2 Ludek Berec Running head: Optimal foraging Send correspondence to: Ludek Berec, Department of Theoretical Biology, Institute of Entomology, Academy

Transcription

1 Bulletin of Mathematical Biology (999) 0, {28 Mixed Encounters, Limited Perception and Optimal Foraging LUDEK BEREC Department of Theoretical Biology, Institute of Entomology, Academy of Sciences of the Czech Republic, and Faculty of Biological Sciences, University of South Bohemia, Branisovska 3, Ceske Budejovice, Czech Republic Key words: diet composition, functional response, individual-based model, partial preferences, sequential and simultaneous encounters /99/ $7.00/0mb99???? c 999 Society for Mathematical Biology

2 2 Ludek Berec Running head: Optimal foraging Send correspondence to: Ludek Berec, Department of Theoretical Biology, Institute of Entomology, Academy of Sciences of the Czech Republic, Branisovska 3, Ceske Budejovice, Czech Republic Tel: , Fax: , I wish to submit the nal manuscript electronically in LaTeX.

3 Mixed Encounters, Limited Perception and Optimal Foraging 3 This article demonstrates how perceptual constraints of predators and their ability to encounter prey both sequentially and simultaneously may inuence the predator attack decisions, diet composition and functional response of a behavioural predator-prey system. Individuals of a predator species are assumed to forage optimally on two prey types and to have exact knowledge of prey population numbers (or densities) only in a neighbourhood of their actual spatial location. Prey individuals are allowed to be encountered by predators either alone or in pairs formed by one individual of each prey type. The system characteristics are inspected by means of a discrete-time, discrete-space, individual-based model of the one-predator two-prey interaction. Model predictions are compared with those ones that have been obtained by assuming sequential encounters of predators with prey only and/or omniscient predators aware of prey population densities in the whole environment. It is shown in the article that the zero-one prey choice rule, optimal for sequential encounters and omniscient predators, shifts to abruptly changing partial preferences for both prey types in the case of omniscient predators faced with both types of prey encounters. The latter, in turn, become gradually changing partial preferences when the predator omniscience is considered only locally. c 999 Society for Mathematical Biology

4 4 Ludek Berec. Introduction Overwhelming majority of works dealing with prey optimal foraging models assume that prey are encountered at random and sequentially by predators, i.e., predators randomly encounter one prey item at a time (Pulliam, 975; Charnov, 976a; Hughes, 979; Gleeson and Wilson, 986; McNamara and Houston, 987; Fryxell and Lundberg, 994; Krivan, 996). Upon such an encounter, predators decide whether to attack that prey or pass it by and wait or search for another one that is possibly more valuable. On the other hand, Waddington and Holden (979) argue that simultaneous encounters, i.e., encounters with two or more prey items at a time, are quite common in nature through, for example, visual abilities of foragers. Upon a simultaneous encounter, predators decide which prey item to attack if any. Mitchell (989, page 46) correctly claims that what matters in the decision process of any forager is its information state which is not necessarily connected to the encounter types, most frequently understood in terms of vision, in a one-to-one manner: `For example, a forager may encounter renewable resources sequentially but forage on them as though they were encountered simultaneously if it can recall the locations of the renewing resources.... Conversely, a forager may encounter several resources simultaneously at a distance, but not be capable of assessing their type... without travelling to each resource and individually inspecting it. If resources are exploited as they are inspected, then the forager must decide its diet strategy based on information similar to that used with sequentially encountered resources.' In what follows, we use the terms sequential and simultaneous encounters while bearing in mind this more general characterization of these concepts. We also use the term mixed encounters for a combination of the two. The idea of simultaneous or mixed encounters has been elaborated upon in a few studies only (Waddington and Holden, 979; Engen and Stenseth, 984b; Stephens et al., 986; Barkan and Withiam, 989). Whereas the rst one assumes zero searching time and the last two works treat the particular case of fully simultaneous encounters (i.e., upon every encounter items of all prey types considered are available to the predator) for two prey types, the study of Engen and Stenseth (984b) is much more general. Not only it treats the case of mixed encounters for any number of prey types, but it covers patch optimal foraging models (Charnov, 976b; Stephens and Krebs, 986), too. This rather high generality is paid, however, by a considerable loss of insight into behaviour of particular subsystems. One of the aims of the present article is to give rigorous predictions of the prey optimal foraging model with two prey types and mixed encounters allowed. If the average rate of net energy gain during foraging is adopted as a currency to be optimized by every predator then, for two prey types, the prey optimal foraging model with sequential encounters predicts that predators

5 Mixed Encounters, Limited Perception and Optimal Foraging 5 should attack every item of more protable prey upon its encounter; moreover, they should attack every item of less protable prey provided that the density of more protable prey is below a certain threshold value, and ignore every item of less protable prey provided that the reverse inequality holds (Schoener, 97; Pulliam, 974; Werner and Hall, 974; Charnov, 976a; Stephens and Krebs, 986; Krivan, 996). The prey model with fully simultaneous encounters predicts, on the other hand, that predators should attack either every item of more protable prey or every item of less profitable prey, the latter if the density of more protable prey is below a certain threshold value (generally dierent from the one for sequential encounters), and the former in the opposite case (Stephens et al., 986; Barkan and Withiam, 989). Three common observations, however, contrast these predictions and prevent us to accept the corresponding models as they stand: partial preferences for both more and less protable prey, and a dependence of the prey choice rule on the population density of less protable prey (Werner and Hall, 974; Davies, 977; Krebs et al., 977; Mittelbach, 98; Rechten et al., 983; Stephens and Krebs, 986). A number of mechanistic explanations have been proposed to reconcile the model predictions with these observations. Predators have been supposed, for example, to misidentify or sample their prey (Krebs et al., 977; Rechten et al., 983), estimate prey encounter rates (McNamara and Houston, 987; Hirvonen et al., 999), have a limited memory capacity (Mangel and Roitberg, 989; Belisle and Cresswell, 997), need a time to correctly recognize prey (Hughes, 979; Houston et al., 980), or have a limited perception range (Berec and Krivan, submitted). Neither of these and other most cited explanations is consistent, at least qualitatively, with all the three observations. We show in the article that considerations of mixed encounters of predators with (two) prey types are sucient to predict them all. For more detailed lists of suggested mechanisms, see McNamara and Houston (987), Mitchell (989), and Belisle and Cresswell (997). Nevertheless, omniscient predators still change their diet synchronously, all at the same moment, thus giving rise to abrupt changes in partial preferences. This population-level behaviour does not seem to be too realistic. In the paper, we use the idea of Berec and Krivan (submitted) that any predator has exact knowledge of prey numbers (or densities) only in a certain neighbourhood of its actual spatial location, and show that the abrupt changes are then substituted by more gradual changes, yet preserving the three required properties cited above. In real systems, the local omniscience of predators can be due to limited abilities in their perception, e.g., a limited detection range of volatile substances released by prey, or limited visual or auditory ranges of predators. Perceptual constraints have been shown to aect various aspects of optimal foraging behaviour such as deviations from the ideal free distribution (Abrahams, 986; Spencer et al., 996; Gray and

6 6 Ludek Berec Kennedy, 994), optimal foraging heights of dierent raptor species hunting for voles (Rice, 983), and optimal strategies to parasitize hosts (Mangel and Roitberg, 989). 2. Behavioural Model and Its Characteristics The discrete-time, discrete-space, individual-based model (IBM) of Berec and Krivan (submitted) is extended here to allow for mixed encounters of predators with their prey. Both individual and spatially explicit characters of the model seem indispensable for an implementation of the idea that foragers are not able to perceive any prey item beyond a distance from the spatial location they actually occupy. The environment is assumed to be homogeneous both across the space and time, and is modelled as a lattice of identical square sites. Time runs in discrete steps. At most one individual of each type (prey, prey 2, predator) is allowed to occupy each site at once; we may think, for example, about territorial, monogamous birds in which at most one male (prey ) and one female (prey 2) occupy a single territory (site). Consequently, population abundances are limited from above by the lattice size. Any predator may thus encounter either a prey individual alone (sequential encounter) or a pair of dierent prey individuals (simultaneous encounter). Every time step, prey individuals are assumed to be randomly distributed across the lattice (may be static or may move) and all the sites are updated simultaneously according to a set of rules, determining behaviour of any single individual on the lattice. Any predator is assumed to be either searching for prey or handling prey. Consider a single predator. Every time step it may nd itself in one of four situations: it can share the lattice site with a prey individual alone, a prey 2 individual alone, with a pair of dierent prey individuals, or the site is free of prey. Behavioural consequences of an elimination of the third possibility by a competition interaction between the prey types and the resulting existence of sequential encounters only have been studied by Berec and Krivan (submitted). For any of the rst three situations, predator decision in respect of attacking a prey has to be specied. If the predator shares the site with a prey i = ; 2 individual alone, it decides to attack that prey with a probability p i. If the predator encounters individuals of both prey types, it decides to attack the prey individual with a probability p 3, whereas it strikes the prey 2 individual with a probability p 32 ; we assume that at most one individual may be attacked at a time so that the inequality p 3 + p 32 has to be satised. These `partial decision probabilities' are allowed to vary with time. If the predator decides to attack a prey i individual, the attack is successful with a probability Pa. i Following a successful attack, the predator handles the prey i item for T i time steps. h

7 Mixed Encounters, Limited Perception and Optimal Foraging 7 During this handling time, the predator cannot attack any other prey. The handled prey is instantaneously removed from the lattice so that it cannot be attacked by any other predator, but is immediately replaced by a new item of the same type, to keep the prey densities xed. Before we derive plausible values for the partial decision probabilities p i and p 3i, we present formulae for the predator diet composition and functional response that are of prime importance below. Consider a single predator individual, making decisions in a series of time steps. The average fraction of prey i items from those that it captures in the series is x i S ip i a x S P a + x 2 S 2P 2 a ; () where S is the number of lattice sites, x i is the (xed) number of prey i = ; 2 individuals (hence x i =S is the xed density of prey i individuals), and = 2 =? x 2 S? x S p + x 2 S p 3 ; (2) p 2 + x S p 32 ; (3) provided that sequential encounters with both prey types as well as simultaneous encounters are possible, i.e., 0 < x i < S; see Appendix for a detailed derivation. Similarly, if we dene the predator functional response to prey i as the number of captured prey i individuals per predator per time step, we get x y i s S ip i a ; (4) y where y s is the number of searching predators and y the total (xed) number of predators. Following Wilson (998), the probability y s =y that a randomly chosen predator is in the searching state can be (after an initial transient period) approximated by the expression =( + T x h S P a + T 2 h approximate predator functional response to prey i thus turns out to be x2 S 2P 2 a ). An + T h x i S ip i a x S P a + T 2 h x2 S 2P 2 a ; (5) a Holling type II functional response if and 2 are independent of x i. We note that the expressions () and (4) can also be derived from the point of view of a single time step and the number of prey i individuals captured at that time step by all the predators present on the lattice. Then, the y s =y term in (4) can be interpreted as the fraction of searching predators at that time step. We also note that there is no interference among predators

8 8 Ludek Berec modelled as would be the case, for example, when the successfully attacked prey are not (instantaneously) substituted by new items or predators are allowed to steal handled prey to conspecics (Stillman et al., 997). The `total decision probabilities' i (i = ; 2), given by the expressions (2) and (3), are the probabilities that a predator will attack a prey i individual upon encounter, without distinguishing whether it is encountered alone or together with the alternative. These probabilities are just the ones that are estimated from experimental observations. Predators specializing on the prey type are characterized by p = p 3 =, p 2 = p 32 = 0, prey generalists by p = p 3 = p 2 =, p 32 = 0, prey 2 generalists have p 3 = 0, p = p 2 = p 32 =, and randomly deciding predators may take arbitrary (but xed) values of p i and p 3i (with p 3 +p 32 ). We note that the expressions formalizing the predator diet composition and functional response in the case of sequential encounters have the same form, with i substituted by the (at that case total) decision probabilities p i (Berec and Krivan, submitted). 3. Optimal Foraging Optimal foraging theory (Stephens and Krebs, 986; Schmitz, 997) assumes that animals choose from among a set of feasible diet compositions the one that maximizes a given currency. The classic prey optimal foraging model assumes that prey are distributed randomly across the environment, and that the currency to be maximized by every predator is the average rate of net energy gain during foraging, E T s + T h ; (6) where T s is, in the above IBM context, the overall number of time steps spent searching when foraging, T h is the overall number of time steps spent handling prey items when foraging, and E is the net amount of energy gained during the total foraging time T s +T h. Following Stephens and Krebs (986), maximization of the expression (6) is equivalent to maximization of R(p ; p 2 ; p 3 ; p 32 ) = E x n + T h S (p ; p 3 )P a x S (p ; p 3 )P a + E 2 x2 n S 2(p 2 ; p 32 )P 2 a + T 2 h? x2 S 2(p 2 ; p 32 )P 2 a ; (7) where the total decision probabilities i are given by the expressions (2) and (3), E i n is the net energy gained from consuming prey i = ; 2, and is the amount of energy lost by a predator per one time step searching. The expression (7) is maximized over the partial decision probabilities p i and p 3i, subject to the constraints 0 p i, 0 p 3i, p 3 + p 32.

9 Mixed Encounters, Limited Perception and Optimal Foraging 9 Assuming, without loss of generality, that prey is more protable than prey 2 (En=T > h E2 n=t 2 ), prey is predicted to be always attacked by h predators when encountered alone (p = ), independently of prey and prey 2 population densities, whereas p 2 = if x < L ; p 2 = 0 if x > L ; p 3 = 0; p 32 = if x + x 2? x x 2 =S < L 2 ; p 3 = ; p 32 = 0 if x + x 2? x x 2 =S > L 2 ; (8) where and E 2 n + T 2 h L = S P a (E T 2? n h E2T ) ; (9) n h L 2 = S (E2 n + T 2 h )? P a (E P 2 n + T ) h a P a (EnT 2? h E2 nt ) : (0) h We do not consider the cases x = L or x +x 2?x x 2 =S = L 2 here because the prey abundances x i can take only non-negative integer values (bounded from above by S) within our IBM framework; therefore, we can always make the threshold values L and L 2 dierent from the values of x and x + x 2? x x 2 =S, respectively, by a negligible change in model parameters. We note that L and L 2 are in units of individuals. Dividing by S gives them the meaning of densities. We make a remark concerning the formulae (9) and (0). The net energy gained from consuming prey i can be expressed as E i n = E i g?i T i, where h h Ei g is the gross energy gained from prey i and i is the energy a forager looses h per one time step spent handling prey i. Let = = h 2, i.e., let searching h for prey and handling prey be equally energy-consuming activities, and let E 2 g = c E g for a constant value c > 0, i.e., let the prey types be perfect substitutes (the gross energy gained from consuming an individual of prey 2 is proportional to that gained from consuming an individual of prey ). Then, c L = S P a (T 2? ct ) ; () h h L 2 = S c? P =P 2 a a P a (T 2? ct ) : (2) h h The same simplications could be achieved if the rate of energy loss during searching is assumed to be negligible with respect to the other terms in the numerator of (7), and E 2 n = c E n for a c > 0. These assumptions are rather standard in optimal foraging literature (Krebs et al., 977; Houston et al., 980; Stephens et al., 986; Schmidt, 998), since measurements of

10 0 Ludek Berec energy gains and losses are very dicult to perform if at all. We use these simplied expressions further on. If we consider the prey 2 abundance xed to a value 0 < x 2 < S, the abundances of prey at which predators change their diet are x L and x (L 2? x 2 )=(? x 2 =S). A necessary and sucient condition for the inequality x x to hold is L > S, L 2 > S and x 2 (L? L 2 )=(L =S? ) which implies x x > S. Hence, both prey thresholds x and x are greater than the lattice size which makes this case behaviourally uninteresting. If x S, p 32 = for all 0 < x < S. If x 0, on the other hand, p 3 = for all the admissible values of x, and similarly for x. The most interesting case is when both of the prey thresholds are `active', i.e., when x > 0 and x < S. Fig. gives an idea of how the space of prey population sizes is divided into regions with dierent predator foraging strategies. Figure around here. The expression x + x 2? x x 2 =S = S x S? x 2 S + x 2 S? x S + x S x 2 ; (3) S deciding on which prey out of the pair encountered should be attacked, is the average number of sites occupied by prey, given that x i individuals of prey i = ; 2 are randomly distributed on the lattice. Hence, if this number is less than L 2, prey 2 will be attacked when encountered in a pair with prey, and prey will be attacked if the average number of sites occupied by prey is greater than L 2. Moreover, by rewriting this expression as x + x 2? x x 2 =S = x + x 2? x S ; (4) the paradox of self-reduction (Engen and Stenseth, 984b; Engen and Stenseth, 984a) can be demonstrated via increasing prey 2 abundance x 2, for a xed prey abundance x : if prey abundances are initially such that p 2 = ; p 3 = 0; p 32 =, then a sole increase in prey 2 abundance x 2 makes this prey more rare in the predator diet provided that conditions change for p 2 = ; p 3 = ; p 32 = 0 to be optimal; see the expressions (8). These expressions also imply that when a predator simultaneously encounters a pair of dierent prey individuals in a lattice site it does not necessarily attack the more protable one. Instead, it may attack the item of less protable

11 Mixed Encounters, Limited Perception and Optimal Foraging prey if such choice is better from the viewpoint of the (long-term) average rate of net energy gain during foraging; see also Stephens et al. (986) and Barkan and Withiam (989). Fig. 2 shows the total decision probabilities (2) and 2 (3) for the optimally foraging predators, as functions of prey abundance x, for a xed prey 2 abundance x 2 and for a particular set of parameters ensuring that both the prey thresholds x and x are active. The interesting observation is not only an appearance of partial preferences for (less protable) prey 2 when prey abundance lies in between the values x and x, but also an appearance of partial preferences for (more protable) prey when its abundance is below the value x. Therefore, contrary to the particular cases of sequential encounters (Schoener, 97; Pulliam, 974; Werner and Hall, 974; Krivan, 996; Berec and Krivan, submitted, and many others) and fully simultaneous encounters (Stephens et al., 986; Barkan and Withiam, 989), partial preferences for both prey types appear in the case of mixed encounters even for the omniscient predators, in a wide range of prey population sizes (or densities x i =S); see also Engen and Stenseth (984b). Moreover, as x is a function of x 2, the optimal diet choice depends on the prey 2 population size/density, too. Figure 2 around here. The eects of the optimal foraging strategy on the predator diet composition and functional response, for a xed prey 2 abundance x 2, are shown in g. 3. Fig. 3A plots the fraction of prey in the diet () for optimally foraging predators against the prey population abundance x, g. 3B plots the expression () against the fraction of prey in the environment, x =(x +x 2 ), and g. 3C depicts the functional response to prey (5) for optimally foraging predators against the prey population abundance x. Figure 3 around here. Although we observe partial preferences for both prey types, the predator diet composition and functional response change abruptly around the prey thresholds x and x, since all the predators change their diet at the same moment. Such a synchrony does not probably work in natural systems; moreover, the assumption on predator omniscience does not seem too realistic by itself, for example, when the lattice size is large.

12 2 Ludek Berec McNamara and Houston (987) and Hirvonen et al. (999) remove the assumption on predator omniscience by letting predators estimate prey densities on the basis of actual time sequence of encounters with prey. Belisle and Cresswell (997) assume that forager can memorize only a number of recently consumed prey items. If all these items are of the same type the forager forgets the relative values of the (two) prey types and consumes prey indiscriminately, until both prey types are again present in the predator memory window. In the framework of sequential encounters both these approaches led to partial preferences for less protable prey. Although these mechanisms could possibly be extended to the case of mixed encounters, too, we adjust here another one that led to the equivalent results and that heavily exploits the spatial character of the above dened IBM (Berec and Krivan, submitted). Predators are supposed to know exact numbers of individuals of each prey type only within a restricted neighbourhood of their actual spatial location (for example, a square of 7 7 lattice sites with the predator in its centre). We have been motivated by a limited detection range of volatile substances released by prey, or limited predator visual or auditory ranges. Let the neighbourhood have the same number N of sites for every predator individual; we refer to it as the N-neighbourhood further on. If we view the N-neighbourhood as an eective environment of the predator, the above derived optimal foraging rule predicts that the predator should always attack prey encountered alone. Moreover, the predator decision to attack prey 2 encountered alone depends on the threshold value () with the size S of the whole environment replaced by the local (or eective) environment size N, i.e., always attack that prey provided that ~x < L 0 and always ignore it if the reverse inequality holds, where ~x is the prey abundance in the N-neighbourhood, and c L 0 = N P a (T 2? ct ) : (5) h h Furthermore, the predator decision of which prey to attack when encountered in a pair of dierent types should depend on the threshold value (2) with S replaced by N, i.e., always attack prey 2 provided that ~x + ~x 2? ~x ~x 2 =S < L 0 2 and always attack prey if the reverse is true, where ~x 2 is the prey 2 abundance in the N-neighbourhood, and L 0 2 = N c? P =P 2 a a P a (T 2? ct ) : (6) h h Obviously, L 0=N = L i i=s, i.e., the critical densities are not changed when going from the global to a local scale. Yet the fundamental observation is that for xed total prey i = ; 2 densities x i, the local abundances ~x i need

13 Mixed Encounters, Limited Perception and Optimal Foraging 3 not be the same for every predator individual, nor for the same individual in dierent time steps (as the spatial distribution of prey may change with time and/or the predator may move to another site). Consider a single predator individual that shares a site with a prey 2 individual alone, and compute the partial decision probability p 2 that the predator will attack that prey item. Clearly, p 2 is the probability that the number ~x of prey individuals in the predator N-neighbourhood is below the threshold value L 0. Let the total prey abundances on the lattice be xed to 0 < x i < S. There are!! S? S? x x 2? (7) possibilities for the distribution of prey individuals on the lattice composed of S sites, conditioned on the event that the focus site is occupied by a prey 2 individual, no prey individual, and the predator. The number of possibilities by which ~x prey individuals might be located in the N- neighborhood is N? ~x! S? N x? ~x! S? x 2?! ; (8) provided that x? (S? N) ~x minfx ; N? g, whereas it is zero otherwise. Consequently, the probability P 2 (~x ; x ) that ~x out of x prey individuals are located in the N-neighbourhood of the predator, sharing the lattice site with a prey 2 individual alone, denes the hypergeometric distribution P (~x ; x ) = 8 >< >:? N?? S?N ~x x?~x? S? x if x? (S? N) ~x minfx ; N? g ; 0 otherwise : (9) Finally, the required probability p 2 that the number of (more protable) prey individuals in the N-neighbourhood is lower than the threshold value L 0 is p 2 (x ) = [L X 0 ] ~x=0 P 2 (~x ; x ) ; (20) where [L 0 ] denotes the largest integer less than L 0 (we assume that L 0 does not take an integer value, to avoid its possible equality to a value of ~x ; otherwise, a negligible change in model parameters can be made to meet this assumption). We note that as N approaches S, p 2 (x ) approaches the zero-one step function; indeed, N = S implies L 0 = L, P 2 (~x ; x ) = if ~x = x, and zero otherwise. In turn, p 2 (x ) = if x < L, and p 2 (x ) = 0 if x > L.

14 4 Ludek Berec Analogously, consider a single predator individual that shares a site with a pair of (dierent) prey individuals, and compute the partial decision probability p 32 of attacking prey 2 out of this pair. Obviously, p 32 is the probability that the condition ~x + ~x 2? ~x ~x 2 =N < L 0 2 is satised. For xed total prey abundances 0 < x i < S, there are!! S? S? (2) x? x 2? possibilities for the distribution of prey individuals on the lattice containing S sites, conditioned on the event that the focus site is occupied by one prey individual, one prey 2 individual, and the predator. The number of possibilities by which ~x prey individuals and ~x 2 prey 2 individuals might be located in the N-neighbourhood is!! S? N S? N N? ~x? x? ~x! N? ~x 2? x 2? ~x 2! ; (22) provided that maxf; x i? (S? N)g ~x i minfx i ; N g, i = ; 2, whereas it is zero otherwise. The probability P 32 (~x ; ~x 2 ; x ; x 2 ) that ~x prey individuals and ~x 2 prey 2 individuals are located in the predator N-neighbourhood thus denes the multihypergeometric distribution P 32 (~x ; ~x 2 ; x ; x 2 ) = 8 >< >:? N? ~x?? S?N? N?? S?N x?~x ~x2? x2?~x2? S?? S? x? x2? if maxf; x i? (S? N)g ~x i minfx i ; N g ; 0 otherwise : (23) Finally, the probability p 32 that ~x + ~x 2? ~x ~x 2 =N < L 0 2 is p 32 (x ; x 2 ) = X A P 32 (~x ; ~x 2 ; x ; x 2 ) ; (24) where the summation is performed over the set A f~x ; ~x 2 : ~x + ~x 2? ~x ~x 2 =N < L 0 2g. Moreover, p 3 (x ; x 2 ) =? p 32 (x ; x 2 ) : (25) Fig. 4 shows the total decision probabilities (2) and 2 (3) as functions of x, for a xed value of x 2 and for locally omniscient predators, i.e., for the partial decision probabilities p =, p 2 given by the expression (20), p 3 =? p 32, and p 32 given by the expression (24). The abrupt changes observed in g. 2 are replaced by gradual transitions over the prey threshold values x and x in g. 4, and the range of x in which partial preferences appear

15 Mixed Encounters, Limited Perception and Optimal Foraging 5 has increased a bit. As N approaches S, p 32 (x ; x 2 ) approaches the zeroone step function; indeed, for N = S, it is L 0 2 = L 2, P 32 (~x ; ~x 2 ; x ; x 2 ) = if ~x = x and ~x 2 = x 2, and zero otherwise. In turn, p 32 (x ; x 2 ) = if x + x 2? x x 2 =S < L 2, and p 32 (x ; x 2 ) = 0 if x + x 2? x x 2 =S > L 2. Figure 4 around here. Fig. 5 is analogous to g. 3, with the total decision probabilities used in g. 2 replaced by those used in g. 4. The abrupt changes in g. 3 are replaced by gradual transitions over the prey threshold values x and x in g. 5. Figure 5 around here. 4. Discussion In this article, we addressed the question of how perceptual limits of predators and their possibility to encounter prey both sequentially and simultaneously inuence the predator attack decisions, diet composition and functional response of the predator-prey system. We have assumed that predators forage optimally on two prey types, that prey individuals are encountered by predators either alone or in pairs composed of dierent prey types, and that every predator has exact knowledge of prey population abundances only in a neighbourhood of their actual spatial position. These features have been implemented quite naturally via a discrete-time, discrete-space, individual-based model. The conventional model of optimally foraging predators for prey distributed randomly in the environment and maximizing their average rate of net energy gain during foraging assumes, among others, that prey items are encountered sequentially, i.e., one at a time, and that predators are omniscient, i.e., they have exact knowledge of prey population densities in the whole environment at any time instant (Schoener, 97; Pulliam, 974; Werner and Hall, 974; Charnov, 976a; Stephens and Krebs, 986; McNamara and Houston, 987; Krivan, 996; Krivan and Sikder, 999, and many others). However, the zero-one rule predicted by this model, i.e., to always attack

16 6 Ludek Berec more protable prey and to attack less protable prey upon each encounter provided that the density of more protable prey is below a threshold value and do not attack it at all if the reverse is true, was not observed in real systems based on this model (Werner and Hall, 974; Davies, 977; Krebs et al., 977; Rechten et al., 983; Mittelbach, 98; Thompson et al., 987, and many others). Although we suspect that some of these works did not set up the experimental conditions properly and some did not process relevant data, we do not doubt that the conventional model is too academic to hold in nature or even under the strictly controlled laboratory conditions. Partial preferences for both prey types and the dependence of predator diet composition on encounter rates with (or population densities of) both prey types are among the most common observations, of which the original model predicts the dependence of optimal diet on density of more protable prey only. Many works that aimed at making this model at least a bit more realistic took the way of removing an assumption from the original model formulation. Some of these works relaxed the assumption on predator omniscience (McNamara and Houston, 987; Mangel and Roitberg, 989; Belisle and Cresswell, 997; Hirvonen et al., 999; Berec and Krivan, submitted) and some the assumption on sequential encounters (Waddington and Holden, 979; Engen and Stenseth, 984b; Stephens et al., 986; Barkan and Withiam, 989). Some of the mechanisms such as nutritional constraints (Pulliam, 975), prey density estimation (McNamara and Houston, 987; Hirvonen et al., 999), limited memory capacity (Belisle and Cresswell, 997; Mangel and Roitberg, 989), or sampling (Krebs et al., 977; Rechten et al., 983) predict partial preferences for less protable prey, some such as misidentication of prey types (Krebs et al., 977; Rechten et al., 983) predict partial preferences for both prey types, and some such as recognition time (Hughes, 979; Houston et al., 980), learning (Hughes, 979; McNair, 98), crypsis (Hughes, 979), or non-poisson encounters (McNair, 979) predict the zero-one rule yet dependent on the encounter rate (or equivalently abundance or density) of less protable prey. We have relaxed both the assumptions on predator omniscience and sequential encounters and showed that this results in predictions of partial preferences for both types of prey as well as their dependence on the abundance of less protable prey. Consideration of mixed (i.e., both sequential and simultaneous) encounters is by no means a standard topic in literature on optimal foraging. Whereas Stephens et al. (986) treat this situation rather vaguely, Engen and Stenseth (984b) form the other extreme and treat it very precisely but too generally: their work covers the patch optimal foraging models, too, and is formulated for an arbitrary number of prey types. Both works thus obscure some interesting quantitative insights. We may nd ourselves somewhere in the middle: we consider two prey types that are randomly distributed

17 Mixed Encounters, Limited Perception and Optimal Foraging 7 in the environment and perform a precise analysis of optimal foraging behaviour by predators. Hence, apart from the more protable prey abundance (or equivalently density, encounter rate or searching time) threshold known from the works on sequential encounters we discover another one that is a consequence of mixed encounters. Whereas the value of this `second' threshold is known from the works on fully simultaneous encounters (Stephens et al., 986; Barkan and Withiam, 989) { and one could be tempted to put these results together without properly revised optimization { the important dierence is that whereas the second threshold is confronted with the more protable prey density (or its equivalent) in these works, the revised optimization for mixed encounters shows that the decision which out of a pair of dierent prey individuals to attack depends on both prey densities via a non-linear combination (8). The most important consequence of the conventional optimal foraging model combined with mixed encounters with prey, which has already been recognized qualitatively by Engen and Stenseth (984b), is that the predator decision to attack prey no more follows the zero-one rule but demonstrates partial preferences for both prey types, even under the assumption of predator omniscience. We note that of particular interest are the partial preferences for more protable prey that are not predicted standardly by other suggested mechanisms (see McNamara and Houston (987) for a review). We have quantied these partial preferences and visualized them for a particular set of parameters in g. 2. The partial preferences for both prey types have, however, a discontinuous course stemming from the synchronous change of predators in their diet, a prediction that seems not to be too realistic. A discontinuity of partial preferences is also predicted, for example, when the assumptions on nutritional constraints (Pulliam, 975) or a limited memory capacity (Belisle and Cresswell, 997) of predators are combined with the conventional prey optimal foraging model. No doubt limited perceptual abilities of real predators do not concord with the standard assumption on predator omniscience, particularly for large habitats. McNamara and Houston (987) and Hirvonen et al. (999) introduce a mechanism by which predators estimate the prey population densities from the actual sequence of their encounters with prey. Berec and Krivan (submitted) assume exact knowledge of prey densities only in a limited predator neighbourhood. Both these mechanisms, derived for the prey optimal foraging model with sequential encounters, change the zero-one rule for less protable prey to a smooth sigmoid-like form of the partial preferences for that prey. In this article, we have been interested in what limited perceptual abilities of predators can bring if partial preferences are predicted already for omniscient predators. We have shown that they change the discontinuous character of partial preferences to a smooth form (see g. 4). We did not inspect a possibility and consequences of applying the mechanism of

18 8 Ludek Berec estimating prey population densities for the model with mixed encounters, but we guess it should be manageable and lead to (at least qualitatively) the same results. We have also demonstrated in the paper that the discontinuity of partial preferences was reected in a discontinuous character of the predator diet composition and functional response (see g. 3) and that the limited perception mechanism smoothed these predator features, too (see g. 5). The predator decision rule based on local prey abundances ~x and ~x 2 and specied by the quantities L 0 and L 0 2, though derived for the homogeneous system in which individuals are randomly distributed across the lattice, seems to be justied for prey individuals that disperse at a limited rate, too. For the fully dynamic individual-based model involving reproduction and mortality processes, spatial pattern occurs on the lattice at low diusive rates (de Roos et al., 99; McCauley et al., 993; Wilson et al., 993) and the environment is no more ne-grained. Yet there is a so-called characteristic spatial scale (or a characteristic area size in the case of two-dimensional lattices) such that if we observe such a system in a window of a smaller scale the dynamics are reminiscent of those of the system in which individuals are randomly distributed on the lattice. The characteristic spatial scale depends on the lattice size and the population dispersal rates, and is closely related to the area visited by individuals during their lifetime (de Roos et al., 99). Hence, the limited-perception-based decision rule might be used even when prey individuals disperse at a limited rate, provided that the size of predator N-neighbourhood is smaller than the characteristic spatial scale. Finally, we would like to stress that we have proted a great deal from a spatially explicit individual-based model that determines the fate of each animal on the lattice. The character of IBM enables us to potentially consider even more with respect to the state of single individuals, e.g., dierent sizes, ages, sex etc., which may, in turn, lead to considerations of state-dependent handling times, perception neighbourhood sizes, probabilities of successful attack etc. Also, population dynamics can be coupled with the foraging behaviour by implementing full life-cycles of the populations involved (de Roos et al., 99; McCauley et al., 993; Wilson et al., 993). Hence, there is still an open area to work on. By constructing a sequence of IBM, each of them having a new feature compared to the previous one, we may inspect how the observed predator or prey characteristics change on the way from simple, strategic models to more realistic but complex, tactical models of ecological interactions (Murdoch et al., 992). Moreover, such an increase in system reality would certainly make the conclusions more credible from the side of experimental ecologists.

19 Mixed Encounters, Limited Perception and Optimal Foraging 9 5. Summary In this article, we show how the local omniscience of predators invoked by their perceptual constraints and the allowance of both sequential and simultaneous encounters with prey may shape the attack decisions, diet composition and functional response of optimal foragers. With the help of an individual-based model of foraging behaviour we show that predators demonstrate gradually changing partial preferences for both prey types considered. These partial preferences are shown, moreover, to depend on the population numbers of both prey. As a result, diet composition and functional response of predators do not demonstrate a discontinuity in the space of prey densities, contrary to the usual predictions of optimal foraging models found in literature. 6. Acknowledgements The author acknowledges the nancial support by the Grant Agency of the Czech Republic (Grant 20/98/P202) and by the Ministry of Education of the Czech Republic (Grant VS 96086). Appendix A. Diet Composition Here we outline a derivation of the diet composition formula (). We show that for a set of partial decision probabilities p, p 2, p 3, and p 32, the fraction of prey i = ; 2 individuals of those captured by a single predator in a series of time steps is x i S ip i a ; (26) with = x S P a? x 2 S + x2 S 2P 2 a p + x 2 p 2 + x S p 3 ; (27) 2 =? x S S p 32 ; (28) where S is the number of lattice sites, x i is the prey i abundance on the lattice, and P i a stands for a probability of successful attack of a prey i item by a predator if the predator decides to attack that prey upon encounter. Given that both prey types are randomly distributed on the lattice at each time step, with each site occupied by at most one prey item of each type, the probability that the predator shares a site with a prey item alone is x =S(? x 2 =S). Then, the probability that the predator will successfully attack that prey is x =S(? x 2 =S)p P, i.e., the probability to encounter the prey item decide to attack it a attack it successfully. Consequently, the mean number of prey items successfully attacked by the predator within any t time steps is t s x =S(? x 2 =S)p P a, where t s t is the number of time steps in which the predator was in the searching state. The probability that the predator shares a site with both prey types is

20 20 Ludek Berec x =S x 2 =S. Therefore, the probability that the predator will successfully attack prey i out of the pair is x =S x 2 =S p 3i P i a. The mean number of prey items encountered in a pair and successfully attacked by the predator within t time steps is t s x =S x 2 =S p 3 P a. Analogous expressions can be derived for prey 2. Hence, the fraction of prey i = ; 2 individuals that are captured by the predator within a series of time steps (regardless of whether prey are encountered alone or in a pair) is given by the expression (26). References Abrahams, M. V. (986). Patch choice under perceptual constraints: a cause for departures from an ideal free distribution. Behavioral Ecology and Sociobiology 9, 409{45. Barkan, C. and M. Withiam (989). Protability, rate maximization, and reward delay: a test of the simultaneous-encounter model of prey choice with Parus atricapillus. American Naturalist 34, 254{272. Belisle, C. and J. Cresswell (997). The eects of a limited memory capacity on foraging behavior. Theoretical Population Biology 52, 78{90. Berec, L. and V. Krivan (submitted). A mechanistic model for partial preferences. Charnov, E. L. (976a). Optimal foraging: attack strategy of a mantid. The American Naturalist 0, 4{5. Charnov, E. L. (976b). Optimal foraging, the marginal value theorem. Theoretical Population Biology 9, 29{36. Davies, N. (977). Prey selection and the search strategy of the spotted ycatcher (Muscicapa striata): a eld study of optimal foraging. Animal Behaviour 25, 06{033. de Roos, A. M., E. McCauley and W. G. Wilson (99). Mobility versus densitylimited predator-prey dynamics on dierent scales. Proceedings of the Royal Society of London B 246, 7{22. Engen, S. and N. C. Stenseth (984a). An ecological paradox: A food type may become more rare in the diet as a consequence of being more abundant. The American Naturalist 24, 352{359. Engen, S. and N. C. Stenseth (984b). A general version of optimal foraging theory: The eect of simultaneous Encounters. Theoretical Population Biology 26, 92{ 204. Fryxell, J. M. and P. Lundberg (994). Diet choice and predator-prey dynamics. Evolutionary Ecology 8, 407{42. Gleeson, S. K. and D. S. Wilson (986). Equilibrium diet: optimal foraging and prey coexistence. Oikos 46, 39{44. Gray, R. and M. Kennedy (994). Perceptual constraints on optimal foraging: a reason for departures from the ideal free distribution? Animal Behaviour 47, 469{ 47. Hirvonen, H., E. Ranta, H. Rita and N. Peuhkuri (999). Signicance of memory properties in prey choice decisions. Ecological Modelling 5, 77{89. Houston, A., J. Krebs and J. Erichsen (980). Optimal prey choice and discrimination time in the great tit (Parus major L.). Behavioral Ecology and Sociobiology 6, 69{75.

21 Mixed Encounters, Limited Perception and Optimal Foraging 2 Hughes, R. (979). Optimal diets under the energy maximization premise: the eects of recognition time and learning. American Naturalist 3, 209{22. Krebs, J. R., J. T. Erichsen, M. I. Webber and E. L. Charnov (977). Optimal prey selection in the great tit (Parus major). Animal Behaviour 25, 30{38. Krivan, V. (996). Optimal foraging and predator-prey dynamics. Theoretical Population Biology 49, 265{290. Krivan, V. and A. Sikder (999). Optimal foraging and predator-prey dynamics, II. Theoretical Population Biology 55, {26. Mangel, M. and B. Roitberg (989). Dynamic information and host acceptance by a tephritid fruit y. Ecological Entomology 4, 8{89. McCauley, E., W. G. Wilson and A. M. de Roos (993). Dynamics of age-structured and spatially structured predator-prey interactions: individual-based models and population-level formulations. The American Naturalist 42, 42{442. McNair, J. N. (979). A generalized model of optimal diets. Theoretical Population Biology 5, 59{70. McNair, J. N. (98). A stochastic foraging model with predator training eects. II. Optimal diets. Theoretical Population Biology 9, 47{62. McNamara, J. M. and A. I. Houston (987). Partial preferences and foraging. Animal Behaviour 35, 084{099. Mitchell, W. (989). Informational constraints on optimally foraging hummingbirds. Oikos 55, 45{54. Mittelbach, G. (98). Foraging eciency and body size: a study of optimal diet and habitat use by bluegills. Ecology 62, 370{386. Murdoch, W. W., E. McCauley, R. M. Nisbet, W. S. C. Gurney and A. M. de Roos (992). Individual-based models: Combining testability and generality, in De Angelis, D. L. and Gross, L. J. (Eds), Individual-based models and approaches in ecology - populations, communities and ecosystems, Chapman & Hall, New York, pp. 8{35. Pulliam, H. R. (974). On the theory of optimal diets. The American Naturalist 08, 57{74. Pulliam, H. R. (975). Diet optimization with nutrient constraints. The American Naturalist 09, 765{768. Rechten, C., M. Avery and A. Stevens (983). Optimal prey selection: why do great tits show partial preferences? Animal Behaviour 3, 576{584. Rice, W. R. (983). Sensory modality: An example of its eect on optimal foraging behavior. Ecology 64, 403{406. Schmidt, K. A. (998). The consequences of partially directed search eort. Evolutionary Ecology 2, 263{277. Schmitz, O. J. (997). Commemorating 30 years of optimal foraging theory. Evolutionary Ecology, 63{632. Schoener, T. W. (97). Theory of feeding strategies. Annual Review of Ecology and Systematics, 369{404. Spencer, H. G., M. Kennedy and R. D. Gray (996). Perceptual constraints on optimal foraging: the eects of variation among foragers. Evolutionary Ecology 0, 33{339. Stephens, D. W. and J. R. Krebs 986. Foraging theory. Princeton University Press, Princeton, NJ. Stephens, D. W., J. F. Lynch, A. E. Sorensen and C. Gordon (986). Preference and protability: Theory and experiment. The American Naturalist 27, 533{553.

22 22 Ludek Berec Stillman, R., J. Goss-Custard and R. Caldow (997). Modelling interference from basic foraging behaviour. Journal of Animal Ecology 66, 692{703. Thompson, D. B., D. F. Tomback, M. A. Cunningham and M. C. Baker (987). Seed selection by dark-eyed juncos (Junco hyemalis): optimal foraging with nutrient constraints? Oecologia 74, 06{. Waddington, K. and L. Holden (979). Optimal foraging: on ower selection by bees. American Naturalist 4, 79{96. Werner, E. E. and D. J. Hall (974). Optimal foraging and the size selection of prey by the bluegill sunsh (Lepomis macrochirus). Ecology 55, 042{052. Wilson, W. G. (998). Resolving discrepancies between deterministic population models and individual-based simulations. The American Naturalist 5, 6{ 34. Wilson, W. G., A. M. de Roos and E. McCauley (993). Spatial instabilities within the diusive Lotka-Volterra system: Individual-based simulation results. Theoretical Population Biology 43, 9{27.

23 Mixed Encounters, Limited Perception and Optimal Foraging 23 Figure legends: Fig.. Division of the space of prey population sizes into regions with dierent optimal foraging strategies of omniscient predators. Parameter values: S = 28 28; P a = 0:3; T = 2; P 2 h a = 0:85; T 2 = 8; c = :2. h Fig. 2. Partial preferences for (more protable) prey (dashed line) and (less protable) prey 2 (solid line) for omniscient predators experiencing mixed encounters, as functions of prey abundance x. Prey 2 abundance is xed to the value x 2 = Other parameter values: S = 6384(2828); P a = 0:3; T = 2; P 2 h a = 0:85; T 2 = 8; c =. h Fig. 3. The average fraction of prey individuals captured by an optimally foraging predator in a series of time steps (or by more predators in a single time step) against the prey population abundance (g. 3A), the same quantity plotted against the fraction of prey in the environment (g. 3B), and the functional response of optimally foraging predators to prey (the average number of successfully attacked prey individuals per predator per time step) against the number of prey in the environment (g. 3C). Prey 2 abundance is xed to the value x 2 = The vertical dotted lines mark locations of the prey threshold values x and x. Parameter values are the same as in g. 2. Fig. 4. Partial preferences for (more protable) prey (dashed line) and (less profitable) prey 2 (solid line) for predators with limited perception neighbourhood and mixed encounters with prey, as functions of prey abundance x. Prey 2 abundance is xed to the value x 2 = Parameter values are the same as in g. 2, N = 49 (77). Fig. 5. The average fraction of prey individuals captured by a locally omniscient predator in a series of time steps (or by more predators in a single time step) against the prey population abundance (g. 5A), the same quantity plotted against the fraction of prey in the environment (g. 5B), and the functional response to prey of predators exhibiting limited perception (the average number of successfully attacked prey individuals per predator per time step) against the number of prey in the environment (g. 5C). Prey 2 abundance is xed to the value x 2 = The vertical dotted lines mark locations of the prey threshold densities x and x. Parameter values are the same as in g. 2, N = 49 (77).