R

— = [email protected] (R) - dn) - fp(N)P, d-P = Pf N) - dp), where r is the plant's intrinsic growth rate, K is the plant's carrying capacity, fN(R) is the functional response of herbivores on plants, p is the conversion rate of plants consumed by herbivores into new herbivores, dN is the herbivore's density-independent mortality rate, fp (N) is the predator's functional response on herbivores, Y is the conversion rate of herbivores consumed by predators into new predators, and dp is the predator's density-independent mortality rate. The variables R, N, and P give the population sizes of plants, herbivores, and predators, respectively.

Type I and Type II Functional Responses with No Fear

The simplest isoclines emerge if both herbivore and predator have type I (linear) functional responses (fN(R) = aNR andfp(N) = apN). In this case, the isoclines are all linear planes (fig. 13.3). Doubling the number of plants doubles the herbivore's harvest rate, and doubling the number of herbivores doubles the predator's harvest rate. This model produces the trophic cascades seen in models of exploitation ecosystems exhibiting the paradox of enrichment (Oksanen and Oksanen 1999).

Figure 13.3. The isoclines of plants, herbivores, and predators in a system where herbivores and predators have type I functional responses. All isoclines are planes. The herbivore's isocline (in the foreground) rises along the plant and predator axes—increasing the standing crop of plants increases the number of predators that the herbivores can support and still maintain zero population growth. The plant's isocline declines linearly along the herbivore and plant axes. As the number of herbivores increases, the system can sustain a smaller standing crop of plants while maintaining zero population growth. The predator's isocline is a plane that is independent of predator and plant abundances. It emerges from a subsistence abundance of herbivores. Above a threshold abundance of herbivores, the predators have a positive population growth rate, while below this threshold they experience a negative population growth rate. The star indicates the equilibrium point (R* = 20, N* = 8, P* = 1). The following parametervalues were used: r = 1, K = 100, on = ap = dN = 0 = y = 0.1, dr = 0.08.

Figure 13.3. The isoclines of plants, herbivores, and predators in a system where herbivores and predators have type I functional responses. All isoclines are planes. The herbivore's isocline (in the foreground) rises along the plant and predator axes—increasing the standing crop of plants increases the number of predators that the herbivores can support and still maintain zero population growth. The plant's isocline declines linearly along the herbivore and plant axes. As the number of herbivores increases, the system can sustain a smaller standing crop of plants while maintaining zero population growth. The predator's isocline is a plane that is independent of predator and plant abundances. It emerges from a subsistence abundance of herbivores. Above a threshold abundance of herbivores, the predators have a positive population growth rate, while below this threshold they experience a negative population growth rate. The star indicates the equilibrium point (R* = 20, N* = 8, P* = 1). The following parametervalues were used: r = 1, K = 100, on = ap = dN = 0 = y = 0.1, dr = 0.08.

Nonlinear isoclines introduce considerable complexity. Isoclines become nonlinear when herbivores must spend time handling the plants they consume (An) and predators must spend time handling the herbivores they consume (hp). The handling time that predators must devote to consuming a captured prey creates a type II (decelerating) functional response. A type II functional response offers the prey safety in numbers and can produce a hump-shaped prey isocline. Figure 13.4 shows such isoclines (surfaces) when both herbivore and predator have type II functional responses to their respective foods. The prey achieve safety in numbers because predators cannot attack a new victim while they are handling a current victim. This safety in numbers weakens the stability of equilibrium points, tends to make the predator-prey dynamics less stable, and increases the likelihood of an unstable equilibrium with nonequi-librial dynamics (as in lynx-hare cycles and weasel-vole cycles). But, even with its nonlinear isoclines and its more complex suite ofdynamic outcomes, the predator-prey interaction is still completely N-driven. Furthermore, the system will continue to exhibit classic trophic cascades (assuming equilibrial population dynamics) and the paradox of enrichment (Rosenzweig 1971).

Figure 13.4. The isoclines of plants, herbivores, and predators in a system where the herbivores and predators have a type II functional response. The predator isocline remains the same as in figure 13.3 (a vertical plane that is independent of plant abundance or predator abundance). The herbivore isocline rises at a decelerating rate along the plant, herbivore, and predatoraxes. With either more plants or more herbivores (safety in numbers), the herbivores can support more predators. The plant isocline takes on a hump shape in the plane defined by the herbivore and plant axes. At low plant abundances, safety in numbers from herbivores dominates the slope (rising portion), whereas at high plant abundances, competition among plants dominates (negatively sloped portion). The star indicates the equilibrium point: R* = 83.2, N* = 8, P* = 0.748. The following parametervalues were used: r = 1, K = 100, aN = aP = 0.166, hN = hp = 0.5, dN = 3 = Y = 0.1, dp = 0.08.

Figure 13.4. The isoclines of plants, herbivores, and predators in a system where the herbivores and predators have a type II functional response. The predator isocline remains the same as in figure 13.3 (a vertical plane that is independent of plant abundance or predator abundance). The herbivore isocline rises at a decelerating rate along the plant, herbivore, and predatoraxes. With either more plants or more herbivores (safety in numbers), the herbivores can support more predators. The plant isocline takes on a hump shape in the plane defined by the herbivore and plant axes. At low plant abundances, safety in numbers from herbivores dominates the slope (rising portion), whereas at high plant abundances, competition among plants dominates (negatively sloped portion). The star indicates the equilibrium point: R* = 83.2, N* = 8, P* = 0.748. The following parametervalues were used: r = 1, K = 100, aN = aP = 0.166, hN = hp = 0.5, dN = 3 = Y = 0.1, dp = 0.08.

Increasing plant density will simply result in more predators with no change in the equilibrium abundance of herbivores.

Introducing Prey Fear Responses

What happens to these systems if the herbivores respond to predators with vigilance? We replacefN(R) with (1 — u*)fN(R), where u* is the herbivore's optimal level ofvigilance; this incorporates the idea that vigilance reduces prey capture. We replacefp(N) with m'N/(k + bu*), where m' gives an individual predator's encounter rate with its prey to incorporate the idea that the prey's vigilance reduces the predator's effectiveness. From the prey's perspective, m = m'P. These substitutions give the model a |l-driven component. The herbivores' vigilance reduces their feeding rate, their fecundity, and their mortality due to predators. The more effectively vigilance reduces predation risk, the more the system will behave as |- rather than N-driven. Figure 13.5 shows the isoclines for such a system.

The herbivore's vigilance strikingly changes the plant and predator isoclines. In the N-driven model, plant isoclines were independent of predator density. Now, an increase in predators increases the number of herbivores that a given plant population can tolerate without declining. This pattern indicates a behavioral trophic cascade in which predators make the herbivores less effective at killing plants. The herbivores strongly influence the plant isocline. Because of vigilance, the herbivores, at low density, have a decreasing effect on plant fitness as predators increase. Plants can tolerate extremely high herbivore abundance so long as there are sufficient predators. The hump of the plant isocline shifts to smaller numbers ofherbivores and disappears completely at high predator densities. Herbivores can magnify the negative effect ofplants on other plants. Increasing the number ofplants increases inter-plant competition and reduces herbivore vigilance, and this reduction in vigilance makes herbivores more effective harvesters. These combined effects warp the contours of the plant isocline into convex folds along the predator-plant axis and along the predator-herbivore axis (fig. 13.5A).

Vigilance affects the herbivore isocline little, except that it increases more steeply along the predator and plant axes. This steepness occurs because vigilant herbivores can manage higher predator numbers as the herbivores exploit increases in plant abundance (fig. 13.5B).

With vigilance, the predator isocline increases with predator density and comes to depend on plant density—in contrast to the N-driven model, in which plant density did not affect the predator isocline. Herbivore vigilance means that predators adversely affect one another because the additional predators make the herbivores more vigilant and less catchable. Therefore, as the number of predators increases, they require a higher standing crop of herbivores to sustain themselves. In addition, the predator isocline increases with plant density. Increasing the abundance of plants makes the herbivores less vigilant and easier to catch, and hence the predators require a lower subsistence number of herbivores. Because of these behavioral effects of vigilance, the predators have a negative direct effect on themselves (which tends to be stabilizing), and the plants have a positive direct effect on the predators, just as the predators have a positive direct effect on the plants (fig. 13.5C).

To an ecologist, a three-trophic-level system with herbivore vigilance would appear to exhibit both "top-down" and "bottom-up" regulation. Increasing plant productivity causes a large increase in plants, a large increase in the numbers of herbivores, and proportionately smaller increases in the number ofpreda-tors. The predators face a larger herbivore population composed ofless catch-able individuals. On the other hand, the seemingly less important predator population (recall the lions and zebras) exerts considerable top-down control via the herbivore's feeding rate and vigilance. Remove the predators and the herbivores will overgraze, not because their population size increases (N-driv-en), but because each less vigilant herbivore now feeds more (| -driven).

Predator density

Predator density

Predator density

Predator density

Predator density

Predator density

To an ecologist, a fear-driven system also looks like "ratio-dependent predation"—the idea that predators experience a zero growth rate at some fixed ratio of prey to predator abundance (see Abrams 1997; Akcakaya et al. 1995). In the classic predator-prey model, the predators merely require a fixed standing crop of prey, independent of the numbers of predators. But when vigilance decreases catchability in response to predation, each predator may require a fixed number of prey (at least as a first-order approximation) to maintain a stable predator population.

Prey fear responses become crucial to debates over Oksanen's exploitation ecosystems (the three-trophic-level model without vigilance and with exploitative competition), top-down versus bottom-up regulation ofecosys-tems, and ratio-dependent models of predator-prey interactions. When herbivores have effective fear responses toward their predators, exploitation ecosystems become intricate. Top-down and bottom-up effects become flip sides of the same vigilance-induced direct effects of predators on plants and vice versa, and approximate ratio-dependent predation becomes the expected outcome of the positively sloped predator isocline (with respect to the predator axis).

Prey vigilance and the arrangement of isoclines in figure 13.5 may resolve a paradox of the classic Rosenzweig and MacArthur (1963) predator-prey model (these authors in their paper fully anticipate this resolution!). Here's the paradox: When predators capture prey very efficiently, they require a low prey density for subsistence. The predator's vertical isocline intersects the prey's isocline in a region where the prey's isocline increases with prey density (positive density dependence of the prey on themselves). The intersection ofthese isoclines creates an unstable equilibrium that can lead to limit cycles, prey extinction followed by predator extinction, or the extinction of the predator. When predators capture prey very inefficiently, they require a high density of prey for subsistence, which creates a stable equilibrium point because the predator's isocline intersects the prey's in a region of negative slope (negative density dependence of prey on themselves). But the predator is now highly susceptible to environmental fluctuations that take the prey population below the subsistence level. Paradoxically, efficient predators produce intrinsic instability in predator-prey dynamics, while inefficient

Figure 13.5. The isoclines of (A) plants, (B) herbivores, and (C) predators for a model in which the herbivores can use vigilance to manage predation risk. This figure shows the isoclines in separate graphs to prevent confusion and to illustrate how strongly herbivore fear responses change the plant and predator isoclines. The model used here directly extends the model used in figure 13.3. This model assumes that herbivores and predators have zero handling times on their respective prey. The star in each graph shows the equilibrium point: R* = 15; N* = 7.847, P* = 0.722. The following parametervalues were used: r = 1, K = 100, aN = 0.15, dN = 0 = y = 0.1, dP = 0.08, m' = 1, k = 0, b = 30. Hence, x = m'/bu and F(dF/d e) = 1/0 = 10.

predators produce vulnerability to extrinsic variability in prey population numbers.

Fear responses by prey can break this paradox. At low predator abundances, the predator can efficiently catch unwary prey. At higher predator numbers, the predator becomes less efficient as the prey become increasingly wary and uncatchable. The high efficiency of predators at low predator numbers buffers the predator from environmental stochasticity, while the inefficiency of predators at higher predator numbers promotes a stable equilibrium and reduces intrinsically unstable dynamics. Rosenzweig and MacArthur (1963), in addition to their "classic" predator-prey model, anticipate these stabilizing effects of prey that respond behaviorally to their predators.

Self-regulation also provides a hypothesis to explain "why big fierce animals are rare" (Colinvaux 1978). Carnivores may represent a sufficient threat to one another to keep their densities low. It seems to take a large prey base to support carnivores (10,000 kg of prey to support 90 kg of carnivore, for instance; Carbone and Gittleman 2002). The word "fierce" suggests a role for the prey and their fear responses. Fierceness is a property of the prey rather than the predator: a predator is fierce because it induces fear in its prey. And, in a highly |l-driven system, the prey's fear responses produce a system in which their fierce predators can and must be rare. From the perspective of ecological energetics, N-driven predator-prey systems support higher densities of predators. The prey compensate for predation risk by higher fecundity that sends energy up the food chain as they feed the predators. In | -driven systems, the prey respond to the presence of predation risk by forgoing fecundity, reducing mortality, and thus sending less energy up the food chain. Fear contributes to the length and to the transfer efficiency of food chains.

13.7 Foraging Games between Prey and Predator

"The deer flees, the wolf pursues" (Bakker 1983). We have considered how prey react to predators with fear responses, but predators need not be passive partners in this interaction. Predators can anticipate and respond to the prey's fear responses. Clever prey and clever predators produce a foraging game of fear and stealth. The abilities of prey and predator to respond to each other contribute to the character and stability of the predator-prey interaction (Abrams 2000). Although relatively few studies have addressed this problem, some recent work has done so (Lima 2002).

Habitat Selection Games

In the conventional ideal free distribution (IFD; seebox 10.1), which measures a forager's fitness in units ofenergy gain, animals should distribute themselves among habitats in a way that equalizes per capita net energy gain (Fretwell and Lucas 1969; Fretwell 1972; Rosenzweig 1981). If habitats vary in food and safety, animals should still equalize the fitness value of net energy gain, but they should discount net energy gain by the cost of predation as given in equation (13.1). In other words, a forager occupying a risky habitat must have a higher harvest rate than one occupying a safer habitat (Moody et al. 1996; Brown 1998; Hugie and Grand 1998; Grand 2002).

Hugie and Dill (1994) expand on the ideal free distribution under predation risk by allowing both prey and predators to select among habitats. The ESS (evolutionarily stable strategy) distribution ofprey and predators creates a spatial paradox of enrichment. In the absence of predators, more productive habitats will harbor more prey as the prey equalize their feeding opportunities. In the presence of predators, however, the prey must balance both food and safety, and the predators must equalize their feeding opportunities among habitats. If predators catch prey with equal ease in all habitats, then equal opportunities for the predators require that each habitat possess the same density of prey. This, in turn, means that the prey in more productive habitats harvest more resources. So, for the prey to have equal opportunity among habitats, there must be more predators in the more productive than in the less productive habitats. The ESS condition on the predators tends to equalize prey abundances independently of habitat productivity for the prey. And the ESS condition on the prey means that predators must bias their activity or distribution toward more productive habitats (fig. 13.6).

We can extend this model by permitting vigilant prey. With vigilant prey, we find a new ESS that requires a higher abundance of more vigilant prey in productive habitats, and a slight shift of predators from the more productive to the less productive habitats. At the new ESS, the more productive habitats have both more prey and more predators than the less productive habitats. In a more productive habitat, the prey are more vigilant, less efficient foragers, and harder to catch. In terms of ecological energetics, the less productive habitats actually have higher ecological efficiencies of transferring energy from one trophic level to the next.

Hugie and Dill's foraging game (1994) can also occur in time (Brown et al. 2001). Instead of productivity or resource availability varying among habitats, we imagine prey resources varying in time. The owls preying on gerbils in the Negev Desert represent such a system (see chap. 12). Afternoon winds redistribute sand and seeds every night, creating a resource that is renewed

Figure 13.6. Density-dependent habitat selection when both predator and herbivore distribute themselves according to an ideal free distribution. In both habitats, plant, herbivore, and predator population dynamics follow the three-trophic-level model with type I functional responses for herbivores and predators to their respective prey. In these graphs, we hold the quality of habitat 1 fixed. The plants in habitat 1 have a carrying capacity of 70. We allow plant dynamics to go to equilibrium, but hold the total numbers of predators (P = 1) and herbivores (N = 6) fixed. The graphs show the distributions of predators (frequency in habitat 1), herbivores (frequency in habitat 1), and plants (density of plants in habitat land habitat 2: R1 and R2, respectively) as the carrying capacity of habitat 2 is increased from 0 to 100. For the herbivore, one line shows its distribution when there are no predators (- pred.) and the other shows its distribution when predators are present in the system (+ pred.). Except for the habitat-specific carrying capacities and the fixed values for N and P, the parameters in both habitats are set to the same values as those in figure 13.4.

Figure 13.6. Density-dependent habitat selection when both predator and herbivore distribute themselves according to an ideal free distribution. In both habitats, plant, herbivore, and predator population dynamics follow the three-trophic-level model with type I functional responses for herbivores and predators to their respective prey. In these graphs, we hold the quality of habitat 1 fixed. The plants in habitat 1 have a carrying capacity of 70. We allow plant dynamics to go to equilibrium, but hold the total numbers of predators (P = 1) and herbivores (N = 6) fixed. The graphs show the distributions of predators (frequency in habitat 1), herbivores (frequency in habitat 1), and plants (density of plants in habitat land habitat 2: R1 and R2, respectively) as the carrying capacity of habitat 2 is increased from 0 to 100. For the herbivore, one line shows its distribution when there are no predators (- pred.) and the other shows its distribution when predators are present in the system (+ pred.). Except for the habitat-specific carrying capacities and the fixed values for N and P, the parameters in both habitats are set to the same values as those in figure 13.4.

nightly (Ben-Natan et al. 2004). The gerbils begin the night with the greatest abundance of seeds, and then they deplete these seeds throughout the night (Kotler et al. 2002). Without predators, clever gerbils would become active after dusk, deplete the seeds to the point at which they could no longer profit energetically, and then return to their burrows and remain dormant. If the owls "knew" this, their activity would track the gerbil activity pattern. But, if owls were most active at dusk, then clever gerbils might want to find a different time for their peak activity. In fact, the ESS distribution of activity by gerbils and owls follows an ideal free distribution in time.

The owl ESS requires a constant level of gerbil activity throughout the night (assuming the gerbils do not vary their vigilance during the night). The gerbil ESS requires an equalization of opportunity throughout the night. In accord with Gilliam and Fraser (1987), the ratio of risk to net feeding rate (| /f rule) must remain constant throughout the night. This requires higher owl activity early in the night and less as the night progresses. Over the course of the night, owl activity should track the level ofseed resources for the gerbils, and gerbil activity should remain relatively constant. Ifthe gerbils also vary their vigilance during the night, then early gerbils will be more numerous and more vigilant. To the owls, this part of the night will offer more, but less catchable, gerbils. In accord with these expectations, gerbil activity does decline as the night progresses, and gerbils behave more apprehensively early in the night (Kotler et al. 2002).

The gerbil-owl foraging game produces distinctive predator and prey isoclines (fig. 13.7). The prey isocline descends steeply from an asymptote along the predator axis. This typically happens in predator-prey models in which the prey have refuges. In this model, the prey have a behavioral refuge. Below a certain level of gerbil activity, it is not profitable for the owls to be active at all. Hence, the gerbils can sustain any number of inactive owls. At gerbil densities beyond this threshold, the gerbil isocline declines almost linearly. Because additional gerbils attract greater owl activity, the gerbils experience greater danger as their numbers increase. The owl isocline rises vertically from the subsistence abundance of gerbils that an owl requires. When there are no owls, the gerbils are most active and most catchable. As the number of owls increases, the owl isocline takes on a positive slope as the gerbils respond by becoming less active and less catchable. The intersection ofthe gerbil and owl isoclines produces a stable equilibrium, as both gerbils and owls have negative direct effects on themselves.

Patch Use Games

In traditional patch use models, the forager seeks patchily distributed and unresponsive prey. Patches are depleted because foragers harvest prey. Charnov et al. (1976) recognized that a predator could also depress patch quality if prey become harder to catch or if prey escape the area. This behavioral resource "depletion" can occur in unusual ways. A Neotropical tree frog lays its eggs out of harm's way in foliage above a pond, but a vine snake can still consume

Figure 13.7. The predatorand prey isoclines forthe gerbil and owl foraging game. The size of the initial pulse of resources (R) in the bottom panel is twice that of the top panel. In both systems, the gerbil isocline declines from an asymptote and then become almost linear as it approaches the prey axis. The owl isocline rises vertically from the prey axis (at this pointthe prey have no fear) and then slopes sharply to the right as the presence of additional predators makes each gerbil less active and catchable. Increasing the size of the resource pulse (bottom panel) increases the equilibrium number of both gerbils and owls.

Prey deriiity

Figure 13.7. The predatorand prey isoclines forthe gerbil and owl foraging game. The size of the initial pulse of resources (R) in the bottom panel is twice that of the top panel. In both systems, the gerbil isocline declines from an asymptote and then become almost linear as it approaches the prey axis. The owl isocline rises vertically from the prey axis (at this pointthe prey have no fear) and then slopes sharply to the right as the presence of additional predators makes each gerbil less active and catchable. Increasing the size of the resource pulse (bottom panel) increases the equilibrium number of both gerbils and owls.

the eggs. Warkentin (1995) showed that the tadpoles, after a certain stage of development, perceive the vibrations caused by an approaching and feeding snake. The tadpoles respond by hatching prematurely. Even as the snake depletes the patch of eggs through consumption, the patch also depletes itself as the tadpoles hatch and escape. When pea aphids (Acyrthosiphon pisum) attack a broad bean (Viciafava) (Guerrieri et al. 1999), the plant secretes pheromones that attract parasitoid wasps (Aphidius ervi). The pea aphids' patch quality declines both through their own herbivory and through increased risk of parasitism.

This kind ofpredator-prey patch use game occurs most commonly when a mobile predator has a larger home range than its prey. Mule deer populations exist in fragmented forest patches in the western montane habitats of North America. Their predator, the mountain lion, requires a home range that encompasses several forest patches. In this game, the mountain lion must decide how long to remain in any given forest patch, and the mule deer must choose vigilance levels in response to their perception of the mountain lion's whereabouts. The prey's information state becomes a critical feature of a game of fear and stealth. The deer would like complete information on the current whereabouts of the mountain lion (prescience), while the lion would prefer to keep the deer ignorant ofits whereabouts. The deer's information strongly influences the behavioral ESS and the subsequent stability of the predator-prey dynamics in this system (Brown et al. 1999).

In the mountain lion's absence, perfectly informed deer can be totally at ease (vigilance, u = 0). When a lion arrives, they can immediately adjust their vigilance to the u* appropriate for the lion's encounter rate and lethality [eq. (13.4) for the optimal level of vigilance] (fig. 13.8). At the other extreme, ignorant deer must adopt a fixed level of vigilance appropriate for the average level of lion proximity. More realistically, the deer's information state lies between these extremes of prescience and total ignorance. Imperfectly informed deer can never be sure when a lion is nearby, but the longer a lion is present (or absent) in their area, the more aware of its presence (or absence) they become. While never completely sure, the deer continuously update their expectation of encountering a lion based on time and direct or indirect cues given by the lion.

Figure 13.8. Three types of learning curves for prey becoming aware of a predator's presence. The x-axis shows the time since the predator's arrival (i.e., the predator arrives at time zero, and negative time refers to time before the predator's arrival). A prescient prey animal knows the predator's whereabouts. Hence, its expectation of encountering the predator is 0 prior to the predator's arrival and then jumps immediately to 1 (standardized to represent one predatorwithin the prey's area) upon the predator's arrival. An ignorant prey animal that never actually knows the predator's whereabouts has a constant expectation of encountering a predator. An imperfectly informed prey animal never actually knows where the predator is, but it can use cues to modify its expectation of a predator encounter. Its baseline expectation of predator encounter is higher than the prescient prey's and lower than the ignorant prey's. (After Brown et al. 1999.)

Figure 13.8. Three types of learning curves for prey becoming aware of a predator's presence. The x-axis shows the time since the predator's arrival (i.e., the predator arrives at time zero, and negative time refers to time before the predator's arrival). A prescient prey animal knows the predator's whereabouts. Hence, its expectation of encountering the predator is 0 prior to the predator's arrival and then jumps immediately to 1 (standardized to represent one predatorwithin the prey's area) upon the predator's arrival. An ignorant prey animal that never actually knows the predator's whereabouts has a constant expectation of encountering a predator. An imperfectly informed prey animal never actually knows where the predator is, but it can use cues to modify its expectation of a predator encounter. Its baseline expectation of predator encounter is higher than the prescient prey's and lower than the ignorant prey's. (After Brown et al. 1999.)

The prey's information state influences the predator's patch use behavior (how long it stays in a given area) and the prey's vigilance tactics. We can describe the value of a patch to the predator by multiplying the number of prey by their catchability. With prescient or ignorant prey, prey catchability stays the same while the predator occupies a patch; it's just that the predator catches ignorant prey more easily than prescient prey at their ESS levels ofvigilance.

When the deer have imperfect information, they must select their baseline level of apprehension (Brown et al. 1999)—the optimal level of vigilance when available information suggests that there is no lion around. This baseline vigilance level acts as the set point for raising vigilance when the deer detect a lion's arrival. If the deer set an excessively high baseline, they waste foraging opportunities when lions are actually absent. If they set too low a baseline, they risk death by reacting too slowly to the arrival and presence of a lion. The deer's ESS baseline level of vigilance will fall somewhere between 0 (the optimal baseline for the prescient prey) and the fixed u* of the ignorant deer. When faced with deer with imperfect information, the lion reduces its chances of capturing a deer simply by spending time in the patch. When the lion first arrives, the deer are using theirbaseline vigilance level and are hence at their most catchable. As the deerbegin to notice the lion, they increase their vigilance and become less catchable (fig. 13.9). As soon as the lion arrives in the patch, patch quality begins to decline for the lion. If patch quality declines to some threshold (set by the overall quality ofthe environment), then the lion should leave the patch and seek another. A lion should spend less time per patch (higher giving-up threshold) in a rich than in a poor environment. A lion should spend more time per patch when the deer have a low baseline level ofapprehension.

The deer's information state and the resulting ESS behaviors for deer and lions influence the predator-prey dynamics and behavioral indirect effects among deer and lions. Prescient prey can result in unstable predator-prey population dynamics (van Balaan and Sabelis 1993, 1999;Brown etal. 1999;Lut-tbeg and Schmitz 2000). Adding more deer reduces their feeding rate and encourages them to be even more vigilant when a lion is present. This contributes to safety in numbers and to the destabilizing ofbehavioral feedbacks. Increasing the number of lions produces few behavioral indirect effects because deer do not become more vigilant; instead, they endure more frequent periods of lion presence. The predator isocline remains roughly vertical, but actually has a slight negative slope. Perfectly informed prey also create a repelling isocline that occurs at higher prey densities than the regular predator isocline. Consequently, the deer-lion model has two unstable equilibria (both interior solutions) and one stable equilibrium point (a corner solution). The corner solution has no lions and the deer at their carrying capacity. Another solution q? 0.20 £

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