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Sit-and-Wait Versus Active-Search Hunting: A Behavioral Ecological Model of Optimal
Search Mode
Cody T. Rossa,c, Bruce Winterhaldera,b
aDepartment of Anthropology, University of California, Davis. United States
bGraduate Group in Ecology, University of California, Davis. United States
cEmail: ctross@ucdavis.edu
Abstract
Drawing on Skellam’s (1958) work on sampling animal populations using transects, we derive a behavioral ecological model of the
choice between sit-and-wait and active-search hunting. Using simple, biologically-based assumptions about the characteristics of
predator and prey, we show how an empirically definable parameter space favoring active-search hunting expands as: 1) the average
rate of movement of prey decreases, or 2) the energetic costs of hunter locomotion decline; the same parameter space narrows as: 3)
prey skittishness increases as a function of a hunter’s velocity, or 4) prey become less detectable as a function of a hunter’s velocity.
Under either search tactic, encounter rate increases as a function of increasing prey velocity and increasing detection zone radius.
Additionally, we investigate the roles of habitat heterogeneity and spatial auto-correlation or grouping of prey on the optimal search
mode of a hunter, finding that habitat heterogeneity has the potential to complicate application of the model to some empirical
examples, while the effects of prey grouping lead to relatively similar model outcomes. As predicted by the model, the introduction
of the horse to the Great Plains and the introduction of the snowmobile to Arctic foraging communities decreased the metabolic
costs of active-search and led to a change in normative hunting strategies that favored active-search in place of sit-and-wait hunting.
Keywords: Foraging Theory, Ambush, Search, Energetics of Hunting, Hunter-Gatherers, Sit-and-Wait
1. Introduction1
1.1. Introduction2
Evolutionary anthropologists have noted a substantial diver-3
sity in hunting strategies across human societies [1,2,3,4,5,4
6,7,8,9], as well as variation in hunting strategies within so-5
cieties [1,10] that depend on the behavior of the prey species,6
individual skills of the hunter, local ecological knowledge, and7
available technology. Likewise, biologists have noted different8
normative hunting styles across species [11,12,13,14,15] and9
ecologies [16,17,18].10
Several simple models have been proposed to explain het-11
erogeneity in hunting strategies in 2-dimensional environments12
[19,20,21,22,23,24,25] in terms of the costs and benefits of13
active-search versus sit-and-wait, but modeling the linkages be-14
tween the biological and ecological context of foraging—prey15
movement velocity, prey skittishness, prey grouping, bioener-16
getics of predator locomotion, and habitat heterogeneity—and17
the differential costs and benefits of each strategy to a hunter18
remains an open problem. In this paper, we attempt to provide19
a more thorough characterization of the linkages between the20
biological and ecological characteristics of predator and prey21
and the costs and benefits of each search mode, by developing22
a model of foraging behavior that is grounded on the optimiza-23
tion of energetic returns and the dynamics of prey movement24
through the environment.25
Our analysis relies heavily on mathematical tools introduced26
by Laing [26], Yapp [27], and Skellam [28] to analyze the prop-27
erties of animal population surveys using transects. These au- 28
thors rely on a physical analogy (the classical kinetic theory 29
of gases) to make inferences about the population density of a 30
species conditional on the count of individuals observed dur- 31
ing a transect walk of specified characteristics. We invert the 32
question and adapt the same tools to investigate encounter rates 33
with prey items observed along a search path conditional on the 34
density and movement of these prey items. We focus on iso- 35
lating the biological properties of predator and prey that might 36
influence the choice of either an active-search hunting strategy 37
(forager velocity >0) or a sit-and-wait hunting strategy (for- 38
ager velocity =0) by a predator. Similar work on search mode 39
optimization has been conducted in 3-dimensional foraging en- 40
vironments [29], see the Conclusions section for a contrast of 41
our findings. 42
We investigate the role of prey species velocity, prey skit- 43
tishness, prey detectability, spatial auto-correlation or group- 44
ing in prey, habitat heterogeneity, and the energetic costs of 45
active-search on the foraging strategy and optimal velocity of a 46
hunter. We begin by outlining the derivation of our model, and 47
then present the analytical results of our model under simple 48
assumptions concerning the characteristics of prey and hunter. 49
We conclude the analysis by assessing empirical predictions of 50
foraging behavior derived from our model using empirical data 51
on human metabolic expenditure, empirical data on average an- 52
imal velocities, and ethnographic accounts of human foraging 53
strategies in the Great Plains and Arctic through technological 54
transitions. Finally, we place our analysis and results in the con- 55
Preprint submitted to Journal of Theoretical Biology July 28, 2015
text of related studies concerning the behavioral ecology of the56
food quest.57
Although we describe the predator in question as a human58
hunter or forager in our prose, our model is general to non-59
human predators. In the empirical analysis, we focus on the60
human case studies for two reasons: 1) there is good data on61
the functions linking energetic expenditure and velocity in hu-62
mans, and 2) novel technological changes such as domesticated63
horses, ATVs, snowmobiles, and motorboats have have dramat-64
ically changed the energy costs and maximal velocity of active-65
search in humans; these changes lead to simple, easily tested66
predictions about hunting strategy change under technological67
expansion.68
1.2. Model Derivation69
Closely following the model derived in Skellam [28, passim],70
we imagine a habitat populated with individuals of a mobile71
prey species—the gas particles of Skellam’s physical analogy—72
moving in arbitrary paths that are not necessarily independent,73
with an average velocity u, whose average density over the74
habitat is ¯
D. Likewise, we imagine a hunter who can chose75
to remain in a fixed location, or move through the habitat on76
an arbitrary path at an average velocity, w, Figure 1(a). The77
hunter carries with him or her a frame of reference, and a con-78
vex contour which outlines the area of his or her visual field, as79
illustrated in Figure 1(b). In contrast to Skellam [28], we do not80
assume that any prey item that enters the visual contour is nec-81
essarily encountered by the hunter, nor do we assume that the82
density of the prey is the same in neighborhood of the hunter83
as elsewhere; we do, however, consider these special cases in84
light of our more general model. For ease of reference, the def-85
initions of these and subsequent parameters and functions are86
collected in Table 1.87
[Table 1 about here.]88
[Figure 1 about here.]89
We measure the velocity of a prey item relative to the hunter90
using the hunter’s frame of reference. Thus, at any a particular91
time, t, a given prey item has a relative velocity characterized by92
a direction, θ(t), and magnitude, v(t); we allow there to be het-93
erogeneity in these values across prey items. We classify prey94
items by their directions and magnitudes such that f(v, θ, t)∂v∂θ95
represents the proportion of prey items in the neighborhood of96
the hunter at time tthat have velocities in the elementary inter-97
val v±1
2∂vand directions in θ±1
2∂θ.98
To calculate the expected encounter rate between the hunter99
and prey items, we draw the contour of the hunter’s convex vi-100
sual field as illustrated in Figure 1(b). We then draw two paral-101
lel tangents to the contour having direction θ; the distance be-102
tween the tangents is then H(θ). We mark offa border of width103
v(t)∂t, and shade the area laying within the tangents, the con-104
tour, and the border width. A prey item with velocity v±1
2∂v105
and direction θ±1
2∂θ can cross into the visual field from outside106
it in the elementary interval of time from tto t+∂tif and only107
if it lies within the shaded area at time t.108
The area of the shaded region is equal to H(θ)v(t)∂t, so the 109
expected number of prey items in the stated class lying inside 110
it will be D(w)f(v, θ, t)∂v∂θH(θ)v(t)∂t, where D(w) is the den- 111
sity of prey items in the neighborhood of the hunter, when the 112
hunter is moving at velocity w. For generality, we consider D113
to be a smooth, strictly decreasing function of the hunter’s av- 114
erage velocity, w, which equals ¯
Dwhen w=0, and has a lower 115
bound at 0. Daccounts for prey skittishness, allowing for a lo- 116
cal decrease in the density of prey to occur as the hunter moves 117
more rapidly, and is thus more conspicuous in the environment. 118
The expected total number of prey items, E(P), entering con- 119
tour from outside in the interval ∂tis given by integrating over 120
all values of vand θ; the expected total number of prey items 121
entering into the contour over the course of the hunt from the 122
start time, TS, to the end time, TE, is given by integrating over 123
t. Thus, E(P) over the course of a hunt is: 124
E(P)=D(w)Z2π
0Z∞
0ZTE
TS
v(t)f(v, θ, t)H(θ)∂v∂θ∂t(1)
By definition, T=TE−TS, and the average velocity of the prey 125
items relative to the hunter, V, is: 126
V=1
TZTE
TSZ2π
0Z∞
0
v(t)f(v, θ, t)∂v∂θ∂t(2)
If, for purposes of analytical tractability, the visual contour is 127
defined to be a circle, then all values of Hare equal to 2 times 128
the visual radius, R, and Equation 1, reduces to: 129
E(P)=D(w)2RVT (3)
As shown by Skellam [28] using a 2-dimensional analogue of 130
Maxwell’s distribution, Vcan also be expressed in terms of the 131
average velocity of the prey items, u, and the hunter, w:132
V=√u2+w2(4)
Finally, we consider the expected number of prey encountered 133
during the hunt, E(Φ), to be: 134
E(Φ)=E(P)ξ(w) (5)
where ξis a smooth, strictly decreasing function of the hunter’s 135
average velocity, w, with a value of 1 when w=0 and a lower 136
bound of 0. From Equation 5, we see that ξmodulates the rate 137
of encounters to reflect the fact that a hunter moving at an in- 138
creased velocity may be more likely to overlook prey items that 139
have crossed into his or her visual contour. While ξ(w) ap- 140
pears to have the same effect on encounters as D(w), they affect 141
the model through different mechanisms, where D(w) reflects 142
changes in the local density of prey as a function of hunter 143
velocity (due to prey skittishness), and ξ(w) further decreases 144
expected encounters (due to the hunter being less able to accu- 145
rately detect prey items as his or her velocity increases). 146
Thus, the full model we will use to investigate the choice of 147
hunting strategies can be written as: 148
E(Φ)=2RT D(w)ξ(w)p(w2+u2) (6)
2
Verbally, this model states that the expected number of prey149
encounters, E(Φ), is equal to the product of twice the radius150
of the detection zone, 2R, the time spent hunting, T, the ef-151
fective local prey density, D(w)ξ(w), and the square root of the152
summed squares of hunter and prey average velocities, wand153
u, respectively. Although different in both scope and deriva-154
tion, our model shows some similarities with previous work on155
foraging strategy in zooplankton in 3-dimensional space [29].156
2. Results157
2.1. Prey Velocity, Energy Expenditure, and Hunting Strategy158
If we assume for the time-being that D(w)=D(0) =¯
Dand159
ξ(w)=ξ(0) =1, we can model the ecological contexts in which160
sit-and-wait hunting will be preferred to active-search hunting,161
when the hunter’s velocity has no effect on local prey density162
or prey detectability. To do so, we investigate the ratio of ex-163
pected caloric returns from prey encounters to metabolic expen-164
diture across hunting strategies. The expected number of prey165
encounters is derived from Equation 6, the caloric content of a166
prey item is defined to be α, and metabolic expenditure is con-167
sidered to be a smooth strictly increasing function, C, of the168
hunter’s average velocity w, which equals 1 when w=0.169
Under these assumptions, sit-and-wait hunting will yield170
higher returns than active-search hunting when:171
2αD(0)ξ(0)RT p(0 +u2)
C(0) >2αD(w)ξ(w)RT p(w2+u2)
C(w)(7)
which is true when:172
u2>w2
ˆ
C2−1(8)
where ˆ
C=C(w)
C(0) is the ratio of energy expenditure rates between173
active-search at average velocity w, and sitting-and-waiting at174
average velocity 0.175
In our numerical analysis, human energy expenditure per176
unit time relative to velocity is expressed in terms of METs177
(Metabolic Equivalents). METs describe the energy required to178
move at velocity, w, under various kinds of exertion (walking,179
jogging, running, rowing, horse riding, snowmobiling) in pro-180
portion to baseline energy consumption during seated rest [30].181
Thus, our parameter ˆ
C=C(w)
C(0) has a direct empirical formula-182
tion. As is shown in Figure 2, decreasing the ˆ
Cratio, repre-183
sented by the shifted indifference curves with identical hunter184
velocities, but differing MET values, expands the area of the185
state space favoring active-search hunting.186
[Figure 2 about here.]187
2.2. Prey Skittishness188
If we relax our assumption that D(w)=D(0) =¯
Dand con-189
sider cases where D(w),D(0), while still assuming ξ(w)=190
ξ(0) =1, we can model the ecological contexts in which sit-191
and-wait hunting will be preferred to active-search hunting,192
when the hunter’s velocity has an effect on local prey density,193
but not prey detectability. We solve Equation 7under this new 194
assumption, which yields: 195
u2>w2
ˆ
C2ˆ
D2−1(9)
where ˆ
D=D(0)
D(w)is the ratio of local prey densities between 196
sitting-and-waiting and active-search at average velocity w.197
If more rapid predator velocity increases the likelihood of 198
alerting prey and thus decreases their local density, and by im- 199
plication the effective encounter rate, then the area of the state 200
space favoring active-search decreases (Figure 2). 201
2.3. Prey Detectability 202
If we relax our assumption that ξ(w)=ξ(0) =1 and consider 203
cases where ξ(w),ξ(0), while assuming D(w)=D(0) =¯
D, we 204
can model the ecological contexts in which sit-and-wait hunting 205
will be preferred to active-search hunting, when the hunter’s ve- 206
locity has an effect on the detectability of prey that come within 207
its visual range, but not local prey density. We solve Equation 208
7under this new assumption, which yields: 209
u2>w2
ˆ
C2ˆ
ξ2−1(10)
where ˆ
ξ=ξ(0)
ξ(w)is the ratio of prey detectability between sitting- 210
and-waiting and active-search at average velocity w.211
Similar to the result in Section 2.2, if more rapid predator 212
velocity decreases the likelihood of detecting prey inside the 213
hunters’s visual radius, and by implication decreases the effec- 214
tive encounter rate, then the area of the state space favoring 215
active-search decreases (Figure 2). 216
2.4. Prey Skittishness and Detectability 217
If we relax both assumptions and consider cases where 218
ξ(w),ξ(0) and D(w),D(0), we can model the ecologi- 219
cal contexts in which sit-and-wait hunting will be preferred to 220
active-search hunting, when the hunter’s velocity has an effect 221
on prey detection and local prey density. We solve Equation 7222
under these new assumptions, which yields: 223
u2>w2
ˆ
C2ˆ
D2ˆ
ξ2−1(11)
ˆ
Dand ˆ
ξinteract multiplicatively, decreasing the scope for 224
active-search hunting as their product increases (Figure 2). 225
2.5. The Effect of Prey Grouping 226
The formulas for the mean encounter rates described above 227
do not require that prey items move independently. Thus, the 228
grouping of prey items—for example, whether prey items live 229
and forage in herds, or live and forage solitarily—has no im- 230
pact on the mean encounter rate under the previous assump- 231
tions. The variance of encounters per unit time, however, will 232
be influenced by the grouping of prey items. If the encounter 233
rate is low, and the prey items move in a random walk, then the 234
number of encounters should be distributed approximately as a 235
3
Poisson variate, where the mean and variance are equal [28]. If236
prey are grouped into clusters composed of Gindividuals, then237
the variance of encounters per unit time will be Gtimes the238
mean number of encounters [28].239
The size of the variance in prey encounters may have an im-240
pact on the decision of an individual to engage in a hunt, partic-241
ularly if potential hunters are attempting to mitigate risk [31].242
However, as long as the tendency of prey items to form into243
groups is independent of the search mode used by the hunter, it244
will have no influence on the preference for one hunting strat-245
egy over the other, even when risk-sensitive models are used to246
analyze the choice of strategies.247
To see why, we adopt the Z-score model from Stephens and248
Charnov [31]. In this model, the strategy that will be preferred249
by an actor is the strategy which minimizes the probability den-250
sity lying below the minimum resources needed for survival,251
Rmin, which is strictly positive.252
Thus, to compare the contexts in which sit-and-wait hunt-253
ing will be preferred to an active-search strategy using a risk-254
sensitive model, we write:255
Ψ(0) −Rmin
Ψ(0)G(0) >Ψ(w)−Rmin
Ψ(w)G(w)(12)
where the mean caloric returns per unit time during sit-and-wait256
hunting, Ψ(0), is given by:257
Ψ(0) =αE(Φ)|w=0
C(0) (13)
Here αis the caloric content of prey collected upon encounter,258
E(Φ)|w=0indicates the value of E(Φ) evaluated when w=0,259
and Gis a function which returns prey group size (≥1) as a260
function of the hunter’s velocity, w. If the grouping behavior261
of prey is independent of hunter velocity then G(0) =G(w),262
and Equation 12 reduces to Equation 7, showing that a hunter’s263
risk-sensitive decision to use sit-and-wait versus active-search264
is unaffected by degree of prey grouping.265
If, on the other hand, prey change their grouping behavior266
in response to the hunter’s velocity, then G(0) ,G(w), and267
Equation 12 becomes:268
G(w)
G(0) >
1−Rmin
Ψ(w)
1−Rmin
Ψ(0)
(14)
and as Ψ(w)→Ψ(0), and the caloric returns expected from269
active-search approach those from sit-and-wait, this inequality270
approaches:271
G(w)>G(0) (15)
which indicates that sit-and-wait hunting is more likely to be272
preferred when active-search hunting leads prey items to be-273
come more congregated. This effect is driven by the congre-274
gation of prey increasing the variance in encounters, leading275
active-searching hunters to face potentially longer periods with276
no prey encounters.277
2.6. The Effect of Environmental Variability (Patchiness) 278
If the environment is characterized by heterogeneity (patchi- 279
ness), then the prey encounter rate for a hunter moving through 280
such an environment (characterized by Ndistinct patches) is 281
given by: 282
E(Φ)|w>0=
N
X
i=1
2Di(wi)ξi(wi)RiTiq(w2
i+u2
i) (16)
where Tiis the time spent in patch i, and all other functions or 283
parameters indexed by iare patch specific values. 284
The prey encounter rate for a hunter using the sit-and-wait 285
strategy in a given patch iover the same interval of time, how- 286
ever, is: 287
E(Φ)|w=0=2Di(0)ξi(0)RiuiT(17)
where the model parameters for a single patch determine the 288
encounter rate. 289
If the hunter moves across patches, he or she can average 290
over the heterogeneity in patch quality within a hunt, while the 291
sit-and-wait hunter is limited to a single patch of a given qual- 292
ity. If the sit-and-wait hunter has no special knowledge of patch 293
quality, then on average, over many hunts, the comparison of 294
sit-and-wait and active-search under heterogeneity reduces to 295
the model under homogeneity. However, if hunters do have 296
specialized knowledge of patch quality and if any of the model 297
parameters have diverging values across patches, then the ef- 298
fects of patch selection may be significant, and even confound 299
the applicability of this model to empirical data. 300
Empirically, hunters are likely to have such special knowl- 301
edge of patch quality. Sit-and-wait hunters often position them- 302
selves in optimal patches, for instance, by hiding in watering 303
holes (crocodiles [32]), or waiting near forest trails or passes 304
(puma [23]). Human foragers, especially, use patch constraints 305
to increase the domain of attraction for sit-and-wait hunting, for 306
instance, by creating man-made walls and cairn lines that funnel 307
prey items into a desired location [33]. We leave the specific so- 308
lution of sit-and-wait versus active-search in a patchy resource 309
environment to another analysis. 310
2.7. Numerical and Empirical Analyses 311
To characterize the functions describing when sit-and-wait 312
hunting will be preferred to active-search hunting, we use a 313
combination of numerical and empirical analysis. We focus on 314
illustrating the behavior of the model using numerical analy- 315
sis and then present a basic empirical test of the model’s pre- 316
dictions using examples from the hunter-gatherer literature on 317
the introduction of transportation technology to foraging pop- 318
ulations. We leave a more complete comparative analysis of 319
hunter-gather search mode for the future. 320
2.7.1. Numerical Analysis - Optimal Search Velocity 321
We assume that the functions ˆ
D(w)ˆ
ξ(w) and ˆ
C(w) take the 322
forms shown in Figure 3(a). We then plot expected return rates 323
as a function of uand w, Figure 3(b). In Figure 3(c), we plot 324
4
slices through the contour map presented in Figure 3(b), in or-325
der to provide a higher resolution image of the model behavior326
at small values of uand w.327
From Figure 3(b), we note that: 1) above a certain prey ve-328
locity (u=3.3, given our specific numerical assumptions in329
this example), it always benefits the forager to reduce velocity330
to zero, 2) below this same threshold, there is an intermediate331
optimal velocity which declines as prey velocity increases, 3)332
even if foragers can move faster than their prey, they may ben-333
efit by sitting-and-waiting rather than actively-searching if they334
are hunting fast-moving prey, and 4) even if the foragers are bi-335
ologically limited to move at slower velocities than their prey,336
there exist locations in the parameter space where active-search337
is favored, Figure 3(c). See Supplementary Materials for Sim-338
ulation Code.339
[Figure 3 about here.]340
2.7.2. Numerical Analysis - Search Mode Changes341
We reproduce a list of METs for various activities published342
by [30] in Table 2, and use these data to calculate the values of343
uat which sit-and-wait hunting would be preferred to active-344
search hunting for various values of ˆ
Dˆ
ξ.345
[Table 2 about here.]346
Figure 4displays an indifference curve for a hunter selecting347
between sit-and-wait (at an energy cost of 1 MET) and active-348
search using various forms of locomotion, as ˆ
Dˆ
ξchanges from349
1→10.350
[Figure 4 about here.]351
Figures 4(a) and 4(b) illustrate how the introduction of trans-352
portation technology such as all-terrain vehicles, snowmobiles,353
and motor-boats might alter the behavior of human foragers.354
Our model would predict foragers to change hunting tactics355
from sit-and-wait to active-search using vehicles for all com-356
binations of prey rate of movement and ˆ
Dˆ
ξratios lying above357
the indifference curve for the previously optimal hunting strat-358
egy and under the indifference curve for vehicle use.359
2.7.3. Bison Hunting and the Introduction of Horses to the360
Great Plains361
Until approximately 2000 years ago, the pedestrian Native362
American hunters of the Great Plains made “exclusive use of363
natural traps” [34, pg. 32], in their hunts for bison, a major364
part of their diet. According to Bamforth, around 2000 years365
ago this tactic intensified, with hunters driving herds which ap-366
peared in topographically-suited valleys into cairn-lined lanes367
leading to visually hidden escarpments, below which the inca-368
pacitated or dead animals were butchered and processed. In our369
terms, these are sit-and-wait tactics, realized through natural370
and, later, human-modified settings constituting traps.371
A dramatic change occurred with the arrival of the horse late372
in prehistory. The frequency of communal drives, jumps, and373
pounds diminished, as the increased velocity and mobility of374
horses allowed mounted hunters, solitary or in small groups, to375
locate bison over a wide range and quickly overtake and har- 376
vest them with the bow or firearms [35]. Among the Hidatsa, a 377
Native American group living along the Missouri River and its 378
tributaries in present-day North Dakota, this shift from sit-and- 379
wait to active-search on horseback is described by Hanson [36,380
pg. 97-101] in terms we would predict: 381
Buffalo hunting on horseback had several ad- 382
vantages over pedestrian methods such as the drive: 383
the mobility and striking speed afforded by mounted 384
hunting allowed groups of hunters to increase their 385
search radius, to report herd locations more quickly 386
to the main camp, and to pursue and surround bison 387
swiftly and efficiently... [Buffalo runners, or horses]... 388
trained for maneuvering in and around buffalo herds, 389
allowing a rider to shoot arrows into an animal at 390
point blank range and yet veer away from the poten- 391
tial danger of a wounded buffalo at precisely the right 392
time... were highly prized and, except in the cases of 393
emergency, were not used for other tasks. 394
2.7.4. Arctic Big Game Hunting and the Rise of the Snowmo- 395
bile 396
Prior to the introduction of the snowmobile and rifle, ar- 397
chaeological data [33,37,38] and oral histories [39,40] indi- 398
cate that Arctic hunters relied heavily on ambush hunting from 399
blinds to capture large ungulates like caribou, muskoxen, and 400
moose. The introduction of the snowmobile, however, dramat- 401
ically changed hunting strategies, as long trips to hunting sites 402
were replaced by fast-paced day trips on snowmobiles [41]. 403
Inuit hunters could not usually overtake moving caribou with 404
dog teams, but the introduction of the snowmobile made such 405
active-search possible [42]. The primary hunting strategy of 406
waiting in sheltered blinds, was replaced almost immediately 407
by active-search and chase on snowmobile, followed by firing 408
on the animals after they reached exhaustion [43]. 409
A similar change is noted in the hunting of Tibetan antelope 410
in the western Chang Tang Nature Reserve, where traditional 411
sit-and-wait strategies have been replaced with active-search 412
using motorcycles and guns [44]. 413
It is interesting to note that before the introduction of rapid 414
transportation technology, sit-and-wait hunting (in blinds) was 415
often the preferred hunting strategy for antelope, deer, and other 416
ungulates, even though intuition would suggest that such ani- 417
mals are typically fast-moving prey items. They are fast prey 418
to be sure—even if their average foraging velocity is lower 419
than their sprinting speed [45,46]—but equally important, they 420
are cursorial ungulates, with highly developed ability to detect 421
moving predators by sound or sight in the open, and typically 422
fairly flat country that they inhabit [47]. As such, the model 423
would suggest that the ˆ
Dˆ
ξratio may empirically become unfa- 424
vorable rapidly if the hunter’s velocity is non-zero. 425
To gain a more rooted understanding of the relationship be- 426
tween our model’s implications and empirical data on hunting 427
strategies and prey velocity, we plot published average rate of 428
movement estimates for various prey species on an indifference 429
curve for hiking at 6 Km/hr on a 5 percent slope carrying 20 430
5
Kg (Figure 5). The intersection of the indifference curves and431
the lines describing prey rate of movement are indicative of the432
ˆ
Dˆ
ξratio above which sit-and-wait hunting would be preferred,433
holding constant prey rate of movement and other model pa-434
rameters.435
[Figure 5 about here.]436
Table 3displays some empirical average rate of movement437
estimates for various prey items corresponding to Figure 5.438
[Table 3 about here.]439
3. Conclusions440
We provide a behavioral ecological model of the choice be-441
tween active-search and sit-and-wait hunting strategies that is442
derived from the biology and ecology of predator and prey; we443
summarize our findings in Table 4. Under the assumptions of444
the model, the choice between sit-and-wait and active-search445
hunting depends on the average rate of movement of the prey446
and at least three aspects of the hunter’s velocity: 1) C(w)
C(0) , the447
ratio of the hunter’s energy expenditure moving at velocity w448
to the energy expenditure at rest, 2) D(0)
D(w), the ratio of local prey449
density when the hunter is at rest to the local density when the450
hunter is moving at velocity w, and 3) ξ(0)
ξ(w), the ratio of the abil-451
ity of the hunter to detect prey when at rest to the ability of the452
hunter to detect prey when moving at velocity w.453
[Table 4 about here.]454
Through numerical methods, we show that decreasing the en-455
ergetic costs associated with elevated velocity should expand456
the area of the state space favoring active-search hunting. We457
use case studies of the effect of the introduction of horses into458
the Great Plains and the introduction the snowmobile into Arc-459
tic hunting communities to test if hunting strategies change as460
the metabolic costs of active-search decrease. In these case461
studies, we find evidence in support of our model.462
However, more wide-ranging empirical work will be needed463
to evaluate the extent to which this model is predictive of hunt-464
ing behavior cross-culturally in humans. Such work should in-465
vestigate the type of search mode used by hunters of a given466
cultural group as a function of the average velocity and temper-467
ament of each of the prey species commonly taken in that cul-468
tural group. Additionally, a wider-ranging cross-cultural anal-469
ysis of the effect of a decrease in the metabolic cost of active-470
search hunting—due to technological intensification—on hunt-471
ing style would help to empirically evaluate our model’s pre-472
dictions in human foragers. Further, we envision an assessment473
of the model through comparative analysis of the use of traps,474
snares, weirs and similar technologies treated as surrogate sit-475
and-wait predators by human and some non-human hunters.476
3.1. The Broader Context477
In an early comprehensive review of foraging theory,478
Schoener [19] distinguished between models developed for sit-479
and-wait foragers and those developed for foragers incurring480
the costs of active-search. Although he identified several math- 481
ematical representations of active-search, including Skellam’s 482
[28], he did not use these to compare the conditions favoring 483
one tactic over the other. Optimal search mode remains to 484
this day insufficiently studied in the behavioral ecology liter- 485
ature [24]. As context for our results, we summarize literature 486
highlights from optimization approaches, and then note the rel- 487
evance of literature on L´
evy walks and game theory models of 488
predator-prey tactics. 489
In a search mode model similar to ours, but based on search 490
in a 3-dimensional space, Gerritsen and Strickler [29] find that 491
active-searching predators are predicted to prey upon slow- 492
moving animals, and ambush predators are predicted to prey 493
upon fast-moving animals. Our major qualitative findings are, 494
in this way, quite similar. It is informative, however, to contrast 495
our encounter models; in the 2-dimensional foraging environ- 496
ment, we have number of encounters in fixed period of time 497
given by: 498
E(Φ)2-dimensions =2RT D(w)ξ(w)p(w2+u2) (18)
Assuming our functional forms for D(w) and ξ(w) can be di- 499
rectly translated to the 3-dimensional case in which density is 500
prey items per unit volume, in a 3-dimensional foraging envi- 501
ronment we have: 502
E(Φ)3-dimensions =πR2T D(w)ξ(w)
6
(w+u)3− |u−w|3
wu (19)
which reduces to: 503
E(Φ)3-dimensions =
πR2T D(w)ξ(w)(3u2+w2)
3u,if u>w,
πR2T D(w)ξ(w)(u2+3w2)
3w,if u<w.(20)
Although the search mode model is not generally solvable for 504
Equation 19, since we have division by zero if w=0, under 505
the assumption that u>w, we can solve the model for the prey 506
velocity uabove which the hunter should elect to sit-and-wait: 507
u2>w2
3( ˆ
Cˆ
Dˆ
ξ−1) (21)
contrasted with the 2-dimensional case: 508
u2>w2
ˆ
C2ˆ
D2ˆ
ξ2−1(22)
Now, comparison of the right-hand sides (RHS) of Equations 509
21 and 22, shows that the RHS of Equation 21 will be bigger 510
than RHS of Equation 22 whenever: 511
ˆ
C2ˆ
D2ˆ
ξ2>3ˆ
Cˆ
Dˆ
ξ−2 (23)
Since ˆ
C,ˆ
D,ˆ
ξ > 1, the left-hand side of Equation 23, will grow 512
faster than the right-hand side as a function of increasing in- 513
puts, meaning that as locomotion becomes more energetically 514
costly ( ˆ
Cincreases) and depresses the effective local density of 515
prey items ( ˆ
Dand/or ˆ
ξincreases), the scope for active-search 516
declines for predators in both 2- and 3-dimensional foraging 517
environments, but the effect will be stronger in 2-dimensional 518
6
environments. As such, ceterus paribus, we might expect to519
see the sit-and-wait strategy emerge more frequently in land-520
dwelling animals foraging primarily in two spatial dimensions,521
than in aquatic, aerial, or other animals foraging in three spatial522
dimensions.523
In an analytical model considering prey density but not524
movement, Norberg [48] showed that a time-minimizing for-525
ager such as an endotherm seeking to meet a set requirement526
in the least amount of time will optimize by shifting to less527
costly, perhaps lower velocity methods as prey density de-528
creases, predator size increases, or as prey grow smaller. Field529
studies have also provided evidence of the importance of prey530
velocity on search mode. Huey and Pianka [20] hypothesized531
that sit-and-wait hunting will be favored when high ranked prey532
choices are mobile, and active-search will be favored when they533
are not. Empirically, they find that the rate of movement of prey534
species is the principle driver of search mode between lizard535
species that specialize in active-search and prey on stationary536
prey items, like termite mounds, and those that specialize in sit-537
and-wait hunting. The same pattern was true of the foraging538
style of snakes, where sit-and-wait snakes specialize in hunt-539
ing active-searching lizards, and active-searching snakes prey540
predominately on lizards that rely on sit-and-wait hunting [20].541
In a model assessing the success of foragers with access to re-542
source sites classified as either Good or Poor with switch prob-543
abilistically occuring from one state to another, Janetos [21]544
found that when active foraging, the rule—move if you experi-545
ence a poor site that day—is favored when relocation is inex-546
pensive, the difference between Good and Poor sites is elevated,547
and sites switch between Good and Poor infrequently. This548
model provides a useful heuristic for describing how a preda-549
tor might effectively select between patches in a heterogeneous550
environment. Our model suggests that the choice between sit-551
and-wait and active-search itself can be patch-specific. As such,552
the development of models that integrate patch selection and553
search mode selection may be of theoretical interest, especially554
in understanding the behavior of predators whose search mode555
is variable in space or time.556
Models have also shown the importance of directionality in557
movement. Using simulation, Scharf et al. [25] find that active-558
search is advantageous when both predator and prey move ran-559
domly, but that the relative advantage of active-search dimin-560
ishes as predator and prey adopt more directional movement.561
So long as prey movement is random, the predator gains advan-562
tage by a modest degree of directional search. If the predator563
is slower than its prey, active-search has little advantage over564
ambush; when the predator’s velocity can exceed that of the565
prey, there is a rapidly increasing advantage to active-search.566
Conversely, ambush becomes more attractive as prey velocity567
increases. As in our model, these results are largely unaffected568
by prey density; however, this model did not assign metabolic569
costs to active-search, so inference about the caloric return rate570
as a function of search velocity is not possible.571
In a broad ranging attempt to determine the relative advan-572
tage of the two foraging modes, Higginson and Ruxton [24]573
model search behavior in a patchy environment as a function of574
prey abundance, size, and distribution over patches, as well as575
clumping of patches, for foragers with the goal of maximizing 576
encounters, avoiding predation, and avoiding depletion of food 577
reserves. Their model does not explicitly consider prey mobil- 578
ity, although the effect of mobility is mimicked by a parameter 579
representing patch transience—assuming resource renewal, de- 580
pletion of a patch requires that prey consumed in one location 581
are offset by prey appearing elsewhere. Patch transience in- 582
creases the attractiveness of sit-and-wait foraging because prey 583
are more likely to “show up” at the site of a sit-and-wait forager, 584
who thereby avoids the energy costs and potential exposure to 585
predation associated with active-search. Increasing search ve- 586
locity (without accounting for its metabolic costs) increases the 587
relative advantage of active-search; increasing prey density fa- 588
vors the sit-and-wait mode. Patch “clumpiness” decreases the 589
advantages of active-search in their state-independent scenar- 590
ios, and increases it in risk-sensitive scenarios when the state of 591
food reserves is taken into account. As in our models, the Hig- 592
ginson and Ruxton [24] model predicts that state-independent 593
choice of foraging mode may change if avoidance of starvation 594
is the immediate goal of the forager. 595
Our findings generally concur with the hypothesis of Huey 596
and Pianka that increasing prey velocity shifts the advantage 597
toward sit-and-wait foraging [20]. However, some of our spe- 598
cific results differ from those of other studies. For instance, 599
Scharf et al. [25, pp. 355] find that “the optimal strategy for 600
predators that cannot move as fast as their prey is the ambush 601
one.” While our results are generally consistent with this pat- 602
tern, insofar as our model does suggest that predators of high 603
velocity prey can often off-load the energetic cost of movement 604
onto their prey, we note a few nuances. We find that at low 605
prey velocities, predators can increase their payoffs by moving, 606
rather than sitting-and-waiting (Figure 3(c)). We also find that 607
for high prey velocities, predators can often increase their pay- 608
offs by sitting-and-waiting, even if they are capable of moving 609
at higher velocities than their prey. And, unlike the results of 610
Higginson and Ruxton [24], who find in their state-independent 611
analyses that increased forager search rate always favors active- 612
search, we show that increasing costs and decreasing search ef- 613
fectiveness at higher velocities will expand the parameter com- 614
binations over which the sit-and-wait mode is advantageous. 615
However, when considering risk sensitivity, our model results 616
match those of Higginson and Ruxton [24] in predicting that 617
prey clumping increases the risk of long intervals without en- 618
countering prey. 619
While there is empirical literature suggesting directionality 620
to prey and/or predator movements (summary in Scharf et al. 621
[25]), evidence suggests that a wide range of searching preda- 622
tors adopt L´
evy or L´
evy-like walks [49,50], including humans 623
[51], while others conform to Brownian movement (examples 624
in [52]). L´
evy walks, are characterized by a distribution of 625
step lengths with randomized turns equally likely in any com- 626
pass direction. At specific parameter settings, L´
evy and Brow- 627
nian walks appear to differ little in outcome, although L´
evy 628
walks can display more heavy-tailed step lengths, and may thus 629
lead to higher efficiency in exploring an environment over a 630
greater range of conditions, especially when prey encounters 631
are rare. This advantage diminishes and may reverse as prey be- 632
7
come more common [49]. L´
evy walks also produce fewer long633
“famine” intervals of no encounters [49]. Sorting out the ap-634
plicability of these various models of movement is difficult be-635
cause detailed quantitative geographic and behavioral descrip-636
tions of search and pursuit for predator-prey systems remain637
“exceedingly rare” [53, pp. 1].638
Finally, insights about search mode may be gained by con-639
sidering tactical responses of predator and prey to each other,640
modeled with a game-theoretic approach [54]. Allowing for641
strategic interactions can change the manner in which predators642
seek prey, and how evasive prey inhabit and move among feed-643
ing patches [55]. These approaches typically are called search,644
ambush, or pursuit-evasion games [56]; a variety of them have645
been proposed and solved [22,57,58]. For instance, consider646
a predator choosing between ambush or active-search that can,647
when in ambush mode, detect prey only when the prey change648
location. Prey in turn can elect to hide in one spot or move649
periodically. Because prey will respond to an ambush preda-650
tor by remaining stationary, they become more susceptible to651
active-search, with the result that the equilibrium predator tac-652
tic is a mix of ambush and active-searching [56]. Likewise,653
a predator that begins a foraging bout with active-search may654
switch to ambush as it narrows the area in which lingering, un-655
exploited prey are isolated. As in our model, Alpern et al. [56]656
allow for the possibility that prey detection declines with in-657
creased predator velocity in their “ambush search” game. In658
a second model that consists of search or hide-and-seek cou-659
pled to pursuit-evasion, Gal and Casas [53] show that prey will660
respond to a predator that visits each possible hiding location661
for a probabilistic capture below a threshold number of times662
by randomizing their positions. However, if the predator re-663
visits particular sites above that threshold, then the prey elect664
the fixed position with the greatest probability of successfully665
evading a pursuit once they are spotted.666
In the present study, we focus on modeling the choice of opti-667
mal search strategy using a model that is grounded on the loco-668
motion dynamics and biological characteristics of predator and669
prey; this places our model in the realm of optimization rather670
than game theory, and allows for an analytical approach in place671
of the simulation models commonly adopted in the analysis672
of structured environmental settings. Except in the localized673
and ephemeral effect that predator velocity has on prey skittish-674
ness and/or detectability once within observational range, we675
assume constant prey density based on instantaneous renewal676
and no effect of hunting on prey distribution [cf. 59]. However,677
under the assumptions of our model, any change in prey density678
for densities >0 will not affect the choice of search mode. We679
model the metabolic cost of active foraging, but do not assign680
fitness consequences arising from hazards such as predation or681
exposure. We also solve the model under the assumption that682
the forager seeks to optimize return rate, but optimization based683
on other constraints is possible as well. For example, the for-684
ager may only have a fixed number of calories that can be ex-685
pended before survival is threatened, and optimal search under686
such a constraint may be different than optimal search under687
our assumptions. Finally, while we examine prey clumping, we688
focus on a lone forager or contiguous foraging group, and thus689
neglect the advantages of information sharing among individu- 690
als searching apart from one another [52]. 691
Other literature on foraging strategy has investigated the role 692
of predator hunger or attributes of prey species other than av- 693
erage rate of movement on hunting strategy [60,61,62,63]; 694
our model, reflecting its assumptions, suggests that prey veloc- 695
ity more than any other characteristic has a significant causal 696
role in choice of search strategy by an associated predator. Fu- 697
ture empirical research on hunting style across a wider range of 698
animal species that contrasts the effects of average prey rate of 699
movement with other predictors will be of importance in the 700
empirical evaluation of our model and others. Additionally, 701
future theoretical work on optimization of our model under a 702
constrained predator energy budget may help to address how 703
predator hunger might alter search mode. 704
The modeling and analysis of algorithms that guide opti- 705
mization of search mode is relevant not only to the food quest 706
of hunter-gatherers or, more generally, predators seeking prey 707
[22], but also to practical matters of attempting to locate crim- 708
inals [64], capture kidnappers [58], and encountering informa- 709
tion in libraries [65] or the internet [66]. 710
4. Acknowledgments 711
We thank the Human Behavioral Ecology Lab at UC Davis 712
for helpful comments on an early draft of this paper, and Sheryl 713
Gerety for help editing the paper. We thank the editors and two 714
anonymous reviewers for constructive criticism that improved 715
the quality of the paper and its connection to previous work. 716
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10
List of Figures 913
1 A Skellam-Inspired Representation of Forager and Prey Movement. Frame (a) Forager and Prey Movement. Ar- 914
bitrary paths over an interval of time for two prey items (dashed lines) and a forager (solid line). The forager is 915
endowed with a visual detection field demarcated by a contour (lightly shaded circle) and a frame of reference 916
(Front, Back, Right, and Left). Each prey item can be described as having a rate of movement relative to the hunter 917
with magnitude, v, and direction, θ, as indicated at the end of the path for prey 1. Frame (b) Prey and the Detection 918
Zone of a Hunter. The forager’s visual detection field is the lightly shaded circle; the dark shaded region represents 919
an area of H(θ)v(t)∂t. Prey items move from their starting positions, ♦, to their ending positions, , over the interval 920
of time ∂t. Prey items can cross into the hunter’s visual range between time tand t+∂tif and only if they lie in the 921
dark shaded region at time t. These illustrations are based on sketches in Skellam [28]. ............... 12 922
2 Effect of Caloric Costs of Active-Search on Hunting Tactic. The area to the lower-left of an indifference curve 923
(with its value of ˆ
Cgiven in METs) represents values of prey rate of movement, u, and local effective density ( ˆ
Dˆ
ξ;924
see definitions in text), where active-search is favored. The area to the upper-right of a curve for a given form 925
of locomotion indicates parameter states at which the sitting-and-waiting tactic (at an energy cost of 1 MET) is 926
favored. As slope and load of the forager moving at a constant velocity increase, his or her metabolic costs (METs) 927
grow, and the parameter space favoring a sit-and-wait tactic expands. Conversely, the lowering of metabolic costs 928
favors active-search. As ˆ
Dˆ
ξincreases, the effective local density of prey items for an active-searching hunter is 1
ˆ
Dˆ
ξ929
times the effective local density for a hunter searching at velocity 0. ......................... 13 930
3 Effect of Relative Predator-Prey Velocity on Search Mode and Optimal Velocity of Active-Search. Frame (a) Ener- 931
getic Costs of Movement and Local Prey Density as Functions of Predator Velocity. We make simple assumptions 932
concerning the functions ˆ
Dˆ
ξ(scaled exponential decline) and ˆ
C(linear growth). Frame (b) Optimal Search Veloc- 933
ity. The gray contours plot the encounter rate experienced by a forager as a function of forager and prey velocities 934
under the functional forms assumed in Frame (a). The solid black lines represent the optimal search velocity; there 935
is a discontinuity at u=3.3, where the hunter is indifferent to active-search at the optimal velocity of w=11.5 and 936
sitting-and-waiting at w=0. The diagonal dashed line represents equal predator and prey velocities; to the upper- 937
left of this line are values of prey velocity greater than forager velocity, and vice versa for the lower-right. Frame 938
(c) Optimal Search Velocity at Small Values of wand u. We plot three slices from the contour plot in Frame(b) for 939
small values of u. The black curves are the return rate contours for u=1, u=2, and u=3, as a function of w. The 940
dashed vertical lines at w=1, w=2, and w=3 intersect their respective ucontours, such that all value of uto the 941
left of the vertical lines have u>w. The dashed horizontal lines show the return rate for u=w. For u=3 we see 942
that the return rate for sitting-and-waiting exceeds the return rate for active-search, under the constraint that u>w.943
However, for slower prey, we see that return rate can frequently be maximized by increasing foraging velocity under 944
the constraint u>w................................................... 14 945
4 Energetic Costs and Transportation Technology. Frame (a) Effect of Search Velocity and Caloric Costs on Hunting 946
Tactic. Increased velocity and lessened metabolic cost to the hunter expand the parameter space favoring active- 947
search. Frame (b) Impact of Mechanical Transportation Technology. Curves (i), (ii) and (iii) are repeated from (a) 948
with the parameter space now showing the consequence of using a snowmobile or motorcycle at search velocities 949
of up to 88 Km/hr (2.5 METs). The introduction of high velocity search at low caloric cost greatly expands the 950
conditions under which active-search is favored, even after accounting for significant decreases in effective local 951
prey density, ˆ
Dˆ
ξ, arising from factors such as increased prey avoidance due to noise or failure of the forager to 952
detect some prey items due to rapid search velocity. The state space represented here is meant to depict model 953
outcomes over a wide range of theoretically possible parameter conditions, without implying that all locations in 954
the state space are attainable (e.g., prey rate of movement) or desirable (forager velocity). .............. 15 955
5 Effect of Prey Rate of Movement on Hunting Tactic. Diminishing rate of movement of prey, shown here via 956
empirically-measured average foraging velocity (see examples in Table 3), favors a shift to active-search on the 957
part of a predator. Conversely, an increase in the average rate of movement of a prey species would favor a shift to 958
the sit-and-wait tactic. The intersection of the indifference curves and the lines describing prey rate of movement 959
are indicative of the ˆ
Dˆ
ξratio above which sit-and-wait hunting would be preferred, conditional on the indicated 960
locomotion method of the hunter. ........................................... 16 961
11
L
R
FB
v
θ
Prey 1
Prey 2
Forager
(a) Forager and Prey Movement
H(
θ
)
θ
v(t)δt
(b) Prey and the Detection Zone of a Hunter
Figure 1: A Skellam-Inspired Representation of Forager and Prey Movement. Frame (a) Forager and Prey Movement. Arbitrary paths over an interval of time for
two prey items (dashed lines) and a forager (solid line). The forager is endowed with a visual detection field demarcated by a contour (lightly shaded circle) and
a frame of reference (Front, Back, Right, and Left). Each prey item can be described as having a rate of movement relative to the hunter with magnitude, v, and
direction, θ, as indicated at the end of the path for prey 1. Frame (b) Prey and the Detection Zone of a Hunter. The forager’s visual detection field is the lightly
shaded circle; the dark shaded region represents an area of H(θ)v(t)∂t. Prey items move from their starting positions, ♦, to their ending positions, , over the interval
of time ∂t. Prey items can cross into the hunter’s visual range between time tand t+∂tif and only if they lie in the dark shaded region at time t. These illustrations
are based on sketches in Skellam [28].
12
Prey Rate of Movement (Km/hr)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
24 6 8 10
D
ξ
Active
Search
Sit &
Wait
Hiking, 7 Km/hr, 5 % slope, carrying 20 Kg (8 METs)
Walking, 7 Km/hr, at surface, no load (5.3 METs)
(a) Effect of Caloric Costs of Active-Search on Hunting Tactic
Figure 2: Effect of Caloric Costs of Active-Search on Hunting Tactic. The area to the lower-left of an indifference curve (with its value of ˆ
Cgiven in METs)
represents values of prey rate of movement, u, and local effective density ( ˆ
Dˆ
ξ; see definitions in text), where active-search is favored. The area to the upper-right
of a curve for a given form of locomotion indicates parameter states at which the sitting-and-waiting tactic (at an energy cost of 1 MET) is favored. As slope and
load of the forager moving at a constant velocity increase, his or her metabolic costs (METs) grow, and the parameter space favoring a sit-and-wait tactic expands.
Conversely, the lowering of metabolic costs favors active-search. As ˆ
Dˆ
ξincreases, the effective local density of prey items for an active-searching hunter is 1
ˆ
Dˆ
ξtimes
the effective local density for a hunter searching at velocity 0.
13
Effective Local Density of Prey Items
Energy Cost of Hunter Movement (METs)
Hunter Velocity (w)
1.0
0.3
0.4
0.7
0.9
0 10 20 30 40 50
2
4
6
8
0.8
0.5
0.6
(a) Energetic Costs of Movement and Local Prey Density as Functions
of Predator Velocity
Return Rate
Hunter velocity (w)
u = 1.0
u = 2.0
u = 3.0
1 2 3 4 5
1.0
1.5
2.0
3.0
2.5
0
(c) Optimal Search Velocity at Small Values of wand u
Prey Velocity (u)
Hunter Velocity (w)
20
0
5
10
15
10 20 30 40 50
1
2
3
4
5
6
8
10
14
3.5
3.295
3
2
u=3.3
w=11.5
0
(b) Optimal Search Velocity
Figure 3: Effect of Relative Predator-Prey Velocity on Search Mode and Optimal Velocity of Active-Search. Frame (a) Energetic Costs of Movement and Local Prey
Density as Functions of Predator Velocity. We make simple assumptions concerning the functions ˆ
Dˆ
ξ(scaled exponential decline) and ˆ
C(linear growth). Frame (b)
Optimal Search Velocity. The gray contours plot the encounter rate experienced by a forager as a function of forager and prey velocities under the functional forms
assumed in Frame (a). The solid black lines represent the optimal search velocity; there is a discontinuity at u=3.3, where the hunter is indifferent to active-search
at the optimal velocity of w=11.5 and sitting-and-waiting at w=0. The diagonal dashed line represents equal predator and prey velocities; to the upper-left of this
line are values of prey velocity greater than forager velocity, and vice versa for the lower-right. Frame (c) Optimal Search Velocity at Small Values of wand u. We
plot three slices from the contour plot in Frame(b) for small values of u. The black curves are the return rate contours for u=1, u=2, and u=3, as a function of
w. The dashed vertical lines at w=1, w=2, and w=3 intersect their respective ucontours, such that all value of uto the left of the vertical lines have u>w. The
dashed horizontal lines show the return rate for u=w. For u=3 we see that the return rate for sitting-and-waiting exceeds the return rate for active-search, under
the constraint that u>w. However, for slower prey, we see that return rate can frequently be maximized by increasing foraging velocity under the constraint u>w.
14
Prey Rate of Movement (Km/hr)
0.0
0.5
1.0
1.5
2.0
2 4 6 8 10
D
ξ
Active
Search
Sit &
Wait
(iii) Snowshoeing, 4 Km/hr (9.5 METs)
(ii) Hiking, 6 Km/hr, 5% slope, carrying 20 Kg (8 METs)
(i) Riding, horseback at trot,13 Km/hr (6.9 METs)
(a) Effect of Search Velocity and Caloric Costs on Hunting Tactic
Prey Rate of Movement (Km/hr)
0
5
10
15
246810
D
ξ
Active Search
Sit &
Wait
(i)
(ii) (iii)
Snowmobile or motorcycle, up to 88 Km/hr (2.5 METs)
(b) Impact of Mechanical Transportation Technology
Figure 4: Energetic Costs and Transportation Technology. Frame (a) Effect of Search Velocity and Caloric Costs on Hunting Tactic. Increased velocity and lessened
metabolic cost to the hunter expand the parameter space favoring active-search. Frame (b) Impact of Mechanical Transportation Technology. Curves (i), (ii) and
(iii) are repeated from (a) with the parameter space now showing the consequence of using a snowmobile or motorcycle at search velocities of up to 88 Km/hr (2.5
METs). The introduction of high velocity search at low caloric cost greatly expands the conditions under which active-search is favored, even after accounting for
significant decreases in effective local prey density, ˆ
Dˆ
ξ, arising from factors such as increased prey avoidance due to noise or failure of the forager to detect some
prey items due to rapid search velocity. The state space represented here is meant to depict model outcomes over a wide range of theoretically possible parameter
conditions, without implying that all locations in the state space are attainable (e.g., prey rate of movement) or desirable (forager velocity).
15
Prey Rate of Movement (Km/hr)
0.0
0.2
0.4
0.6
0.8
1.0
2 4 6 8 10
D
ξ
Red fox (Vulpes vulpes)
Pintail Duck (Anas Acuta)
White-tailed deer (Odocoileus virginianus)
Spanish Goat (Carpa hircus)
Gadwall Duck (Anas strepera)
Elk (Cervus elaphus)
Mule Deer (Odocoileus hemionus)
Sit &
Wait
Active
Search
Hiking, 6 Km/hr, 5% slope, carrying 20 kg (8 METs)
(a) Effect of Prey Rate of Movement on Hunting Tactic
Figure 5: Effect of Prey Rate of Movement on Hunting Tactic. Diminishing rate of movement of prey, shown here via empirically-measured average foraging
velocity (see examples in Table 3), favors a shift to active-search on the part of a predator. Conversely, an increase in the average rate of movement of a prey species
would favor a shift to the sit-and-wait tactic. The intersection of the indifference curves and the lines describing prey rate of movement are indicative of the ˆ
Dˆ
ξratio
above which sit-and-wait hunting would be preferred, conditional on the indicated locomotion method of the hunter.
16
List of Tables 962
1 Definitions and Limits of Parameters and Functions. .................................. 18 963
2 Velocity and Energetic Costs. The velocity (in Km/hr) and energy costs (in METs) of various activities, as well 964
as the prey velocities (u) above which sit-and wait hunting would be favored to active-search for each indicated 965
activity, under ˆ
Dˆ
ξratios of 1, 1.25, 1.5, 2, and 5. Backpacking is walking up a 5 percent slope with 20Kg of supplies. 19 966
3 Average Rate of Movement (in Km/hr) of Several Animal Species in Natural Environments. ............ 20 967
4 Summary of Model Results. .............................................. 21 968
17
Table 1: Definitions and Limits of Parameters and Functions.
Symbol Type Lower Limit Upper Limit Definition
wParameter 0 ·Hunter’s average velocity.
uParameter 0 ·Prey’s average velocity.
t· · · Time.
v(t) Function 0 ·Relative velocity of prey item to hunter at time t.
θ(t) Function · · Relative direction of prey item to hunter at time t.
H(θ) Function 0 ·Distance between tangents in Figure 1(a).
¯
DParameter 0 ·Average density of prey items in the wider environment.
D(w) Function 0 ¯
DLocal density of prey items when hunter is moving at velocity w.
Ts· · · Start time of hunt.
Te· · · End time of hunt.
TParameter 0 ·Duration of hunt.
RParameter 0 ·Visual radius of hunter.
VParameter 0 ·Average relative velocity of hunter and prey items.
ξ(w) Function 0 1 A function, decreasing with increasing w, that translates between prey
crossing into the detection zone and prey items that are actually detected.
E(P)·0·Expected number of prey items
crossing into the detection zone during a hunt.
E(Φ)·0·Expected number of prey items
encountered during a hunt.
αParameter 0 ·Caloric value of a prey item.
C(w) Function 1 ·Hunter’s energy expenditure as a function of w.
G(w) Function 1 ·Group size of prey items as a function of w.
Ψ(w) Function 0 ·Expected caloric returns from hunting as a function of w.
Rmin Parameter 0 ·Minimum caloric input per unit time needed to survive.
18
Table 2: Velocity and Energetic Costs. The velocity (in Km/hr) and energy costs (in METs) of various activities, as well as the prey velocities (u) above which
sit-and wait hunting would be favored to active-search for each indicated activity, under ˆ
Dˆ
ξratios of 1, 1.25, 1.5, 2, and 5. Backpacking is walking up a 5 percent
slope with 20Kg of supplies.
Activity Km/hr METs u ˆ
D=1uˆ
D=1.25 uˆ
D=1.5uˆ
D=2uˆ
D=5
Resting 0 1 - - - - -
Walking 3 1.8 2 1.5 1.2 0.9 0.3
Walking 5 3.2 1.6 1.3 1.1 0.8 0.3
Walking 7 5.3 1.3 1.1 0.9 0.7 0.3
Jogging 9 8.8 1 0.8 0.7 0.5 0.2
Jogging 11 11.2 1 0.8 0.7 0.5 0.2
Running 13 12.9 1 0.8 0.7 0.5 0.2
Running 15 14.6 1 0.8 0.7 0.5 0.2
Backpacking 6 8 0.8 0.6 0.5 0.4 0.2
Backpacking 7 9.6 0.8 0.6 0.5 0.4 0.2
Backpacking 8 11.6 0.7 0.6 0.5 0.3 0.1
Backpacking 10 13.1 0.7 0.6 0.5 0.4 0.1
Backpacking 11 15.5 0.7 0.6 0.5 0.4 0.1
Rowing 4 5.5 0.7 0.6 0.5 0.4 0.1
Rowing 8 10.3 0.8 0.6 0.5 0.4 0.2
Rowing 12 13.5 0.9 0.7 0.6 0.4 0.2
Rowing 16 16.4 1 0.8 0.7 0.5 0.2
Rowing 20 19.1 1 0.8 0.7 0.5 0.2
Snowshoeing 4 9.5 0.4 0.3 0.3 0.2 0.1
Horsebacking-Walk 6 3.2 2.1 1.7 1.4 1 0.4
Horsebacking-Trot 13 6.9 1.9 1.5 1.3 0.9 0.4
Horsebacking-Gallop 44 8.6 5.2 4.1 3.4 2.6 1
Snowmobile/Cart 88 2.5 38.4 29.7 24.3 18 7.1
19
Table 3: Average Rate of Movement (in Km/hr) of Several Animal Species in Natural Environments.
Species Name Location Average Velocity (Km/hr) Source
Cervus elaphus Elk Oregon, USA 0.162 [45]
Odocoileus hemionus Mule Deer Oregon, USA 0.126 [45]
Odocoileus
virginianus White-tailed Deer Texas, US 0.36 [67]
Carpa hircus Spanish Goats (Domestic) Texas, US 0.306 [67]
Vulpes vulpes Red Fox Bristol, UK 0.438 [68]
Anas strepera Gadwall Duck North Carolina, US 0.21 [69]
Anas acuta Pintail Duck North Carolina, US 0.57 [69]
20
Table 4: Summary of Model Results.
Variable Direction of
Change
Has this
Consequence
Notes
Prey’s Average
Velocity, u↓+Active-Search As average prey velocity declines, the
parameter space favoring active-search
expands.
Prey Skittishness
as a Function of w,
ˆ
D=D(0)
D(w)
↑ − Active-Search As the hunter moves more rapidly and
is thus easier for prey to detect, local
density of prey in the neighborhood of
the hunter declines, decreasing the pa-
rameter space in which active-search is
favored.
Prey Detectability
as a Function of w,
ˆ
ξ=ξ(0)
ξ(w)
↓ − Active-Search As the hunter moves more rapidly and
is thus more likely to overlook prey
items, the effective local density of prey
in the neighborhood of the hunter is di-
minished, again decreasing the param-
eter space in which active-search is fa-
vored.
Hunter’s Energy
Expenditure as a
Function of w,
ˆ
C=C(w)
C(0)
↑ − Active-Search As hunter locomotion becomes exceed-
ingly costly in terms of energy expen-
diture as a function of w, the parameter
space favoring active-search shrinks.
Group Size of Prey
Items as a Function
of w,G(w)
↑+Risk of longer
periods of no
encounters with
active-search
This result only occurs if clumping is
a consequences of searching actively;
otherwise, clumping does not affect
choice of hunting strategy.
Caloric Value of a
Prey Item, α
The caloric value of prey items is un-
related to the optimal foraging strategy
under the assumptions of this model.
An increase in αincreases caloric re-
turn rate under both search modes.
Hunter’s Visual
Radius, R
The visual radius of the hunter is un-
related to the optimal foraging strategy
under the assumptions of this model.
An increase in Rincreases the en-
counter rate under both search modes.
Hunt Duration, TThe duration of the hunt is unrelated
to the optimal foraging strategy under
the assumptions of this model. An in-
crease in Tincreases total encounters
under both search modes.
Prey Density, ¯
DThe overall density of prey is unrelated
to the optimal foraging strategy under
the assumptions of this model. An in-
crease in ¯
Dincreases the encounter rate
under both search modes.
+Active-Search :=expands parameter space in which forager will elect active-search;
−Active-Search :=expands parameter space in which forager will elect sit-and-wait.
21
