The evolution of spatial memory

Mathematical Biosciences 242 (2013) 25–32

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The evolution of spatial memory Matt Grove ⇑ ACE, University of Liverpool, William Hartley Building, Brownlow Street, Liverpool L69 3GS, United Kingdom

a r t i c l e

i n f o

Article history: Received 3 May 2012 Received in revised form 26 November 2012 Accepted 28 November 2012 Available online 13 December 2012 Keywords: Foraging Search Patch density Detection distance

a b s t r a c t Models of foraging behaviour often assume either that animals are searching for resources, and therefore have no prior knowledge of resource locations, or that they are effectively omniscient, with a comprehensive knowledge of their habitat. By contrast, few attempts have been made to examine the actual conditions under which spatial memory will provide net benefits to foragers. To redress this balance, a model is developed that relates the sensory acuity of the forager and key indices of resource structure to the expected foraging efficiency via calculation of inter-patch distances. Efficiencies of ‘ignorant’ and ‘prescient’ foragers are examined in order to derive sets of conditions under which natural selection will favour the evolution of spatial memory capabilities. Results suggest that when resources are densely distributed or sensory acuity is high, spatial memory for resource locations provides no increase in efficiency over that of an ‘ignorant’ forager encountering resources at random. Ó 2012 Elsevier Inc. All rights reserved.

1. Introduction Models of foraging behaviour often assume either that foragers have no information about the spatial distribution of the resources they seek or, at the other extreme, that they are omniscient with regard to the locations of those resources [33,19,21,22,68–70,83,6]. This dichotomy is paralleled by a distinction between the optimization of search behaviour, often implicitly considered to be governed by an evolved algorithm, and the pursuit of efficient routes between multiple resource patches, often explicitly considered to be a cognitive task [61,25,5]. Optimal search models make similar assumptions at the individual level to the population-level diffusion models employed by landscape ecologists (e.g. [31,89]), with individual agents as essentially mindless automata repetitively following simple routines. Specifically, search behaviour – as the term suggests – assumes no knowledge of resource locations. At the other end of the spectrum, combinatorial optimization problems exemplified by the ‘travelling salesman’ algorithm assume a complete knowledge of all the locations to be visited during a given bout [3,29]. Models at both extremes have provided valuable insights into animal foraging patterns, as have occasional intermediate models considering ‘incomplete information’ (e.g. [53]), yet there has been little focus on the value of spatial memory or spatial knowledge to an animal in a given environmental setting. It is generally assumed that spatial memory is of positive value to foraging animals in that it could increase foraging efficiency, ⇑ Tel.: +44 0 151 7945056; fax: +44 0 151 7945057. E-mail address: [email protected] 0025-5564/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.mbs.2012.11.011

reducing the possibility of energy budget shortfalls and starvation. However, cognitive abilities can also bring considerable costs, not least in the maintenance of the metabolically expensive neural substrate [1,48]. The capacity to remember the locations of resources, then, will only be under positive selection in circumstances under which the benefits outweigh the costs. Certain circumstances can be ruled out as unsuitable for spatial memory evolution on simple, a priori grounds. For example, there would be no value in recalling the locations of resources that are fully depleted during foraging and do not regenerate; in such cases, the learning of more generalised cues to the likely presence of resources would be of greater value. On a more general level, however, the modelling presented below suggests that there are simple relations between habitat structure and the sensory capabilities of animals that can be used to predict the presence or absence of selection for spatial memory. 2. The model 2.1. The initial model To determine the conditions under which spatial memory would increase foraging performance, we can examine the patch encounter rates of animals with and without spatial memory (‘prescient’ and ‘ignorant’ animals, respectively) in habitats of varying resource density. Specifically, mean inter-patch distances of prescient and ignorant foragers can be compared under identical resource densities to quantify the extent to which spatial memory improves foraging performance in a given situation. Throughout, a number of simplifying assumptions are made so as to retain analytical tractability; these assumptions are enumerated here.

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(1) The maintenance of spatial memory is associated with a metabolic cost; the structure and magnitude of this cost are unimportant, but it is assumed that a prescient animal pays this cost whilst an ignorant animal does not. (2) It will benefit both prescient and ignorant foragers to move to the nearest possible resource patches, as this reduces the energetic burden associated with travel. (3) Patches are distributed at random and all patches are of equal energetic value. (4) Prescient and ignorant foragers derive the same net energetic benefit from consuming any given patch. (5) The foraging environment is energetically and spatially stable; patches are regarded either as non-depleting or as rapidly replenishing. (6) Spatial memory will be under selection if and only if the advantages accrued from its possession (in terms of reduced energetic expenditure in foraging) are greater than the costs incurred in its maintenance. (7) Selective pressures other than those arising from the foraging problem as outlined in the model are not considered.

and Evans [20] equation is therefore adopted here as being a fair approximation of the inter-patch distances experienced by prescient foragers. In this simple initial model, then, the performance of an ignorant animal is dependent upon just two variables – the density of resources and the detection distance of the animal – whilst the performance of the prescient animal is dependent upon patch density alone. This difference makes explicit the fact that only the ignorant animal is ‘searching’ for resources; the prescient animal is simply moving between known locations. To evaluate the conditions under which spatial memory will be favoured, we let a = lk  ls, and note that for a = 0,

d ¼ D0:5

ð1Þ

The effects of relaxing these assumptions are discussed further in Section 3.2. The mean inter-patch distance for an ignorant forager is assumed to be proportional to the reciprocal of patch density, D, 1 and is given as ls ¼ 2dD where d is the perceptual range or detection distance of the animal [89,47]. The notion of detection distance is commonly used in gas models of foraging behaviour [30,4,42– 44], and is best thought of as the radius of a circular perceptual range centred on the forager. Detection distance is likely to be limited by several factors, including the visual acuity of the animal itself, elements of topography such as relief, and the openness of the habitat in terms of vegetation cover. Environment- and speciesspecific assessments of these factors would be needed in empirical applications of the model. To model the mean inter-patch distance for a prescient forager, we make use of the finding of [20] (see also [16]) that the mean distance from a single patch to its nearest neighbour is lk ¼ 2p1 ffiffiDffi. This equation, like that for the ignorant forager, assumes

This latter relationship is plotted under varying D and d in Fig. 1, and describes the isocline at which ignorant and prescient foragers perform equally in terms of mean inter-patch distances. Below this line, with a negative and Dd2 < 1, spatial memory will be favoured by selection, whilst above it, with a positive and Dd2 > 1, there is no benefit to the forager of being prescient. The isocline plotted in Fig. 1 shows the conditions in which spatial memory will be favoured under the assumptions of the initial model. The sections below expand this model to take into account some important additional considerations. The initial model considers only mean inter-patch distances of the ignorant and prescient foragers, assumes a forager visits only one patch during a foraging bout, and regards patches as equivalent to single points in the plane. An important lesson arising from risk-sensitive foraging theory [81,50] is that foragers should also be sensitive to the variance of any variable that affects foraging performance. The model is thus expanded to account for risk-sensitive behaviour via an examination of variance in inter-patch distances in Section 2.2. Section 2.3 develops the model further to examine the prevalent case in which a forager must visit more than one patch during a single foraging bout, introducing patch number as a variable in the modelling process. Finally, Section 2.4 examines the effects of varying patch size on the relative performance of ignorant and prescient foragers.

that patches are randomly distributed with density D, an assumption that will be discussed further when interpreting the results. Calculating the true minimal distance that a forager could traverse when visiting a series of patches in known locations would involve use of an algorithm that solves the travelling salesman problem; it would also require a complete specification of the environment and an estimate of the number of patches required, and is generally considered to be beyond the capabilities of even highly intelligent animals [3,34]. The logic adopted here, that a prescient forager will simply move to the nearest patch each time, is supported by numerous theoretical and empirical studies of foraging dynamics. In an extensive simulation study, Anderson [3] found that what he termed the ‘one-step look-ahead’ algorithm (see [2]) achieved average inter-patch distances that were less than or equal to those produced by a series of algorithms that involved a greater degree of advance planning. He concluded that movement distances described here by the Clark and Evans [20] equation would be an evolutionarily parsimonious solution for a prescient forager. Higgins and Strauss [46] reported a similar result, and in a discriminant function analysis employing a list of path descriptors for random walks devised by Bell [8], they found that inter-patch distance paths of one-step look-ahead movement were misclassified as travelling salesman paths in 44% of cases. In addition to these theoretical results, one-step look-ahead matches the empirically documented behaviour of bees [65,75], rats [15], sheep [41] and primates including humans [61,25,86]. The Clark

Fig. 1. The a = 0 isocline at which ignorant and prescient foragers experience equal mean inter-patch distances divides the graph into regions in which selection for spatial memory should be present or absent.

M. Grove / Mathematical Biosciences 242 (2013) 25–32

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2.2. The effects of variance Researchers in both investment economics and evolutionary biology have come to associate variance with risk. Mean–variance analysis as initiated by [59] (see also [55]) suggests that investors should favour stocks that display a balance of high means and low variances in returns. The development of risk-sensitivity theory [67,17,72,73,81] was a vital adjunct to basic optimal foraging theory, and has received substantial empirical support [51,7]. The economic theory that the expectation (=mean) should be reduced by some function of the variance finds a direct parallel in the ‘variance discounting’ approach to risk-sensitive foraging [17,72,73]. Both approaches note that variance has a negative effect over the long-term in multiplicative processes (sensu [13] [1738]), and that optimization therefore involves reducing the variance as far as possible. The variance discounting approach involves penalising the mean by some function of the variance to account for the negative effect of variance on long-term optimisation. Real [72,73] and Caraco [17], via independent derivations, both arrived at the expression V(l) = l  pr2, with p a positive constant, as the ideal function. The most commonly used form in both biology and economics has p ¼ 21l [88,80], and it is this form that is adopted in what follows. In an empirical study of stock performance, Young and Trent [88] found that this expression approximated the geometric mean with an average error of less than 0.5%, even when the majority of distributions considered had significant skewness and kurtosis. This finding is of interest here as it is often the geometric mean that is chosen for optimisation in long-term evolutionary scenarios, where high variance is considered particularly detrimental. In addition, note that this expression involves rescaling the variance by a function of the mean, and approach that some contemporary risk theorists, via use of the coefficient of variation, have suggested is a better reflection of perceived risk in both animals and humans than is the raw variance (see [52,78,60,85]. This traditional variance discounting expression is used in situations in which the aim is to maximise the currency under consideration - thus the variance function is subtracted from the mean. In the current scenario, however, the currency under consideration – inter-patch distance – is optimised at its smallest possible value. As such, the variance discounting expression is here amended to 2

V þ ðlÞ ¼ l þ 2rl, so as to provide an analogous ’penalty’ to the foraging animal for long term variance (or risk) in inter-patch distances. The ignorant forager’s inter-patch distances are exponentially dis1 1 and variance ð2dDÞ tributed with mean 2dD 2 , whilst the prescient forager’s inter-patch distances are normally distributed with mean p [20,16]. Substituting these terms into the modand variance 4 4pD

1 ffiffiffi p 2 D

ified variance discounting formula, setting b ¼ V þ ðlk Þ  V þ ðls Þ, and solving for d when b = 0 yields

3p dV þ ¼ pffiffiffiffi Dðp þ 4Þ

ð2Þ

Equation (2) thus provides a modified version of equation (1) that takes into account the variance associated with the two inter-patch distance distributions, and thus balances the risks associated with ignorant and prescient foraging. Fig. 2 plots the b = 0 isocline via equation (2), yielding a line further from the abscissa than the p . This indicates that, to deal with a = 0 isocline by the factor p3þ4 the higher variance in inter-patch distances encountered by the ignorant animal, its detection distance must be raised yet further if it is to perform on a par with the prescient animal in a habitat of given patch density. The region between the isoclines described by Eqs. (1) and (2) can therefore be described as a region of risk sensitivity with regard to inter-patch distance distributions, and may

Fig. 2. The b = 0 isocline is further from the abscissa than the a = 0 isocline by the p ; the region between these lines is regarded as one of risk-sensitivity. factor p3þ4

be populated by both risk-prone ignorant animals and risk-averse prescient animals. The theme of risk-sensitivity is expanded in the discussion below; before this can be assessed in its true context, however, we must broach the situation in which animals visit multiple patches per foraging bout. 2.3. Multiple patches The initial model deals only with average inter-patch distances whilst the modified, risk-sensitive model described above accounts for both the mean and variance of the inter-patch distance distributions. Given that both mean and variance are shown to be important, however, we must also consider the possibility that the actual number of patches visited during a foraging bout affects the location of the isocline equating the performance of ignorant and prescient animals. The rationale for considering this possibility relates again to the issue of risk raised above. An animal that utilises a small number of patches per day is likely to experience a higher daily variance in energy expended travelling between those patches than will an animal visiting a large number of patches per day. In other words, the greater the number of patches visited per day, the closer the daily mean inter-patch distance will be to the true mean of the underlying inter-patch distance distribution. In these terms, the strategy of visiting multiple patches during the course of a bout might itself be described as a risk-sensitive (specifically, risk-averse) solution, particularly for an ignorant animal. The model is easily extended to deal with the use of multiple patches. In visiting n patches, the total distance moved by an ignorant animal will be described by the probability density function n1

f ðxÞ ¼ 2dD ð2dDxÞ e2dDx , which corresponds to an Erlang distribuðn1Þ! n n and variance ð2dDÞ tion with mean 2dD 2 . Since the sum of n Gaussian

distributions with mean l and variance r2 is another Gaussian P P with mean ni¼1 li and variance ni¼1 r2i , the mean of the total distance travelled for the prescient forager is simply 2pn ffiffin, and the varpÞ. With n as a parameter, equalizing the performance of iance nð4 4pD the ignorant and prescient foragers and solving for d yields

pð2n þ 1Þ dVþ ðnÞ ¼ pffiffiffiffi Dð2np þ 4  pÞ

ð3Þ

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Fig. 3. (a) A 3D iso-surface depicting the relationship between number of patches (n), patch density (D), and detection distance (d). (b) A contraction of (a) along the n scale highlighting isoclines for foragers visiting 2, 3, 6, and 14 patches. The a = 0 (long dashed) and b = 0 (short dashed) lines are retained for comparison. Note that as n increases, isoclines move away from the b = 0 isocline and towards the a = 0 isocline.

Values of d calculated via Eq. (3) for a range of (D, n) values are plotted in Fig. 3. Fig. 3a shows a mesh plot of the detection distance required for an ignorant animal to forage as efficiently as a prescient animal; Fig. 3b presents a contraction of Fig. 3a along the n axis. The first result that becomes clear from these figures is that as the number of patches utilised increases, the relationship between d and D becomes more like that described by equation [1] and less like that described by Eq. (2). This result is in accord with the notion that an animal utilising a greater number of patches per foraging bout reduces the risk associated with the inter-patch distance distribution. As found above, to perform at the same efficiency level as a prescient forager when utilising a single patch, a risk-sensitive ignorant p relaforager must increase its detection distance by the factor p3þ4 tive to a risk-insensitive ignorant forager. This increase is due to the greater variance associated with the exponential inter-patch distance distribution of the ignorant forager. However, as n increases to two this factor falls to 3p5pþ4, and continues to fall as the number of patches utilised increases. With increasing n, both numerator and denominator increase linearly with slope 2p, becoming proportionally closer in value such that dV + (n) ? 1 as n ? 1. Fig. 3 suggests that, as the number of patches visited during a foraging bout increases, so the variance between bouts becomes less important in determining the value of d that equalises performance of the ignorant and prescient foragers. This shift is rooted in the fact that as an ignorant animal utilises more patches per bout, the total distance travelled per bout, 1 T, will increase as T / cn, with c ¼ 2dD , whilst the standard deviation of total distance travelled, ST, will increase only as ST / cn0:5 . The net effect of these scaling factors is that the coefficient of variation of total distance travelled, CVT, decreases with higher n, as CVT = 2cn0.5, meaning that variance becomes relatively smaller as n increases. This logic suggests that a high-n strategy is one that reduces variance between bouts by reducing variance within bouts, a mechanism that should have wide applicability in foraging theory. The important issue of optimisation within and between bouts is discussed below in relation to the distinction between ‘time minimisers’ and ‘energy maximisers’ [76] and the ecological ‘grain’ [57,58]; first, however, the model requires a final addition to deal with differing patch sizes.

2.4. Variable patch size Modifying the model to take into account varying patch size involves only an adjustment of the ignorant forager’s inter-patch distance distribution, due to the fact that larger patches will be easier to find. Prescient foragers will still move directly to patches regardless of their size, and inter-patch distances are still calculated as the distances between the centres of patches. Thus patch size has an effect only on the detection of patches by searching animals. By analogy with the gas model approach [47] we denote patch radius s, and set the modified mean and variance of the ignorant for1 1 ager’s inter-patch distance distribution as 2ðdþsÞD and ð2ðdþsÞDÞ 2 respectively. The isocline at which the performance of ignorant and prescient foragers is equalised is now given by

pð2n þ 1Þ dV þ ðn; sÞ ¼ pffiffiffiffi s Dð2np þ 4  pÞ

ð4Þ

Plots of Eq. (4) are shown in Fig. 4a and b. It comes as no surprise that increasing s reduces the value of d at which the ignorant and prescient foragers perform equally, yet note that the reduction of d by s has a more tangible effect at higher resource densities. This effect occurs because large patches at high density rapidly begin to fill the landscape such that the resource base is almost continuous. Formally, the maximum s for a given D is maxðsjDÞ ¼ ðD1p Þ0:5 , beyond which patches are not only continuous but must begin to overlap. The area of Fig. 4a for which d = 0 results from a continuous resource base, a situation which might be applicable to grazing animals. In this situation, the extent to which s < ðD1p Þ0:5 becomes a potentially useful index of habitat fragmentation. Finally, empirical assessments of patch density and size will be carried out over some bounded, regular area A. When the number np of patches in that area are calculated, patch density may be caln culated simply as Ap ; however, this metric does not depend upon s. A more accurate formulation in certain instances, particularly when s is large relative to A, might be to calculate resource density (as opposed to patch density) simply as the proportion of the area P ps2 in which resources occur, thus D ¼ ni¼1 Ai . This way of calculating D assumes approximately circular patches and a maximum value

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Fig. 4. (a) A 3D iso-surface depicting the relationship between patch radius (s), patch density (D), and detection distance (d). (b) A contraction of (a) along the s scale highlighting isoclines for foragers feeding on patches of radius 0.1, 0.2, and 0.4. The a = 0 (long dashed) and b = 0 (short dashed) lines are retained for comparison.

of D = 1 without patch overlap, but could be easily amended to accommodate various alternative situations. 3. Discussion 3.1. Discussion of primary results The single most important result of the above modelling process is the delineation of those sets of circumstances, expressed in terms of patch density, detection distance, patch number, and patch size, that will and will not favour the evolution of spatial memory. Though results have thus far been presented in terms of equations for d, we can rearrange equation [4] into a more generic form by suggesting that when

Dðd þ sÞ2 ð2np þ 4  pÞ2

p2 ð2n þ 1Þ2

>1

ð5Þ

there will be no selection for spatial memory. This conclusion rests upon the important assumption that spatial memory, like any cognitive process, incurs some cost to the memorizer, such that when the performance of ignorant and prescient foragers is equivalent, evolution will favour ignorance as the more economical solution. Though it is difficult to specify a particular cost structure, the assumption is that spatial memory is reliant on metabolically expensive neural tissue (e.g. [1,48]. Ultimately, however, the model only requires that the use of spatial memory makes the prescient strategy more costly than the ignorant strategy; this cost need not be large, it need only be present. Spatial memory will thus be under selection in the long term if it provides individuals with even a slight net energetic advantage via reducing their expenditure in travelling between patches. It is unfortunate that research papers seeking to establish the circumstances under which learning and memory are likely to be beneficial are rare relative to those that simply assume a benefit in any and all circumstances (see discussion in [77]). We should remain cognizant of (e.g. as [90]) conclusion that ‘‘the situations in which learning confers a net selective benefit on its possessors may indeed be few, and in attributing ‘obvious’ advantages to learning we may merely be reflecting an unwarranted anthropocentric bias’’. The clearest generic trend to emerge from the modelling presented above confirms previous suggestions that ‘‘the

richer the environment, the less costly it is to be uninformed’’ [66] (see also [41]). Indeed, both higher patch densities and larger patch sizes reduce the differences in inter-patch distances between prescient and ignorant foragers. Perhaps the most interesting novel conclusions suggested by the model, however, relate to the effects of n in relation to the parameter space between the isoclines described by Eqs. (1) and (2). To recap, Eq. (2) is essentially a ‘variance discounted’ alternate of Eq. (1), whilst equation (3) is a variance discounted formula that also takes into account the number of patches visited, n. Increasing n in Eq. (3) gradually maps equation (2) back into Eq. (1) as n ? 1. Variance discounting is often employed in behavioural ecology using the logic, borrowed directly from investment economics, that variance is synonymous with risk. However, another facet to this form of analysis is that the variance discounting equation, in the 2 form V ¼ l  2rl, is a direct estimator of the geometric mean. A mapping from Eqs. (1) and (2) is therefore a mapping from the arithmetic to the geometric mean of the inter-patch distance distribution, raising immediately the issue of the spatio–temporal scale over which foraging should be optimised [87,54]. A useful example of this involves the notion of ‘grain’ [57,58]. At the basic level, Levins [58] argues that ‘‘if patches are large enough so that the individual spends his whole life in a single patch, the grain is coarse, while if patches are small enough so that the individual wanders among many patches the environment is fine-grained’’. At a more general level, Levins [57] assumes that fine-grained environments are homogeneous in time but heterogeneous in space, and that the arithmetic mean is the index to optimise in such environments. Conversely, coarse-grained environments are heterogeneous in time but homogeneous in space, and necessitate optimisation of the geometric mean. In the more realistic scenario in which environments are heterogeneous in both time and space, the degree to which results resemble the arithmetic or geometric calculations will depend upon ‘‘the relative predominance of spatial or temporal heterogeneity’’ [57]. The use of the geometric mean when environments are temporally variable and the arithmetic mean when they are spatially variable is in fact a central tenet of many genetic models (e.g. [28,23,40,45]), and in Levins’ terms it allows us to reinterpret the result of increasing n as a shift from the relative predominance of temporal variation towards the relative predominance of spatial variation. More specifically, it represents a shift towards the

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greater importance of within-bout variation relative to betweenbout variation in inter-patch distances. As discussed above, visiting a greater number of patches per bout reduces the coefficient of variation of inter-patch distances within bouts, but it necessarily also reduces the variance in total distance covered between bouts. This reduces the difference between the arithmetic and variance discounting formulas because the variance itself is reduced. Animals visiting very large numbers of patches per bout therefore witness very little variance between bouts, making this a risk-averse and potentially very successful evolutionary strategy in the long-term [80,79,56]. If n governs a continuum of foraging strategies, how might these relate to existing conceptual models of the foraging process? Perhaps the most enduring of these is the distinction between time minimisers and energy maximisers [45,76]. Time minimisers are constrained by a threshold energetic requirement, with optimisation involving the achievement of that requirement in the shortest time possible. Energy maximisers, by contrast, are constrained by a maximum time threshold, with optimisation involving the maximisation of energetic gain within that period. In as far as energy maximisers visit a greater number of patches, they will also be risk minimisers, being more likely to receive a value close to the mean net intake. If per-patch benefit, b, is a constant and the cost of locomotion, c, is a constant per unit distance, with mean inter-patch distance over many bouts given by  d, a forager visiting a large number of patches will minimize the per-bout difference between P nðb  c dÞ and n1 ni¼1 b  cdi ; by doing so, the forager will ensure that bouts falling short of the mean per-bout energetic return do so by the smallest margin possible. A particularly useful application of Schoener’s model is provided by Strier [82] in her consideration of the foraging strategies of the Atelid primates. The three genera comprising the Atelin subfamily, Ateles, Brachyteles, and Lagothrix, are frugivorous, have long day ranges, large home ranges, and fluid grouping patterns, whilst the single genera of the subfamily Alouattinae, Alouatta, are folivorous, have short day ranges, small home ranges, and cohesive grouping patterns. Strier [82] identifies the Atelins as energy maximisers and Alouatta as time minimizers. In terms of the above discussion, we could argue that folivory is a low risk strategy relative to frugivory, in that the latter relies on a dispersed, heterogeneous resource base, an argument that is supported by evidence that all four Atelid genera become more folivorous in more seasonal environments. The strategy of visiting multiple patches, which we here equate with energy maximization, then stands alongside group fluidity as a risk-reduction mechanism for the predominantly frugivorous Atelins [43]. Note also that it is when seasonality – an index of temporal heterogeneity – increases that risk is elevated to levels where even the Atelins fall back on the more homogeneous, low energy diet of green vegetation. Thus far it has been assumed that a given species will be either prescient or ignorant, with spatial memory abilities conditioned by an evolutionary history of subsistence on a particular resource base within a particular habitat type. However, we might also ask when a forager might deploy spatial memory or, of equal importance, what about its home range a forager should remember. Many results, both theoretical and empirical, suggest that even prescient foragers do not remember detailed records of their environments but, rather, that they remember generic locations. This implies that the most accurate way to draw the distinction between ignorant and prescient foragers is to suggest that while the former will be searching at all scales, the latter will be searching only at small scales. This notion accords well with the popular model of ‘area-restricted search’ [9,10], in which foragers switch between two behavioural modes: rapid, straight-line movement between known locales and slower, more sinuous movement in the expected or detected presence of a resource. Together, these two modes ‘‘enable

an animal to increase the time spent in more favourable areas of the environment and to decrease the time spent travelling between them’’ [9]. Of particular relevance to the model presented here are two recently published simulation models of animal movement that rely on composite or ‘multi-scaled’ random walks [11,12,35,37,36]. Benhamou [11] considers a Composite Brownian Walk (CBW) that combines two exponential step length distributions representing within- and between-patch movements, the logic being that animals use different movement patterns when searching for and then searching within patches. Gautestad and Mysterud [35] simulate a Multi-scaled Random Walk (MRW) describing a mixture of short-term, local, ‘tactical steps’ and less frequent but more farranging ‘strategic displacements’. A pertinent difference between the models is the explicit reliance in [35,37] simulations upon a memory for past locations, with directed steps at scales beyond the detection distance necessarily relying on this memory for information. Both Benhamou [11] and Gautestad and Mysterud [35] allow that detection distance may be critical in determining the point at which an animal switches between locomotor modes, and we can suggest that an increase in d will equate the two models. Mathematically, infinite d is equivalent to a spatial memory of the entire habitat; therefore, increasing d makes ignorant foragers capable of prescient movements to distant locations, in much the way that spatial memory does in the Gautestad and Mysterud [35] model. This equivalence should have important ramifications for agent-based models of animal foraging and memory capabilities (Grove in prep). 3.2. Relaxation of assumptions and implementation of empirical tests Section 2.1 began with a list of assumptions highlighting the situations in which the current model applies. Some of these are more restrictive than others, but all are necessary to retain the mathematical tractability of the model. In this section, the effects of relaxing each of these assumptions are considered, along with modelling procedures that could be used to study these effects in more detail. The section concludes by considering a number of important empirical tests of the model as presented above. The issue of the metabolic cost of spatial memory is briefly dealt with in Section 3.1 (above), but it should be stressed once again here that it is the existence rather than the magnitude of a cost to spatial memory that is important, as a cost of any magnitude will lead to the relationship posited in the model. However, a number of other factors could alter the relative value of spatial memory, and these relate primarily to relaxation of the other assumptions listed in Section 2.1. Firstly, alterations to the structure of the resource base could alter the relative value of spatial memory, particularly when patches are depleting or mobile. When patches are depleting (and replenish slowly or not at all) the primary value of spatial memory may be in remembering which patches have previously been visited; however, if the same territory if visited by multiple foragers or foraging groups independently, depletion may occur without the knowledge of the focal individual, leading to an erroneous (and potentially costly) memory profile for that individual. There is some evidence that territorial primates monitor phenological cycles (e.g. [62], and that corvids respond to the feeding practices of other agents (e.g. [26,27], but the detailed study of either situation would equate to a considerable expansion of the analytical model proposed here, and would ideally be situated within an agent-based simulation. The other assumptions that could be profitably relaxed relate to the equal value and random distribution of the resource patches in the model. The assumption of ‘complete spatial randomness’ (CSR) is fundamental to most analytical spatial models [47] and spatial statistics (e.g. [32]), but could be relaxed via simulation to include

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uniformly distributed and aggregated patch distribution patterns. The continuum between uniformity and aggregation (with CSR as a midpoint) is usefully represented by the R-scale [20,24]. Variation in the value of resource patches has been incorporated in some previous agent-based simulations of primate foraging (e.g. [14,71]), and a number of simple heuristics have been developed. For example, [14] employed a ‘maximum efficiency metric’ in which their agents moved between patches by dividing the value of each patch in their knowledge base by the distance to it and moving to the patch with the highest score. This simple algorithm led to output that matched empirical data from spider monkeys relatively well. In this situation, the value of spatial memory would clearly be greater than it is in the current model, as it would increase overall efficiency rather than simply reducing travel distances. Although all the modelling developments outlined above would be of considerable value, the most important next step in this process will to be to test the predictions of the model outlined in Section 2, to assess any shortcomings or potential confounds in the model as it stands. This process is currently underway, and involves at its most basic level an attempt to populate Fig. 4 using data on the detection distances and resource densities experienced by various taxa. Thankfully, gas models are relatively common in empirical research on foraging theory (e.g. [74,30,49,42]), and require estimates of detection distance, so d as an empirical variable is relatively abundant. Less fortunately, population density is often used as a proxy for resource density (e.g. [18,91], in a format not amenable to conversion into D for current purposes. Most longestablished field sites do however maintain data on the resource base in terms of density and size of specific resources, providing data which can ultimately be assessed in relation to both D and s. The number of patches an animal visits during a bout is of course an apparently random variable even in empirical situations, but mean estimates are not hard to come by. Populating Fig. 4 with empirical data on a series of taxa will provide only predictions of spatial memory abilities in those taxa. These must ultimately be tested against extant or newly gathered data on the actual spatial memory abilities of those taxa. The latter will necessarily often come from captive studies, though some pioneering studies of spatial cognition in the wild have been published in the past decade [63,64,62]. Further verification involves the measurement of neural structures subserving spatial memory, such as the hippocampus and associated complex; animals predicted to have better spatial memory abilities would also be predicted to possess greater relative hippocampal volumes. A number of considerations need to be addressed when assessing empirical data in the context of this model, and they are briefly summarised here. Firstly, whilst D is entirely extrinsic to the animal (an animal can only actively alter the D it experiences via relocation), d is not entirely intrinsic to the animal, being a more complex variable dependent on a combination of sensory acuity, habitat openness, and other environmental and topographic features. In heavily forested environments, for example, animals will not be able to see as far as their usual levels of visual acuity suggest. Furthermore, in many animals resources will be detected via olfaction or other sensory mechanisms, making assessment of detection distance a difficult and multi-faceted problem. Finally, it should be stressed that different species may perceive the same habitat in very different ways, and that such differences in perception will necessarily impact upon spatial cognition. Animals perceive their environments in terms of affordances [39,38], which will necessarily be very different for animals of different sizes, locomotor and sensory capabilities, and dietary specializations. In the terms of [84], the umwelt of each species is distinct, even when many species share what is, to us, a single environment.

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