation of past encounters in favour of recent ones provided rapid responses to new prey environments, although this was achieved with the cost of relatively high error rate. Foragers with low devaluation rate, however, only slowly approached the correct estimate, because they were averaging over more than the current patch. Therefore their error rates were also high, especially in small patches.
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3.2. Memory and prey choice To demonstrate how prey availability estimation affects prey choice, one example was run in a fluctuating environment (see above) for both de-
Fig. 1. Panels (A) and (B) give the estimates pˆm of p =0.65 by foragers with low (a =0.01) and high (a =1) memory devaluation rates in a patch with 100 encounters (mean and the 0.025 and 0.975 percentiles of the approximate normal distribution of the estimate). Length of the memory window m is equal to k. Panel (C) graphs the error rate as a measure of accuracy in estimating p by the two memory types.
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Fig. 2. A schematic illustration of the simulation procedure in a variable environment with four consecutive patches. (A) Simulated sequences of prey encounters for the more profitable (black triangles) and the less profitable prey type (white triangles). (B) Simulated estimates of the better prey type by an individual forager with a low memory devaluation rate (a = 0.01). (C) Simulated estimates of the better prey type by an individual forager with a high memory devaluation rate (a = 1).
valuation rates with the actual encounter sequence shown (Fig. 2). If the critical threshold proportion, l1, of the better prey type for specialisation in that type is, for example, l1 =0.5 (Fig. 2), then with an estimate ] 0.5 the forager decides to ignore the less profitable prey item, but when B0.5, the forager is a generalist taking both prey types when encountered. Hence specialisation would be a correct decision in the first and third prey patch (Fig. 2) where p= 0.7, while it would be incorrect in patches two and four with p= 0.2. As all estimates by the individual with a low memory devaluation rate were above the
threshold value throughout the first prey patch (Fig. 2B), all decisions were correct. In the next patch, due to erroneous estimates, poorer prey was not taken, although it would have been beneficial. In the end of the second patch, however, due to continuous but slow updating of the estimate, pˆm, the forager started to make correct decisions to generalise. Because it underestimated the availability of the better prey it missed to specialise in the third patch, but correctly behaved as a generalist in the fourth patch (Fig. 2B). Although the pˆm by the individual with a high memory devaluation rate varied widely around
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the correct value in patch one, it mostly remained above the threshold value (Fig. 2C). Therefore the forager made correct decisions to specialise for most prey encounters. In the second patch it correctly chose a generalist tactic, as the estimates remained below 0.5. In the third patch the forager quickly adjusted its estimate and succeeded to specialise with only few occasional errors. In the fourth patch the forager was a generalist, except a few errors leading to specialisation in the end of the patch (Fig. 2C). The actual outcome of foraging, in terms of correct prey choice decisions and thus higher energy gain for a given memory devaluation rate, depends on both the structure of the environment and the threshold value for specialisation in the better prey. As was shown here, even a highly erroneous estimate may produce the right decision, if the threshold density is not too close to the density of the better prey type. Exponentially weighted averaging proved to be an efficient way to process past and present information in environments with small scale fluctuations. In general, it turned out that low memory devaluation rate was superior in estimating prey availability in a stable and high devaluation rate in a variable prey environment. Therefore it would be advantageous for an individual to be able to adjust its information usage in response to large scale changes in the environment. This clearly requires a feedback from the environment and success of the memory employed. An individual’s experience reveals some previously unknown attribute of the environment and this presumably leads to a change in behaviour.
3.3. Adjusting memory to experience from foraging success Adjusting memory devaluation rate to match the current environment would result in better estimates of prey availability yielding higher energy rewards. We simulated prey choice behaviour and memory adjustments in variable and stable environments, where patch size and the proportion of the better prey type were random variables from a known distribution.
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First, patch sizes were random draws from uniform distribution between 5 and 50, Uniform(5,50). In the variable environment the proportion of the better prey type pi in the different patches were drawn randomly from Uniform(0.3,0.8), while in the stable environment pi were drawn randomly from Uniform(0.5,0.6). Foraging individuals with a given initial memory devaluation rate (either low, a=0.01, or high, a= 1.0) were then let to forage in the two environments. The threshold level for specialisation in the most rewarding prey was set to pi = 0.55, which was also the global average of the proportion of the better prey type in the foraging environment. To make correct decisions on prey choice, when pi ] 0.55 and pˆm ] 0.55, the individual should ignore type 2 prey items upon encounter, while with pi B 0.55 and pˆm B 0.55 both prey types should be taken indiscriminately. Wrong diet decisions follow if pi B 0.55 (generalise) and pˆm ] 0.55 (specialise), or pi ] 0.55 (specialise) and pˆm B 0.55 (generalise). In both environments, each forager was first let to go through 50 prey patches. The forager could evaluate its performance in foraging decisions by scoring the net energy intake, which is directly related to the proportion of correct prey choice decisions (according to the energy maximisation rule), for this set of patches (Score 1), henceforth called cycle. After the first cycle the forager was allowed to adjust its initial memory devaluation rate by 5% to a random direction. Using the new devaluation rate the forager continued for another cycle of 50 prey patches, and now scored the energy intake obtained during the second cycle (Score 2). Before starting the next cycle of 50 patches, the forager compared the two previous scores. If Score 1 BScore 2, the parameter a was adjusted by 5% to the same direction as in the previous adjustment (either increased or decreased devaluation rate) between cycles 1 and 2, otherwise the direction of the adjustment was reversed. Thereafter, a similar comparison between scores of the two latest cycles and the respective adjustment in memory was repeated until the end of the 300 cycle run. Four different settings were used: an initially slow and fast memory devaluation rate in a fluctuating and in a more stable foraging
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Fig. 3. Development of high (a=1, A and C) or low (a= 0.01, B and D) initial memory devaluation rates of foragers in a variable (A and B) and in a more stable (C and D) patchy environment for 300 cycles of 50 patches. For each setting five randomly selected trajectories are shown. The insets give mean and 95% confidence limits for 100 replicated runs. The foragers were allowed to adjust their memory devaluation rate after every cycle to the direction evaluated as better in terms of net energy intake during the two most recent cycles. Energy obtained is related to the proportion of correct prey choice decisions (according to energy maximization).
environment. Altogether 100 replicated runs were done with each of the four settings. In the more variable environment the individuals starting with a relatively high memory devaluation rate (a=1) did not much change their memory during foraging (Fig. 3A). However, foragers starting with a relatively low devaluation rate (a=0.01) gradually shifted towards higher
devaluation rates and ended up with a devaluation rate fluctuating around a = 0.5 to 1.0 after some 80–100 memory refreshment cycles (Fig. 3B). Because no buffer system was used in the memory refreshment procedure, there is much small scale fluctuation in individual values of a over the course of the 300 memory refreshment cycles (Fig. 3A).
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Fig. 3. (Continued)
In the low variance environment the foragers with relatively high initial memory devaluation rate (a= 1.0) gradually reduced their devaluation rate with successive refreshment cycles resulting in fluctuating around a low rate of a =0.01 after almost 200 cycles (Fig. 3C). No such change was observable with individuals initially starting with a low memory devaluation rate (a = 0.01), though even here local fluctuations were rather high (Fig. 3D).
4. Discussion We assumed an individual forager’s memory quality to contribute to prey density estimates, which in turn would result in correct or false decisions on optimal prey choice. The examples show that foraging performance, which was based on optimal decisions on prey choice, was dependent on both memory properties as well as environmental variability. This interplay between
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memory quality and structure of the environment would be an essential selective factor. This was further indicated in the last simulation where we found that when the foragers were allowed to alter their memory they gradually attained a memory devaluation rate best matched to the current environment. In these investigations we used a memory model with exponential weighting the past events. The memory parameter determines how past and recent experience is weighted. Although models of this kind perhaps do not perfectly account for memory function (Kacelnik et al., 1987; Devenport and Devenport, 1994), they give a useful framework to analyse the integration of previous and present experience in behavioural allocations and during learning. In accordance with the nature of our memory devaluation model, foragers show much variation in how they integrate and weight past experience (Real, 1992; Shettleworth and Plowright, 1992; Mackney and Hughes, 1995). Many observations indicate that individuals often devaluate past experience, sometimes responding only to the most recent events (Kacelnik and Todd, 1992; Cuthill et al., 1994). Acquiring information during foraging and somehow referring to the past experience gives an individual the advantage to more efficiently utilise the food resources in the environment and may therefore be of crucial importance for the decisions on prey choice (Krebs and Inman, 1992). We found that as well in estimation of prey availability as in actual prey choice decisions, individuals devaluating past experience with a high rate, performed better in the more variable environment. In contrast, foragers with low memory devaluation rate were more successful in stable environments, or in environments with smaller variance in patch quality. The simulations also indicated that if foragers are able to use the information they gather from the environment and from their own success in foraging decisions, they can adjust their memory to better match the characteristics of the environment. This accords with theories on information use in foraging suggesting that memory window should be adjusted to the stability of the environment, either through evolution or through individual experience (Cuthill et al., 1990; Real, 1991). In much the same
way as in this study with prey choice, McNamara and Houston (1987a), modelling patch use by foragers, found that with increasing predictability of the environment, an optimal forager should give more weight to the observations in the more distant past. On the other hand, based on a model used to examine patch estimation Valone (1992a) concluded that the degree of environmental variation has little influence on the optimal size of the memory window. Effects of memory quality on optimal prey choice have not been modelled before, but empirical studies related to food selection have given somewhat unambiguous results (Shettleworth and Plowright, 1992; Mackney and Hughes, 1995). The difference in memory qualities in terms of energy intake would depend on the combination of three factors: accuracy of the estimation, detailed structure of the prey environment and the threshold availability of the better prey for specialisation in that prey type. In accord with our simulation results, Shettleworth and Plowright (1992) showed that pigeons having experience with different degrees of environmental predictability appeared to change the length of the memory window. In addition, they found that pigeons reacted disproportionately to very recent experience and accepted more of the poorer food items than expected by the optimal prey choice model. This was due to an equal probability of accepting a poor item after a poor item as after a good item rather than responding only to a correlate of density of good items (Shettleworth and Plowright, 1992). The optimal prey model predicts that acceptance of the poorer of two prey types should depend only on the density of the better type (Stephens and Krebs, 1986). Our results and those by Shettleworth and Plowright (1992) support suggestions that variance or error in the predator’s estimate of the relevant environmental parameters is one source of deviations from optimality predictions in studies of prey selection (Stephens and Krebs, 1986; McNamara and Houston, 1987b). What animals learn about the state of the environment may affect their subsequent foraging decisions (Krebs and Inman, 1992). Therefore, error or variability in the forager’s estimate of prey type availability
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may be one of the reasons for partial preferences frequently found in tests of the model of optimal prey choice (McNamara and Houston, 1987b). Such inaccuracies in prey estimates could be related to the individual’s ability to acquire and process information. Incomplete information may cause the foragers to fail to maximise their rate of energy intake. In our model the estimate of p was updated at every encounter (Killeen, 1981), which is a more realistic approach than many others (Kacelnik et al., 1987). If a forager is to track changes in the environment, its information gathering and processing system should be sensitive to small scale changes and should employ a continuous updating procedure. This is evident for example in studies on sampling behaviour (Stephens, 1987; Shettleworth et al., 1988; Krebs and Inman, 1992). However, we found that individuals starting with low memory devaluation rate in the high variance environment could faster adjust their memory to match the new situation than individuals with high initial rate in the low variance environment. This was mainly because individuals in the latter situation were constrained by their high devaluation rate, which quickly responded also to stochastic variations in prey proportions. Thus, the forager may make wrong decisions on prey choice because it encounters a brief run-of-abad-luck, as has been noticed also in patch departure decisions (Inoue, 1983; Valone, 1992b). These findings emphasise the importance of including information processing and learning in modelling behavioural ecology of individual foraging. Capability to assess patch quality and prey density or proportion is an essential constituent of any behavioural decision based on these environmental attributes. We suggest that the most effective predators are those capable to adjust their devaluation rate of the past to match the quality of the current environment. Besides optimal prey choice, decisions on search tactics, for example, usually are dependent on prey density (Inoue, 1983; Hirvonen, 1998), or an individual’s estimate of it. There are clear differences among different species in how they use past experience (Real, 1993). Recent empirical evidence (Blanckenhorn, 1991) also indicates high variation among individ-
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uals’ capacity to assess patch profitability and among their spatial memory and learning ability, which are connected to foraging success contributing directly to individual fitness (Blanckenhorn, 1991). To figure out how flexibly individual foragers are able to change the usage of past and present information is an important direction for future research. If individual foragers differ considerably in energy intake rate and foraging efficiency, there might be large differences in the consequent growth rates of the individuals. This might mean that the individuals with the highest growth rates would achieve maturation earlier than their conspecifics. If the properties allowing high rates of intake are heritable, during some generations the mean age of maturation decreases. Such a shift in life-history characteristics could also affect population dynamics, and the relationship between the forager population and their prey, and thus community dynamics. On the other hand, intensive selective predation may change prey population dynamics (Fryxell and Lundberg, 1997) and prey community structure (Lazzaro, 1987). Here we studied the effects of memory properties on prey choice of solitary foragers, but in many natural situations individual foraging decisions and performance are dependent on the properties and performance of other individuals (Ranta et al., 1995; Ranta and Rita, 1998). Therefore, to increase realism, interactions between foraging individuals should be integrated into models of optimal foraging, including also the consequences at levels higher than the individual. This can be achieved by applying individual-based modelling.
Appendix A. Characteristics of the memory model As the expectation of Ik is E[Ik ]= p for all k, E[pˆ ] =E
n
k=0
k=0
% wkIk = % wkE[Ik ]= p % wk = p
k=0
(A1) Thus, for any choice of the memory parameter a, the estimate pˆ of the true population proportion p is unbiased. As the variance of Ik is:
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V[Ik ] =p(1− p),
(A2)
we have, using the independency assumption: V[pˆ ] =V
% wkIk k=0
n
= % w 2kV[Ik ] k=0
= p(1−p)
1− e − a 1+e − a
(A3)
For any positive a, V[pˆ ]B p(1 − p). Variance of pˆ approaches zero as a approaches zero. As a increases, the variance of the estimator pˆ increases and finally approaches the variance of the most recent indicator p(1−p) as a approaches infinity. Large values of a correspond to a high devaluation rate and thus fast reactions to even occasional indicator values in the most recent encounters.
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H. Hir6onen et al. / Ecological Modelling 115 (1999) 177–189 state operant behavior. Elsevier/North Holland, Amsterdam, pp. 21 – 34. Krebs, J.R., Inman, A.J., 1992. Learning and foraging: individuals, groups, and populations. Am. Nat. 140, S63–S84. Krebs, J.R., Stephens, D.W., Sutherland, W.J., 1983. Perspectives in optimal foraging. In: Brush, A.H., Clark, G.A. (Eds.), Perspectives in ornithology. Cambridge University Press, NY, pp. 165 – 216. Lazzaro, X., 1987. A review of planktivorous fishes: Their evolution, feeding behaviours, selectivities, and impacts. Hydrobiologia 146, 97–167. Lemon, W.C., 1993. Heritability of selectively advantageous foraging behaviour in a small passerine. Evol. Ecol. 7, 421 – 428. Mackney, P.A., Hughes, R.N., 1995. Foraging behaviour and memory window in sticklebacks. Behaviour 132, 1241– 1253. Marchetti, K., Price, T., 1989. Differences in the foraging of juvenile and adult birds: the importance of developmental constraints. Biol. Rev. 64, 51–70. McNair, J.N., 1983. A class of patch use strategies. Am. Zool. 23, 303 – 313. McNamara, J.M., 1982. Optimal patch use in a stochastic environment. Theor. Pop. Biol. 21, 269–285. McNamara, J.M., Houston, A.I., 1987. Memory and the efficient use of information. J. Theor. Biol. 125, 385–395. McNamara, J.M., Houston, A.I., 1987. Partial preferences and foraging. Anim. Behav. 35, 1084–1099. McNamara, J.M., Houston, A.I., Weisser, W.W., 1993. Combining prey choice and patch use—what does ratemaximizing predict? J. Theor. Biol. 164, 219–238. Milinski, M., Regelmann, K., 1985. Fading short-term memory for patch quality in sticklebacks. Anim. Behav. 33, 678 – 680. Persson, L., Diehl, S., Eklo¨v, P., Christensen, B., 1997. Flexibility in fish behaviour: consequences at the population and community levels. In: Godin, J.G.J. (Ed.), Behavioural ecology of teleost fishes. Oxford University Press, Oxford, pp. 316 – 343. Petersen, J.H., DeAngelis, D.L., 1992. Functional response and capture timing in an individual-based model: Predation by northern squafish (Ptychocheilus oregonensis) on juvenile salmonids in the Columbia River. Can. J. Fish. Aquat. Sci. 49, 2551 –2565. Provenza, F.D., Cincotta, R.P., 1993. Foraging as a self-organizational learning process: accepting adaptability at the expense of predictability. In: Hughes, R.N. (Ed.), Diet selection: an interdisciplinary approach to foraging behaviour. Blackwell Scientific Publications, Oxford, pp. 78– 101. Ranta, E., Nuutinen, V., 1985. Foraging by the smooth newt (Triturus 6ulgaris) on zooplankton: functional responses and diet choice. J. Anim. Ecol. 54, 275–293. Ranta, E., Rita, H., Peuhkuri, N., 1995. Patch exploitation, group foraging, and unequal competitors. Behav. Ecol. 6, 1–5. .
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Ranta, E., Rita, H., 1998. Competition in a group of equal foragers. Am. Nat. 152, 71 – 81. Real, L.A., 1991. Animal choice behaviour and the evolution of cognitive architecture. Science 253, 980 – 986. Real, L.A., 1992. Information processing and the evolutionary ecology of cognitive architecture. Am. Nat. 140, S108 – S145. Real, L.A., 1993. Toward a cognitive ecology. Trends Ecol. Evol. 8, 413 – 417. Rice, J.A., Miller, T.J., Rose, K.A., Crowder, L.B., Marschall, E.A., Trebitz, A.S., DeAngelis, D.L., 1993. Growth rate variation and larval survival: Inferences from an individual-based size-dependent predation model. Can. J. Fish. Aquat. Sci. 50, 133 – 142. Samu, F., 1993. Wolf spider feeding strategies: optimality of prey consumption in Pardosa hortensis. Oecologia 94, 139 – 145. Scheffer, M., Baveco, J.M., DeAngelis, D.L., Lammens, E.H.R.R., Shuter, B., 1995. Stunted growth and stepwise die-off in animal cohorts. Am. Nat. 145, 376 – 388. Shettleworth, S.J., Krebs, J.R., Stephens, D.W., Gibbon, J., 1988. Tracking a fluctuating environment: a study of sampling. Anim. Behav. 36, 87 – 105. Shettleworth, S.J., Plowright, C.M.S., 1992. How pigeons estimate rates of prey encounter. J. Exp. Psychol. 18, 219 – 235. Shettleworth, S.J., Reid, P.J., Plowright, M.S., 1993. The psychology of diet selection. In: Hughes, R.N. (Ed.), Diet selection: an interdisciplinary approach to foraging behaviour. Blackwell Scientific Publications, Oxford, pp. 56 – 77. Stephens, D.W., 1987. On economically tracking a variable environment. Theor. Pop. Biol. 32, 15 – 25. Stephens, D.W., 1993. Learning and behavioral ecology: incomplete information and environmental predictability. In: Papaj, D.R., Lewis, A.C. (Eds.), Insect Learning: ecological and evolutionary perspectives. Chapman and Hall, NY, pp. 195 – 218. Stephens, D.W., Krebs, J.R., 1986. Foraging theory. Princeton University Press, Princeton. Sutherland, W.J., 1996. From individual behaviour to population ecology. Oxford University Press, Oxford. Uchman´ski, J., Grimm, V., 1996. Individual-based modelling in ecology: what makes the difference? Trends Ecol. Evol. 11, 437 – 441. Valone, T.J., 1992. Patch estimation via memory windows and the effect of travel time’. J. Theor. Biol. 157, 243 – 251. Valone, T.J., 1992. Information for patch assessment: a field investigation with black-chinned hummingbirds. Behav. Ecol. 3, 211 – 222. Valone, T.J., Brown, J.S., 1989. Measuring patch assessment abilities of desert granivores. Ecology 70, 1800 – 1810. Werner, E.E., 1992. Individual behavior and higher-order species interactions’. Am. Nat. 140, S5 – S32. Zaret, T.M., 1980. Predation and freshwater communities. Yale University Press, New Haven.
Significance of memory properties in prey choice decisions Heikki Hirvonen *, Esa Ranta, Hannu Rita, Nina Peuhkuri Integrati6e Ecology Unit, Di6ision of Population Biology, Department of Ecology and Systematics, PO Box 17, FIN-00014, Uni6ersity of Helsinki, Helsinki, Finland
Abstract To forage efficiently in a spatially and temporally heterogeneous environment requires that an individual’s information from the immediate past is combined with information from the more distant past to track environmental change. We made use of a model involving exponentially devaluating weights for past events to emulate behaviour of the individual’s memory. As the devaluation rate increases, more weight is given to the most recent events. First, performance of individuals with different memory properties was tested in simulations in which two prey types with different profitabilities were available in different proportions. In a structurally stable prey environment a low memory devaluation rate gave better estimation of prey proportions than a high memory devaluation rate. In a highly variable environment, on the contrary, individuals with high devaluation rate could more quickly correct their estimates as prey availability changed, although this was achieved with the cost of high error rate of the estimate. Second, the ability to reliably assess relative abundances of the prey types proved to increase an individual’s success in prey choice (according to the decision rules by the optimal prey choice model). Third, in further simulations individuals were allowed to adjust their memory devaluation rate according to experience from their success in prey choice decisions in previous patches. We found that there was no need to adjust a high devaluation memory in a highly variable environment, but foragers starting with low devaluation value rather rapidly shifted to high devaluation rates. In a relatively stable environment the situation was reversed and finally all foragers used low devaluation rates. These results imply that the variation in estimation efficacy of prey availability may be critical in terms of optimal prey choice and thus memory properties should be included in examinations of prey choice. Including individual variation in foraging performance in individual-based models could increase our understanding of the consequences of these differences at the population and community levels. © 1999 Elsevier Science B.V. All rights reserved. Keywords: Foraging theory; Learning; Memory; Optimal foraging; Prey selection
1. Introduction * Corresponding author. Tel.: + 358-9-1917374; fax: + 3589-1917492; e-mail: [email protected].
‘‘It’s a poor sort of memory that only works backwards’’ White Queen to Alice in ‘Through the
0304-3800/99/$ - see front matter © 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 8 0 0 ( 9 8 ) 0 0 1 9 1 - 4
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Looking-Glass’ by Lewis Carroll, MacMillan, London, 1896. Understanding the patterns and ways in which animals utilize food resources is fundamental to the study of any animal population and community. Resource intake rate for individual foragers may be very different, even when the resource is plentiful (Sutherland, 1996). This is due to the fact that foraging efficiency may differ considerably between classes of animals within the population and between individuals within a class. Class differences have been related to, for example, size or age (Marchetti and Price, 1989; Hirvonen and Ranta, 1996a). However, even between individuals within a class there may be big differences in rates of energy intake (Ranta and Nuutinen, 1985; Ehlinger, 1989). This might be because of differences in individual abilities in utilizing the prey available, but also because individuals utilize different prey species or sizes of those available (Holbrook and Schmitt, 1992). These sort of differences can have a genetic base (Lemon, 1993), or maybe due to learning (Hughes, 1997).
1.1. Optimal foraging and indi6idual-based modelling When having access to the same prey environment, different predator individuals may adopt different strategies to utilize the food resource. Optimal foraging theory (Stephens and Krebs, 1986) explains how foragers maximizing their net energy intake may do this in different types of prey environments. How individuals make their prey choice decisions has profound consequences on their own energetic input and thereby on the output, i.e. growth, development, and body reserves, which in turn influence age of maturation and reproductive success (Calow, 1994), but also on the dynamics of the prey populations and structure of the prey community (Persson et al., 1997). For example, size-selective predation has been shown to regulate the structure of zooplankton communities (Zaret, 1980; Lazzaro, 1987) and may have a major impact on prey population dynamics (Fryxell and Lundberg, 1997). Fryxell and Lundberg (1997) constructed a model using individuals with two different prey
choice strategies, those taking the prey indiscriminately when encountered and those adopting the optimal prey choice strategy (Charnov, 1976a; Stephens and Krebs, 1986). When prey abundances are low, there are no differences between the impacts of the two strategies, but as prey abundances increase, differences in prey selection may cause profound differences on dynamics and stability (Fryxell and Lundberg, 1997). This example shows that an individual-based approach has the potential to increase our understanding of fundamental processes in population ecology. However, in the above model individuals adopting a certain foraging strategy were considered similar to each other in terms of foraging efficiency and prey choice. This is certainly not the case in nature, recent empirical and theoretical work has indicated that many properties of an individual influence its feeding rate and diet (Blanckenhorn, 1991; Krebs and Inman, 1992) and these in turn affect population interactions (Werner, 1992; Persson et al., 1997). Real individual-based models (Huston et al., 1988; DeAngelis and Matsinos, 1996; Uchman´ski and Grimm, 1996) take the advantage of individual differences to better capture and understand population and community dynamics. Foraging theories, on the other hand, have emphasized individuals as the units of selection (Ce´zilly et al., 1991) and hence the significance of optimal behaviour on individual fitness, but have paid little attention to the consequences of individual foraging decisions on dynamics of populations and communities. However, combining individualbased modelling and foraging theory would prove a productive approach to proceed. Individualbased modelling has recently been used to simulate, for example, fish predation and population growth (DeAngelis et al. 1991; Petersen and DeAngelis, 1992; Rice et al., 1993; Scheffer et al., 1995). Still, Breck’s study (Breck, 1993) on first-year growth of bluegill sunfish is probably the only one involving optimal foraging theory, at least to some extent, in individual-based modelling. One of the major properties of a foraging individual is its ability to estimate prey abundance, which is dependent on its cognitive abilities (Krebs and Inman, 1992). Therefore any differences in these
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abilities might also affect prey selectivity and consequently the rate of net energy intake, which is often used as a measure of individual fitness, but can also influence prey population dynamics. Thus far, differences in individual cognitive abilities have mostly been neglected in studies of prey choice. When aiming at building individual-based models including foraging behaviour, the effect of individual properties on prey choice and proportional prey mortality should be integrated into model construction. As a first step to this direction, we first present a model of forager memory, and then show how memory quality affects prey choice decisions of individuals adopting the same foraging strategy, optimal prey choice. Finally we discuss the implications of differences in individual memory properties for population and community consequences.
2. Foraging and memory Functional models of foraging are concerned with behaviours that represent goals, but usually they do not consider the adaptive processes involved in attaining these goals (Provenza and Cincotta, 1993). For example, conventional foraging models assume that individual foragers maximise the net rate of energy gain while foraging and regard this as a correlate of individual fitness (Stephens and Krebs, 1986). These models often assume environments as static and that foraging individuals have complete information of the global properties of the environment, and consequently, of the model parameters. Indeed, foraging success of individuals has been shown to increase individual fitness (Blanckenhorn, 1991). However, to forage efficiently in a spatially and temporally heterogeneous environment requires that individuals are able to acquire and integrate different sources of information from their environment. Therefore a fundamental question about behavioural ecology of foraging is how such information from the immediate past is combined with information from the more distant past to track environmental change. As foraging individuals are not omniscient, the way this information is acquired is still unclear (Shettleworth et al.,
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1993) and is one of the most challenging questions in the study of foraging behaviour (Krebs et al., 1983; Krebs and Inman, 1992; Stephens, 1993). To take advantage of the heterogeneous environment an individual forager should be able to assess differences in resource distribution among patches (Milinski and Regelmann, 1985; Valone, 1992a) and the resource abundance of specific patches (McNamara, 1982; McNamara et al., 1993). Depending on an individual’s cognitive properties, two main outcomes are possible. First, if we assume that the individual forager is not capable to acquire patch-specific information, all patches should be treated identically (McNair, 1983; Valone, 1992b). On the other hand, if the individual can assess resource density instantaneously and accurately, foraging decisions should —in terms of energy maximisation— generally follow the predictions set by theories on prey selection and patch usage (Charnov, 1976a,b; Stephens and Krebs, 1986). Foraging individuals in nature probably lie somewhere between the two extremes, which means that they should be, at least to some extent, capable of assessing patch quality and resource availability within patches. The accuracy needed to meet the rate maximising principle, however, depends on the decision process used (Valone and Brown, 1989). Natural selection would thus be expected to favour individuals with memory update systems that permit efficient exploitation of available food resources. Empirical evidence suggests that foraging individuals may be capable to assess prey density, e.g. by using encounter rates (Hughes, 1979; Jaeger et al., 1982; Shettleworth and Plowright, 1992; Samu, 1993). Foragers may also be able to track changes in the prey environment in the course of foraging (Shettleworth et al., 1988; Cuthill et al., 1990, 1994; Hirvonen and Ranta, 1996b). Such tracking skills require some type of memory, time perception and learning. To track changes in a useful way, individuals must distinguish between local stochastic fluctuations in parameter values from more significant directional movement. Two approaches have been used to model declining reliability of information over time. Discarding is used in memory window models to estimate environmental parameters over a range of experiences,
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dropping the oldest events as new ones are added to the estimate (Cowie, 1977; Valone, 1992a). Another approach involves devaluating outdated information with relative weighting of past and present experience (Kacelnik et al., 1987; Devenport and Devenport, 1994). The way to track environmental change with minimal memory load is to form the current estimate of the environmental parameter by using an exponentially weighted moving average, EWMA (Killeen, 1981; Dow and Lea, 1987; McNamara and Houston, 1987a). Here we make use of a model involving exponentially devaluating weights for past events to emulate behaviour of an individual’s memory. We then investigate how memory devaluation influences (i) foragers’ estimation of prey availability, and (ii) their subsequent performance in prey choice decisions in terms of maximising net rate of energy gain in different types of environments. As there should be an optimal memory window length or devaluation factor that depends on the predictability of the environment, we tested (iii) how memory devaluation rate of individual foragers changes in the course of foraging through a series of patches in different types of environments. 3. A model for memory The treatment here is based on the optimal prey model (Stephens and Krebs, 1986; Crawley and Krebs, 1992), except that we relax the complete information assumption. We assume that a forager discovers a food patch and commences encountering the prey items, which are of two types, 1 (the more profitable) and 2 (the less profitable), in a random sequence with given average encounter rates. The decision rule is that the better prey type should be always taken upon encounter. The decision to take the poorer prey depends on the encounter rate, l1, of the better type only. If it is high enough to promote specialisation in the better type (in terms of net rate of energy gain), the poorer prey should be ignored, regardless of its encounter rate, l2. Thus the decision to specialise or generalise is based on a threshold encounter rate with the better prey type, which is set by the model (Stephens and Krebs, 1986).
Let Ik be the indicator for the type of the kth prey item, if it is of the more profitable type 1, Ik = 1, otherwise Ik = 0. The index k runs from 0 to . The indicator I0 is for the current prey item and Ik for the prey k items earlier. Throughout the current treatment the occurrences of the prey types are assumed to be independent. The proportion p= P(Ik = 1) of the more profitable prey in the patch is an indicator of availability of that prey type and it is estimated by:
pˆ = (1− e − a) % e − akIk
(1a)
k=0
The parameter a\ 0 and describes the devaluation behaviour of the memory by determining the relative weight of recent and past experience (note that the weights wk = (1− e − a)× e − ak sum up to unity). Small values of a indicate that somewhat equal weight is given to the memory of past experience, while with large a only the most recent encounters are weighted. Basic properties of the memory model are treated in Appendix. The Eq. (1a) assumes an indefinitely long foraging history. Obviously, this is not always the case. Therefore the Eq. (1a) has to be modified to acknowledge that the forager already has encountered —in addition to the current prey— a limited number, m, of prey items. The estimate of the more profitable prey items in the patch, based on the m+1 prey encountered, is: pˆm =
m 1− e − a % e − akIk − (m + 1)a 1− e k=0
(1b)
Note, that in the summation the index m affects the memory window length. This, in turn, is dependent on the value of the parameter a (however, practice shows that values of m\ 100 are of little importance for Eq. (1b)). In our model the estimate of p is updated upon every encounter.
3.1. Memory in estimating prey a6ailability The model Eq. (1b) was used to explore how individual foragers with low (a=0.01) and high (a= 1.0) devaluation rates of the past events perform in estimating availability (the proportion, p= P(Ik = 1)) of the more profitable prey type in a stable and a variable environment with two
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types of prey. The appearance of the two prey types in our simulations is random and sequential, as is assumed by the optimal prey model (Stephens and Krebs, 1986). The first assessment was run in a prey environment where the proportion of the better prey type was p =0.65, with a total of 100 encounters. For both the low and high memory devaluation rates the procedure was repeated for 100 times. As a measure of performance in estimation of the availability of the better prey type we scored mean of 100 foragers with 95% confidence limits of the mean. The second measure of the forager’s performance, called the error rate, is the proportion of the estimates falling outside 910% margin of p. The second evaluation of memory performance of foragers with either a low or a high memory devaluation rate was run in an environment where two patch types alternated as follows: the first patch included 50 encounters with p= 0.7; the second 50 encounters, p= 0.2; the third 20 encounters, p = 0.7; and the fourth 100 encounters with p = 0.2. Simulations in this environment were also run for 100 times. In the structurally stable environment foragers with low memory devaluation rate performed better in estimating prey proportions (Fig. 1A). Error rate of their estimates also decreased with a decelerating rate with increasing number of encounters with prey (Fig. 1C). In contrast, estimates of p by foragers with high memory devaluation rate tended to fluctuate around the real value with rather broad 95% confidence limits (Fig. 1B) and error rate of their estimate remained high through the whole sequence of encounters (Fig. 1C). In the fluctuating environment, however, it appeared that foragers with high memory devaluation rate performed better in estimation of p. High devaluation of past encounters in favour of recent ones provided rapid responses to new prey environments, although this was achieved with the cost of relatively high error rate. Foragers with low devaluation rate, however, only slowly approached the correct estimate, because they were averaging over more than the current patch. Therefore their error rates were also high, especially in small patches.
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3.2. Memory and prey choice To demonstrate how prey availability estimation affects prey choice, one example was run in a fluctuating environment (see above) for both de-
Fig. 1. Panels (A) and (B) give the estimates pˆm of p =0.65 by foragers with low (a =0.01) and high (a =1) memory devaluation rates in a patch with 100 encounters (mean and the 0.025 and 0.975 percentiles of the approximate normal distribution of the estimate). Length of the memory window m is equal to k. Panel (C) graphs the error rate as a measure of accuracy in estimating p by the two memory types.
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Fig. 2. A schematic illustration of the simulation procedure in a variable environment with four consecutive patches. (A) Simulated sequences of prey encounters for the more profitable (black triangles) and the less profitable prey type (white triangles). (B) Simulated estimates of the better prey type by an individual forager with a low memory devaluation rate (a = 0.01). (C) Simulated estimates of the better prey type by an individual forager with a high memory devaluation rate (a = 1).
valuation rates with the actual encounter sequence shown (Fig. 2). If the critical threshold proportion, l1, of the better prey type for specialisation in that type is, for example, l1 =0.5 (Fig. 2), then with an estimate ] 0.5 the forager decides to ignore the less profitable prey item, but when B0.5, the forager is a generalist taking both prey types when encountered. Hence specialisation would be a correct decision in the first and third prey patch (Fig. 2) where p= 0.7, while it would be incorrect in patches two and four with p= 0.2. As all estimates by the individual with a low memory devaluation rate were above the
threshold value throughout the first prey patch (Fig. 2B), all decisions were correct. In the next patch, due to erroneous estimates, poorer prey was not taken, although it would have been beneficial. In the end of the second patch, however, due to continuous but slow updating of the estimate, pˆm, the forager started to make correct decisions to generalise. Because it underestimated the availability of the better prey it missed to specialise in the third patch, but correctly behaved as a generalist in the fourth patch (Fig. 2B). Although the pˆm by the individual with a high memory devaluation rate varied widely around
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the correct value in patch one, it mostly remained above the threshold value (Fig. 2C). Therefore the forager made correct decisions to specialise for most prey encounters. In the second patch it correctly chose a generalist tactic, as the estimates remained below 0.5. In the third patch the forager quickly adjusted its estimate and succeeded to specialise with only few occasional errors. In the fourth patch the forager was a generalist, except a few errors leading to specialisation in the end of the patch (Fig. 2C). The actual outcome of foraging, in terms of correct prey choice decisions and thus higher energy gain for a given memory devaluation rate, depends on both the structure of the environment and the threshold value for specialisation in the better prey. As was shown here, even a highly erroneous estimate may produce the right decision, if the threshold density is not too close to the density of the better prey type. Exponentially weighted averaging proved to be an efficient way to process past and present information in environments with small scale fluctuations. In general, it turned out that low memory devaluation rate was superior in estimating prey availability in a stable and high devaluation rate in a variable prey environment. Therefore it would be advantageous for an individual to be able to adjust its information usage in response to large scale changes in the environment. This clearly requires a feedback from the environment and success of the memory employed. An individual’s experience reveals some previously unknown attribute of the environment and this presumably leads to a change in behaviour.
3.3. Adjusting memory to experience from foraging success Adjusting memory devaluation rate to match the current environment would result in better estimates of prey availability yielding higher energy rewards. We simulated prey choice behaviour and memory adjustments in variable and stable environments, where patch size and the proportion of the better prey type were random variables from a known distribution.
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First, patch sizes were random draws from uniform distribution between 5 and 50, Uniform(5,50). In the variable environment the proportion of the better prey type pi in the different patches were drawn randomly from Uniform(0.3,0.8), while in the stable environment pi were drawn randomly from Uniform(0.5,0.6). Foraging individuals with a given initial memory devaluation rate (either low, a=0.01, or high, a= 1.0) were then let to forage in the two environments. The threshold level for specialisation in the most rewarding prey was set to pi = 0.55, which was also the global average of the proportion of the better prey type in the foraging environment. To make correct decisions on prey choice, when pi ] 0.55 and pˆm ] 0.55, the individual should ignore type 2 prey items upon encounter, while with pi B 0.55 and pˆm B 0.55 both prey types should be taken indiscriminately. Wrong diet decisions follow if pi B 0.55 (generalise) and pˆm ] 0.55 (specialise), or pi ] 0.55 (specialise) and pˆm B 0.55 (generalise). In both environments, each forager was first let to go through 50 prey patches. The forager could evaluate its performance in foraging decisions by scoring the net energy intake, which is directly related to the proportion of correct prey choice decisions (according to the energy maximisation rule), for this set of patches (Score 1), henceforth called cycle. After the first cycle the forager was allowed to adjust its initial memory devaluation rate by 5% to a random direction. Using the new devaluation rate the forager continued for another cycle of 50 prey patches, and now scored the energy intake obtained during the second cycle (Score 2). Before starting the next cycle of 50 patches, the forager compared the two previous scores. If Score 1 BScore 2, the parameter a was adjusted by 5% to the same direction as in the previous adjustment (either increased or decreased devaluation rate) between cycles 1 and 2, otherwise the direction of the adjustment was reversed. Thereafter, a similar comparison between scores of the two latest cycles and the respective adjustment in memory was repeated until the end of the 300 cycle run. Four different settings were used: an initially slow and fast memory devaluation rate in a fluctuating and in a more stable foraging
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Fig. 3. Development of high (a=1, A and C) or low (a= 0.01, B and D) initial memory devaluation rates of foragers in a variable (A and B) and in a more stable (C and D) patchy environment for 300 cycles of 50 patches. For each setting five randomly selected trajectories are shown. The insets give mean and 95% confidence limits for 100 replicated runs. The foragers were allowed to adjust their memory devaluation rate after every cycle to the direction evaluated as better in terms of net energy intake during the two most recent cycles. Energy obtained is related to the proportion of correct prey choice decisions (according to energy maximization).
environment. Altogether 100 replicated runs were done with each of the four settings. In the more variable environment the individuals starting with a relatively high memory devaluation rate (a=1) did not much change their memory during foraging (Fig. 3A). However, foragers starting with a relatively low devaluation rate (a=0.01) gradually shifted towards higher
devaluation rates and ended up with a devaluation rate fluctuating around a = 0.5 to 1.0 after some 80–100 memory refreshment cycles (Fig. 3B). Because no buffer system was used in the memory refreshment procedure, there is much small scale fluctuation in individual values of a over the course of the 300 memory refreshment cycles (Fig. 3A).
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Fig. 3. (Continued)
In the low variance environment the foragers with relatively high initial memory devaluation rate (a= 1.0) gradually reduced their devaluation rate with successive refreshment cycles resulting in fluctuating around a low rate of a =0.01 after almost 200 cycles (Fig. 3C). No such change was observable with individuals initially starting with a low memory devaluation rate (a = 0.01), though even here local fluctuations were rather high (Fig. 3D).
4. Discussion We assumed an individual forager’s memory quality to contribute to prey density estimates, which in turn would result in correct or false decisions on optimal prey choice. The examples show that foraging performance, which was based on optimal decisions on prey choice, was dependent on both memory properties as well as environmental variability. This interplay between
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memory quality and structure of the environment would be an essential selective factor. This was further indicated in the last simulation where we found that when the foragers were allowed to alter their memory they gradually attained a memory devaluation rate best matched to the current environment. In these investigations we used a memory model with exponential weighting the past events. The memory parameter determines how past and recent experience is weighted. Although models of this kind perhaps do not perfectly account for memory function (Kacelnik et al., 1987; Devenport and Devenport, 1994), they give a useful framework to analyse the integration of previous and present experience in behavioural allocations and during learning. In accordance with the nature of our memory devaluation model, foragers show much variation in how they integrate and weight past experience (Real, 1992; Shettleworth and Plowright, 1992; Mackney and Hughes, 1995). Many observations indicate that individuals often devaluate past experience, sometimes responding only to the most recent events (Kacelnik and Todd, 1992; Cuthill et al., 1994). Acquiring information during foraging and somehow referring to the past experience gives an individual the advantage to more efficiently utilise the food resources in the environment and may therefore be of crucial importance for the decisions on prey choice (Krebs and Inman, 1992). We found that as well in estimation of prey availability as in actual prey choice decisions, individuals devaluating past experience with a high rate, performed better in the more variable environment. In contrast, foragers with low memory devaluation rate were more successful in stable environments, or in environments with smaller variance in patch quality. The simulations also indicated that if foragers are able to use the information they gather from the environment and from their own success in foraging decisions, they can adjust their memory to better match the characteristics of the environment. This accords with theories on information use in foraging suggesting that memory window should be adjusted to the stability of the environment, either through evolution or through individual experience (Cuthill et al., 1990; Real, 1991). In much the same
way as in this study with prey choice, McNamara and Houston (1987a), modelling patch use by foragers, found that with increasing predictability of the environment, an optimal forager should give more weight to the observations in the more distant past. On the other hand, based on a model used to examine patch estimation Valone (1992a) concluded that the degree of environmental variation has little influence on the optimal size of the memory window. Effects of memory quality on optimal prey choice have not been modelled before, but empirical studies related to food selection have given somewhat unambiguous results (Shettleworth and Plowright, 1992; Mackney and Hughes, 1995). The difference in memory qualities in terms of energy intake would depend on the combination of three factors: accuracy of the estimation, detailed structure of the prey environment and the threshold availability of the better prey for specialisation in that prey type. In accord with our simulation results, Shettleworth and Plowright (1992) showed that pigeons having experience with different degrees of environmental predictability appeared to change the length of the memory window. In addition, they found that pigeons reacted disproportionately to very recent experience and accepted more of the poorer food items than expected by the optimal prey choice model. This was due to an equal probability of accepting a poor item after a poor item as after a good item rather than responding only to a correlate of density of good items (Shettleworth and Plowright, 1992). The optimal prey model predicts that acceptance of the poorer of two prey types should depend only on the density of the better type (Stephens and Krebs, 1986). Our results and those by Shettleworth and Plowright (1992) support suggestions that variance or error in the predator’s estimate of the relevant environmental parameters is one source of deviations from optimality predictions in studies of prey selection (Stephens and Krebs, 1986; McNamara and Houston, 1987b). What animals learn about the state of the environment may affect their subsequent foraging decisions (Krebs and Inman, 1992). Therefore, error or variability in the forager’s estimate of prey type availability
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may be one of the reasons for partial preferences frequently found in tests of the model of optimal prey choice (McNamara and Houston, 1987b). Such inaccuracies in prey estimates could be related to the individual’s ability to acquire and process information. Incomplete information may cause the foragers to fail to maximise their rate of energy intake. In our model the estimate of p was updated at every encounter (Killeen, 1981), which is a more realistic approach than many others (Kacelnik et al., 1987). If a forager is to track changes in the environment, its information gathering and processing system should be sensitive to small scale changes and should employ a continuous updating procedure. This is evident for example in studies on sampling behaviour (Stephens, 1987; Shettleworth et al., 1988; Krebs and Inman, 1992). However, we found that individuals starting with low memory devaluation rate in the high variance environment could faster adjust their memory to match the new situation than individuals with high initial rate in the low variance environment. This was mainly because individuals in the latter situation were constrained by their high devaluation rate, which quickly responded also to stochastic variations in prey proportions. Thus, the forager may make wrong decisions on prey choice because it encounters a brief run-of-abad-luck, as has been noticed also in patch departure decisions (Inoue, 1983; Valone, 1992b). These findings emphasise the importance of including information processing and learning in modelling behavioural ecology of individual foraging. Capability to assess patch quality and prey density or proportion is an essential constituent of any behavioural decision based on these environmental attributes. We suggest that the most effective predators are those capable to adjust their devaluation rate of the past to match the quality of the current environment. Besides optimal prey choice, decisions on search tactics, for example, usually are dependent on prey density (Inoue, 1983; Hirvonen, 1998), or an individual’s estimate of it. There are clear differences among different species in how they use past experience (Real, 1993). Recent empirical evidence (Blanckenhorn, 1991) also indicates high variation among individ-
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uals’ capacity to assess patch profitability and among their spatial memory and learning ability, which are connected to foraging success contributing directly to individual fitness (Blanckenhorn, 1991). To figure out how flexibly individual foragers are able to change the usage of past and present information is an important direction for future research. If individual foragers differ considerably in energy intake rate and foraging efficiency, there might be large differences in the consequent growth rates of the individuals. This might mean that the individuals with the highest growth rates would achieve maturation earlier than their conspecifics. If the properties allowing high rates of intake are heritable, during some generations the mean age of maturation decreases. Such a shift in life-history characteristics could also affect population dynamics, and the relationship between the forager population and their prey, and thus community dynamics. On the other hand, intensive selective predation may change prey population dynamics (Fryxell and Lundberg, 1997) and prey community structure (Lazzaro, 1987). Here we studied the effects of memory properties on prey choice of solitary foragers, but in many natural situations individual foraging decisions and performance are dependent on the properties and performance of other individuals (Ranta et al., 1995; Ranta and Rita, 1998). Therefore, to increase realism, interactions between foraging individuals should be integrated into models of optimal foraging, including also the consequences at levels higher than the individual. This can be achieved by applying individual-based modelling.
Appendix A. Characteristics of the memory model As the expectation of Ik is E[Ik ]= p for all k, E[pˆ ] =E
n
k=0
k=0
% wkIk = % wkE[Ik ]= p % wk = p
k=0
(A1) Thus, for any choice of the memory parameter a, the estimate pˆ of the true population proportion p is unbiased. As the variance of Ik is:
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V[Ik ] =p(1− p),
(A2)
we have, using the independency assumption: V[pˆ ] =V
% wkIk k=0
n
= % w 2kV[Ik ] k=0
= p(1−p)
1− e − a 1+e − a
(A3)
For any positive a, V[pˆ ]B p(1 − p). Variance of pˆ approaches zero as a approaches zero. As a increases, the variance of the estimator pˆ increases and finally approaches the variance of the most recent indicator p(1−p) as a approaches infinity. Large values of a correspond to a high devaluation rate and thus fast reactions to even occasional indicator values in the most recent encounters.
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