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Optimal foraging and the traveling salesman
THEORETICAL
POPULATION
Optimal
24, 145-159 (1983)
BIOLOGY
Foraging
and the Traveling
Salesman
D. JOHN ANDERSON* Department of Zoology NJ-IS. University of Washington. Seattle, Washington 98195 Received December 7, I98 I
Optimal foraging models are examined that assume animals forage for discrete point resources on a plane and attempt to minimize their travel distance between resources. This problem is similar to the well-known traveling salesman problem: A salesman must choose the shortest path from his home office to all cities on his itinerary and back to his home oftice again. The traveling salesman problem is in a class of enigmatic problems, called NP-complete, which can be so difficult to solve that animals might be incapable of finding the best solution. Two major results of this analysis are: (I) The simple foraging strategy of always moving to the closest resource site does surprisingly well. More sophisticated strategies of “looking ahead” a small number of steps, choosing the shortest path. then taking a step, do worse if all the resource sites are visited, but do slightly better (less than 10%) if not all the resource sites are visited. (2) Short cyclical foraging routes resulted when resources were allowed to renew. This is suggested as an alternative explanation for “trap-lining” in animals that forage for discrete, widely separated resources.
Animals are faced with decisions of where and when to move. The outcomes of these decisions can be of crucial importance. Ultimately, they may determine energy intake, exposure to predation, degree of competition and mating success. In this paper I address one aspect of this problem: What are the optimal patterns of movement in animals foraging for discrete resources on a plane? This problem is related to an important class of problems, such as the traveling salesman problem, that are extremely difficult to solve. The problem may be so difficult to solve that animals are simply not capable of finding the best solution. For this reason the performance of a class of simple, less than optimal (but plausible), models of foraging will be compared to the optimal pattern of movement for a variety of resource distributions types. In addition I examine the effects of renewable * Present address: Solaster Software Corporation, P. 0. Box 15216, Seattle, Wash. 98115.
14.5 0040-5809183$3.00 Copyright 0 1983 by Academic Press, Inc. All rights of reproduction in any form reserved.
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resources on the long-range behavior of animals using these simple decision rules for foraging. The models presented here, like other optimal foraging models, make the following assumptions (Schoener, 1971; Pyke ef al., 1977: Gould and Lewontin, 1979; Lewontin, 1979): Darwinian natural selection continuously tends to increase the fitness of organisms within certain constraints. Foraging behaviors are passedfrom generation to generation, having either a genetic or cultural basis, or some combination thereof. They exhibit variation, so there is a range of possible behaviors upon which selection acts. Fitness is maximized when the net energy gain is maximized. This is true even for animals with fixed energy requirements, becauseforaging time saved can be invested in other useful activities, some of which are also related to fitness. According to these assumptions animals should exhibit behaviors that maximize their energy intake. Pyke ef al. (1977) grouped the situations to which optimal foraging models have been applied into four categories: “( 1) choice by an animal of which food types to eat (i.e., optimal diet); (2) choice of which patch type to feed in (i.e., optimal patch type); (3) optimal allocation of time to different patches; and (4) optimal patterns and speed of movements.” The models presented in this paper are in the fourth category, which historically has received the least attention. Published models of optimal movement patterns represent food resources as points on a plane. (Pyke, 1974, 1978a,b; Cody 1971, 1974). Animals are assumedto move with certain probabilities straight ahead, backwards, to the right, or to the left. Once a direction of movement is chosen, an animal moves to the closest resource site in that movement direction. Both Cody and Pyke argue that the maximum rate of energy intake, constrained by the above capabilities of movement, is obtained when path overlap is minimized, so that the animal does not waste energy visiting already depleted resources. They then proceed to determine the parameters that minimize overlap. If animals are assumed to follow movement rules besides those proposed by Pyke and by Cody, even higher energy intake can be obtained. In the extreme case, for point resources, the maximum energy intake is obtained by visiting all resource sites and minimizing travel cost. Travel costs can be important for animals such as bees. According to Heinrich (1979, p. 109) “a bee spends most of its time traveling, and all the efficiency that it may derive from its flight motor is of little avail if that traveling does not achieve given objectives. When moving between food sources, bees try to keep flight time and distance to a minimum.” This foraging problem is a variation of the traveling salesman problem: A salesman travels from his office to a number of cities and back to his home office. The goal is to find the shortest path that contains all the cities (Graham, 1978, p. 197). The only difference between the standard traveling salesman problem and this foraging problem
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is the last step-the animal does not necessarily return to the original starting resource site. Much that is known about the traveling salesman problem has implications for foraging animals. The traveling salesman problem is in a class of enigmatic problems, called “NP-complete,” which are among the more difficult problems to solve (Karp, 1975; a formal definition of NPcomplete, too long to present here, can be found in Garey and Johnson, 1979). They resist analytic solution and no known methods exist to solve them efficiently when the number of cities is large. Perhaps the simplest way to solve the traveling salesman problem is to examine all possible paths. For a C city problem there are (C - l)! possible paths to examine. This amounts to 6.08 x 106’ potential paths to examine for a 50-city problem. Even if one route could be examined in the time it takes light to travel 1 cm, it would take 6.43 x lOa centuries to complete the search! This so-called combinatorial explosion characterizes all NP-complete problems; even the most clever known methods require a worst-case completion time which grows exponentially as a function of the number of cities examined. With current methods, for example, it is only practical to solve the traveling salesman problem for fewer than about 100 cities. Strong circumstantial evidence suggests that efficient methods to solve these problems do not exist. By efficient I mean a method that terminates within a time bounded by a polynomial in the “length” of the input (i.e.. a finite sum of terms, each term being the number of cities or resource sites raised to a constant power). If an efficient, or polynomial-time, method did exist for any of these NP-complete problems, then it could be used to solve all NP-complete problems in polynomial time (Reingold et. al. 1977. p. 405). Efficient methods are thought not to exist because such a large number of classic unsolved problems are NP-complete and no polynomial-time method has been found for any of them (Garey and Johnson, 1979). NP-complete problems include a host of combinatorial optimization problems such as scheduling problems, computational graph problems and bin packing problems (Garey and Johnson, 1979). Undoubtedly there are other NP-complete or NP-hard (at least as hard as NP-complete) problems that animals must face. Animals may be physically unable to find the exact optimum for some of the combinatorial problems they face. This could have implications for optimization theory in ecology. There is some hope, however. These results on the intractability of combinatorial problems apply to worst-case situations. They do not rule out the possibility of methods which efficiently solve these problems most of the time, but not always. Also, methods may exist which efficiently find approximately optimal solutions. These approaches have had mixed success (Karp, 1976). For the traveling salesman problem, Christofides (1976) has developed an efficient approximate method which is guaranteed to yield a
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tour whose length is less than 1.5 times the length of the optimal tour. No other approximate method is known to have a lower guaranteed bound (Gary and Johnson, 1979). In this paper I examine approximation methods for the solution of the foraging problem. Some, but not all, plausible foraging behaviors will be examined and compared with the optimal solution.
THE MODEL
I assume that an animal is foraging on a finite plane, in this case the unit square. Randomly scattered over the plane are points representing food resource sites. By random I mean that the X and Y coordinate of each resource site was independently drawn from a uniform distribution over the unit interval. Other distributions of resource sites will be examined later. An animal is initially placed on one of the food resource sites and allowed to forage according to certain rules of movement (described below) until all the resources are consumed. Food resources are assumed to have equal energy values which are known to the animal or arbitrary energy values which are unknown to the animal. In either case the best an animal can do is visit all resource sites with equal preference. Initially, I assume that each resource, once visited, is completely consumed and is never renewed. Therefore an animal has nothing to gain by visiting a resource more than once. Either the resource disappears or the animal marks or in some way remembers it to avoid returning. Later these assumptions will be relaxed, allowing resources to be renewed. A cost is associated with traveling that is proportional to the total distance moved. The goal, as in many other optimal foraging models is to maximize net energy intake. This is obtained when the animal visits all resources, irrespective of their energy content, and minimizes the distance traveled, assuming constant speed of movement. Furthermore it is assumed that the cost of traveling to any resource site is less than the energy gained by consuming the resource. Therefore it always pay to visit every resource site. Without this assumption it is only possible to explore the problem with a very limited number of resources, for reasons explained below.
LOOK-AHEAD
STRATEGIES
One of the simplest foraging strategies that can be envisioned is always moving to the nearest resource site that has not been eaten. A more sophisticated strategy which I will call “L-step look-ahead” was introduced by Altmann (1974, pp. 24&241). It works like this: The animal contem-
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plates (looks ahead at) all possible paths from its current location that visit L of the resource sites, chooses the path with the shortest total length, moves to the first resource site in the chosen path and then repeats the process. If L is larger than the number of remaining resources the animal only considers movement to the remaining resources sites. This strategy is similar to how a game might be played. The player considers the potential paths with L steps, chooses the best one, makes a move and then repeats the process. Animals would not need to examine explicitly all possible paths L steps away. Instead, the animal might choose an initial path containing L steps, possibly by using the nearest neighbor rule. Next the animal begins considering alternate paths. If, while considering a path containing fewer than L steps, the distance already traveled would be greater than the initial path, then all paths containing this partial path need not be considered-they will necessarily be further than the initial path since all distances are positive. If an L-step path is found that is shorter than the initial path, its distance is used when considering remaining partial paths. Using this process, which I will call “path pruning,” combinations of paths involving far-away resources need not be considered. Only “reasonable” paths involving nearby resources need to be examined. Even when some paths can be eliminated in this way, the problem remains NP-complete and exhibits the familiar combinatorial explosion. Furthermore, using path pruning to simplify the problem requires that the cost of traveling to a resource site be less than the energy content of that resource. Without path pruning it would be impossible to carry out the calculations presented here on computers that were available to me. When the animal looks ahead one step in the future (one-step look-ahead) it always moves to the nearest resource site. If there are R resources and the animal employs R-step or greater look-ahead, then it will always choose the shortest, or best possible path. R-Step look-ahead exhibits the combinatorial explosion, making it impractical for large numbers of resources. When L is varied, L-step look-ahead displays a range of possible strategies, from moving to the closest resource point (a simple task) to the best possible strategy (a difficult task). Approximate methods like one-step look-ahead can do very poorly. Figure 1 illustrates a situation where one-step look-ahead is nearly twice as bad as the optimal strategy. However, in nature these situations rarely occur. A better approach would be to compare the average behavior of various foraging strategies for resource site distributions that animals are likely to encounter. To quantify the average behavior of different look-ahead strategies I will use D(L, R, V), the average total distance traveled for a particular combination of L (amount of look-ahead), R (number of resources), and V (number of resources points visited).
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FIG. I. An example illustratinghow poorly one-step look-ahead can perform. Figure la shows a path through resource points using one-step look-ahead that produces a toal distance of 128 units. Figure lb shows another path through the same resource points with a distance of 67, almost half the distance produced using one-step look-ahead.
Simulations were carried out to estimate D as follows: an animal was randomly placed on one of the R resource sites, allowed to forage using the L-step look-ahead rule, with path pruning, until it had visited a total of V resource sites. This procedure was repeated many times with different resource locations and D was estimated as the average total distance the animal traveled per repetition. Consider first the case when the animal visits every resource site (V = R). To compare various strategies, the ratio of total distance traveled using onestep look-ahead to total distance traveled using L-step look-ahead (D(1, R, R)/D(L, R, R)) is plotted in Fig. 2. Only certain ranges of parameters are shown because of the excessive computation time for the other cases. 1.1 -
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L (Look-Ahead) FIG. 2. A comparison of various look-ahead strategies when the animals visit all resource sites (V = R). The distance ratio, plotted on the Y axis, is the ratio of the total distance traveled using one-step look-ahead to the total distance traveled using L-step look-ahead (D(1, R, R)/D(L, R, R)). The larger the ratio the more efficiently the animal forages. Notice that simple one-step look-ahead does better than 2-. 3-, and 4-step look-ahead. Four different numbers of resource sites (R), placed randomly on the plane, were used. Each point represents the average of 500 replicate simulations.
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There are two interesting results. Focusing first on the results for 10 resource sites, one-step look-ahead does surprisingly well-only about 8.5% worse than 9-step look-ahead (the best possible strategy). Two-step lookahead does worse than one-step look-ahead and it is not until 5step lookahead that there is any improvement over one-step look-ahead. For 6-step look-ahead with 10 resource sites, when an animal makes its first decision about which resource site to visit, there are 60,480 ([R - l]!/[R - 1 -L]!) different steps possible; that is, only 6 times less than the number using 9step look-ahead. The 6-step look-ahead, therefore, requires that many more steps be examined than one-step look-ahead with only a slight benefit of reducing travel distance. In an evolutionary sense, if animals started with one-step look-ahead and look-ahead could only evolve through small increases, then selection would not favor these small increases. If large jumps in look-ahead could occur, however, they would be favored by selection. When the number of resources increase, these trends are magnified. Twostep look-ahead does even worse than one-step look-ahead, higher amounts of look-ahead take even longer to improve and the number of possible steps grows large even faster (Fig. 2). With only 5-step look-ahead and 64 resource sites there are already 843,461,640 steps possible. Increasing the number of resource sites produces successively smaller changes in the curves shown in Fig. 2, until the change observed by increasing the number of resource sites from 46 to 64 in negligible. The surprising result that one-step look-ahead does as well as or better than 2-, 3-, 4- and 5-step look-ahead requires an explanation. When watching a hypothetical animal employ 2-step look-ahead it becomes clear that the animal skips a few nearby resource sites early in the tour for the sake of saving a little distance, but later must go back to visit many widely distributed resource sites that it left behind. This can be seenusing paper and pencil to draw some sample tours. With 2-step look-ahead the underlying distribution of distances traveled during the first and middle steps are almost identical. The distribution of distance of the last step is sharply skewed towards longer distances. How do the various look-ahead strategies perform when only part of the resource sites are visited (i.e., V < R)? To answer this question, Fig. 3a shows the ratio of total distance traveled using one-step look-ahead to total distance traveled using 2-step look-ahead (D(1, R, V)/D(2, R, V)) as a function of V, the number of resource sites visited (R = 10, 20, 40, and 60 resource sites). Figure 3b is the same as Fig. 3a except it compares one-step look-ahead to 5-step look-ahead (D(1, R, v)/D(5, R, V)). Some interesting results emerge: first, 2-step look-ahead, even though it does significantly worse than one-step look-ahead when all the resource sites are visited, can actually do slightly better if the animal visits at least the first few resource
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FIG. 3. A comparison of various look-ahead strategies assuming the animal does not visit all the resource sites (V < R). Shown on the Y axis is the ratio of average distance traveled using one-step look-ahead to the average total distance traveled using 2-step look-ahead (Fig. 3a) or 5step look-ahead (Fig. 3b). A ratio larger than one is more efficient than one-step look-ahead and a ratio less than one is less efficient than one-step look-ahead. Figure 3a. calculated using 2-step look-ahead, shows that if more than the first few resource sites but fewer than about 75% of the total are visited, then 2-step look-ahead out performs one-step look-ahead by a small amount (less than 4%). For 5-step look-ahead (Fig. 3b) this advantage is increased, but remains below about 10“6. Each curve, constructed from the average of 500 replicates, shows the results for a different number of randomly placed resource sites.
sites and not more than 75 % of the total number. The benefit of using 2-step look-ahead over one-step look-ahead is small, at most only about 4%. Using 5-step look-ahead, which is probably more than most animals are capable (Fig. 3b), increases this benefit to at most IO%, but usually less, depending upon the number of resource sites visited. Summarizing, small increases in look-ahead over one-step look-ahead will pay off if not all the resource sites are visited; but, the payoff is small, less than about 10%. Under certain circumstances R-step look-ahead, the optimal strategy to
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visit all the resource sites, does worse than one-step look-ahead if not all the resource sites are visited. In particular, for 10 resource sites, one-step lookahead out performs 9-step look-ahead if 8 or fewer resource sites are visited. The difference is largest, about 12 941,when four or fewer resource sites are visited. The particular foraging path is highly dependent upon the location of food resource sites. The random distribution from which resource locations were drawn is only one highly idealized possibility. To determine how these results depend upon the distribution of food resource sites, two alternative distributions were used in the simulations. Pielou (1977, p. 159) has pointed out that most spatial distributions in nature are clumped rather than random or even. Therefore. the first distribution. which will be called “randomGaussian.” consists of clumps of resource sites where the location of each clump is chosen randomly on the plane and the resource locations within each clump were chosen from a circular normal, or twodimensional Gaussian distribution in which the X and Y standard deviations are the same. The second distribution, which will be called “even-Gaussian” differs from the random-Gaussian in that the location of each clump, rather than being drawn at random, was placed on an evenly spaced N x N lattice. This lattice was centered on the unit square such that the distance between the center of each clump and its neighbors or the edge was l/(N + I ).
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L (Look-Ahead) FIG. 4. This figure illustrates how look-ahead strategies perform under different kinds of resource distributions. As in Fig. 2, the Y axis shows the ratio of one-step look-ahead to Lstep look-ahead (D(1, R. R)/D(L. R, R)) and the X axis shows amount of look-ahead. Four different cases are illustrated, representing 10 and 40 resource sites, random-Gaussian or evenGaussian distributions (see text), different numbers of clumps and different amounts of clumping. Notice how under these various conditions the qualitative results obtained earlier (shown in Fig. 2) remain unchanged. The degree of clumping was adjusted by varying the standard deviation of the circular normal distribution from which the resource sites were sampled. Each point on the graph was constructed from the average of 500 replicates.
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Simulations were carried out for a variety of parameters. including different numbers of clumps, different numbers of resource sites. and varying degrees of resource site clumping. Resource site clumping was changed by varying the standard deviation of the circular normal distribution from which locations were drawn. The results of some sample cases are shown in Fig. 4. Qualitatively, the results are similar to those obtained earlier (both in Figs. 2 and 3) using randomly distributed resource sites.
RENEWABLE RESOURCES
Given that the animal uses some form of look-ahead strategy to determine its movements among resource sites, how do renewable resources affect its long-term ranging behavior? We have assumed that once a resource is consumed it is not renewed and the animal never returns. Now we let resources renew over time. Furthermore, we give the animal a memory so it can remember the order of the last M resource sites it visited (not including the resource site it is currently visiting), where M is less than the number of total resource sites. We assume that the animal does not revisit the last M resource sites it can remember because they have less food than resource sites visited long ago. This simple strategy of not returning to recently visited resource sites seems most reasonable when the animal knows the location of resources. but cannot judge the energy content until it has visited the resource. Although this is not the optimal strategy under all forms of resource renewal, it is most appropriate when resources gradually improve over a period of time. Pyke (198 1) has observed that hummingbirds avoid resources just previously visited. As mentioned before, we assume that the energy content of resources are variable and unknown to the animal before they are visited, or the animal knows that all resources have the same energy content. The case in which resources renew completely after a fixed period of time, known to the animal. is addressed below. In the original model, without renewable resources, the animal stopped foraging after it had visited all the resource sites. The new model, with renewable resources, allows the animal to continue foraging forever since there may always be another resource site to visit. This occurs whenever M. the memory size, is 2 or fewer than R, the number of resource sites. Because this model is deterministic an animal presented with the same set of resource sites at two different times will always choose the same resource site to visit next. Consider the case where these two times are the present and some past time. If, after the next resource site is visited, the same resource sites are available as in the past time plus one step, then the animal will again repeat its choice of which resource site to visit next. This process can
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lead to cycles in which the animal continually traces out the same path forever. Cyclical behavior of this kind is similar to “trap-lining” (Manning, 1956; Janzen, 1971), a foraging behavior in which animals repeatedly trace out a similar foraging path. To guarantee that a cycle will repeat foreover requires that the resource sites the animal chooses be the same as those in a previous path. This occurs only when the resource sites are transferred from the memory to the pool that the animal chooses from in exactly the same order as in the previous time, guaranteeing that the set of resource sites the animal chooses from is always the same at both times. Thus, the state of the memory, that is, the particular permutation of resource sites remembered, must be the same as some previous state before a cycle can be guaranteed to repeat forever. If the animal never runs out of resource sites to visit (M < R - 1) it will always ultimately end up in a cycle. This is because the memory has a finite number of states, but an infinite number of foraging steps are possible since there is always another resource site to visit. Eventually the memory will repeat a previous state, causing the animal to retrace its path forever. Given that a cycle will always occur, how long does it take for an animal to get into a cycle? The number of steps that an animal takes before it has completely traced out one cycle that will repeat forever is less than or equal to the number of states in the memory. The numer of states in the memory can be very large; 10 resource sites with a 5-step memory has 30.240 states: 20 resource sites with a l&step memory has 1.2 x 10” states. Potentially, cycles may not happen for a long time. Simulations were used to determine the actual number of steps an animal takes before traveling through a complete cycle. Figure Sa shows the average number of resource sites visited when the first cycle completes versus memory size. Even when the number of states in the memory is large, the actual number of resource sites visited when the first cycle completes, in most cases, is small by comparison. Figure 5a shows only the average number of resource sites visited versus memory length. The actual underlying distribution, from which the mean was calculated, is strongly skewed towards early times (few resource sites visited). Even if animals do not behave deterministically according to the rules of the model, but rather change their rules periodically or make errors, cycles may still occur, they just will not repeat forever. A foraging animal that ends up in a cycle may exit the cycle when it changes foraging rules or makes a mistake, only to end up in a different cycle soon afterwards. After a cycle ultimately occurs what is its length? That is, how many resource sites are visited when traversing the cycle once? The maximum possible cycle length is less than the number of different states the memory may take, which can potentially be very large. Figure 5b shows average cycle length versus number of steps in the memory for various combinations
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of parameters. For all the cases examined the most common cycle length is relatively small, even when the number of states in the memory is large. The underlying distribution of cycle lengths from which the mean was calculated is highly variable in shape, sometimes sharply skewed, at other times very flat. Increasing look-ahead from one-step to 3-step had almost no effect on the length of a cycle. Apparently, under these assumptions, animals quickly get locked into locally high resource densities, where the size of the locality depends on the memory size or renewal rate (the larger the memory the larger the locality). Other resource site distributions, such as the random-Gaussian and evenGaussian examined earlier, did not produce radical differences in the time at
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which cycles occurred or their length. In general these clumped distributions increased the time at which cycles appeared and their length. by less than a factor of two-certainly less than that suggested by the large number of states in the memory. Now consider an alternative model in which resources, once fed upon, are unusable for a known, fixed period of time, after which they are completely renewed. In this case the animal should not return to a resource until after the renewal time has passed. This model is quite similar to the previous model. If, in the last model. the animal remembered all resource sites during the last T units of time, where T is the time for a resource to renew, it would be identical. In this model only the resources not eaten in the past T units of time can be visited. In the latter model only resource sites not visited A4 steps ago can be visited. Not surprisingly, the results for this model, shown in Fig. 6, are quite similar. Cycles occur relatively early and their length is relatively short in comparison to the number of possible states in the system. Trap-lining has been observed in diverse animals such as bees and primates, that forage on widely distributed discrete resources (Janzen. 1971: Cluton-Brock, 1975). The results presented here suggest that trap-lining might occur as a natural consequence of animals attempting to minimize travel distance when their resources renew. Janzen (197 l), however. has observed situations in which trap-lining bees could shorten their foraging path by visiting resource sites in a different order. He suggests the order that resource sites are discovered determines the order that resource sites are visited in a trap-lining route. In all the problems studied here, resource sites have been assumed to lie on a plane. The goal was to minimize travel distance among the resource sites, thereby maximizing net energy intake. This assumption puts certain constraints on distances between resource sites, for example. the distance from resource site A to resource site B is equal to the distance from B to A. In general it is possible to reparameterize the system and replace distance with overall cost. Cost could incorporate not only distance, but handling time, differential energy costs (i.e., going up hill might be more costly than going down hill), etc. Furthermore, net energy gained (or lost) could be used instead of cost. Net energy would incorporate all the above costs plus the energy gained from feeding upon the resource. When distances on a plane are replaced with energy lost or gained, constraints on distances between planar resource sites disappear. For example, the energy change or cost in traveling from A to B is not necessarily the same as the reverse direction. from B to A. In this situation it is easy to construct sample cases of nonplanar resource sites, in which the nearest neighbor rule (in terms of cost or energy) does worse than with planar resources distributions. When cost is used to replace distance in this way it is no longer possible to use path
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FIG. 6. The properties of foraging cycles that occur when resources have a renewal time after being eaten are shown. Figure 6a shows the average number of resource sites visited when the first cycle completes versus resource renewal time. A renewal time of one is the amount of time it takes the animal to travel one unit of distance. or the width of the unit square in which it forages. The animal was assumed to be traveling at a constant velocity and handling time was assumed to be negligible compared to travel time. Figure 6b shows the average cycle length versus renewal time. The number of resource sites, look-ahead and number of replicates are the same as Fig. 5.
pruning. Without path pruning only a limited number of resource sites can be examined because of combinatorial explosion. Whether the results presented here generalize to situations where path pruning cannot be applied remains an open question. ACKNOWLEDGMENTS I thank Charles Janson, whose concurrent work on similar problems inspired this work. Montgomery Slatkin provided substantial advice and support for this project. Stuart Altmann. William Galway, Raymond Huey, Gordon Orians, Mary Power. Montgomery Slatkin.
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Christopher Stinson, Thomas Schoener. and anonymous reviewers provided advice and constructive criticism.
REFERENCES ALTMANN, S. A. 1974. Baboons, space, time, energy. Amer. Zool. 14. 221-248. CHRISTOFIDES,N. 1976. Worst-case analysis of a new heuristic for the traveling salesman problem. Abstract. Symposium on new directions and recent results in algorithms and complexity, Carnegie-Mellon Univ. CI.UTTON-BROCK, T. H. 1975. Ranging behavior of red colobus (Colobus baclirts fephrosceles) in the Gombe National Park. Anim. Behau. 23. 706-722. CODY, M. L. 1971. Finch flocks in the Mohave Desert, Theor. Pop. Biol. 2, 142-158. CODY, M. L. 1974. Optimization in ecology, Science 183, 1156-I 164. GAREY. M. R., AND JOHNSON,D. S. 1979. “Computers and Intractability: A Guide to the Theory of NP-Completeness,” Freeman, San Fransisco. GOULD, S. J.. AND LEWONTIN,R. C. 1979. The spandrels of San Marco and the panglossian paradigm: A critique of the adaptationist programme. Proc. R. Sot. London Ser. B 205. 58 l-598. GRAHAM. R. L. 1978. Combinatorial scheduling theory, in “Mathematics Today: Twelve Informal Essays” (L. A. Steen, Ed.). pp. 183-211, Springer-Verlag, New York. HEINRICH,B. 1979. “Bumblebee Economics,” Harvard Univ. Press, Cambridge, Mass. JANZEN,D. H. 1971. Euglossine bees as long-distance pollinators of tropical plants, Science 171, 203-205.
KARP, R. M. 1975. On the computational complexity of combinatorial problems. Netivorks 5. 45-68. KARP, R. M. 1976. The probabilistic analysis of some combinatorial search algorithms. Or “Proceedings of a Symposium on New Directions and Recent Results in Algorithms and Complexity” (J. F. Traub, Ed.), pp. l-19, Academic Press. New York. LEWONTIN,R. C. 1979. Fitness, survival and optimality, in “Analysis of Ecological Systems” (D. J. Horn et al., Eds.), pp. 3-21, Ohio State Univ. Press, Columbus. MANNING, A. 1956. Some aspects of the foraging behaviour of bumblebees. Behmiour 9. 164-201. PIELOU,E. C. 1977. “Mathematical Ecology,” Wiley, New York. PYKE. G. H. 1974. “Studies in the Foraging Efficiency of Animals,” Ph.D. Thesis. University of Chicago. PYKE, G. H. 1978a. Optimal foraging: Movement patterns of bumblebees between inflorescences, Theor. Pop. Biol. 13, 72-98. PYKE, G. H. 1978b. Are animals efficient harvesters?, Anim. Behac. 26, 24 l-250. PYKE, G. H. 1981. Optimal foraging in hummingbirds: Rule of movement between inflorescences, Anim. Behau. 29, 889-896. PYKE. G. H.. PULLIAM. H. R.. AND CHARNOV,E. L. 1977. Optimal foraging: A selective review of theory and tests, Quarr. Reu. Biol. 52, 137-154. REINGOLD.E. M., NIEVERGELT,J., AND DEO, N. 1977. “Combinatorial Algorithms: Theory and Practice,” Prentice-Hall, Englewood Cliffs, N. J. SCHOENER.T. W. 1971. Theory of feeding strategies, Annu. Rec. Ecol. SW. 2. 369-404.
POPULATION
Optimal
24, 145-159 (1983)
BIOLOGY
Foraging
and the Traveling
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D. JOHN ANDERSON* Department of Zoology NJ-IS. University of Washington. Seattle, Washington 98195 Received December 7, I98 I
Optimal foraging models are examined that assume animals forage for discrete point resources on a plane and attempt to minimize their travel distance between resources. This problem is similar to the well-known traveling salesman problem: A salesman must choose the shortest path from his home office to all cities on his itinerary and back to his home oftice again. The traveling salesman problem is in a class of enigmatic problems, called NP-complete, which can be so difficult to solve that animals might be incapable of finding the best solution. Two major results of this analysis are: (I) The simple foraging strategy of always moving to the closest resource site does surprisingly well. More sophisticated strategies of “looking ahead” a small number of steps, choosing the shortest path. then taking a step, do worse if all the resource sites are visited, but do slightly better (less than 10%) if not all the resource sites are visited. (2) Short cyclical foraging routes resulted when resources were allowed to renew. This is suggested as an alternative explanation for “trap-lining” in animals that forage for discrete, widely separated resources.
Animals are faced with decisions of where and when to move. The outcomes of these decisions can be of crucial importance. Ultimately, they may determine energy intake, exposure to predation, degree of competition and mating success. In this paper I address one aspect of this problem: What are the optimal patterns of movement in animals foraging for discrete resources on a plane? This problem is related to an important class of problems, such as the traveling salesman problem, that are extremely difficult to solve. The problem may be so difficult to solve that animals are simply not capable of finding the best solution. For this reason the performance of a class of simple, less than optimal (but plausible), models of foraging will be compared to the optimal pattern of movement for a variety of resource distributions types. In addition I examine the effects of renewable * Present address: Solaster Software Corporation, P. 0. Box 15216, Seattle, Wash. 98115.
14.5 0040-5809183$3.00 Copyright 0 1983 by Academic Press, Inc. All rights of reproduction in any form reserved.
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resources on the long-range behavior of animals using these simple decision rules for foraging. The models presented here, like other optimal foraging models, make the following assumptions (Schoener, 1971; Pyke ef al., 1977: Gould and Lewontin, 1979; Lewontin, 1979): Darwinian natural selection continuously tends to increase the fitness of organisms within certain constraints. Foraging behaviors are passedfrom generation to generation, having either a genetic or cultural basis, or some combination thereof. They exhibit variation, so there is a range of possible behaviors upon which selection acts. Fitness is maximized when the net energy gain is maximized. This is true even for animals with fixed energy requirements, becauseforaging time saved can be invested in other useful activities, some of which are also related to fitness. According to these assumptions animals should exhibit behaviors that maximize their energy intake. Pyke ef al. (1977) grouped the situations to which optimal foraging models have been applied into four categories: “( 1) choice by an animal of which food types to eat (i.e., optimal diet); (2) choice of which patch type to feed in (i.e., optimal patch type); (3) optimal allocation of time to different patches; and (4) optimal patterns and speed of movements.” The models presented in this paper are in the fourth category, which historically has received the least attention. Published models of optimal movement patterns represent food resources as points on a plane. (Pyke, 1974, 1978a,b; Cody 1971, 1974). Animals are assumedto move with certain probabilities straight ahead, backwards, to the right, or to the left. Once a direction of movement is chosen, an animal moves to the closest resource site in that movement direction. Both Cody and Pyke argue that the maximum rate of energy intake, constrained by the above capabilities of movement, is obtained when path overlap is minimized, so that the animal does not waste energy visiting already depleted resources. They then proceed to determine the parameters that minimize overlap. If animals are assumed to follow movement rules besides those proposed by Pyke and by Cody, even higher energy intake can be obtained. In the extreme case, for point resources, the maximum energy intake is obtained by visiting all resource sites and minimizing travel cost. Travel costs can be important for animals such as bees. According to Heinrich (1979, p. 109) “a bee spends most of its time traveling, and all the efficiency that it may derive from its flight motor is of little avail if that traveling does not achieve given objectives. When moving between food sources, bees try to keep flight time and distance to a minimum.” This foraging problem is a variation of the traveling salesman problem: A salesman travels from his office to a number of cities and back to his home office. The goal is to find the shortest path that contains all the cities (Graham, 1978, p. 197). The only difference between the standard traveling salesman problem and this foraging problem
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is the last step-the animal does not necessarily return to the original starting resource site. Much that is known about the traveling salesman problem has implications for foraging animals. The traveling salesman problem is in a class of enigmatic problems, called “NP-complete,” which are among the more difficult problems to solve (Karp, 1975; a formal definition of NPcomplete, too long to present here, can be found in Garey and Johnson, 1979). They resist analytic solution and no known methods exist to solve them efficiently when the number of cities is large. Perhaps the simplest way to solve the traveling salesman problem is to examine all possible paths. For a C city problem there are (C - l)! possible paths to examine. This amounts to 6.08 x 106’ potential paths to examine for a 50-city problem. Even if one route could be examined in the time it takes light to travel 1 cm, it would take 6.43 x lOa centuries to complete the search! This so-called combinatorial explosion characterizes all NP-complete problems; even the most clever known methods require a worst-case completion time which grows exponentially as a function of the number of cities examined. With current methods, for example, it is only practical to solve the traveling salesman problem for fewer than about 100 cities. Strong circumstantial evidence suggests that efficient methods to solve these problems do not exist. By efficient I mean a method that terminates within a time bounded by a polynomial in the “length” of the input (i.e.. a finite sum of terms, each term being the number of cities or resource sites raised to a constant power). If an efficient, or polynomial-time, method did exist for any of these NP-complete problems, then it could be used to solve all NP-complete problems in polynomial time (Reingold et. al. 1977. p. 405). Efficient methods are thought not to exist because such a large number of classic unsolved problems are NP-complete and no polynomial-time method has been found for any of them (Garey and Johnson, 1979). NP-complete problems include a host of combinatorial optimization problems such as scheduling problems, computational graph problems and bin packing problems (Garey and Johnson, 1979). Undoubtedly there are other NP-complete or NP-hard (at least as hard as NP-complete) problems that animals must face. Animals may be physically unable to find the exact optimum for some of the combinatorial problems they face. This could have implications for optimization theory in ecology. There is some hope, however. These results on the intractability of combinatorial problems apply to worst-case situations. They do not rule out the possibility of methods which efficiently solve these problems most of the time, but not always. Also, methods may exist which efficiently find approximately optimal solutions. These approaches have had mixed success (Karp, 1976). For the traveling salesman problem, Christofides (1976) has developed an efficient approximate method which is guaranteed to yield a
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tour whose length is less than 1.5 times the length of the optimal tour. No other approximate method is known to have a lower guaranteed bound (Gary and Johnson, 1979). In this paper I examine approximation methods for the solution of the foraging problem. Some, but not all, plausible foraging behaviors will be examined and compared with the optimal solution.
THE MODEL
I assume that an animal is foraging on a finite plane, in this case the unit square. Randomly scattered over the plane are points representing food resource sites. By random I mean that the X and Y coordinate of each resource site was independently drawn from a uniform distribution over the unit interval. Other distributions of resource sites will be examined later. An animal is initially placed on one of the food resource sites and allowed to forage according to certain rules of movement (described below) until all the resources are consumed. Food resources are assumed to have equal energy values which are known to the animal or arbitrary energy values which are unknown to the animal. In either case the best an animal can do is visit all resource sites with equal preference. Initially, I assume that each resource, once visited, is completely consumed and is never renewed. Therefore an animal has nothing to gain by visiting a resource more than once. Either the resource disappears or the animal marks or in some way remembers it to avoid returning. Later these assumptions will be relaxed, allowing resources to be renewed. A cost is associated with traveling that is proportional to the total distance moved. The goal, as in many other optimal foraging models is to maximize net energy intake. This is obtained when the animal visits all resources, irrespective of their energy content, and minimizes the distance traveled, assuming constant speed of movement. Furthermore it is assumed that the cost of traveling to any resource site is less than the energy gained by consuming the resource. Therefore it always pay to visit every resource site. Without this assumption it is only possible to explore the problem with a very limited number of resources, for reasons explained below.
LOOK-AHEAD
STRATEGIES
One of the simplest foraging strategies that can be envisioned is always moving to the nearest resource site that has not been eaten. A more sophisticated strategy which I will call “L-step look-ahead” was introduced by Altmann (1974, pp. 24&241). It works like this: The animal contem-
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plates (looks ahead at) all possible paths from its current location that visit L of the resource sites, chooses the path with the shortest total length, moves to the first resource site in the chosen path and then repeats the process. If L is larger than the number of remaining resources the animal only considers movement to the remaining resources sites. This strategy is similar to how a game might be played. The player considers the potential paths with L steps, chooses the best one, makes a move and then repeats the process. Animals would not need to examine explicitly all possible paths L steps away. Instead, the animal might choose an initial path containing L steps, possibly by using the nearest neighbor rule. Next the animal begins considering alternate paths. If, while considering a path containing fewer than L steps, the distance already traveled would be greater than the initial path, then all paths containing this partial path need not be considered-they will necessarily be further than the initial path since all distances are positive. If an L-step path is found that is shorter than the initial path, its distance is used when considering remaining partial paths. Using this process, which I will call “path pruning,” combinations of paths involving far-away resources need not be considered. Only “reasonable” paths involving nearby resources need to be examined. Even when some paths can be eliminated in this way, the problem remains NP-complete and exhibits the familiar combinatorial explosion. Furthermore, using path pruning to simplify the problem requires that the cost of traveling to a resource site be less than the energy content of that resource. Without path pruning it would be impossible to carry out the calculations presented here on computers that were available to me. When the animal looks ahead one step in the future (one-step look-ahead) it always moves to the nearest resource site. If there are R resources and the animal employs R-step or greater look-ahead, then it will always choose the shortest, or best possible path. R-Step look-ahead exhibits the combinatorial explosion, making it impractical for large numbers of resources. When L is varied, L-step look-ahead displays a range of possible strategies, from moving to the closest resource point (a simple task) to the best possible strategy (a difficult task). Approximate methods like one-step look-ahead can do very poorly. Figure 1 illustrates a situation where one-step look-ahead is nearly twice as bad as the optimal strategy. However, in nature these situations rarely occur. A better approach would be to compare the average behavior of various foraging strategies for resource site distributions that animals are likely to encounter. To quantify the average behavior of different look-ahead strategies I will use D(L, R, V), the average total distance traveled for a particular combination of L (amount of look-ahead), R (number of resources), and V (number of resources points visited).
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FIG. I. An example illustratinghow poorly one-step look-ahead can perform. Figure la shows a path through resource points using one-step look-ahead that produces a toal distance of 128 units. Figure lb shows another path through the same resource points with a distance of 67, almost half the distance produced using one-step look-ahead.
Simulations were carried out to estimate D as follows: an animal was randomly placed on one of the R resource sites, allowed to forage using the L-step look-ahead rule, with path pruning, until it had visited a total of V resource sites. This procedure was repeated many times with different resource locations and D was estimated as the average total distance the animal traveled per repetition. Consider first the case when the animal visits every resource site (V = R). To compare various strategies, the ratio of total distance traveled using onestep look-ahead to total distance traveled using L-step look-ahead (D(1, R, R)/D(L, R, R)) is plotted in Fig. 2. Only certain ranges of parameters are shown because of the excessive computation time for the other cases. 1.1 -
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L (Look-Ahead) FIG. 2. A comparison of various look-ahead strategies when the animals visit all resource sites (V = R). The distance ratio, plotted on the Y axis, is the ratio of the total distance traveled using one-step look-ahead to the total distance traveled using L-step look-ahead (D(1, R, R)/D(L, R, R)). The larger the ratio the more efficiently the animal forages. Notice that simple one-step look-ahead does better than 2-. 3-, and 4-step look-ahead. Four different numbers of resource sites (R), placed randomly on the plane, were used. Each point represents the average of 500 replicate simulations.
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There are two interesting results. Focusing first on the results for 10 resource sites, one-step look-ahead does surprisingly well-only about 8.5% worse than 9-step look-ahead (the best possible strategy). Two-step lookahead does worse than one-step look-ahead and it is not until 5step lookahead that there is any improvement over one-step look-ahead. For 6-step look-ahead with 10 resource sites, when an animal makes its first decision about which resource site to visit, there are 60,480 ([R - l]!/[R - 1 -L]!) different steps possible; that is, only 6 times less than the number using 9step look-ahead. The 6-step look-ahead, therefore, requires that many more steps be examined than one-step look-ahead with only a slight benefit of reducing travel distance. In an evolutionary sense, if animals started with one-step look-ahead and look-ahead could only evolve through small increases, then selection would not favor these small increases. If large jumps in look-ahead could occur, however, they would be favored by selection. When the number of resources increase, these trends are magnified. Twostep look-ahead does even worse than one-step look-ahead, higher amounts of look-ahead take even longer to improve and the number of possible steps grows large even faster (Fig. 2). With only 5-step look-ahead and 64 resource sites there are already 843,461,640 steps possible. Increasing the number of resource sites produces successively smaller changes in the curves shown in Fig. 2, until the change observed by increasing the number of resource sites from 46 to 64 in negligible. The surprising result that one-step look-ahead does as well as or better than 2-, 3-, 4- and 5-step look-ahead requires an explanation. When watching a hypothetical animal employ 2-step look-ahead it becomes clear that the animal skips a few nearby resource sites early in the tour for the sake of saving a little distance, but later must go back to visit many widely distributed resource sites that it left behind. This can be seenusing paper and pencil to draw some sample tours. With 2-step look-ahead the underlying distribution of distances traveled during the first and middle steps are almost identical. The distribution of distance of the last step is sharply skewed towards longer distances. How do the various look-ahead strategies perform when only part of the resource sites are visited (i.e., V < R)? To answer this question, Fig. 3a shows the ratio of total distance traveled using one-step look-ahead to total distance traveled using 2-step look-ahead (D(1, R, V)/D(2, R, V)) as a function of V, the number of resource sites visited (R = 10, 20, 40, and 60 resource sites). Figure 3b is the same as Fig. 3a except it compares one-step look-ahead to 5-step look-ahead (D(1, R, v)/D(5, R, V)). Some interesting results emerge: first, 2-step look-ahead, even though it does significantly worse than one-step look-ahead when all the resource sites are visited, can actually do slightly better if the animal visits at least the first few resource
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FIG. 3. A comparison of various look-ahead strategies assuming the animal does not visit all the resource sites (V < R). Shown on the Y axis is the ratio of average distance traveled using one-step look-ahead to the average total distance traveled using 2-step look-ahead (Fig. 3a) or 5step look-ahead (Fig. 3b). A ratio larger than one is more efficient than one-step look-ahead and a ratio less than one is less efficient than one-step look-ahead. Figure 3a. calculated using 2-step look-ahead, shows that if more than the first few resource sites but fewer than about 75% of the total are visited, then 2-step look-ahead out performs one-step look-ahead by a small amount (less than 4%). For 5-step look-ahead (Fig. 3b) this advantage is increased, but remains below about 10“6. Each curve, constructed from the average of 500 replicates, shows the results for a different number of randomly placed resource sites.
sites and not more than 75 % of the total number. The benefit of using 2-step look-ahead over one-step look-ahead is small, at most only about 4%. Using 5-step look-ahead, which is probably more than most animals are capable (Fig. 3b), increases this benefit to at most IO%, but usually less, depending upon the number of resource sites visited. Summarizing, small increases in look-ahead over one-step look-ahead will pay off if not all the resource sites are visited; but, the payoff is small, less than about 10%. Under certain circumstances R-step look-ahead, the optimal strategy to
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visit all the resource sites, does worse than one-step look-ahead if not all the resource sites are visited. In particular, for 10 resource sites, one-step lookahead out performs 9-step look-ahead if 8 or fewer resource sites are visited. The difference is largest, about 12 941,when four or fewer resource sites are visited. The particular foraging path is highly dependent upon the location of food resource sites. The random distribution from which resource locations were drawn is only one highly idealized possibility. To determine how these results depend upon the distribution of food resource sites, two alternative distributions were used in the simulations. Pielou (1977, p. 159) has pointed out that most spatial distributions in nature are clumped rather than random or even. Therefore. the first distribution. which will be called “randomGaussian.” consists of clumps of resource sites where the location of each clump is chosen randomly on the plane and the resource locations within each clump were chosen from a circular normal, or twodimensional Gaussian distribution in which the X and Y standard deviations are the same. The second distribution, which will be called “even-Gaussian” differs from the random-Gaussian in that the location of each clump, rather than being drawn at random, was placed on an evenly spaced N x N lattice. This lattice was centered on the unit square such that the distance between the center of each clump and its neighbors or the edge was l/(N + I ).
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L (Look-Ahead) FIG. 4. This figure illustrates how look-ahead strategies perform under different kinds of resource distributions. As in Fig. 2, the Y axis shows the ratio of one-step look-ahead to Lstep look-ahead (D(1, R. R)/D(L. R, R)) and the X axis shows amount of look-ahead. Four different cases are illustrated, representing 10 and 40 resource sites, random-Gaussian or evenGaussian distributions (see text), different numbers of clumps and different amounts of clumping. Notice how under these various conditions the qualitative results obtained earlier (shown in Fig. 2) remain unchanged. The degree of clumping was adjusted by varying the standard deviation of the circular normal distribution from which the resource sites were sampled. Each point on the graph was constructed from the average of 500 replicates.
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Simulations were carried out for a variety of parameters. including different numbers of clumps, different numbers of resource sites. and varying degrees of resource site clumping. Resource site clumping was changed by varying the standard deviation of the circular normal distribution from which locations were drawn. The results of some sample cases are shown in Fig. 4. Qualitatively, the results are similar to those obtained earlier (both in Figs. 2 and 3) using randomly distributed resource sites.
RENEWABLE RESOURCES
Given that the animal uses some form of look-ahead strategy to determine its movements among resource sites, how do renewable resources affect its long-term ranging behavior? We have assumed that once a resource is consumed it is not renewed and the animal never returns. Now we let resources renew over time. Furthermore, we give the animal a memory so it can remember the order of the last M resource sites it visited (not including the resource site it is currently visiting), where M is less than the number of total resource sites. We assume that the animal does not revisit the last M resource sites it can remember because they have less food than resource sites visited long ago. This simple strategy of not returning to recently visited resource sites seems most reasonable when the animal knows the location of resources. but cannot judge the energy content until it has visited the resource. Although this is not the optimal strategy under all forms of resource renewal, it is most appropriate when resources gradually improve over a period of time. Pyke (198 1) has observed that hummingbirds avoid resources just previously visited. As mentioned before, we assume that the energy content of resources are variable and unknown to the animal before they are visited, or the animal knows that all resources have the same energy content. The case in which resources renew completely after a fixed period of time, known to the animal. is addressed below. In the original model, without renewable resources, the animal stopped foraging after it had visited all the resource sites. The new model, with renewable resources, allows the animal to continue foraging forever since there may always be another resource site to visit. This occurs whenever M. the memory size, is 2 or fewer than R, the number of resource sites. Because this model is deterministic an animal presented with the same set of resource sites at two different times will always choose the same resource site to visit next. Consider the case where these two times are the present and some past time. If, after the next resource site is visited, the same resource sites are available as in the past time plus one step, then the animal will again repeat its choice of which resource site to visit next. This process can
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lead to cycles in which the animal continually traces out the same path forever. Cyclical behavior of this kind is similar to “trap-lining” (Manning, 1956; Janzen, 1971), a foraging behavior in which animals repeatedly trace out a similar foraging path. To guarantee that a cycle will repeat foreover requires that the resource sites the animal chooses be the same as those in a previous path. This occurs only when the resource sites are transferred from the memory to the pool that the animal chooses from in exactly the same order as in the previous time, guaranteeing that the set of resource sites the animal chooses from is always the same at both times. Thus, the state of the memory, that is, the particular permutation of resource sites remembered, must be the same as some previous state before a cycle can be guaranteed to repeat forever. If the animal never runs out of resource sites to visit (M < R - 1) it will always ultimately end up in a cycle. This is because the memory has a finite number of states, but an infinite number of foraging steps are possible since there is always another resource site to visit. Eventually the memory will repeat a previous state, causing the animal to retrace its path forever. Given that a cycle will always occur, how long does it take for an animal to get into a cycle? The number of steps that an animal takes before it has completely traced out one cycle that will repeat forever is less than or equal to the number of states in the memory. The numer of states in the memory can be very large; 10 resource sites with a 5-step memory has 30.240 states: 20 resource sites with a l&step memory has 1.2 x 10” states. Potentially, cycles may not happen for a long time. Simulations were used to determine the actual number of steps an animal takes before traveling through a complete cycle. Figure Sa shows the average number of resource sites visited when the first cycle completes versus memory size. Even when the number of states in the memory is large, the actual number of resource sites visited when the first cycle completes, in most cases, is small by comparison. Figure 5a shows only the average number of resource sites visited versus memory length. The actual underlying distribution, from which the mean was calculated, is strongly skewed towards early times (few resource sites visited). Even if animals do not behave deterministically according to the rules of the model, but rather change their rules periodically or make errors, cycles may still occur, they just will not repeat forever. A foraging animal that ends up in a cycle may exit the cycle when it changes foraging rules or makes a mistake, only to end up in a different cycle soon afterwards. After a cycle ultimately occurs what is its length? That is, how many resource sites are visited when traversing the cycle once? The maximum possible cycle length is less than the number of different states the memory may take, which can potentially be very large. Figure 5b shows average cycle length versus number of steps in the memory for various combinations
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Memory Size of the characteristics of foraging cycles that occur when the animal FIG. 5. A summary does not revisit the last M resource sites it fed at. Figure 5a shows the average time that cycles occur: that is, the total number of resource sites visited when the first cycle completes as a function of M. the memory size. Figure 5b shows the average length of cycles versus M. Various combinations of look-ahead and number of resource sites are shown. each point was calculated as the average of 500 replicates.
of parameters. For all the cases examined the most common cycle length is relatively small, even when the number of states in the memory is large. The underlying distribution of cycle lengths from which the mean was calculated is highly variable in shape, sometimes sharply skewed, at other times very flat. Increasing look-ahead from one-step to 3-step had almost no effect on the length of a cycle. Apparently, under these assumptions, animals quickly get locked into locally high resource densities, where the size of the locality depends on the memory size or renewal rate (the larger the memory the larger the locality). Other resource site distributions, such as the random-Gaussian and evenGaussian examined earlier, did not produce radical differences in the time at
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which cycles occurred or their length. In general these clumped distributions increased the time at which cycles appeared and their length. by less than a factor of two-certainly less than that suggested by the large number of states in the memory. Now consider an alternative model in which resources, once fed upon, are unusable for a known, fixed period of time, after which they are completely renewed. In this case the animal should not return to a resource until after the renewal time has passed. This model is quite similar to the previous model. If, in the last model. the animal remembered all resource sites during the last T units of time, where T is the time for a resource to renew, it would be identical. In this model only the resources not eaten in the past T units of time can be visited. In the latter model only resource sites not visited A4 steps ago can be visited. Not surprisingly, the results for this model, shown in Fig. 6, are quite similar. Cycles occur relatively early and their length is relatively short in comparison to the number of possible states in the system. Trap-lining has been observed in diverse animals such as bees and primates, that forage on widely distributed discrete resources (Janzen. 1971: Cluton-Brock, 1975). The results presented here suggest that trap-lining might occur as a natural consequence of animals attempting to minimize travel distance when their resources renew. Janzen (197 l), however. has observed situations in which trap-lining bees could shorten their foraging path by visiting resource sites in a different order. He suggests the order that resource sites are discovered determines the order that resource sites are visited in a trap-lining route. In all the problems studied here, resource sites have been assumed to lie on a plane. The goal was to minimize travel distance among the resource sites, thereby maximizing net energy intake. This assumption puts certain constraints on distances between resource sites, for example. the distance from resource site A to resource site B is equal to the distance from B to A. In general it is possible to reparameterize the system and replace distance with overall cost. Cost could incorporate not only distance, but handling time, differential energy costs (i.e., going up hill might be more costly than going down hill), etc. Furthermore, net energy gained (or lost) could be used instead of cost. Net energy would incorporate all the above costs plus the energy gained from feeding upon the resource. When distances on a plane are replaced with energy lost or gained, constraints on distances between planar resource sites disappear. For example, the energy change or cost in traveling from A to B is not necessarily the same as the reverse direction. from B to A. In this situation it is easy to construct sample cases of nonplanar resource sites, in which the nearest neighbor rule (in terms of cost or energy) does worse than with planar resources distributions. When cost is used to replace distance in this way it is no longer possible to use path
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FIG. 6. The properties of foraging cycles that occur when resources have a renewal time after being eaten are shown. Figure 6a shows the average number of resource sites visited when the first cycle completes versus resource renewal time. A renewal time of one is the amount of time it takes the animal to travel one unit of distance. or the width of the unit square in which it forages. The animal was assumed to be traveling at a constant velocity and handling time was assumed to be negligible compared to travel time. Figure 6b shows the average cycle length versus renewal time. The number of resource sites, look-ahead and number of replicates are the same as Fig. 5.
pruning. Without path pruning only a limited number of resource sites can be examined because of combinatorial explosion. Whether the results presented here generalize to situations where path pruning cannot be applied remains an open question. ACKNOWLEDGMENTS I thank Charles Janson, whose concurrent work on similar problems inspired this work. Montgomery Slatkin provided substantial advice and support for this project. Stuart Altmann. William Galway, Raymond Huey, Gordon Orians, Mary Power. Montgomery Slatkin.
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Christopher Stinson, Thomas Schoener. and anonymous reviewers provided advice and constructive criticism.
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