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Foraging, predation hazard and patch use in grey squirrels
Anita. Behav., 1987, 35, 1804-1813
Foraging, predation hazard and patch use in grey squirrels J O N A T H A N A. N E W M A N & T H O M A S C A R A C O Behavioral Ecology Group, Biological Sciences, State University of New York, Albany, New York 12222, U.S.A.
Abstract. Two patches of food, one large and one small, were placed at equal distances from protective
cover. Grey squirrels, Sciurus earolinensis, responded to the difference in food quantity, so that aggregation sizes matched predictions based on relative patch size, as well as those based on feeding rates within patches. The large patch was then moved farther from cover, where the probability of predation per unit foraging time should be greater. Aggregation sizes in the large patch decreased significantly, and overcrowding in the small patch resulted in reduced average feeding rates. The experiment was next repeated with the amount of food in each patch doubled. Increased distance to cover again resulted in decreased aggregation sizes in the larger patch, but average feeding rates did not decline in the small groups that were observed. Squirrels responded to relative, but not necessarily absolute, differences in food availability. When patches differed in both food availability and the hazard of predation, each attribute exerted a significant effect on the pattern of patch use.
The distribution of competing animals among patches of a resource should depend on the way local competitor density influences each individual's access to the resource (Parker 1978, 1984; Rosenzweig 1981; Pulliam & Caraco 1984; Clark & Mangel 1986). Fretwell (1972) discussed two models predicting equilibrium frequencies of competitors in patches, the Ideal Free Distribution and the Ideal Despotic Distribution. The ideal free distribution assumes that each individual (of a particular phenotype, Parker & Sutherland 1986) in a patch has equal access to the resources found there. The ideal despotic distribution assumes that individuals arriving earlier in a patch may suppress later arrivals' access to resources (e.g. through territoriality). This asymmetry in access to resources may imply that the later arrivals shift to a less-preferred habitat. Ideal Free Distribution
The ideal free distribution assumes that all identical competitors obtain the same amount of resource and so achieve the same fitness. In many, but not all, cases the predicted distribution of individuals across patches has density ratios that match resource ratios (Parker 1984; Pulliam & Caraco 1984). Fagen (1987) presents a general analysis of this habitat-matching rule. Suppose that the function W(Rdn~) gives the fitness of each
of the niidentical individuals in patch i. Ri is the rate of resource input at patch i. Fagen (1987) requires only that W is bounded, single-valued, and increases strictly monotonically as a function of (Ri/ n~). Then the assumption that all individuals have equal fitness at equilibrium, i.e. W(Ri/n~)= W(Rj/ nj), implies that ndnj = Ri/Ri for any fitness function W with the three properties mentioned. If a total of p patches are occupied, the habitat-matching rule predicts relative densities P
p
(nil Z ni)~ (Ri/ ~. Ri) i=1
(1)
i=i
The ideal free distribution is usually tested in this form, with the Rz known or manipulated experimentally. If fitness always decreases as m increases, the ideal free distribution must be a stable Nash equilibrium (Pullam & Caraco 1984; Parker & Sutherland 1986; cf. Clark & Mangel 1986). That is, if all individuals conform to the ideal free distribution, an individual deviating unilaterally from the equilibrium incurs a reduction in fitness. Since the ideal free distribution requires equal fitness for identical competitors, it corresponds to an evolutionarily stable strategy (Milinski 1979; Parker 1984; Pulliam & Caraco 1984). Several studies report observations of the habitat-matching version (equation 1) of the ideal free distribution (e.g. Davies & Halliday 1978; Parker
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Newman & Caraco: Patch use by squirrels 1978; Milinski 1979, 1984; Harper 1982; Ens & Goss-Custard 1984; Godin & Keenleyside 1984). Lefebvre (1983) successfully applied the ideal free distribution and noted that habitat matching often occurs even though the ideal free distribution's assumption of identical competitors is violated. For example, per caput resource consumption may vary with dominance rank (Harper 1982) or with variation in foraging ability (Godin & Keenleyside 1984), but density ratios still approximate resource ratios. Parker & Sutherland (1986) summarize empirical studies of the ideal free distribution and derive equilibrium models of patch use for some cases where individuals differ in competitive ability. Our results concern two aspects of the ideal free distribution. (1) Following Lefebvre (1983), we suggest that individuals' decision rules (sensu Pyke 1984) may induce density ratios matching resource ratios even when the relationship between per caput resource consumption and local density predicts another patch-occupation pattern at equilibrium. (2) Several tests of the ideal free distribution have involved a food resource (see Parker & Sutherland 1986). In nature, an increase in foraging opportunity sometimes coincides with an increase in the hazard of predation (probability of being killed per unit foraging time), so that patch occupation and resource use become problems with conflicting objectives (e.g. Mittelbach 1981; Martindale 1982; Cerri & Fraser 1983; Leger et al. 1983; Lima et al. 1985). We expect patch-use patterns to depend on spatial variation in food availability, variation in the hazard of predation, and the covariation between food availability and the hazard of predation. Ideal Despotic Distribution Fretwell (1972) introduces the ideal despotic distribution by considering the example of a territorial species. Once a habitat has been filled by territory holders, further arrivals will find the habitat less suitable. Consequently, beyond a certain density, later-arriving individuals wilt not have access to the same amount of resource. An analogous situation can arise when food within a patch occurs at a finite number of point sources, since some individuals may obstruct other foragers' access to food. In this context we consider equilibrium aggregation sizes in patches where feeding
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rates depend on local density in a manner roughly consistent with the assumptions of Fretwell's (1972) ideal despotic distribution. Consider two patches, designated A and B. Let nA and na identify the number of foragers in the respective patches, and let n = nA+nB. We represent the number of food sources in the patches/cA and kB. Throughout, we designate the patch with more food as patch A; i.e. kA > kB. If ni ki, the first ki individuals feed at rate S and the remaining ( n i - ki) individuals feed at diminished rate s (i.e. S > s). Feeding rates exhibit density dependence only when n~>k~. F o r small aggregations this assumption may be more realistic than assuming that food intake is proportional to kilns. We let F(ni) represent the average individual feeding rate in patch i. Our assumptions imply that
F(ni) = S, when I _ ki
(2)
Assuming that the patches do not differ in the mortality hazard due to predation, an individual should seek to feed at rate S (if possible). For a given n, we define equilibrium states (nA*, nB*) as those patch-occupation patterns where no individual should be tempted to switch patches in order to obtain a greater feeding rate. For n kAq-kB, the states (hA*, na*) are neutrally stable; no individual can increase its feeding rate by switching patches (as long as all others stay put). When n=kA+kB, the single equilibrium (hA* =kA, nB*=kB) is stable. In this case any individual switching patches incurs a lower feeding rate (as long as all others stay put). Note that nA*/nB* =kA/ kB when n = kA + kB. For a given n we assume that each equilibrium state (when multiple equilibria exist) is equally probable. We calculate the expected number of foragers in each patch, E[nA*] and E[nB*], under this assumption. For clarity we consider n increasing through four intervals. (1) When n < kB, each forager has unobstructed access to a food source even if all choose the smaller patch. Consequently, all individuals feed at rate S, and any feasible aggregation pattern is a possible equilibrium. Then E [nA*]= E [nB*] = n/2
(3)
(2) When kB < n < kA, the total number of foragers exceeds the number of food sources in patch
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Animal Behaviour, 35, 6
B, but not in patch A. Then density dependence can reduce the average feeding rate in patch B, but not in patch A. All individuals will feed at rate S as long as nB
(4) E [nB*]= kB/2 (3) When kA < n_< kA+ kB, the total number of foragers exceeds the number of food sources in the larger patch. Density dependence can reduce the average feeding rate in both patches. However, all individuals can still feed at rate S if neither n~ exceeds the corresponding kt. Possible equilibria range from (n--kB, kB) to (kA, n--kA); the foragers simply avoid too large an aggregation in either patch. Then E [hA*]= (n + kA -- kB)/2
(5) E [nB*]= (n + kB -- kA)/2 When n = kA+ kB, equation (5) is equivalent to the habitat-matching form of the ideal free distribution. (4) When n>kA+kB, the total number of foragers exceeds the total number of food sources. The first n foragers feed at rate S. The remaining individuals' access to food is obstructed, and they feed at the reduced rate s. Any state where nA> kA and nB>_kB is an equilibrium, and the E[ni*] increase as n increases. Paralleling our comments on the ideal free distribution, we next assume that the larger patch (A) is located farther from cover and consequently is more exposed to predators than is patch B. If survival depends on the product of the probability of not starving and the probability of avoiding predation, efficient survival strategies may imply patterns of patch occupation differing from the equilibria predicted by feeding rates. Increasing danger in patch A suggests that, for a given n, more individuals should feed in patch B. Efficient strategic responses might imply that individuals accept a decreased feeding rate (if patch B becomes 'overcrowded') to increase their probability of escaping an attack by a predator. In any case, we can anticipate that nA will decline as distance to cover increases. Additionally, we might speculate that individuals use patches in a manner consistent with the equilibration of survival probabilities. That is,
the change in aggregation sizes presumably accompanying the increased mortality hazard in patch A might induce a new equilibrium where survival probabilities in the two patches are the same for any given n. For example, a forager in patch A might obtain a required amount of food in a relatively short time. A forager in patch B would require more time to obtain the same amount of food, but its mortality hazard per unit would be lower than in patch A. In this paper we first examine the distribution of grey squirrels, Sciurus carolinensis, between patches differing in food availability, but not distance from protective cover. We then compare this distribution to that observed when the richer patch is located farther from cover, where exposure to predators is presumably greater. Although grey squirrels give alarm calls, they do not depend on social groups to avoid predation. Furthermore, empirical evidence suggests that an individual's behaviour while foraging depends very strongly on its distance to cover (Lima et al. 1985; Lima & Valone 1986). Therefore, grey squirrels should be reasonable subjects for this sort of experiment.
METHODS
We conducted the study from late June to early October 1985, in an open, rectangular field (75 x 25 m) on the campus of the State University of New York at Albany. The field is bordered on three sides by trees, mainly black locust, Robinia pseudoacacia. The grey squirrels used the trees as protective cover. A campus road runs along the fourth side of the field. Data were collected 6 days per week from 0530 to 0800 hours. Traffic on the road was light at that time of day. Potential predators seen at the study site included red-tailed hawks, Buteojamaicensis, great horned owls, Bubo virginianus, barred owls, Strix varia, and domestic dogs and cats. Foraging squirrels ran to the trees when they sighted any of these animals. However, we saw only one actual attack (by a domestic cat) during the study; the squirrel escaped. Our experimental patches contained sunflower seeds, Helianthus annuus. We chose sunflower seeds because the squirrels ate them in the patches, rather than caching them. We placed food in aluminium pans (18.5 x 9.8 x 5.7 cm). The pans held enough seeds to avoid any effects of food depletion during
Newman & Caraco: Patch use by squirrels any day's observation; each pan always contained some seeds at the completion of data collection. The pans were large enough for two squirrels to feed simultaneously at a single pan, but this occurred only rarely. Consequently, the number of pans in a patch corresponded to k~ of the previous section. Each day we made available a larger (A) and a smaller (B) patch. Within a patch the pans were arranged so that the inter-pan distance (when a patch contained more than one pan) was a constant 0.65 m. The positions of the patches (left versus right) were alternated randomly throughout the study to control for position preferences unrelated to distance from cover. We collected data with a portable microcomputer equipped with an event-recorder program. Whenever a change in the state (hA, nB) occurred, whether or not n changed, the program recorded the new state and the time of the transition to the nearest second. For each n from one to five, the total time spent in each feasible state (i.e. where nA+na=n) was summed for each day. These data allowed calculation of the average number of squirrels occupying patch A, conditional on the value of n. We rarely saw more than five squirrels, in total, occupying the patches at any one time. Hence, most of our analyses are restricted to n_< 5. Many more squirrels could be found in the area, and there was turnover in the occupants of the patches during each day's observation. We estimated feeding rates from both direct observation and analysis of super-8 movies. We counted the number of seeds eaten by a focal animal in 1 min. For each measure we noted the patch size (k~), the number of squirrels in the patch (ni), and distance from cover. We divided our study into two parts. The important difference between parts 1 and 2 is that patch sizes were doubled in part 2. If squirrels follow the habitat-matching rule (equation 1), we can predict their distribution using only the relative patch sizes. Consequently, doubling patch sizes should not affect the patch-use pattern at a given distance to cover. However, if squirrels follow the rules of the feeding-rate model (equations 3, 4 and 5), they will respond to a change in the absolute patch sizes, at least for certain values of n. Under that model, doubling patch sizes could affect the patch-use pattern for a given n and distance to cover.
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Part 1
Patch A contained four pans of food, attached at the corners of a piece of plywood measuring 0.7 • 0.7 m and placed on the ground. Patch B contained one pan of food, attached in the middle of a similar piece of plywood. Initially, both patches were placed 5 m from cover. We sampled until our records of time in the various states yielded six representative probability distributions of the feasible states for each n from one to five. For a given n we observed one such distribution each day, but a distribution was considered representative only if the system remained in that particular distribution's most common state for at least 50 s (simply our rule of thumb to ensure a reasonably long sampling period). More than 6 days were required to obtain six representativ e distributions in every case. For some values of n our data yielded more than six representative distributions. Next, we moved patch A to 15 m from cover (leaving B at 5 m), and recorded six representative state distributions for each n. The process was then repeated with patch A at 5 m from cover, and again with patch A at 15 m from cover.
Part 2
During this phase of the study, patch A contained eight food pans, and patch B contained two pans. We recorded six representative distributions of states for n from one to five with both patches 5 m from cover, and then with patch A moved to 15 m from cover. Hence, the sample sizes were onehalf of those recorded in part 1. In both parts 1 and 2, the ratio of kA to kB was 4.
RESULTS Table I lists the average feeding rates (seeds consumed/min) observed during both parts of the study. Data for patch A are partitioned according to distance from cover. The table contains four (n~,ki) combinations where n~>ki, and in each of these cases there are two entries. The larger feeding rate refers to the squirrel(s) with unobstructed access to a feeding pan. The smaller rate refers to the squirrel(s) lacking direct access to a feeding pan (i.e. they were obstructed by squirrels with direct
Animal Behaviour, 35, 6
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Table I. Average individual feeding rates (seeds/min) observed during the study Feeding rates
ni Partl 1 2
PatchA(5m, k = 4 ) 20.13 _+1-23 19.15___2.42
Patch A (15 m, k = 4 ) 20.5 ___3 . 6 4 20"41+1'67
3
18.12___2.12
21'79__+3'19
4 5 Part2 1 2 3
17.41__+2.59 18'15_+3"07(S) 1.39_+2.4 (s) PatchA(5m, k = 8 ) 17.56_+ 1.34 17.59• 19.3-+ 1.7
21.7+3.18 --Patch A (15 m, k = 8 ) 17.1 +2.01 19.87_+2.62 19.6___2.53
4 5 6
17.58+3.0 20.22_+2.89 21.05__+ 1.76
18.82+ 1.65 21.46+ 1-49 19.72+__6.7
PatchB(5m, k = l ) 16.34___0-98 16.07+4'47 (S) 1.01 + 1.78 (s) 13.01+2'15 (S) 0.87+0.57 (s) ---PatchB(5m, k = 2 ) 18.49+ 1.69 18.14_+2.14 18.14_+2.06(S) 3.54_+4-75 (s)
---
Entries are means_95% confidence intervals, rti is the number of individuals within a patch. In double entries, S refers to unobstructed foragers, and s refers to obstructed foragers. 5 m, 15 m: distance from cover; k: number of food sources in each patch.
access). The following analysis suggests that observed feeding rates correspond roughly to the assumptions expressed in equation (2). We subjected the feeding rates observed in patch A to a two-way A N O V A adjusted for unequal sample sizes; data from the two parts of the study were analysed separately. Since individual feeding rates can depend on both competitive effects and advantageous social interactions (e.g. Caraco 1979; Pulliam & Caraco 1984), we took aggregation size (nA) and distance to cover as the treatments. In part 1 of the study we had sufficient data for values of nA < 4 (see Table I). During part 1, neither aggregation size (F3,73=0"76, Ns), distance to cover (FI,73 = 1.83, Ns), nor their interaction (F3,73= 1.56, MS) caused significant variation in feeding rates. In part 2, we had sufficient data for values of nA < 6. During part 2, neither distance to cover (Fi,75=0'04, NS) nor the na-distance interaction (F5,75= 1.03, Ns) influenced feeding rates significantly. However, the variation due to aggregation size was significant (Fs,ys = 4.91, P < 0.025). Average feeding rates for these unobstructed foragers increased as nA increased. In part 1 of the study, feeding rates of unobstructed squirrels in patch B varied significantly
across aggregation size ( A N O V A , F2,49=4"73, P < 0.05). The average feeding rate declined as nB increased, contrasting with the single significant effect referred to in the preceding paragraph. We found no significant variation across aggregation size for unobstructed squirrels in patch B during part 2 of the study (F2,32=0"05, Ns). When we considered the feeding rates of obstructed squirrels, we detected no significant variation among the four mean feeding rates ( A N O V A , F3,45=2'6, Ns). Five of the seven F-tests for unobstructed squirrels failed to detect significant variation in average feeding rates, and our single test for obstructed squirrels does not indicate significant variation. Collectively, these results lend reasonable support for the assumptions motivating equation (2) and, hence, justify comparing patch-occupancy patterns to the predictions of equations (3-5). For both parts 1 and 2, Table II lists two sets of predictions and two sets of observations for hA. F o r n = 1 through 5, the table gives the following. (1) The first column shows the predictions for E[nA*] obtained from the model of equilibrated feeding rates, equations (3-5). (2) The second column shows the values for na predicted by the habitat-matching rule, equation
Newman & Caraco: Patch use by squirrels Table II. Predicted and observed mean aggregation sizes in patch A* Predicted nA n
Observed nA
Rate
Match
5m
15 m
0.5 1.5 2.5 3.5 4.0
0.8 1-6 2.4 3.2 4.0
0.77 1.47 2.04 2.98 3.81
0.13 0,66 1~50 2.39 3-09
0-5 1.0 2'0 3-0 4.0
0.8 1'6 2.4 3.2 4.0
0.55 l '48 2-23 2.78 3.55
0.53 0-48 1.13 2.29 2.92
Part 1
1 2 3 4 5 Part 2
1 2 3 4 5
Predictions follow from the feeding-rate model (designated by Rate) and from habitat matching (designated by Match). Predictions refer to average nA with patch A 5 m from cover (third column). The number of feeding pans was doubled in part 2. n is the total number of foragers. * A complete explanation of each column is given in the text.
(1). Since kA/kB was 4 in both parts of our study, habitat-matching simply predicts that nA = 0-8n. (3) The third column shows the average nA values when both patches were 5 m from cover. (4) The fourth column shows the average nA values observed when patch A was 15 m from cover. Both the feeding-rate model and habitat matching predict the average nA values observed when the two patches were equally distant iu cover. F o r part 1 (kA=4, k B = l ) , the correlation between predictions of the feeding-rate model and the observations is r=0"985 (P<0.05). The corresponding correlation for the habitat-matching rule is r = 0 . 9 9 6 ( P < 0 ' 0 5 ) . For part 2 (kA=8, kB=2), the correlations are r=0.981 ( P < 0 ' 0 5 ) for the feeding-rate model and r = 0"996 (P < 0.05) for the habitat-matching rule. The two models' predictions hardly differ, and these data do not discriminate between the models. The growth and decline of the aggregations showed considerable variability. Additionally, our observations revealed that the squirrels rarely switched between patches (sensu Houston & M c N a m a r a 1987) during a foraging bout, Most changes of state involved entry into or departure from the system.
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Table II suggests that distance to cover influences aggregation size; each of the 10 average nA values declined when distance to cover was increased from 5 to 15 m. We conducted a two-way A N O V A on the average daily nA values for each n from 1 to 5. We used the method of unweighted means (Neter & Wasserman 1974) because sample sizes were smaller in part 2. Treatments were distance to cover and total amount of food (i.e, part 1 versus part 2). In none of the five cases did average nA values vary significantly between parts 1 and 2 of the study (values for F1,33 ranged from 0.33 to 1-67, all Ns). Hence, aggregation sizes did not respond to a change in the absolute difference in patch size. The correlations noted above are presumably a response to relative patch sizes. In all five cases, average nA values were significantly smaller at the greater distance to cover (values for Ft,33 ranged from 11-0 to 27.3, all P < 0.005). Presumably, the squirrels responded to a change in the mortality hazard due to predation. For solitary squirrels (i.e. n = 1) we found a significant interaction between treatments (Fi,33=9.0, P < 0-01); distance to cover influenced patch use in part 1, but not in part 2, In the other four cases we failed to detect a significant interaction (values for F1,33 ranged from 0-1 to 2.66, all NS). We next return to the data on feeding rates. We compare foraging in patch A versus patch B when distance to cover was equal, and we ask whether the decreases in nA associated with the greater distance to cover influenced individual feeding rates. The feeding rates listed in Table I were estimated when a given number of individuals occupied a particular patch, without regard to the number of individuals in the other patch. However, we can use those data to define between-patch comparisons of feeding rates for different values of n. Combining the data in Table I and the relative frequencies of the states (hA, n~ln), we estimated an average feeding rate for an individual in patch A, and an individual in patch B, for each n_< 5. We simply calculated an overall patch-specific rate by weighting the rates for the feasible nl by the proportion of time the system spent in the associated state (see Fig. 1). We then took ratios of the overall rate in patch A to the corresponding overall rate in patch B for each n and distance to cover. We treated parts 1 and 2 separately. Figure 2A shows the ratios of the overall feeding rates (for both parts of the study) for data collected when both patches were 5 m from cover. Figure 2B shows the ratios of the
Animal Behaviour, 35, 6
1810
DISTANCE FROM COVER 5m 15m 0.8-
n=2
A
n=2
B
2,0 .o_
0.6-
"~ 1,5
iiiiiiiiiiiii~ 0.2 0.8
n=3
O
n=3
1,0
o
o
o
[
I
I
2
3
O
o
o
o
o
~,0.6 -
go.4 90.2 ~0.8
I
I
I
I
4
5
r
2
_t 3
1 I_ 4 5
Total foragers (n) Figure 2, Feeding rate ratios. For a given total number of foragers (n = hA-'}-na), we calculated a weighted average feeding rate for each patch. Dividing the estimate for patch A by the estimate for patch B gives the ratio plotted. Closed circles refer to part I. Open circles refer to part 2, when the number of feeding pans was doubled. (A) Data collected when both patches were 5 m from cover. (B) Data collected when patch A was 15 m from cover.
n=4
n:4
~
::::::2::::
0,6
~
1~0.4 0,2 n=5
n=5
0,8'
0.6 0,4'
i:!:i:!:i:~:i
iiiiiiiiiiiiii
_
0.2! 0 1 2 3 4 5 0 1 2 3 4 5
lqA Figure 1. Frequency distributions of aggregation sizes in patch A. Data are taken from part 1 of the study (four food pans in patch A, one in patch B). For n = 2 through 5 (where n is the number of foragers), the figure shows the relative frequencies of aggregation sizes in patch A (hA) for both distance-to-cover treatments.
overall feeding rates for data collected when patch A was 15 m from cover. In part 1 of the study, feeding rates in patch A consistently exceeded those in patch B by about 20% (Fig. 2A). Since patch B was seldom overcrowded when the patches were equally distant from cover (Fig. 1), we cannot attribute the pattern to obvious aspects of patch choice. In part 2, feeding rates in patch A were nearly identical to those in patch B when both patches were 5 m from cover (Fig. 2A). In this case, feeding rates for n _<5 clearly equilibrated between patches, as would be assumed by any model of identical competitors when n < ka +kB. Given the limitations of our data,
we attribute the apparent difference in ratios between parts 1 and 2 to chance. Figure 2B shows feeding-rate ratios, for parts 1 and 2, calculated from data collected when the larger patch was farther from cover. In part 1 of the study, feeding rates in patch A always exceeded those in patch B; for n = 5 the average feeding rate in patch A was more than double the average rate in patch B. Previously, we noted that nA values declined significantly when patch A was moved away from cover. Additionally, we found that feeding rates did not vary significantly with distance to cover. Therefore, the greater feeding rates in patch A shown in Fig. 2B for part 1 arise largely because patch B was overcrowded (Fig. 1). This pattern may represent a trade-off between foraging objectives and safety from predation, since individuals closer to protective cover incur a lower feeding rate. However, in part 2 (10 total feeding pans), feeding rates in patch A do not tend to exceed the rates in patch B (Fig. 2B). The difference arises simply because the same numbers of squirrels had access to twice as many feeding pans in part 2 of the study. Above, we noted that average nA values varied significantly between the distances to cover, but not between the two parts of the study. Increased distance to cover implies the same changes in aggregation sizes in both part 1 and part
Newman & Caraco: Patch use by squirrels 2, but this particular difference affects feeding rates only when patch B contains a single feeding pan. If larger values of n had been observed during the second part of the study, we might have found additional evidence of a trade-off between foraging rate and predation hazard. When distances to cover were equal, feeding rates in the two patches were roughly equal (Fig. 2A). However, this does not necessarily imply that the feeding-rate model, equation (3), better predicts patch use than does a habitat-matching 'rule of thumb' (Krebs et al. 1983). The correlations for mean aggregation sizes (discussed above, both patches at 5 m) indicate that both habitat matching and the feeding-rate model are plausible explanations of the squirrels' behaviour. That is, the correlations suggest that patch use may respond to either relative patch size (habitat matching, equation 1) or absolute patch size (feeding rates, equation 3). We later noted that mean nA values did not vary significantly as total food (i.e. part 1 versus part 2) varied; the ANOVA results imply that patch use did not depend on absolute patch sizes. We therefore prefer the interpretation suggesting that the squirrels chose to feed in the experimental patches so that density ratios (hA~riB) matched relative patch size (kA/k~); they responded more to relative than to absolute patch sizes. We suspect that a similar rule held when the patches differed in both amount of food and distance to cover. Since mean nA values did not differ significantly between parts 1 and 2, the density ratios hA~riB should be the same in both parts of the study (for a given distance to cover). We already noted that the relative numbers of feeding pans (4/t) accurately predicted ng values when both patches were 5 m from cover. In part 1, when patch A was 15 m from cover, the average density ratio observed ([~(nA/nB)]/5) was close to 5/6. Therefore, the nA values in part 2, when patch A was 15 m from cover, should equal 5n/ (5+6)=0-45n. The observed average nA values (Table II: part 2, patch A at 15 m) correlate significantly with predictions of the 5/6 density ratio (r=0'956, P<0-05). Since habitat matching worked well when the patches were equally exposed to predators, this last correlation should not be surprising. Squirrels may use a rule such that density ratios match patchvalue ratios, where patch value increases with patch size (resource abundance) and decreases with exposure to predators.
181l
DISCUSSION Our analysis assumes that greater distance to cover increases the mortality hazard due to predation. We did not verify this assumption prior to the study. However, a great deal of evidence supports our assertion (e.g. Barnard 1980; Caraco et al. 1980; Underwood 1982; Leger et al. 1983; Lima et al. 1985; Lima & Valone 1986). The probability of escaping a detected predator should decrease as the distance to cover increases for those animals dependent on cover for safety. Our results show that squirrels alter their use of patches to avoid the danger of predation, and that they sometimes sacrifice foraging efficiency to do so. A number of recent studies have documented qualitatively similar responses (e.g. Milinski & Heller 1978; Mittelbach 1981; Cerri & Fraser 1983; Edwards 1983; Werner et al. 1983; Lima et al. 1985). A variety of foraging problems in nature presumably involve simultaneous objectives of avoiding predation and averting starvation. Mangel & Clark (1986) discuss some interesting methods for analysing problems of this type. We interpreted our results as indicating that squirrels used a general habitat-matching rule that responded to relative differences in the amount of food and exposure to predators. A relative scaling, invariant with a doubling of the size of each patch, does not contradict concepts drawn from the psychology of perception (Hinde 1970; Fagen 1987). Several simple mechanistic rules are not consistent with our data. For example, increased danger in patch A might imply that foragers accept a given constant reduction in feeding rate by using patch B more often, or that foragers reduce time in patch A by a constant amount. We can reject both of these possibilities. The most consistent pattern in our data is found in the density ratios hA~riB. When the patches were equidistant from cover, a density ratio of 4/1 described the data in both parts of the study. When patch A was moved farther from cover, a density ratio closer to 1/1 described the data in both parts 1 and 2. Whatever the squirrels' decision rule may be, patch A was clearly devalued when distance to cover increased. Parker & Sutherland (1986), emphasizing that individuals may vary significantly in competitive ability, point out that several field studies report habitat matching when other spatial distributions would also allow feeding rates to equilibrate.
Animal Behaviour, 35, 6
1812
P a r k e r & S u t h e r l a n d (t986) suggest t h a t h a b i t a t m a t c h i n g provides a n efficiem 'rule o f t h u m b ' (see Lefebvre 1983). O u r results indicate t h a t squirrels in small aggregations use a h a b i t a t - m a t c h i n g rule that is (1) effective even before all individuals' feeding rates become density-dependent, a n d (2) sensitive to b o t h foraging economics a n d the h a z a r d of predation. The n u m b e r of studies d o c u m e n t i n g simultaneous effects o f feeding rate and p r e d a t i o n h a z a r d on spatial distributions has increased rapidly. In this p a p e r we assumed identical individuals, an obvious oversimplification. Individual variation in the trade-off between foraging and avoiding predators m i g h t depend o n easily identified attributes such as sex, age, or d o m i n a n c e status. A t a different level, co-occurring species might exhibit predictable contrasts in the way foraging requirements and averting predators influence the use o f space (Pulliam & Mills 1977).
ACKNOWLEDGMENTS C. Paradise assisted in the field. C. P. L. B a r k a n provided technical guidance, and M. W i t h i a m offered advice on statistics. C o m m e n t s by D. W. Stephens and an a n o n y m o u s reviewer were very helpful during revision of the manuscript. We t h a n k N S F for grant BNS 84-18714.
REFERENCES Barnard, C. J. 1980. Equilibrium flock size and factors affecting arrival and departure in feeding house sparrows. Anim. Behav., 28, 503-511. Caraco, T. 1979. Time budgeting and group size: a theory. Ecology, 60, 618-627. Caraco, T., Martindale, S. & Pulliam, H. R. 1980. Avian time budgets and distance to cover. Auk, 97, 872-875. Cerri, R. D. & Fraser, D. F. 1983, Predation and risk in foraging minnows: balancing conflicting demands. Am. Nat., 121, 552-561. Clark, C. W. & Mangel, M. 1986. The evolutionary advantages of group foraging. Theor. Pop. Biol., 30, 45-75. Davies, N. B. & Halliday, T. R. 1978. Competitive mate searching in common toads, Bufo bufo. Anita. Behav., 27, 1253-1267. Edwards, J. 1983. Diet shifts in moose due to predator avoidance. Oecologia (Berl.), 60, 185-189. Ens, B. & Goss-Custard, J. D. 1984. Interference among oystercatchers Haematopus ostralegus, feeding on mussels Mytilus edulis on the Exe Estuary. J. Anita. Ecol., 53, 217-231.
Fagen, R. 1987. A generalized habitat matching rule. Evol. Ecol., 1, 5-10. Fretwell, S. D. 1972. Populations in a Seasonal Environment. Princeton, New Jersey: Princeton University Press. Godin, J.-G. J. & Keenleyside, M. H. A. 1984. Foraging on patchily distributed prey by cichlid fish (Teleostei: Cichlidae): a test of the ideal free distribution theory. Anita. Behav., 32, 120 131. Harper, D. G. C. 1982. Competitive foraging in mallards: 'ideal free' ducks. Anita. Behav., 30, 574~584. Hinde, R. A. 1970. Animal Behavior. A Synthesis" of Ethology and Comparative Psychology. New York: McGraw-Hill. Houston, A. I. & McNamara, J. M. 1987. Switching between resources and the ideal free distribution. Anim. Behav., 35, 301-302. Krebs, J. R., Stephens, D. W. & Sutherland, W. J. 1983. Perspectives in optimal foraging. In: Perspectives in Ornithology (Ed. by G. A. Clark & A. H. Brush), pp. 245-281. New York: Cambridge University Press. Lefebvre, L. 1983. Equilibrium distribution of feral pigeons at multiple food sources. Behav. Ecol. Sociobiol., 12, 11-17. Leger, P. W., Owing, D. H. & Cross, R. C. 1983. Behavioral ecology of time allocation in California ground squirrels (Spermophilus beecheyi): microhabitat effects. J. comp. Psychol., 97, 283-291. Lima, S. L. & Valone, T. J. 1986. Influence of predation risk on diet selection: a simple example in the grey squirrel. Anim. Behav., 34, 536-544. Lima, S. L., Valone, T. J. & Caraeo, T. 1985. Foragingefficiency-predation-risk trade-off in grey squirrels. Anita. Behav., 33, 155-165. Mangel, M. & Clark, C. W. 1986. Towards a unified foraging theory. Ecology, 67, 1127-1138. Martindale, S. 1982. Nest defense and central place foraging: a model and experiment. Behav. Ecol. SociobioL, 10, 85-89. Milinski, M. 1979. An evolutionarily stable feeding strategy in sticklebacks. Z. Tierpsychol., 51, 3640. Milinski, M. 1984. Competitive resource sharing: an experimental test of a learning rule for ESSs. Anita. Behav., 32, 238-242, Milinski, M. & Heller, R. 1978. Influence of a predator on the optimal foraging behaviour of sticklebacks (Gasterosteus aeuleatus L.). Nature, Lond., 275, 642-644. Mittelbaeh, G. E. 198 I. Foraging efficiency and body size: a study of optimal diet and habitat use in bluegills. Ecology, 62, 1370-1386. Neter, J. & Wasserman, W. 1974. Applied Linear Statistical Models. Homewood, Illinois: R. D. Irwin. Parker, G. A. 1978. Searching for mates. In: Behavioural Ecology." An Evolutionary Approach (Ed. by J. R. Krebs & N. B. Davies), pp. 214~244. Oxford: Blackwell Scientific Publications. Parker, G. A. 1984. Evolutionarily stable strategies. In: Behavioural Ecology: An Evolutionary Approach. 2nd edn (Ed. by J. R. Krebs & N. B. Davies), pp. 30-62. Oxford: Blackwell Scientific Publications. Parker, G. A. & Sutherland, W. J. 1986. Ideal free distributions when individuals differ in competitive
Newman & Caraco: Patch use by squirrels ability: phenotype-limited ideal free models. Anim. Behav., 34, 1222-1242. Pulliam, H. R. & Caraco, T. 1984. Living in groups: is there an optimal group size?: In Behavioural Ecology: An Evolutionary Approach. 2rid edn (Ed. by J. R. Krebs & N. B. Davies), pp. 122-148. Oxford: Blackwell Scientific Publications. Pulliam, H. R. & Mills, G. S. 1977. The use of space by wintering sparrows. Ecology, 58, 1393-1399. Pyke, G. H. 1984. Optimal foraging theory: a critical review. A. Rev. Ecol. Syst., 15, 523-573.
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Rosenzweig, M. L. 1981. A theory of habitat selection. Ecology, 62, 327 335. Underwood, M. L. 1982. Vigilance behaviour in grazing African antelopes. Behaviour, 79, 81-107. Werner, E. E., Gilliam, J. F., Hall, D. J. & Mittelbach, G. G. 1983. Experimental test of the effects of predation risk on habitat use in fish. Ecology, 64, 1540-1548.
(Received 2 May 1986; revised 22 November 1986; MS. number: A4764)
Foraging, predation hazard and patch use in grey squirrels J O N A T H A N A. N E W M A N & T H O M A S C A R A C O Behavioral Ecology Group, Biological Sciences, State University of New York, Albany, New York 12222, U.S.A.
Abstract. Two patches of food, one large and one small, were placed at equal distances from protective
cover. Grey squirrels, Sciurus earolinensis, responded to the difference in food quantity, so that aggregation sizes matched predictions based on relative patch size, as well as those based on feeding rates within patches. The large patch was then moved farther from cover, where the probability of predation per unit foraging time should be greater. Aggregation sizes in the large patch decreased significantly, and overcrowding in the small patch resulted in reduced average feeding rates. The experiment was next repeated with the amount of food in each patch doubled. Increased distance to cover again resulted in decreased aggregation sizes in the larger patch, but average feeding rates did not decline in the small groups that were observed. Squirrels responded to relative, but not necessarily absolute, differences in food availability. When patches differed in both food availability and the hazard of predation, each attribute exerted a significant effect on the pattern of patch use.
The distribution of competing animals among patches of a resource should depend on the way local competitor density influences each individual's access to the resource (Parker 1978, 1984; Rosenzweig 1981; Pulliam & Caraco 1984; Clark & Mangel 1986). Fretwell (1972) discussed two models predicting equilibrium frequencies of competitors in patches, the Ideal Free Distribution and the Ideal Despotic Distribution. The ideal free distribution assumes that each individual (of a particular phenotype, Parker & Sutherland 1986) in a patch has equal access to the resources found there. The ideal despotic distribution assumes that individuals arriving earlier in a patch may suppress later arrivals' access to resources (e.g. through territoriality). This asymmetry in access to resources may imply that the later arrivals shift to a less-preferred habitat. Ideal Free Distribution
The ideal free distribution assumes that all identical competitors obtain the same amount of resource and so achieve the same fitness. In many, but not all, cases the predicted distribution of individuals across patches has density ratios that match resource ratios (Parker 1984; Pulliam & Caraco 1984). Fagen (1987) presents a general analysis of this habitat-matching rule. Suppose that the function W(Rdn~) gives the fitness of each
of the niidentical individuals in patch i. Ri is the rate of resource input at patch i. Fagen (1987) requires only that W is bounded, single-valued, and increases strictly monotonically as a function of (Ri/ n~). Then the assumption that all individuals have equal fitness at equilibrium, i.e. W(Ri/n~)= W(Rj/ nj), implies that ndnj = Ri/Ri for any fitness function W with the three properties mentioned. If a total of p patches are occupied, the habitat-matching rule predicts relative densities P
p
(nil Z ni)~ (Ri/ ~. Ri) i=1
(1)
i=i
The ideal free distribution is usually tested in this form, with the Rz known or manipulated experimentally. If fitness always decreases as m increases, the ideal free distribution must be a stable Nash equilibrium (Pullam & Caraco 1984; Parker & Sutherland 1986; cf. Clark & Mangel 1986). That is, if all individuals conform to the ideal free distribution, an individual deviating unilaterally from the equilibrium incurs a reduction in fitness. Since the ideal free distribution requires equal fitness for identical competitors, it corresponds to an evolutionarily stable strategy (Milinski 1979; Parker 1984; Pulliam & Caraco 1984). Several studies report observations of the habitat-matching version (equation 1) of the ideal free distribution (e.g. Davies & Halliday 1978; Parker
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Newman & Caraco: Patch use by squirrels 1978; Milinski 1979, 1984; Harper 1982; Ens & Goss-Custard 1984; Godin & Keenleyside 1984). Lefebvre (1983) successfully applied the ideal free distribution and noted that habitat matching often occurs even though the ideal free distribution's assumption of identical competitors is violated. For example, per caput resource consumption may vary with dominance rank (Harper 1982) or with variation in foraging ability (Godin & Keenleyside 1984), but density ratios still approximate resource ratios. Parker & Sutherland (1986) summarize empirical studies of the ideal free distribution and derive equilibrium models of patch use for some cases where individuals differ in competitive ability. Our results concern two aspects of the ideal free distribution. (1) Following Lefebvre (1983), we suggest that individuals' decision rules (sensu Pyke 1984) may induce density ratios matching resource ratios even when the relationship between per caput resource consumption and local density predicts another patch-occupation pattern at equilibrium. (2) Several tests of the ideal free distribution have involved a food resource (see Parker & Sutherland 1986). In nature, an increase in foraging opportunity sometimes coincides with an increase in the hazard of predation (probability of being killed per unit foraging time), so that patch occupation and resource use become problems with conflicting objectives (e.g. Mittelbach 1981; Martindale 1982; Cerri & Fraser 1983; Leger et al. 1983; Lima et al. 1985). We expect patch-use patterns to depend on spatial variation in food availability, variation in the hazard of predation, and the covariation between food availability and the hazard of predation. Ideal Despotic Distribution Fretwell (1972) introduces the ideal despotic distribution by considering the example of a territorial species. Once a habitat has been filled by territory holders, further arrivals will find the habitat less suitable. Consequently, beyond a certain density, later-arriving individuals wilt not have access to the same amount of resource. An analogous situation can arise when food within a patch occurs at a finite number of point sources, since some individuals may obstruct other foragers' access to food. In this context we consider equilibrium aggregation sizes in patches where feeding
1805
rates depend on local density in a manner roughly consistent with the assumptions of Fretwell's (1972) ideal despotic distribution. Consider two patches, designated A and B. Let nA and na identify the number of foragers in the respective patches, and let n = nA+nB. We represent the number of food sources in the patches/cA and kB. Throughout, we designate the patch with more food as patch A; i.e. kA > kB. If ni
F(ni) = S, when I _
(2)
Assuming that the patches do not differ in the mortality hazard due to predation, an individual should seek to feed at rate S (if possible). For a given n, we define equilibrium states (nA*, nB*) as those patch-occupation patterns where no individual should be tempted to switch patches in order to obtain a greater feeding rate. For n
(3)
(2) When kB < n < kA, the total number of foragers exceeds the number of food sources in patch
1806
Animal Behaviour, 35, 6
B, but not in patch A. Then density dependence can reduce the average feeding rate in patch B, but not in patch A. All individuals will feed at rate S as long as nB
(4) E [nB*]= kB/2 (3) When kA < n_< kA+ kB, the total number of foragers exceeds the number of food sources in the larger patch. Density dependence can reduce the average feeding rate in both patches. However, all individuals can still feed at rate S if neither n~ exceeds the corresponding kt. Possible equilibria range from (n--kB, kB) to (kA, n--kA); the foragers simply avoid too large an aggregation in either patch. Then E [hA*]= (n + kA -- kB)/2
(5) E [nB*]= (n + kB -- kA)/2 When n = kA+ kB, equation (5) is equivalent to the habitat-matching form of the ideal free distribution. (4) When n>kA+kB, the total number of foragers exceeds the total number of food sources. The first n foragers feed at rate S. The remaining individuals' access to food is obstructed, and they feed at the reduced rate s. Any state where nA> kA and nB>_kB is an equilibrium, and the E[ni*] increase as n increases. Paralleling our comments on the ideal free distribution, we next assume that the larger patch (A) is located farther from cover and consequently is more exposed to predators than is patch B. If survival depends on the product of the probability of not starving and the probability of avoiding predation, efficient survival strategies may imply patterns of patch occupation differing from the equilibria predicted by feeding rates. Increasing danger in patch A suggests that, for a given n, more individuals should feed in patch B. Efficient strategic responses might imply that individuals accept a decreased feeding rate (if patch B becomes 'overcrowded') to increase their probability of escaping an attack by a predator. In any case, we can anticipate that nA will decline as distance to cover increases. Additionally, we might speculate that individuals use patches in a manner consistent with the equilibration of survival probabilities. That is,
the change in aggregation sizes presumably accompanying the increased mortality hazard in patch A might induce a new equilibrium where survival probabilities in the two patches are the same for any given n. For example, a forager in patch A might obtain a required amount of food in a relatively short time. A forager in patch B would require more time to obtain the same amount of food, but its mortality hazard per unit would be lower than in patch A. In this paper we first examine the distribution of grey squirrels, Sciurus carolinensis, between patches differing in food availability, but not distance from protective cover. We then compare this distribution to that observed when the richer patch is located farther from cover, where exposure to predators is presumably greater. Although grey squirrels give alarm calls, they do not depend on social groups to avoid predation. Furthermore, empirical evidence suggests that an individual's behaviour while foraging depends very strongly on its distance to cover (Lima et al. 1985; Lima & Valone 1986). Therefore, grey squirrels should be reasonable subjects for this sort of experiment.
METHODS
We conducted the study from late June to early October 1985, in an open, rectangular field (75 x 25 m) on the campus of the State University of New York at Albany. The field is bordered on three sides by trees, mainly black locust, Robinia pseudoacacia. The grey squirrels used the trees as protective cover. A campus road runs along the fourth side of the field. Data were collected 6 days per week from 0530 to 0800 hours. Traffic on the road was light at that time of day. Potential predators seen at the study site included red-tailed hawks, Buteojamaicensis, great horned owls, Bubo virginianus, barred owls, Strix varia, and domestic dogs and cats. Foraging squirrels ran to the trees when they sighted any of these animals. However, we saw only one actual attack (by a domestic cat) during the study; the squirrel escaped. Our experimental patches contained sunflower seeds, Helianthus annuus. We chose sunflower seeds because the squirrels ate them in the patches, rather than caching them. We placed food in aluminium pans (18.5 x 9.8 x 5.7 cm). The pans held enough seeds to avoid any effects of food depletion during
Newman & Caraco: Patch use by squirrels any day's observation; each pan always contained some seeds at the completion of data collection. The pans were large enough for two squirrels to feed simultaneously at a single pan, but this occurred only rarely. Consequently, the number of pans in a patch corresponded to k~ of the previous section. Each day we made available a larger (A) and a smaller (B) patch. Within a patch the pans were arranged so that the inter-pan distance (when a patch contained more than one pan) was a constant 0.65 m. The positions of the patches (left versus right) were alternated randomly throughout the study to control for position preferences unrelated to distance from cover. We collected data with a portable microcomputer equipped with an event-recorder program. Whenever a change in the state (hA, nB) occurred, whether or not n changed, the program recorded the new state and the time of the transition to the nearest second. For each n from one to five, the total time spent in each feasible state (i.e. where nA+na=n) was summed for each day. These data allowed calculation of the average number of squirrels occupying patch A, conditional on the value of n. We rarely saw more than five squirrels, in total, occupying the patches at any one time. Hence, most of our analyses are restricted to n_< 5. Many more squirrels could be found in the area, and there was turnover in the occupants of the patches during each day's observation. We estimated feeding rates from both direct observation and analysis of super-8 movies. We counted the number of seeds eaten by a focal animal in 1 min. For each measure we noted the patch size (k~), the number of squirrels in the patch (ni), and distance from cover. We divided our study into two parts. The important difference between parts 1 and 2 is that patch sizes were doubled in part 2. If squirrels follow the habitat-matching rule (equation 1), we can predict their distribution using only the relative patch sizes. Consequently, doubling patch sizes should not affect the patch-use pattern at a given distance to cover. However, if squirrels follow the rules of the feeding-rate model (equations 3, 4 and 5), they will respond to a change in the absolute patch sizes, at least for certain values of n. Under that model, doubling patch sizes could affect the patch-use pattern for a given n and distance to cover.
1807
Part 1
Patch A contained four pans of food, attached at the corners of a piece of plywood measuring 0.7 • 0.7 m and placed on the ground. Patch B contained one pan of food, attached in the middle of a similar piece of plywood. Initially, both patches were placed 5 m from cover. We sampled until our records of time in the various states yielded six representative probability distributions of the feasible states for each n from one to five. For a given n we observed one such distribution each day, but a distribution was considered representative only if the system remained in that particular distribution's most common state for at least 50 s (simply our rule of thumb to ensure a reasonably long sampling period). More than 6 days were required to obtain six representativ e distributions in every case. For some values of n our data yielded more than six representative distributions. Next, we moved patch A to 15 m from cover (leaving B at 5 m), and recorded six representative state distributions for each n. The process was then repeated with patch A at 5 m from cover, and again with patch A at 15 m from cover.
Part 2
During this phase of the study, patch A contained eight food pans, and patch B contained two pans. We recorded six representative distributions of states for n from one to five with both patches 5 m from cover, and then with patch A moved to 15 m from cover. Hence, the sample sizes were onehalf of those recorded in part 1. In both parts 1 and 2, the ratio of kA to kB was 4.
RESULTS Table I lists the average feeding rates (seeds consumed/min) observed during both parts of the study. Data for patch A are partitioned according to distance from cover. The table contains four (n~,ki) combinations where n~>ki, and in each of these cases there are two entries. The larger feeding rate refers to the squirrel(s) with unobstructed access to a feeding pan. The smaller rate refers to the squirrel(s) lacking direct access to a feeding pan (i.e. they were obstructed by squirrels with direct
Animal Behaviour, 35, 6
1808
Table I. Average individual feeding rates (seeds/min) observed during the study Feeding rates
ni Partl 1 2
PatchA(5m, k = 4 ) 20.13 _+1-23 19.15___2.42
Patch A (15 m, k = 4 ) 20.5 ___3 . 6 4 20"41+1'67
3
18.12___2.12
21'79__+3'19
4 5 Part2 1 2 3
17.41__+2.59 18'15_+3"07(S) 1.39_+2.4 (s) PatchA(5m, k = 8 ) 17.56_+ 1.34 17.59• 19.3-+ 1.7
21.7+3.18 --Patch A (15 m, k = 8 ) 17.1 +2.01 19.87_+2.62 19.6___2.53
4 5 6
17.58+3.0 20.22_+2.89 21.05__+ 1.76
18.82+ 1.65 21.46+ 1-49 19.72+__6.7
PatchB(5m, k = l ) 16.34___0-98 16.07+4'47 (S) 1.01 + 1.78 (s) 13.01+2'15 (S) 0.87+0.57 (s) ---PatchB(5m, k = 2 ) 18.49+ 1.69 18.14_+2.14 18.14_+2.06(S) 3.54_+4-75 (s)
---
Entries are means_95% confidence intervals, rti is the number of individuals within a patch. In double entries, S refers to unobstructed foragers, and s refers to obstructed foragers. 5 m, 15 m: distance from cover; k: number of food sources in each patch.
access). The following analysis suggests that observed feeding rates correspond roughly to the assumptions expressed in equation (2). We subjected the feeding rates observed in patch A to a two-way A N O V A adjusted for unequal sample sizes; data from the two parts of the study were analysed separately. Since individual feeding rates can depend on both competitive effects and advantageous social interactions (e.g. Caraco 1979; Pulliam & Caraco 1984), we took aggregation size (nA) and distance to cover as the treatments. In part 1 of the study we had sufficient data for values of nA < 4 (see Table I). During part 1, neither aggregation size (F3,73=0"76, Ns), distance to cover (FI,73 = 1.83, Ns), nor their interaction (F3,73= 1.56, MS) caused significant variation in feeding rates. In part 2, we had sufficient data for values of nA < 6. During part 2, neither distance to cover (Fi,75=0'04, NS) nor the na-distance interaction (F5,75= 1.03, Ns) influenced feeding rates significantly. However, the variation due to aggregation size was significant (Fs,ys = 4.91, P < 0.025). Average feeding rates for these unobstructed foragers increased as nA increased. In part 1 of the study, feeding rates of unobstructed squirrels in patch B varied significantly
across aggregation size ( A N O V A , F2,49=4"73, P < 0.05). The average feeding rate declined as nB increased, contrasting with the single significant effect referred to in the preceding paragraph. We found no significant variation across aggregation size for unobstructed squirrels in patch B during part 2 of the study (F2,32=0"05, Ns). When we considered the feeding rates of obstructed squirrels, we detected no significant variation among the four mean feeding rates ( A N O V A , F3,45=2'6, Ns). Five of the seven F-tests for unobstructed squirrels failed to detect significant variation in average feeding rates, and our single test for obstructed squirrels does not indicate significant variation. Collectively, these results lend reasonable support for the assumptions motivating equation (2) and, hence, justify comparing patch-occupancy patterns to the predictions of equations (3-5). For both parts 1 and 2, Table II lists two sets of predictions and two sets of observations for hA. F o r n = 1 through 5, the table gives the following. (1) The first column shows the predictions for E[nA*] obtained from the model of equilibrated feeding rates, equations (3-5). (2) The second column shows the values for na predicted by the habitat-matching rule, equation
Newman & Caraco: Patch use by squirrels Table II. Predicted and observed mean aggregation sizes in patch A* Predicted nA n
Observed nA
Rate
Match
5m
15 m
0.5 1.5 2.5 3.5 4.0
0.8 1-6 2.4 3.2 4.0
0.77 1.47 2.04 2.98 3.81
0.13 0,66 1~50 2.39 3-09
0-5 1.0 2'0 3-0 4.0
0.8 1'6 2.4 3.2 4.0
0.55 l '48 2-23 2.78 3.55
0.53 0-48 1.13 2.29 2.92
Part 1
1 2 3 4 5 Part 2
1 2 3 4 5
Predictions follow from the feeding-rate model (designated by Rate) and from habitat matching (designated by Match). Predictions refer to average nA with patch A 5 m from cover (third column). The number of feeding pans was doubled in part 2. n is the total number of foragers. * A complete explanation of each column is given in the text.
(1). Since kA/kB was 4 in both parts of our study, habitat-matching simply predicts that nA = 0-8n. (3) The third column shows the average nA values when both patches were 5 m from cover. (4) The fourth column shows the average nA values observed when patch A was 15 m from cover. Both the feeding-rate model and habitat matching predict the average nA values observed when the two patches were equally distant iu cover. F o r part 1 (kA=4, k B = l ) , the correlation between predictions of the feeding-rate model and the observations is r=0"985 (P<0.05). The corresponding correlation for the habitat-matching rule is r = 0 . 9 9 6 ( P < 0 ' 0 5 ) . For part 2 (kA=8, kB=2), the correlations are r=0.981 ( P < 0 ' 0 5 ) for the feeding-rate model and r = 0"996 (P < 0.05) for the habitat-matching rule. The two models' predictions hardly differ, and these data do not discriminate between the models. The growth and decline of the aggregations showed considerable variability. Additionally, our observations revealed that the squirrels rarely switched between patches (sensu Houston & M c N a m a r a 1987) during a foraging bout, Most changes of state involved entry into or departure from the system.
1809
Table II suggests that distance to cover influences aggregation size; each of the 10 average nA values declined when distance to cover was increased from 5 to 15 m. We conducted a two-way A N O V A on the average daily nA values for each n from 1 to 5. We used the method of unweighted means (Neter & Wasserman 1974) because sample sizes were smaller in part 2. Treatments were distance to cover and total amount of food (i.e, part 1 versus part 2). In none of the five cases did average nA values vary significantly between parts 1 and 2 of the study (values for F1,33 ranged from 0.33 to 1-67, all Ns). Hence, aggregation sizes did not respond to a change in the absolute difference in patch size. The correlations noted above are presumably a response to relative patch sizes. In all five cases, average nA values were significantly smaller at the greater distance to cover (values for Ft,33 ranged from 11-0 to 27.3, all P < 0.005). Presumably, the squirrels responded to a change in the mortality hazard due to predation. For solitary squirrels (i.e. n = 1) we found a significant interaction between treatments (Fi,33=9.0, P < 0-01); distance to cover influenced patch use in part 1, but not in part 2, In the other four cases we failed to detect a significant interaction (values for F1,33 ranged from 0-1 to 2.66, all NS). We next return to the data on feeding rates. We compare foraging in patch A versus patch B when distance to cover was equal, and we ask whether the decreases in nA associated with the greater distance to cover influenced individual feeding rates. The feeding rates listed in Table I were estimated when a given number of individuals occupied a particular patch, without regard to the number of individuals in the other patch. However, we can use those data to define between-patch comparisons of feeding rates for different values of n. Combining the data in Table I and the relative frequencies of the states (hA, n~ln), we estimated an average feeding rate for an individual in patch A, and an individual in patch B, for each n_< 5. We simply calculated an overall patch-specific rate by weighting the rates for the feasible nl by the proportion of time the system spent in the associated state (see Fig. 1). We then took ratios of the overall rate in patch A to the corresponding overall rate in patch B for each n and distance to cover. We treated parts 1 and 2 separately. Figure 2A shows the ratios of the overall feeding rates (for both parts of the study) for data collected when both patches were 5 m from cover. Figure 2B shows the ratios of the
Animal Behaviour, 35, 6
1810
DISTANCE FROM COVER 5m 15m 0.8-
n=2
A
n=2
B
2,0 .o_
0.6-
"~ 1,5
iiiiiiiiiiiii~ 0.2 0.8
n=3
O
n=3
1,0
o
o
o
[
I
I
2
3
O
o
o
o
o
~,0.6 -
go.4 90.2 ~0.8
I
I
I
I
4
5
r
2
_t 3
1 I_ 4 5
Total foragers (n) Figure 2, Feeding rate ratios. For a given total number of foragers (n = hA-'}-na), we calculated a weighted average feeding rate for each patch. Dividing the estimate for patch A by the estimate for patch B gives the ratio plotted. Closed circles refer to part I. Open circles refer to part 2, when the number of feeding pans was doubled. (A) Data collected when both patches were 5 m from cover. (B) Data collected when patch A was 15 m from cover.
n=4
n:4
~
::::::2::::
0,6
~
1~0.4 0,2 n=5
n=5
0,8'
0.6 0,4'
i:!:i:!:i:~:i
iiiiiiiiiiiiii
_
0.2! 0 1 2 3 4 5 0 1 2 3 4 5
lqA Figure 1. Frequency distributions of aggregation sizes in patch A. Data are taken from part 1 of the study (four food pans in patch A, one in patch B). For n = 2 through 5 (where n is the number of foragers), the figure shows the relative frequencies of aggregation sizes in patch A (hA) for both distance-to-cover treatments.
overall feeding rates for data collected when patch A was 15 m from cover. In part 1 of the study, feeding rates in patch A consistently exceeded those in patch B by about 20% (Fig. 2A). Since patch B was seldom overcrowded when the patches were equally distant from cover (Fig. 1), we cannot attribute the pattern to obvious aspects of patch choice. In part 2, feeding rates in patch A were nearly identical to those in patch B when both patches were 5 m from cover (Fig. 2A). In this case, feeding rates for n _<5 clearly equilibrated between patches, as would be assumed by any model of identical competitors when n < ka +kB. Given the limitations of our data,
we attribute the apparent difference in ratios between parts 1 and 2 to chance. Figure 2B shows feeding-rate ratios, for parts 1 and 2, calculated from data collected when the larger patch was farther from cover. In part 1 of the study, feeding rates in patch A always exceeded those in patch B; for n = 5 the average feeding rate in patch A was more than double the average rate in patch B. Previously, we noted that nA values declined significantly when patch A was moved away from cover. Additionally, we found that feeding rates did not vary significantly with distance to cover. Therefore, the greater feeding rates in patch A shown in Fig. 2B for part 1 arise largely because patch B was overcrowded (Fig. 1). This pattern may represent a trade-off between foraging objectives and safety from predation, since individuals closer to protective cover incur a lower feeding rate. However, in part 2 (10 total feeding pans), feeding rates in patch A do not tend to exceed the rates in patch B (Fig. 2B). The difference arises simply because the same numbers of squirrels had access to twice as many feeding pans in part 2 of the study. Above, we noted that average nA values varied significantly between the distances to cover, but not between the two parts of the study. Increased distance to cover implies the same changes in aggregation sizes in both part 1 and part
Newman & Caraco: Patch use by squirrels 2, but this particular difference affects feeding rates only when patch B contains a single feeding pan. If larger values of n had been observed during the second part of the study, we might have found additional evidence of a trade-off between foraging rate and predation hazard. When distances to cover were equal, feeding rates in the two patches were roughly equal (Fig. 2A). However, this does not necessarily imply that the feeding-rate model, equation (3), better predicts patch use than does a habitat-matching 'rule of thumb' (Krebs et al. 1983). The correlations for mean aggregation sizes (discussed above, both patches at 5 m) indicate that both habitat matching and the feeding-rate model are plausible explanations of the squirrels' behaviour. That is, the correlations suggest that patch use may respond to either relative patch size (habitat matching, equation 1) or absolute patch size (feeding rates, equation 3). We later noted that mean nA values did not vary significantly as total food (i.e. part 1 versus part 2) varied; the ANOVA results imply that patch use did not depend on absolute patch sizes. We therefore prefer the interpretation suggesting that the squirrels chose to feed in the experimental patches so that density ratios (hA~riB) matched relative patch size (kA/k~); they responded more to relative than to absolute patch sizes. We suspect that a similar rule held when the patches differed in both amount of food and distance to cover. Since mean nA values did not differ significantly between parts 1 and 2, the density ratios hA~riB should be the same in both parts of the study (for a given distance to cover). We already noted that the relative numbers of feeding pans (4/t) accurately predicted ng values when both patches were 5 m from cover. In part 1, when patch A was 15 m from cover, the average density ratio observed ([~(nA/nB)]/5) was close to 5/6. Therefore, the nA values in part 2, when patch A was 15 m from cover, should equal 5n/ (5+6)=0-45n. The observed average nA values (Table II: part 2, patch A at 15 m) correlate significantly with predictions of the 5/6 density ratio (r=0'956, P<0-05). Since habitat matching worked well when the patches were equally exposed to predators, this last correlation should not be surprising. Squirrels may use a rule such that density ratios match patchvalue ratios, where patch value increases with patch size (resource abundance) and decreases with exposure to predators.
181l
DISCUSSION Our analysis assumes that greater distance to cover increases the mortality hazard due to predation. We did not verify this assumption prior to the study. However, a great deal of evidence supports our assertion (e.g. Barnard 1980; Caraco et al. 1980; Underwood 1982; Leger et al. 1983; Lima et al. 1985; Lima & Valone 1986). The probability of escaping a detected predator should decrease as the distance to cover increases for those animals dependent on cover for safety. Our results show that squirrels alter their use of patches to avoid the danger of predation, and that they sometimes sacrifice foraging efficiency to do so. A number of recent studies have documented qualitatively similar responses (e.g. Milinski & Heller 1978; Mittelbach 1981; Cerri & Fraser 1983; Edwards 1983; Werner et al. 1983; Lima et al. 1985). A variety of foraging problems in nature presumably involve simultaneous objectives of avoiding predation and averting starvation. Mangel & Clark (1986) discuss some interesting methods for analysing problems of this type. We interpreted our results as indicating that squirrels used a general habitat-matching rule that responded to relative differences in the amount of food and exposure to predators. A relative scaling, invariant with a doubling of the size of each patch, does not contradict concepts drawn from the psychology of perception (Hinde 1970; Fagen 1987). Several simple mechanistic rules are not consistent with our data. For example, increased danger in patch A might imply that foragers accept a given constant reduction in feeding rate by using patch B more often, or that foragers reduce time in patch A by a constant amount. We can reject both of these possibilities. The most consistent pattern in our data is found in the density ratios hA~riB. When the patches were equidistant from cover, a density ratio of 4/1 described the data in both parts of the study. When patch A was moved farther from cover, a density ratio closer to 1/1 described the data in both parts 1 and 2. Whatever the squirrels' decision rule may be, patch A was clearly devalued when distance to cover increased. Parker & Sutherland (1986), emphasizing that individuals may vary significantly in competitive ability, point out that several field studies report habitat matching when other spatial distributions would also allow feeding rates to equilibrate.
Animal Behaviour, 35, 6
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P a r k e r & S u t h e r l a n d (t986) suggest t h a t h a b i t a t m a t c h i n g provides a n efficiem 'rule o f t h u m b ' (see Lefebvre 1983). O u r results indicate t h a t squirrels in small aggregations use a h a b i t a t - m a t c h i n g rule that is (1) effective even before all individuals' feeding rates become density-dependent, a n d (2) sensitive to b o t h foraging economics a n d the h a z a r d of predation. The n u m b e r of studies d o c u m e n t i n g simultaneous effects o f feeding rate and p r e d a t i o n h a z a r d on spatial distributions has increased rapidly. In this p a p e r we assumed identical individuals, an obvious oversimplification. Individual variation in the trade-off between foraging and avoiding predators m i g h t depend o n easily identified attributes such as sex, age, or d o m i n a n c e status. A t a different level, co-occurring species might exhibit predictable contrasts in the way foraging requirements and averting predators influence the use o f space (Pulliam & Mills 1977).
ACKNOWLEDGMENTS C. Paradise assisted in the field. C. P. L. B a r k a n provided technical guidance, and M. W i t h i a m offered advice on statistics. C o m m e n t s by D. W. Stephens and an a n o n y m o u s reviewer were very helpful during revision of the manuscript. We t h a n k N S F for grant BNS 84-18714.
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(Received 2 May 1986; revised 22 November 1986; MS. number: A4764)