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1 The University of Chicago Ecological Consequences of the Trade-Off between Growth and Mortality Rates Mediated by Foraging Activity Author(s): Earl E. Werner and Bradley R. Anholt Reviewed work(s): Source: The American Naturalist, Vol. 142, No. 2 (Aug., 1993), pp Published by: The University of Chicago Press for The American Society of Naturalists Stable URL: Accessed: 26/08/ :20 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. The University of Chicago Press, The American Society of Naturalists, The University of Chicago are collaborating with JSTOR to digitize, preserve and extend access to The American Naturalist.

2 Vol. 142, No. 2 The American Naturalist August 1993 ECOLOGICAL CONSEQUENCES OF THE TRADE-OFF BETWEEN GROWTH AND MORTALITY RATES MEDIATED BY FORAGING ACTIVITY EARL E. WERNER* AND BRADLEY R. ANHOLTtt *Department of Biology, University of Michigan, Ann Arbor, Michigan 48109; tdepartment of Biology, Queen's University, Kingston, Ontario K7L 3N6, Canada Subbmitted October 23, 1991; Rev,ised June 29, 1992; Accepted July 10, 1992 Abstract.-Animals are frequently faced with trade-offs created by the fact that both resource acquisition and risk of mortality increase with activity, for example, with foraging speed or time spent foraging. We develop models predicting adaptive responses for both foraging speed and proportion of time active when individual growth rate and mortality risk are functions of these variables. Using the criterion that animals should minimize the ratio of mortality to growth rates, we show that, when both growth and mortality rates are linear with activity levels, the latter should be either maximal or minimal depending on resource level. If growth rate is a decelerating function of activity, then speed or time active should decrease with increases in resources, handling time, or the effect of activity on mortality rate. By contrast, if mortality rate unrelated to activity increases, then activity rate also should increase. We also develop predictions for cases in which time horizon is critical using a dynamic programming framework. The general patterns of predicted activity responses are similar to the time-invariant analytical solutions, but foraging speed is reduced relative to the analytical solutions when time remaining is long or when accumulated reserves are high. This effect is ameliorated when accumulated reserves (size) increase resource capturefficiency or reduce mortality risk. If resources decline with time (e.g., because of competition) optimal foraging speeds are also higher than predicted by the analytical solutions. We discuss the relation of our predictions to previous models and the available empirical evidence. The majority of available data appear to be consistent with our models, and in some cases quantitative comparisons are quite close. Finally, we discuss the implications of our results for ontogenetichanges in behavior and for population- and community-level phenomena, particularly the role of activity responses in competitive interactions and indirect effects and patterns of coexistence among competitors. A hallmark of the modern behavioral ecology program is the attempto understand how animals make decisions under conflicting demands or, alternatively, to understand how patterns in species behavior reflecthe resolution of such trade-offs in evolutionary terms. This article concerns the consequences of tradeoffs in growth and mortality rates mediated through rates of activity while foraging. The ubiquity and importance of the trade-off is simply stated. Searching for and harvesting resources requires movement for most animals. Movement, however, usually increases encounterates with or detection by predators. These relationships lead to a fundamental trade-off between growth rate and risk of t Present address: Department of Zoology, University of Toronto, Erindale College, 3359 Mississauga Road N., Mississauga, Ontario L5L 1C6, Canada. Am. Nat Vol. 142, pp ? 1993 by The University of Chicago /93/ $ All rights reserved.

3 CONSEQUENCES OF GROWTH-MORTALITY TRADE-OFFS 243 predation as functions of activity level (Abrams 1990, 1991). This trade-off is likely an important determinant of animalife-styles in many taxa. Further, there is growing evidence that animals are able to adaptively "balance" positive and negativ effects of such trade-offs through changes in behavior (see review in Lima and Dill 1990), which allows us to experimentally test ideas concerning the adaptive balance. Similar trade-offs with other types of resourcesuch as mates or nesting sites probably exist (see, e.g., Magnhagen 1991), and the models we present here could be adapted to fit those circumstances as well. There is substantial empirical evidence demonstrating relationships between activity and growth rate and activity and mortality rate. More active individuals encounter food at a greaterate and grow faster than less active individuals, both within and between species (e.g., frogs: Pough and Taigen 1990; tadpoles: Skelly and Werner 1990; stoneflies: Walde and Davies 1984; beetles: Richards 1984; fish: Grant and Noakes 1987). More active individuals (or species) are often better competitors and expropriate resources from less active individuals (e.g., triclads: Reynoldson et al. 1981; tadpoles: Woodward 1982, 1983; Morin 1983; Morin and Johnson 1988; Lawler 1989; Werner 1991, 1992a). Increasing activity, however, clearly increases vulnerability to predators in many taxa (e.g., zooplankton: Wright and O'Brien 1982, 1984; benthic invertebrates: Ware 1973; tadpoles: Lawler 1989; E. E. Werner, unpublishedata; damselflies: McPeek 1990b; see also review in Sih and Moore 1989). In addition, it has been documented that prey movement enhances or is required for prey detection and attack by many predators (e.g., octopus: Wodinsky 1971; lizards: Burghardt 1964; snakes: Herzog and Burghardt 1974; odonates: Corbet 1962; Pritchard 1965). These examples reflecthe broad applicability of the trade-off that we address here. In this article we develop a collection of models that predict adaptive responses in activity rate when resource gain and vulnerability to predation are both functions of activity. We examine two complementary elements of activity rate, first treating movement speed and then developing models to account for the proportion of time spent active. These initial models do not incorporate time horizons, so we also explore the consequences of time constraints, for example, seasonality, using a dynamic programming framework. Finally, we consider the consequences of these trade-offs for species interactions and dynamics. MODEL DEVELOPMENT There are two components of activity that influence growth and mortality rates: speed while foraging and the proportion of time spent foraging. Movement speed while foraging can affect both the encounterate with predators and the rate at which food items are encountered and collected. Similarly, the probability that an animal will be active at any given moment (over longer time periods, the fraction of available time spent foraging) will also directly affect predation risk and feeding rates. Either measure of activity could be important in different systems, and they may co-vary in important ways. We argue that speed and proportion of time active need not be identically affected by food and predator densities. Below, we justify some simple relations between these components of activity

4 244 THE AMERICAN NATURALIST and mortality and growth rates that enable us to explore adaptive responses in activity level. Animals that respond adaptively to an activity-mediated trade-off should adjust activity levels to balance growth and risk to maximize fitness. Given specified relations between mortality ([) and growth (g) rates and components of activity, we determine optimal activity levels by minimizing the ratio 1L/g. We employ this fitness criterion because we are interested individual behavior and changes in behavior with size over ontogeny. This allows us to speculate about behavioral responses in the context of the life cycle. Gilliam (1982; see also Werner and Gilliam 1984; Stephens and Krebs 1986) has shown that, in a size-structured, continuously reproducing population in which birth and death rates are functions of individual size, juveniles maximize fitness by minimizing the ratio of mortality to growth rates ([/g) if the population is stationary. The criterion is more complicated for reproductive stages (containing a term for the ratio of presento future reproduction; Gilliam 1982), if the population is growing or declining (containing a term for population growth rate; Gilliam 1982), or if time constraints on lifehistory events are incorporated (e.g., seasonal effects on size at metamorphosis; Ludwig and Rowe 1990). Here we limit our considerations to the simple case of juveniles in a stationary population. It is intuitively clear that minimizing l/g allows accrual of each unit of growth at minimu mortality cost and therefore maximizes the probability of reaching reproductive size. Abrams (1982, 1990) has approached similar questions in a more general context using different fitness criteria. He has generally assumed tha time active (or foraging effort) has positive effects on reproductive rates but negativeffects on survival rates and has determined optimal activity times by maximizing the difference between birth and death rates or by maximizing reproduction times survival rate. Abrams (1983, 1991) has also examined some of these questions in the context of different life histories (semelparity vs. iteroparity). Many of his results enable us to compare solutions arrived at by the [/g fitness criterion. Activity and Mortality Rates We characterize the relation between foraging speed (s) and the probability of detection by a predator (and consequent mortality rate) by the relation (s) = q + msx. (1) The constant q represents the background mortality rate, that is, the mortality experienced unrelated to movement. If q = 0, then the optimal course is to always be completely inactive. Because this implies immortality, we discard this possibility as biologically uninteresting. We constrain q to be greater than zero for all situations discussed below. The constant m represents the increase in mortality with foraging speed, and x is related to factors such as changes in detectability with speed. If x = 0, then mortality rate is constant regardless of speed, if 0 < x < 1, then increase in mortality risk decelerates with speed, if x = 1, then the relation between speed and mortality rate is linear, and if x > 1, then risk (owing to increases in detection or capture probabilities) increases with speed. In this way we can separate the probability of encounter (within detection range) from the probability of detection and capture when within encounterange,

5 CONSEQUENCES OF GROWTH-MORTALITY TRADE-OFFS 245 which can change with speed (see, e.g., Gendron and Staddon 1984). For the majority of our analyses we use the case in which x = 1, that is, in which encounterate with predators is linearly related to speed. We employ equation (1) because it provides a good approximation to a number of different situations. We are interested the behavior of prey, that is, decisions concerning at which speed to move. In the two-dimensional case with a circular perceptual field, the number of predators encountered by a prey when both are moving at characteristic velocities, if we assume that the direction of predator movement is randomly and uniformly distributed with respec to prey movement direction, is Z 2rN v2, (2) where Z is the encounte rate of a prey individual with the predator, r is the perceptual radius of the predator, N is predator density, and v and s are average foraging speeds of the predator and prey, respectively (Skellam 1958). If prey are moving much faster than the predator (in the extreme, a sit-and-wait predator and moving prey), then encounterate is directly proportional to prey speed: Z = 2rNs. The constants (2rN) can be collapsed into the proportionality constant m in equation (1), and we can use the linear case (x = 1). Similar considerations apply in the three-dimensional case (Gerritsen and Strickler 1977). If the predator is also moving, the linear relation between prey speed and encounterate with predators does not hold. Figure 1 presents the relation between prey movement speed and encounte rate for different predator movement speeds in the twodimensional case. It is clear that, if the predator moves slowly relative to the prey, then the linear case is a good approximation, especially at higher prey speeds. As predator speed increases, however, the relationship becomes more curvilinear with prey speed, and encounterates increase. Note that there are large regions of lower prey speeds when predator speed is high in which increases in prey speed have very little effect on encounterates (see also Gerritsen and Strickler 1977). Thus, there is a strong asymmetry; if prey speeds are much higher than predator speeds, then changes in prey speed have the potential to greatly affect encounterates with the predator. However, if prey speeds are low relative to the predator's, then changes in prey speed have littleffect on encounterates. It can be seen from figure 1 that, if prey adjustments in speed are over a relatively narrow range, the linear form of equation (1) will still provide a reasonable fit to the relation. This is not the case if prey adjust speed over a wide range that encompasses the predator'speed. These relations, however, can be closely approximated by equation (1) with x > 1. Given that the question here concerns behavior of the prey and not that of the predator, we assume a characteristic movement rate of the predator and submerge the effect of the predator speed in equation (1). In the case of proportion of time active, we assume that the number of encounters with predators is directly related to the proportion of time active (p) for some fixed s. Thus, the mortality relation is l(p) = q + msp, (3)

6 246 THE AMERICAN NATURALIST LU v 16 LU z 0 z LU 0 4 v PREY SPEED FIG. 1.-The effect of predator speed (v) on the linearity of the relationship between prey speed and predator encounterate. The relationship is more curvilinear at high predator and low prey speeds, but not markedly so. where m is again a constant of proportionality, and 0? p? 1. We consider only the linear case as it is less likely that mortality rate accelerates-with time active, as may be the case with speed of movement (but see Abrams 1982). Activity and Growth Rate In the simplest case resource acquisition is a linear function of speed, f(s) = z + ks. (4) The constant z represents the rate of resource acquisition when the animal is inactive (e.g., sit-and-wait tactic), and k is the increase in resource acquisition with foraging speed. Often the relationship between growth rate and speed will be curvilinear, increasing at a decelerating rate to an asymptote. A variety of considerations can lead to such a shape (Taylor 1984). We will consider two here. When an individual cannot handle (subdue and consume) prey at the same time as search for additional prey, the relationship between speed and growth rate (proportional to food intake rate f(s)) will be described by a Type II functional response (Holling 1959), f(s) = sr?+shr' (5)

7 CONSEQUENCES OF GROWTH-MORTALITY TRADE-OFFS 247 where R is resource density, and h is the handling time. We have not incorporated the energeticosts of movement into this formulation. Energy expenditures often increase as a power of speed, especially in the aquatic environment (Weihs 1973). If costs are subtracted from gain, growth rates will decline at high movement rates. We do not incorporate energy costs here because the optimal activity that balances mortality rate will lie on the ascending limb of the relation of growth versus movement speed. A simple saturating curve allows us to understand the factors that will affec this solution. However, the costs of movement would necessarily affect exact quantitative predictions. The gain over some longer period of time will be the foraging rate as determined by speed f(s) times the proportion of time active. At low resource levels it is possible that the gain relation will be linear over the entire range of possible activity times. In many cases, however, growth with time active will increase with diminishing returns because of various processing constraints, particularly digestion times, gut capacity, and decline in feeding motivation as gut fullness increases (see Sjoberg 1980; Taylor 1984). We characterize these limitations with an asymptotic function that reflects these constraints, pf(s) (6) g()=n + f3pf(s)~ 6 The maximum rate of growth (proportional to steady state throughput of the gut) is equal to 1/p3, and the initial slope of the relation (the efficiency of conversion of prey capture to growth) is equal to 1/X. RESULTS We first consider short-term foragingain as a function of movement speed with no constraints other than handling time. Resource Acquisition and Mortality Rates Are Both Linear Functions of Movement Speed When we differentiate the quotient of the linear mortality function (equation [1]; i.e., x = 1) and the growth function (equation[4]) with respect to s, we find that the optimal movement speed (s*) depends on the relative slopes and intercepts of the 1L(s) andf(s) functions. When the ratio of the intercepts (qlz) is equal to the ratio of the slopes (mlk), then all s's have equal fitness. If qlz < mlk, the optimal speed is zero, and when qlz > mlk, the optimal speed is the physiological maximum. Clearly, the same result applies to proportion of time active when it is linearly related to growth rate. It is easy to see from this formulation how transient changes in predation risk or resource availability could lead to individuals' switching foraging modes from active search to sit-and-wait. If predation risk increases relative to growth rate, then animals should switch to ambushing their prey, whereas if resource density increases (thereby increasingrowth rate) the animal should use active search to find food. Abrams (1982) arrives at a similar conclusion by maximizing the difference between linear birth and death rates. This result is also consistent with models of foraging mode that do not incorporate

8 x =O0.5 x =1 x= 2 a e v b m v~~ \I- lv h I R SPEED FIG. 2.-The effect of changes in the parameters of eqq. (1) and (5) (x, q, m, h, R) and their effect on optimal speed (s*, i.e., where pjf is minimized). In graphs a-d, mortality increases as the square root of speed (s05); in e-h, mortality increases linearly with speed (s); and in i-i, mortality increases as the square of speed. Parameter values for adjacent lines differ by a factor of four. The thick line has identical parameter values in each panel: q = 0.4, m = 0.02, h = 0.05, R = 20. Graphs a, e, and i show the variation in background mortality rate (q). Graphs b, f, and j show the variation in slope of the mortality relation (m). Graphs c, g, and k show the variation in handling time (h). Graphs d, h, and I show the variation in resource level (R). Values of the parameter increase higher up in the figure in graphs a-c, e-g, and i-k; values decrease higher in the figure for graphs d, h, and 1. Only two lines are shown in graph i; the missing line is off the scale.

9 CONSEQUENCES OF GROWTH-MORTALITY TRADE-OFFS 249 mortality costs (see, e.g., Ware 1975; Norberg 1981; Speakman 1986; Dunbrack and Giguere 1987) as well as with some empirical observations (Huey and Pianka 1981). Resource Acquisition Rate Is a Decelerating Function and Predator Encounter Rate Is a Linear Function of Movement Speed Differentiating the quotient of equations (1) and (5) with respecto s and setting it to zero, we obtain 'X+1 +? - (x- )x_ q h(x1) mhrx O(a 0. xis (7a) For the case in which x = 0, the solution is trivial; the animal should always move at the maximum speed because mortality rate is constant andf(s) increases monotonically with s. Therefore, VL(s)lf(s) is minimized at the extremum. For the case in which x = 1, that is, encounterate with predators is directly proportional to speed, the optimal speed, s*, is given by q 5* mhr= (7b) Examining the second derivative indicates that equation (7b) is a relative minimum. We also consider the cases in which x = 2 and x = 0.5, simply to illustrate the qualitative changes in predictions if animals are more or less likely to be detected when moving quickly (see, e.g., Corbet 1962; Burghardt 1964; Pritchard 1965; Wodinsky 1971; Herzog and Burghardt 1974). The relationships among speed, fitness, and the variables discussed above are illustrated in figure 2. There are three effects to consider: How does s* change with variation in the parameters? How does fitness (inversely proportional to,.lf) at s* change with variation in the parameters? How sensitive is fitness to deviations from s*? Increases in x lead to lower s* and reduce fitness (higher,ulf). As x increases, the cost of deviations above s* are increasingly expensive in units of fitness, but there is littl effect of changes in x on movement speeds below s*. Deviations below s* are always expensive. For a given value of x, increases in q promote an increased optimal movement rate. In contrast, increases in m, h, and R reduce the optimal movement speed. However, increases in q, m, and h lower fitness, whereas increases in R raise fitness. As shown in equation (7b), when x = 1, changes in s* as a result of changes in the parameters are on the order of the square root of the change in the parameter value. That is, a doubling of q increases s* by a factor of V/, but a doubling of m, h, or R decreases s* by a factor of \/27 When x > 1, the predicted change in s* is less than the square root of the change in parameter value and, conversely, when x < 1, s* is predicted to change by more than the square root of the change in parameter value. If movement increases the probability of detection and capture (x > 1) when a predator is encountered, the optimal movement speed is slightly reduced compared to when this effect is absent (x = 1), and fitness is lower at s* in all cases because of the higher mortality cost of movement. The reverse is true when

10 250 THE AMERICAN NATURALIST increased movement reduces the probability of detection and capture (x < 1). The fitness cost associated with positive deviations from s*, however, are much more severe when x > 1 because of the higher mortality cost of movement. Thus, we would expect that, in situations in which detection by predators increases with speed, prey should more closely match s*. As q increases, optimal speed should increase (fig. 2a, e, i). This may seem counterintuitive at first but reflects the fact that the higher the mortality rate experienced or accumulated when inactive and not growing, the greater the premium for growing quickly through a stage. Thus, the interpretation of q is critical. If zero movement is a refuge from predators (q is independent of predator density), then increasing the density of predators alters only m and should select for reduced speed (fig. 2b, f, j). However, if density of predators affects primarily q and not the slope (m), then increasing predators can have the effect of increasing optimal movement speed. Because overall mortality increases with increases in q, fitness declines (increased pif). Fitness costs of deviations from s* are severe below s* in all cases, but slight above s*, especially when x < 1. If q is variable in space or time and animals are tracking a longer-term average of background mortality, then we expect that animals should err in the direction of foraging speeds greater than s* for the long-termean because of the asymmetrical fitness costs of deviations above and below s*. Changes in q have little effect on the fitness cost of foraging speeds in excess of s* (the lines are parallel). Increasing the effect of foraging speed on mortality (fig. 2b, f, j) will select for a decrease in the optimal movement speed because prey capture becomes more expensive (in units of mortality) as speed is increased. This prediction is a consequence of the decelerating functional response curve. Recall from equation (1) that m can be interpreted directly as changes in the density of predators or in the radius of encounter. Increases in m reduce fitness, but the decline in fitness is less than for that of variation in q because speed is reduced instead of increased in response to increases in the parameter (fig. 2b, f, j). The forager can do nothing to mitigate the effect of q except to grow to a relatively invulnerable stage but can mitigate the effects of m by reducing movement speed while searching for food. Changes in handling time for resources (fig. 2c, g, k) have effects on s* and fitnessimilar to those caused by changes in m. The change in fitness, though, has its primary effecthrough prey capture rather than mortality. Because s* declines with increases in handling time, fewer prey are captured, which exaggerates the effect of reduced prey capture. The sensitivity of fitness to foraging speeds that deviate from s* is similar to those with m. Increasing resource density (fig. 2d, h, 1) does not change the asymptote of the growth curve but increases the initial slope, and thus Ft/f is minimized at a lower risk level. This predicte decrease in movement speed is counter to the usual predictions of optimal movement speed as a function of resource density (Ware 1975; Norberg 1981; Pyke 1981; Speakman 1986; Dunbrack and Giguere 1987), in which increased resource levels are predicted to lead to higher foraging rates. In contrasto the above parameters, fitness at s* increases with increased resource density. Positive deviations from s* have slightly greater fitness conse-

11 CONSEQUENCES OF GROWTH-MORTALITY TRADE-OFFS 251 quences at high resource densities than at low levels, but in general positive deviations are much less costly than negative deviations. Similarly, we expect that incorporating the metabolicosts of movement would make activity more expensive in terms of reduced growth, and deviations from s* would have more serious fitness consequences than those illustrated in figure 2. Growth Rate Is a Decelerating Function and Predator Encounter Rate Is a Linear Function of Time Active We now ask how the animal should maximize fitness relative to feeding period or fraction of the day active at a characteristic movement rate. Mortality is linear as in equation (3), and growth is realized as in equation (6). The optimal time active (p*) has many similarities to the solution for movement speed: pf = s)x (8) Again, as q increases, time active should increase, and as mortality rate when active (ms) or feeding rate (f(s); determined by resource density, handling time, and movement speed) increases, time active should decrease (see Abrams 1990 for a similaresult with a different fitness criterion). As digestive constraints are relaxed (throughput, 1/3, increased), time active should increase, and as the efficiency of conversion of feeding rate to growth (1/I) increases, time active should decrease. Patterns in optimal time active and sensitivity of fitness to deviations from p* are similar to those explored in figure 2. The above solutions indicate some general relations affecting optimal movement speed and time active. Factors that increas either the slope of the mortality rate function or the initial slope of the growth function (slope when both movement speed and proportion of time active are zero) all decrease optimal movement speed or activity times. In contrast, increasing the elevation of either relation, that is, increasing the background mortality maximum growth rate, selects for increased movement speed or activity time. In the case of optimal movement speed, s* = /(qh)i(mr). Both q and 1/h determine the elevations of the mortality and functional response curves, respectively, while m and R determine the slopes. Optimal time active can be rearranged such that it is proportional to (q/f3)/(m/x). Again, the quantities that determine the elevations of the curves are in the numerator; those that determine the slopes are in the denominator. As noted above, parameters that affect optimal speed or time active in the same direction can have opposite effects on fitness. Time Constraints The minimization of p/g assumes that there is an infinite time horizon and that returns are realized immediately. For organisms that live in a seasonal environment where the timing of development and reproduction is constrained, these assumptions are not appropriate. We examined thc sensitivity of the p/g criterion

12 252 THE AMERICAN NATURALIST when the payoff is realized at the end of a fixed time interval by using a dynamic programming framework. We examine this sensitivity in the context of movement speed; the general results will apply to time active as well, since parameters affect both in similar ways. The dynamic program incorporated the Type II functional response for speed of movement (eq. [5]). The rates of return for foraging were deterministic. The probability of survival from time interval to time interval was the zero term of the Poisson distribution (e-1(s)), where VL(s) was calculated from equation (1) with x = 1. Each individual had a state variable (X) that represented reserves already accumulated up to a maximum capacity, Xmax. Individuals that had no reserves were dead. A small metabolicost (c) was included so that animals had to forage or die. The magnitude of the cost did not change the patterns reported; only the positioning of the curves changed, so this cost was fixed in the simulations. Metabolic costs would be expected to vary with foraging speed and change the curvature of the gain function. We did not include this level of realism because it does not bear on the question of time horizons, although it would be necessary for precise quantitative predictions. We interpret the state variable, X, as size (accumulated reserves), with the time horizon being, for example, a season, time to pupation, or metamorphosis. However, X might also be reasonably interpreted as gut fullness (or conversely hunger) over a diel feeding cycle. The dynamic programming equation then was where F(X; t; T) = max EH, {e - (q +,ins)[f(x'; t + 1; T)]}, (9) s sr x, Xs X + hsr -c, (10) F is the objective function, X' is the reserves at a given speed, T is the final time when fitness is realized, EWv is expected fitness (where w represents fitness), and t is time remaining to the final time. If there are n time steps left until T, the expected fitness at T of an individual with X food reserves will depend on the foraging speed during every time interval from T - n to T - 1. When there is a choice among a behavioral alternatives then the vector that maximizes fitness must be chosen among a' alternative vectors for each of the Xmax states. Testing all possible combinations of speed at each time step for each value of the state variable (X) rapidly becomes impossible, even for the largest supercomputer. We can cut this problem down to a manageable size by using Bellman's (1957) backward iteration algorithm (see Mangel and Clark 1988 for a clear explanation of the algorithm and its uses in the context of behavioral ecology). Optimal movement speed was determined by using the golden section search method (Press et al. 1988) for discrete values of the state variable. The expected return from s* between the discrete values of the state variable when t = 1 was estimated by linear interpolation (Mangel and Clark 1988). Having determined the importance of the time constraint and delayed payoffs,

13 CONSEQUENCES OF GROWTH-MORTALITY TRADE-OFFS 253 we examined two modifications of the basic dynamic program. First, we asked how the optimal behavior pattern changes if resource acquisition and mortality risk are size-dependent (as assumed by the Rig criterion). Second, we asked whether the optimal behavior pattern changes if the availability of resources declines through time, for example, because of competition. We investigated the effect of size-dependent foraging success by using a model that assumed that prey capture was a power function of accumulated reserves (X) so that f(s) oa XO5. This is approximately equivalento dividing h by X. We investigated size-dependent mortality risk by reducing both the background mortality and the increase in mortality with speed by X. This increased the loge of survival between time intervals by X. Preliminary simulations showed that foraging speed could increase or decrease as X increased. We investigated the change in s* when X = 0 and X = 100 when we systematically varied the strength of the size dependence of foraging success and mortality risk. We also compared the vector of optimal foraging speeds when resources decline uniformly through time with that predicted for constant resources with the same time remaining. Our justification is straightforward. In a large population, the foraging of a single individual will have a tiny effect on total resources. So preempting resources by foraging at a higherate will not appreciably affecthe remainder of the population, whereas the mortality costs will be high. Simulation Results No size dependencies.-with one time unit left, individuals with few reserves move at higher speeds than predicted by the p/g criterion. The optimal speed of movement from the simulations (s*) is -N 1/mhR. Movement speed declines with both increasing reserves and time remaining until the payoff is realized (fig. 3A). Individuals with large reserves move at slower speeds than predicted by the p/g criterion because the fitness increment from additional resources with increased foraging speed is only that realized from immediate foraging. However, the decrement in expected future fitness due to mortality is the loss not only of the resources being gathered at the moment but also of all those resources that have been gathered in previous time periods. Thus, an individual with twice as many reserves has twice as much to lose by foraging a risky way. Individuals near the upper bound of reserves are even less active because the benefits of additional foraging cannot be realized when reserves reach maximum capacity. Figure 3 portrays the region of state variable space below this upper bound. An individual with many time periods left has an expectation of how many resources can be gathered in the time remaining until the payoff is realized. Foraging in a risky way immediately increases reserves but decreases the likelihood of realizing the payoff at time T. Thus, individuals should forage at a minimal rate and increase the rate of foraging as time available declines. With one exception, changes in the parameters of the Vi and f curves alter behavior in the same way as predicted by the analytical model. Increasing the mortality rate with speed, the handling time, or the resource density all lead to a reduction in the optimal movement speed. Moreover, the scaling is also the same as in equation (7b). A fourfold change in these variables leads to a twofold

14 5 4 3 'Ii ~ ~ ~ A FIG. 3. Optimal movement speed as a function of reserves and time units remaining until the payoff is realized. A, Prey capture and mortality are independent of reserves. B, Prey capture increases as the square root of reserves. C, Mortality risk declines as the square 254

15 6 4 3 C D root of reserves. D, Both prey capture and mortality risk are functions of reserves. Paramete values are q = 0.2, m = 0.04, h = 0.4, and R =

16 U) ON0 ir) UC C)~ 0q ~ ~~~~U 0,0 P- C\ CO CO ON~ (ON z ~~~~~ON C)ON \ONa U - U) ONOON C C\ON ON....- C)~~~~~. H - U)c~r C 0 0 H ~~~ ONONOCNC)ON ON C) 0 C\ C C\ N O H U) ONO C)ON ON ON D~UUC ONOC)ON ON ON U) ONOONONON -~~~~~~C CD ~. m - xx0 -~~~~~~~

17 CONSEQUENCES OF GROWTH-MORTALITY TRADE-OFFS 257 change in speed. However, changes in background mortality have no effect on s* at any level of gut fullness or time left. This is consistent with our interpretation that sensitivity to q reflects the value of growing through vulnerable size classes when growth and mortality are size-dependent. This version of the dynamic program does not have size dependences (implicit in the >/g criterion) built into it. The sensitivity of expected fitness to deviations from s* depends on present reserves and the parameter values used (table 1). Deviations from s* have a larger effect on individuals with few reserves than on those with large reserves. Individuals with large reserves reduced their expected fitness by less than 1% (2.5% in one case) even when they moved at double the optimal speed. The absolute change in speed is small, however, because s* is also small. For individuals with no reserves, traveling at half the optimal speed reduced expected fitness by 9% and traveling at double the optimal speed reduced expected fitness by 16%. Sensitivity to movement speeds above s* are more serious when the increase in mortality with speed is high, and for higher values of the background mortality as well. Conversely, sensitivity is reduced at movement speeds below s* for high values of these parameters. The expected fitness of individuals with few resources is less sensitive to deviations from s* when handling time or resource density is large. It is interesting that individuals with large reserves were most sensitive to deviations from s* when resources were abundant. Traveling at twice the optimal speed reduced expected fitness by 2.5%. In summary, when growth and mortality rates are independent of size, there are strong effects of time left until the payoff (t) and reserves (X) on s*, but there is little cost of deviations from the optimal behavior. Size Dependence Making either foraging or survival rates, or both, moderate functions of size (accumulated reserves; X) reduces the sensitivity of optimal speed to size and time remaining (fig. 3B-D), although s* is identical at the origin in all cases. If foraging success is strongly size-dependent but survival is not, then s* again declines at large size (fig. 4). When growth and survival are both functions of body size, s* increases with body size, especially when the power of the foraging efficiency is -0.5 (fig. 4). When the powers of both size dependences were equal to 0.5 (fig. 3D), the increase in s* at time step 20 (s* changes only slightly for values of t greater than 20) was nearly 100% (1.47 when X = 0 vs when X = 100). The sensitivity of s* to the parameters is approximately that of the analytical solution (Vq/mhR) when there is only one time unit remaining until the payoff. Even when there are many time units until the payoff, s* increases with increases in q and decreases at higher values of m, h, and R. The fitness consequences of deviations from s* are more severe when prey capture and mortality risk are both size-dependent (table 2). Expected fitness is more sensitive to deviations both above and below s* at high values of m and less sensitive to deviations away from s* for high values of q, h, and R. In general, small individuals are more sensitive than large ones, except at high values of m, in which case there is a greater cost in relative fitness for large individuals moving at twice the optimal speed.

18 258 THE AMERICAN NATURALIST I~~~~~It FIG. 4.-Log1o of the ratio of optimal foraging speed when reserves are large (X = 100) to that when reserves are at the minimum (X = 0) as a function of the power of the size parameters for foraging efficiency and mortality risk. A value of zero denotes no change in s*. The four solid circles show the parts of the parameter space depicted in fig. 3. Resource Depletion When growth and mortality rates were size-independent, resource depletion through time caused s* to be higher than if the current resource availability had continued until time T (table 3). This is consistent with the low rate of foraging when there are many time units left until the payoff is realized. Because the expectation of gain in the remaining time is less, there is less to be lost (and relatively more to be gained) by increasing activity levels and risking predation. Individuals with large reserves do not increase their speed as much in the face of declining resources as do those with fewer resources. There is also less of an effect when prey capture and mortality are size-dependent. When the marginal mortality rate was higher, the increase in s* in the face of declining resources was higher. But increases in handling time or resource density reduced the difference. There was no effect of variation in background mortality for the sizeindependent case, but there was an increase in the size-dependent case. Again, this is consistent with the notion of growing through a vulnerable stage; future opportunities for growth when resources are declining will be limited so growth should occur early.

19 d ) C) \C W 00~~~~~~~~0 0)~~~~~~~~~0 ~~~~0 0CDW)\ o 0) z 0)) 0) 0000 r 00 H00 0 ~~~~~~~~~~~~~~~~~~~~0 ON\OO 0)v 0 0) a)l a) 00 C,\ 00 S -s 2 o ~~~~~~~~~~~~~~~~~~~0c ~~~~~~~ 0 ~ C\0 C\C\~ a~ ) ~: 4-0 z CD C t 0) CI)~ ~ ~~C z u~ 0)~~~~~~~~~~0 0)~~~~~~~~~0 E~.-Q cf) 0) C0)O S~~~~ CI) C) = = (0).- rlrlrlrl0 C 0 I C4 Oct~

20 260 THE AMERICAN NATURALIST TABLE 3 CHANGE IN OPTIMAL SPEED IN THE FACE OF DECLINING RESOURCES Parameters and Reserves Size-Independent Size-Dependent Fig. 3: x q: x m: x h: x R: NOTE.-With 10 time units left until the payoff, the optimal speed was calculated with constant future resources and resources declining uniformly by 10% of the initial resource level per time unit. The effect of declining resources was determined for low and high levels of reserves, two values for each of the parameters, and the size-dependent and size-independent cases. The initial parameters were those used in fig. 3, and then each of the parameters was doubled in turn. DISCUSSION Behavioral Consequences Most animals have behavioral control over fundamental aspects of life-style such as activity schedules and choice of habitat. There is extensivevidence that behavioral adjustments, for example, in movement speed, time active, or habitat use, in part occur in response to changes in perceived risk of predation (see review in Lima and Dill 1990). Similarly, animals alter these behaviors in response to variation in food quantity, quality, and spatial distribution (see Stephens and Krebs 1986 for theory and examples). Because changes in predation risk and resource abundance affect growth potential and the probability of mortality differently, adaptive responses require a complex balancing of the gains from different foraging behaviors with the expected risk of predation (Gilliam 1982; Sih 1982; Abrahams and Dill 1989). There is little doubt that these fundamental conflicts are confronted by most mobile animals and that they have significant fitness consequences. The family of models that we present provide some predictions of adaptive responses in components of activity with changes in resources or resource acquisition capacity, elements of mortality risk, and constraints time horizon. Moreover, the fitness criterion is derived from considerations of size and growth, and, since size dramatically alters many of these relations (Werner and Gilliam 1984), the adaptive balance is intimately intertwined with changes in size over ontogeny. Previous theoretical work on adaptive control of activity has largely considered

21 CONSEQUENCES OF GROWTH-MORTALITY TRADE-OFFS 261 the two components, speed and time active, in isolation. Considerations of optimal movement speed have concentrated the speed that maximizes the rate of net energy gain, that is, energy intake minus the metabolicosts of movement (Ware 1975; Norberg 1981; Speakman 1986; Dunbrack and Giguere 1987). These analyses in general predic that movement speed should increase with resource density. An important contrast, then, is that our model predicts that, when predation risk is incorporated and gain is curvilinear, speed can decrease with resource density (see also Abrams 1990, 1991). When both rate of resource acquisition and mortality risk increase linearly with speed, foragershould switch from the sit-and-waitactic to active search when resources increase. In addition to risk, metabolic models of optimal speed typically do not include constraints due to digestive time or feeding motivation (related to gut fullness), which also lead to decreases in speed with increases in resources. The adaptive control of time devoted to resource acquisition has relevance to a wide variety of important ecological questions. Abrams (1982, 1990) has considered how control of time active alters the functional response of a forager and the consequences to different life histories (Abrams 1983, 1991) and to food web interactions (Abrams 1984). He has generally assumed that time active (or foraging effort) has positive effects on reproductive rates but negativeffects on survival rates and has determined optimal activity times by maximizing the difference between birth and death rate or reproduction times survival rate. Using an array of different-shaped birth rate functions, he has shown that time active or foraging effort can increase or decrease with resource level and that it is more likely to decrease as the birth rate function becomes more convex (Abrams 1991). He also shows that life history (iteroparity vs. semelparity) may affecthe nature of the response. When death rate is directly proportional to time active and reproductive rate is a saturating function of time active, his results are similar to ours (Abrams 1990). He has not examined the effects of individual growth or other effects of size or age structure. It is very important to isolate foraging speed from proportion of time spent foraging; the two are often conflated in the empirical literature (McLaughlin 1991). Time active and foraging speed may also themselves be traded off in response to environmental changes. For example, there may be important constraints trading off speed and time active. First, the efficacy of reducing speed in the presence of a predator is limited by the relative speeds of predator and prey. Only when relative speed of the prey is high compared to that of the predator can the prey have much effect on encounterates by adjusting travel speed (eq. [2]; Gerritsen and Strickler 1977). Second, it is likely that the incremental cost of predation risk is higher with increases in speed than with increases in time active. If detectability increases with speed, as will often be the case (at least over some range of speeds), this will favor adjustments in time active because the incremental costs in risk for time active are more likely linear. Third, the incremental costs in terms of metabolic expenditures can be greater for increases in speed than for increases in time active. For example, the energeticost of movement in the aquatic environment generally increases with the second power of speed (Weihs 1973). The incremental cost of time active is likely close to linear, so this would heavily

22 262 THE AMERICAN NATURALIST favor extending time active over increasing speed. The case is not as clear for terrestrial organisms in which the energeticost of movement is roughly linear with speed (Taylor et al. 1982). Day-night differences also set additional boundaries on our expectations. Clearly, factorsuch as thermal constraints of prey or the activity or efficacy of predators change with day and night. Thus, it is easy to visualize, for example, that aquatic animals vulnerable to visual predatorsuch as fish would favor increasing time active as resources declined as long as this increase occurred at night when visual predators were relatively ineffective. As time active increased to encompass the entire night period, however, the increased incremental cost of speed might be favored rather than crossing the day-night boundary, because of the large increase in danger of being active during the day. The sort of models that we have proposed could be usefully extended to examine the very important question of day and night patterns of activity. We are aware of only one published study in which both time active and movement speed were measured in a contex that permits us to evaluate some of our predictions. Kohler and McPeek (1989) quantified both components of activity with Baetis mayfly and Glossosoma caddis fly larvae while manipulating the presence of a predator (sculpins Cottus), resource levels, hunger, and time of day. Baetis was more vulnerable to the predator and adjusted both components of activity in the presence of the predator as predicted: it spent less time active (on the top of the substrate where food was presented), and moved more slowly in the presence of the predator. Resources were varied over 1.8-fold and both species reduced movement rate as resource density increased, as predicted. Contrary to predictions, both species spent more time active as resources increased. Thus, there was a strong inverse relation between time active and movement rate. It is plausible, then, that both species were trading off or mutually adjusting time active and movement speed as discussed above. Changes in movement speed were not simply a consequence of feeding on denser resources; movement speed between patches of food also declined with resource level. It is difficult to determine whether overall activity or effort increased or decreased with an increase in resources; for example, the product of time active and movement speed is about the same at both food levels. The sculpin is a visual, diurnal predator, and time of day therefore should principally affect predation risk. The responses of the highly vulnerable Baetis supporthis assumption; it reduced time active during the day but not at night in the presence of the predator. There was no effect of day or night on movement speed. We speculated above that time active at night would be the preferred activity adjustment with visual predators, and the animals again may be responding in ways that weigh time active versus movement speed. When hunger was increased (decreased reserves), Baetis increased time active as predicted by the dynamic programming model. There again was no effect on movement rates, though we would expect that responses would be similar to those for movement speed. It is possible that the model does not adequately representhe situation or that the species are again mutually adjusting time active and movement speed, preferentially increasing time active either because this is the safer, or less meta-

23 CONSEQUENCES OF GROWTH-MORTALITY TRADE-OFFS 263 bolically expensive, parameter to adjust. Glossosoma exhibited parallel responses to Baetis, but the responses generally took the form of proportion of population active rather than time active. When we know the factor by which resources or predators have been increased in experiments, we can quantitatively compare results to model predictions. For example, if resources or predation risk are multiplied by a factor, A, then speed should decrease by a factor I/iV (cf. eq. [7b]). In Kohler and McPeek's (1989) study, the high food level was 1.83 times that of the low food level, and so speed should decrease by a factor of Using an average of the proportional decrease in all of the speed measures presented, we found the reduction in speed at the high resource levels corresponds very well to predictions: 0.72 for Baetis and 0.79 for Glossosoma. We cannot compare the predictions for increase in mortality risk because estimates of the magnitude of increase are not available. We also have data from a number of experiments with anuran tadpoles in which time active at different resource levels was estimated by quantifying the fraction of the population swimming at an instant in time (see, e.g., Skelly and Werner 1990; Werner 1991, 1992a). If feeding or growth rate is approximately proportional to resource levels over the range considered (e.g., eq. [8]) or h is very small and speed does not change, then p also should decrease by the factor 1/ V as above. In a competition study (Werner 1992a) with the wood frog (Rana sylvatica) and northern leopard frog (Rana pipiens), activity levels were quantified at two resource levels that differed by a factor of Given the above assumptions, time active should decrease by a factor of The results for the wood frog were very close to predictions, with activity decreasing by a factor of 0.79 at the highe resource level; those for the leopard frog were not as close at These data are encouraging and indicate that the responses of animals of widely divergentaxa often closely match those predicted by our models. A number of additional predictions from the above models have been empirically confirmed for single factorstudied in isolation. For example, regardless of whether we are concerned with movement rate or time active, variation in the risk of mortality with activity affects predictions in the same way. Increasing the mortality risk of additional activity (e.g., increasing predator density or reducing cover) should select for a reduction in movement rate and proportion of time active. Conversely, reducing predator density should lead to an increase in activity. Considerablexperimental evidence of responses by prey to predators conform to this prediction, the majority of which can be interpreted as some measure of time active (reviewed in Lima and Dill 1990). Factors that alter prey vulnerability should have similar effects on activity. Several experimental studies also report a decline in search activity with increases in resource density, which conforms to the predictions here. For example, a series of studies by Formanowicz and colleagues indicate a substantial decrease in search activity with increases in resource density in taxa as divergent as dytiscid and hydrophilid beetle larvae (Formanowicz 1982; Formanowicz and Brodie 1989), centipedes (Formanowicz and Bradley 1987), scorpions (Formanowicz et al. 1991), and matamata turtles (Formanowicz et al. 1989). These results are unconfounded by differences in number of prey eaten, and therefore responses

24 264 THE AMERICAN NATURALIST are not due to satiation effects or processing constraints. Johansson (1991) reports a similar pattern with damselflies. Walde and Davies (1984) show that stoneflies increase the proportion of time spent moving with initial increases in density of prey but then reduce time moving at higher prey densities. It has also been shown (Werner 1992a) that activity is low in wood and leopard frog larvae when resources are high and increases as resources decline. A clear counterexample is provided by Huey and Pianka (1981), who show that the gecko Ptenopus garrulus switched from sit-and-wait tactics to widely foraging during termite swarming periods, which would be predicted if f(s) and mortality curves were linear. Thus, the results generally appear to conform to the models that incorporate risk tradeoffs (and do not conform to models that ignore mortality risk), but it is difficult to relate these results directly to model predictions. Typically, the data are the number of moves per unit time (or in some cases distance moved), and samples are taken at fairly long intervals (or, in the case of Werner 1992a, the number of individuals moving in a population). Thus, it is not possible to isolate the mode of response, that is, the speed of movement or time spent active, though in many cases the data would suggest something closer to the time spent active. More care is required in future tests of these ideas to clearly separate these two components of activity. The predictions of the analytical models concerning variation in background mortality rate are particularly intriguing; increases in background mortality should lead to increases in either movement rate or time spent active, or in both. Fraser and Gilliam (1987) reported that guppies and Rivulus hartii populations found in the presence and absence of piscivorouspecies reacted to the presence of the predator in an unexpected manner. In both species, populations found in the presence of the predator were more "tenacious," or bolder, in maintaining feeding rates when exposed to the predator. It is possible that the chronically higher mortality rates that these populations are exposed to have selected for higher activity rates in the presence of the predator if a substantial fraction of this mortality is unrelated to activity level, that is, if there is an elevated background mortality. An alternative is that these populations have been selected for lower vulnerability to predation and therefore have lower m. Incorporating time constraints does not significantly alter the sensitivity of the optimal movement speed to the parameters. Increasing the mortality rate with speed, handling time, or resource density reduces optimal movement speed, while increasing the background mortality increases optimal movement speed (when growth and/or survival rates are size-dependent). For any individual, optimal behavior will depend not only on these parameters but will also be contingent the present state (the result of previous behavior) and expectations of future payoff. The payoff depends on both the present state (X) and time remaining (t) until the payoff is realized. Individuals with large reserves or long periods to gather those reserves have a higher expected payoff. All of this expected payoff can be lost by foraging a risky way, while only a small increment to the payoff can be gained. In the size-dependent case, this tendency for reduced activity is opposed by increased efficiency of gathering resources and reduced risk of mortality. When foraging efficiency is very strongly size-dependent, the expectation

25 CONSEQUENCES OF GROWTH-MORTALITY TRADE-OFFS 265 of future payoff is very high and optimal foraging speed again declines with size. Increasing the size dependence of survivaleads to increased foraging speed at large sizes, and this is strikingly true at moderate values of the size dependence of foraging success. This suggests that when mortality risk declines with size, we should observe that animals increase foraging activity over ontogeny, and there is evidence for this prediction (Stein and Magnuson 1976; Lawler 1989; E. E. Werner, unpublishedata). However, many prey are so much smaller than their predators (e.g., zooplankton and planktivorous fish) that increases in size will not decrease mortality risk (and may even increase it). In this situation we would expect that activity would decline with body size. The tube-dwelling chironomid Chironomus tentans fits this pattern, with large individuals being much less active than small ones (Macchiusi and Baker 1991). The p/g criterion was explicitly developed to examine the consequences of decisions in size-structured populations (Gilliam 1982; Werner and Gilliam 1984). Thus, one of the important implications of our results is the inferences that can be made about expected changes in behavior with ontogenetic or developmental size changes. Size affects virtually of the parameters we have been concerned with. Size affects encounterates with prey by changing encounteradius (see, e.g., Eggers 1977; Wright and O'Brien 1984); size affects vulnerability to predators as size refuges are approached (see, e.g., Kusano 1981; Werner and Gilliam 1984); size affects the maximum attainable speeds of animals (see, e.g., Peters 1983) and the scaling of resource capture abilities (see, e.g., Werner and Gilliam 1984; Sebens 1987); and size affects constraintsuch as digestive times or throughput rates (see, e.g., Townsend and Calow 1981). Size also clearly affects the temporal distribution of strategies when there are time constraints. For example, vulnerability of many animals declines with size, which has prompted much speculation on selection for growth rates (Wilbur 1980; Werner and Gilliam 1984). If background mortality is relatively independent of predators and prey size, then increases in prey size over ontogeny will principally reduce m, and therefore activity should increase with size. Lawler (1989) presents evidence that hatchlings in four anuran larvae were largely immobile but became more active as they became larger. Both bullfrog and green frog tadpoles increase activity levels over a size range from 0.05 to 1 g (E. E. Werner, unpublished data). Stein and Magnuson (1976) demonstrated the same pattern in crayfish correlated with changes in their vulnerability to fish. If an animal can become more active as it gets larger, this may also contribute to its competitive ability (Werner 1991) and thus may be a component of the size-related competitive advantage of larger individuals reported in some taxa (see, e.g., Wilbur 1984). If changes in size, however, principally reduce background mortality, then activity could decrease with size. In cases such as with zooplankton, for which predation risk can increase with size, time active in the epilimnetic zone may decrease with size over ontogeny. Changes in activity levels over ontogeny will be an excellent place to examine our predictions, since the confounding effects of interspecific comparisons are eliminated. Maximum sustainable speed increases as a direct function of length or size in many organisms (Peters 1983). Thus, the range in capability to adjust speed and

26 266 THE AMERICAN NATURALIST its relation with time active is greatly extended as an animal grows. A detailed analysis of how this should occur will have to explicitly examine the way metabolic costs increase with size as well. Resource-processing abilities will also change with size. Handling times often will decrease with size if the consumer does not shifto much larger prey, and gut throughput (proportional to 1/V) will increase with size. Both of these changes will also select for an increase in activity levels and therefore will reinforce the expected trend due to the decrease in m. The exact scaling relations for these parameters will be needed to separate their effect on ontogenetic activity patterns. Population and Community Consequences The trade-offs that we have explored have extensive implications for species dynamics and species interactions (Abrams 1984, 1990). For example, for predators to regulate prey numbers, predation rates must be density-dependent (reviewed in Sinclair 1989). Consequently, the functional response is an essential componento inferences about the stability of predator-prey relations. If the mortality and growth rate functions of an animal are approximately linear with activity as in equations (1) and (4), then small changes in prey density can cause a shift in the foraging mode from sit-and-wait to active search, with an increase in prey capture rate far larger than the increase in prey density (see also Abrams 1982). Population regulation may be possible over this density. Though the model predicts a saltatory shift in foraging mode by the individual, the functional response determined from a collection of predators will more closely match the Type III curve because of individual differences either in the parameters of the growth and mortality curves or in individual estimates of how these parameters are changing (or both) (Stephens 1985). Such effects of adaptive behavioral responses in time active on the functional response have been extensively explored by Abrams (1982, 1990). The most widely employed functional response, the Holling disk equation, assumes that time foraging, capture success, and handling time are constant and independent of prey density. Abrams has examined the consequences of animals' adaptively varying these parameters with prey density to the shape of the functional response. Such behavioral adjustments can greatly alter details of the Type II functional response or convert it to a stabilizing form. These studies suggest alternative mechanisms for the traditionally recognized shapes of the functional response. Consideration of the manner in which speed and time active may be mutually adjusted-as we have indicated here-simply adds to these possibilities. Studies of the manner in which animals adaptively vary parameters of the functional response will lead to more useful and predictive predator-prey models (Abrams 1990). That is, we will be able to construct mechanistic models of how the shape of the functional response is determined by adaptive behavior of the predator in the face of changes in prey densities-adaptive behavior that clearly occurs widely in predators. Of course, these behaviors then affect growth, birth, and death rates of both the predators and their prey and thus have considerable importo the dynamics of both. Because activity is a trait associated with competitive ability as well as risk to

27 CONSEQUENCES OF GROWTH-MORTALITY TRADE-OFFS 267 predation, the trade-off that we have explored in this article is likely central to the interaction of competition and predation in many communities. Many experimental studies have demonstrated how the interaction of competition and predation can determine the species composition of communities or the distribution of coexisting species among habitats (see, e.g., Paine 1966; Hall et al. 1970; Lubchenco 1978). These studies generally suggesthat the interactions are predicated on trade-offs; for example, preferred prey are often superior competitors. If the mechanisms underlying trade-offs in competitive ability and vulnerability to predators are general, they can provide relatively simple constructs for predicting community organization among groups of species with similar life-styles. Tradeoffs mediated through activity may be a very general such mechanism. Species (or larger individuals, for that matter) with lower vulnerability to predators can be more active when predators are abundant and thus have greater access to resources than species (or small individuals) whose mortality is a stronger function of activity. This could lead to competitive displacement when the relatively invulnerable species depletes resources. If this invulnerability is bought at some cost (e.g.; armament, shells, toxins), the balance of competition and predation can subtly shift with changes in predator densities and resources so that alternatively one or the other species is favored or coexistence occurs (see, e.g., Vance 1978). The importance of working out the details and predictive basis of the mechanisms behind these trade-offs-as we have done here for activity-is that these details then provide the mechanistic basis for predicting the specific quantitative consequences of the trade-off that then can be incorporated into higher-level models that predic the consequences of the interaction of competition and predation in communities. Alternatively, the trade-off we have examined can lead to separation of species into different habitats in which more active species dominate the habitats where predators are scarce, but less active species dominate those where predators are common. There are empirical data on several systems that appear to illustrate these points. For example, anuran species often replace each other along the gradient of permanento temporary waters. The former usually contain higher densities of predators (see, e.g., Woodward 1983; D. K. Skelly, unpublished manuscript). Temporary pond species are generally described as more active and better competitors than permanent pond species (see, e.g., Woodward 1982), but they are more vulnerable to predators (see, e.g., Kats et al. 1988; E. E. Werner and M. A. McPeek, unpublished manuscript). It has been shown (Werner 1991, 1992a) that relative activity levels are associated with competitive outcomes in two pairs of ranid species and with their vulnerability to an odonate predator (E. E. Werner, unpublishedata). Similarly, the more active Hyla versicolor emerges from cattle tank competition experiments 15 d earlier at a much greater size than does Hyla (syn. Pseudacris) crucifer but also is more vulnerable to predators (Lawler 1989). Thus, changing predator densities or types along the habitat gradient likely alters the balance of competition and predation such that species replacements occur; that is, selection for differences in activity may account for some of the diversity that we see in anuran larvae on the permanent to temporary water gradient.

28 268 THE AMERICAN NATURALIST The potential generality of this scenario is illustrated by studies of species of damselflies segregated between ponds with and without fish along the same gradient (McPeek 1990a, 1990b). Enallagma species found in the absence of fish were more active, and McPeek found that not only did the two groups of species differ in the frequency of walks, swims, and so on, but they also differed strikingly in walking speeds when moving. Species found with fish moved extremely slowly, often cautiously moving one leg at a time. McPeek (1990a, 1990b) presents experimental evidence that these behavioral differences are the primary determinant of the differences in species distributions between the two habita types because of the interactions with different predator types. G. A. Wellborn (unpublished manuscript) has noted similar differences between amphipod populations in fish and fishless ponds. Thus, these systems in many ways parallel the case of the anurans on the same gradient. Ecologists have long investigated the role of single morphological physiological traits in determining differences in species distributions on environmental gradients. It appears that, despite the inherent flexibility of behavior, simple behavioral traits organized around trade-offs are important as well. Some of the most important consequences of adaptive behavioral responses to interactions among species are the indirect effects or higher-order interactions that result. These indirect effects may be mediated through changes in abundance due to differences in vulnerability to predators and to competitive abilities related to activity rates (numerical indirect effect; Miller and Kerfoot 1987). Alternatively, facultative behavioral responses, for example, responses in the presence of a predator, can cause qualitative changes in the nature of competitive interactions among prey (behavioral indirect effect or higher-order interactions; Miller and Kerfoot 1987). The ease with which behavior can be adjusted in the presence of a predator suggests that the activity rate trade-off could lead to widespread higher-order interactions (Werner 1992b). For example, if species respond directly to a predator by changing activity rate and this response differs among species, then the nature of the competitive relation changes. It has been shown (Werner 1991) that such an interaction results in a higher-order interaction that changes competitive interactions between two anuran larvae in the nonlethal presence of a predator. Moreover, competitive interactions will generally lower resource levels in the environment and therefore should select for higher speeds or increased time active. Consequently, the absolute mortality rate on the population due to predation should concomitantly increase. Thus, competitive interactions may indirectly affect predation rates through at least two avenues: by lowering resources, growth rates are reduced, which prolongs the time spent in vulnerable sizes; by increasing activity, mortality rates are also increased. In the absence of manipulative experiments, it may appear that competitive outcomes are mediated by resources because little predation occurs. However, when predators are absent, quite a different outcome is possible. McNamara and Houston (1987) have pointed out that conclusions about the nature of population regulation cannot be drawn on the basis of the source of mortality for precisely the same reason. Abrams (1991) has shown theoretically that indirect effects arising from adaptive adjustments of time active by a consumer can be comparable in magnitude to or

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