Flight initiation distance is the distance separating predator and prey when escape begins. The optimal flight initiation distance occurs where expected postencounter fitness is maximized, which depends on the prey's initial fitness, benefits obtainable by not fleeing, energetic escape costs, and expected fitness loss due to predation risk. In current optimal escape theory, prey die when contacted by a predator. We explore effects of variable lethality, L, the probability of being killed on contact. Optimal flight initiation distance increases as lethality increases, matching expectations that prey should not flee when contact entails no fitness loss but should be increasingly wary as expected fitness loss on contact increases. Addition of lethality improves the ability of optimal escape theory to predict effects of factors affecting escape ability. Autotomy, the voluntary shedding of tails or other expendable parts as a last-ditch defense to permit escape, provides an example. After autotomy, running speed decreases in many prey, lethality increases because autotomy cannot be used again until the lost part has regenerated, and ability to obtain benefits may decrease due to reduced social status and foraging ability. These changes favor longer flight initiation distance but lowered initial fitness after autotomy has the opposite effect. Optimal escape theory including a lethality term clarifies how autotomy may lead to increase or decrease in flight initiation distance depending on the balance of its multiple effects. Effects of additional factors that may alter multiple parameters of the model, including age, sex, reproductive condition, injury, disease, and parasitism are discussed.
Current optimal escape theory predicts escape decisions based on economic considerations about effects of predation risk, current fitness, and cost of escaping (Cooper and Frederick 2007,a). We extend it to account for effects of variation in the lethality of predators on escape behavior and show how the conceptual framework of the extended model allows nuanced interpretations about effects of states or events that have highly complex effects on fitness components affecting escape decisions. Autotomy, a defense in which an expendable body part is detached as a defense of last resort when a prey is overtaken by a predator, is considered in detail as an example. The extended model makes explicit several effects of autotomy, some of which are opposite, limiting the utility of predictions based on single effects of autotomy, such as reduced escape speed or inability to use autotomy in subsequent encounters with predators. We briefly note other states that may alter more than one of the economic bases of escape decisions.
Optimal escape theory was recently developed to predict flight initiation distance, which is the distance separating a prey from an approaching predator, based on effects of the prey's initial fitness, predation risk, and costs of escaping on expected fitness after the encounter (Cooper and Frederick 2007,a). These costs of escaping correspond to benefits that might be obtained by not fleeing but must be forgone to escape. Like an earlier graphical escape model (Ydenberg and Dill 1986), the optimal escape model assumes that a prey has detected an approaching predator and does not flee immediately but monitors its approach (Cooper 2008). Consideration of the effects of starting distance, which is the distance from prey when the predator begins to approach, led to recognition of a zone beyond which prey do not detect a predator or may detect it but not react because risk is at background level (Blumstein 2003; Stankowich and Coss 2006). In optimal escape theory, an encounter cannot begin beyond this distance.
Optimal escape theory predicts that prey initiate escape attempts at a distance that maximizes the prey's expected lifetime fitness after the encounter (Cooper and Frederick 2007,a). In the Ydenberg and Dill (1986) model, which is an economic, but not optimality, model, the predicted flight initiation distance occurs where cost of fleeing, primarily opportunity cost, is equal to cost of not fleeing. Prey can make more profitable escape decisions under the optimal escape model than the earlier model. Their predicted flight initiation distances differ quantitatively. However, the 2 models make similar qualitative predictions about effects of many factors that influence predation risk and cost of escaping on flight initiation distance (Cooper and Frederick 2007,a). In some cases, especially related to large reproductive benefits, the predictions may differ qualitatively.
The optimal escape model has the advantages of permitting the prey to make better escape decisions, allowing prey to accept being killed if benefits to lifetime fitness are large enough and being mathematically explicit (Cooper and Frederick 2007,a). Because it is explicit, it can be modified to apply in situations not considered in its original form. Because effects of fleeing and not fleeing are specified and the prey's initial fitness is taken into account, optimal escape theory provides a conceptual framework for understanding how single factors that alter multiple traits relevant to escape in complex ways may alter escape decisions.
Flight initiation distance is predicted to increase as predation risk increases (Ydenberg and Dill 1986; Cooper and Frederick 2007,a). It decreases as opportunity cost, the loss of benefit that might be gained if the prey does not flee, increases. Other factors that affect escape decisions are the prey's fitness at the beginning of the predator–prey encounter (initial fitness) and escape costs, which are costs due to energetic expenditure and risk of injury due to escape efforts.
In one version of optimal escape theory, fitness benefits that may be gained by not fleeing are lost if the prey is killed (Cooper and Frederick 2007,a). These include all nonreproductive benefits and reproductive benefits lost if the parent is killed, such as fertilizations obtained during a predator–prey encounter that do not contribute to lifetime fitness because the offspring cannot survive without the parent. In this model, the distance between predator and prey is d, the expected fitness for a prey at a given distance from the predator is F(d), the prey's fitness at the outset of the encounter is F0, the benefit that the prey gains by delaying escape until the predator has approached to a given distance is, B(d), and the escape cost is E(d). The model represents the probability of being killed as e−cd, c > 0, with corresponding probability of survival, P(d) = 1 − e−cd. The benefit function is calculated as a proportion of the maximum benefit, B*, that can be obtained by not fleeing, that is, when d = 0. This function is the product of B* and , where dd is the distance at which the prey detects the predator and the exponent n affects the rate of accumulation of benefits. We have slightly modified the benefit function from its form in Cooper and Frederick (2007,a), which reverses the concavity for n ≠ 1 but otherwise bears a qualitatively similar relationship to d. The energetic escape cost function is E(d) = fdm, where f and m are constants. Expected fitness at a particular distance from predator is
Lethality of predator
Predators that successfully overtake prey may kill the prey, but often prey escape even after being physically contacted by predators. Indeed, the possibility of surviving after being overtaken by the predator is the basis of numerous antipredatory defenses, including thanatosis (letisimulation = death feigning), deimatic behavior (startling displays; Cott 1940; Ruxton et al. 2004), body armor (Losos et al. 2002; Caro 2005), behaviors, and morphological traits that deflect attacks to armored or expendable parts (Cott 1940; Cooper 1998a, 1998b; Ruxton et al. 2004; Caro 2005) and physiological defenses such as venom resistance (Caro 2005), retaliatory use of weapons, strength, and speed (Edmunds 1974; Caro 2005).
The degree of lethality differs among pairs of predator and prey species and, within a given predator and prey species combination, among pairs of different individual predators and prey. In our original model (Cooper and Frederick 2007,a), predators were considered to be completely lethal, so that prey all die when d = 0. However, prey contacted by predators sometimes escape. At the opposite extreme, an approaching animal might never cause death or any decrease in fitness even at d = 0 and would not be expected to elicit escape. Between the extremes, predators may have continuously varying degrees of lethality, and it may be anticipated that prey take this into account while assessing risk leading to escape decisions.
To model effects of lethality, we multiply the risk of being contacted by the predator by the constant L, which is the proportion of encounters in which contact proves fatal. The survival function in the model becomes (1 − Le−cd). The optimal flight initiation distance is
The effect of lethality is as might be expected: flight initiation distance is zero for a nonpredator or would-be predator that can never harm the prey and increases as lethality increases up to the maximum value associated with a uniformly lethal predator (L = 1.0). In Figure 1, the top line in each panel shows the fitness that might be attained by a prey that has not fled, which is the sum of its initial fitness and fitness gained during the encounter. The bottom curve shows the probability of survival, which declines appreciably farther from the predator as lethality increases. The solid middle curve gives expected fitness in relation to distance. The vertical line is drawn from the highest point on the fitness curve to emphasize the connection between the optimal flight initiation distance, d*, and the maximum expected fitness.
The predation risk and expected loss of fitness are directly proportional to lethality. We have presented L as probability of death on contact, but predators sometimes have important sublethal effects, notably injury (Arnold 1988; Heithaus et al. 2002; McCarthy and Dickey 2002). Effects of sublethal predation on escape decisions can be included in the model by using a more general interpretation of L as the expected fitness loss on contact with the predator, including lethal and sublethal contact. Indirect nonlethal effects of contact, such as behavioral compensation and vigilance, that impact prey fitness after encounters with predators (Lima and Dill 1990; Lima 1998; Cresswell 2008) are not modeled.
Because the probability of being killed is the product of L and e−cd, joint variation of L and c strongly affects the optimal flight initiation distance. Optimal flight initiation distance is the greatest when L is large and c is small, and decreases as L decreases and c increases (Figure 2). Thus, not only the rate constant for probability of capture included in our previous model but also the predator lethality are predicted to strongly influence escape decisions. Because the optimal flight initiation distance reaches zero at −lnL/c using the linear benefit function (n = 1), setting the lower bound of F′ = −lnL/c excludes negative optimal flight initiation distances.
Empirical effects of autotomy
Because a prey that has autotomized a body part cannot escape by using autotomy again unless and until the body part has been regenerated, lethality increases after autotomy. With the addition of the lethality term, the optimal escape model can be used to understand how multiple effects of autotomy on model parameters can lead to increase, decrease, or no change in flight initiation distance. Before discussing predictions of the model, we briefly review our knowledge of costs and benefits of autotomy and its effects on escape behavior.
Autotomy is interesting to ecologists, behaviorists, and evolutionary biologists because this defense is an effective means of escaping even when overtaken by a predator, yet is costly in many ways and has varying effects on subsequent antipredatory behavior (Arnold 1984, 1988; Fleming et al. 2007; Bateman and Fleming 2009). Some investigators have predicted that flight initiation should increase after autotomy because sprint speed is reduced after loss of body parts (e.g., Punzo 1982; Guffey 1999; Bateman and Fleming 2005; Cooper et al. 2009). The prediction of longer flight initiation distance has been verified for striped plateau lizards, Sceloporus virgatus (Cooper 2007; Cooper and Wilson 2008), but the opposite effect occurred in ground skinks, Scincella lateralis (Formanowicz et al. 1990). The shorter flight initiation after autotomy in ground skinks was attributed to a change in escape strategy toward greater reliance on crypsis due to immobility to reduce probability of being detected (Formanowicz et al. 1990).
Because autotomy has several effects relevant to escape decisions, increase or decrease in flight initiation distance is possible after autotomy without change in escape strategy. We discuss the combined effects of autotomy on subsequent escape ability, fitness, and ability to succeed in activities that enhance fitness. The multiple effects of autotomy may lead to failure of predictions based solely on reduced speed or other single factors, making it difficult to predict the effect of autotomy on flight initiation distance.
Increase in lethality is a major consequence of autotomy (Congdon et al. 1974; Cooper and Vitt 1985, 1991; Downes and Shine 2001; Cooper and Vitt 1991). Once a structure such as a lizard's tail has been autotomized, capability to use autotomy as a defense may be lost permanently unless and until regeneration occurs. The large increase in lethality is likely to have a major impact, increasing the optimal flight initiation distance. Some prey, such as sea stars and lizards (Arnold 1988; Fleming et al. 2007; Cooper and Smith, 2009) exhibit economy of autotomy, autotomizing only the portion of the appendage distal to the point contacted by the predator. Economy of autotomy is predicted to reduce effects of autotomy on lethality, fitness, running speed, and trade-offs between predation risk and fitness-enhancing activities.
Due to injury and loss of energy stored in the shed part, expected fitness declines after autotomy. Clark's (1994) asset protection principle states that prey with greater assets protect them more carefully, implying that flight initiation distance increases with expected fitness. Optimal escape theory (Cooper and Frederick 2007,a) incorporates this effect, predicting shorter flight initiation distance by prey having lowered fitness after autotomy. With the exception of energy lost in the autotomized part, energetic cost of escape for a prey that suffers autotomy may be identical to that for prey that escapes without resorting to autotomy. If so, the escape cost function for prey deploying autotomy is identical to that for nonautotomizing prey but with a large upward step at d = 0 when the tail or other body part is lost. This cost of autotomy incurred in one encounter is carried over to later encounters as a decrease of initial fitness.
Other effects of autotomy may counteract or increase the effect of decreased fitness on flight initiation distance. In many species of lizards and arthropods, loss of tails or appendages is accompanied by a large decrease in maximum running speed (Ballinger et al. 1979; Punzo 1982; Guffey 1999; Bateman and Fleming 2005; Cooper et al. 2009), which increases risk of being overtaken by a predator. This effect is predicted to lead to increased flight initiation distance (Ydenberg and Dill 1986; Cooper and Frederick 2007,a). Some lizards that store large amounts of lipid in their tails run faster after losing their tails (Daniels 1983). For such lizards, decrease in predation risk is predicted to favor shortened flight initiation distance.
Benefits obtainable while a predator approaches may be strongly influenced by autotomy. Flight initiation distance decreases greatly during social encounters (Cooper 1997, 1999; Díaz-Uriarte 1999) in intact individuals that may gain by defeating rivals and courting females. However, in some lizards, autotomy causes a large decrease in social status (Fox and Rostker 1982; Fox et al. 1990), leaving little potential to increase fitness by engaging in social behavior. In these species, autotomized individuals are predicted not to trade predation risk against social benefits in a manner leading to decreased flight initiation distance.
Flight initiation distance of intact individuals decreases in the presence of food, indicting a trade-off between predation risk and foraging opportunity (Cooper 2000; Cooper and Pérez-Mellado 2004; Cooper et al. 2006). Foraging skills decline after autotomy in some species (mantis shrimp, Berzins and Caldwell 1983; crabs, Smith and Hines 1991; spiders, Riechert 1988; Brueseke et al. 2001; opiliones, Guffey 1999; sea stars, Barrios et al. 2008), indicating that benefits obtainable during a predator's approach are reduced. In such prey, smaller decrease in flight initiation distance in the presence of food is predicted for autotomized than intact prey.
However, the importance of food may increase due to greater risk of starvation after autotomy (Daniels 1984). Reproductive output and growth may be negatively impacted by the initial energy loss and subsequent allocation of available energy among them and regeneration, requiring more energy (Arnold 1988; Fleming et al. 2007; Bateman and Fleming 2009). Therefore, optimal escape theory predicts shorter flight initiation distance by autotomized prey. Foraging abilities and nutritive needs, thus, can have opposing effects on flight initiation distance, causing flight initiation distance to increase, be unchanged, or decrease after autotomy.
Energetic costs of escape due to locomotion during brief predator–prey encounters not involving long chases are considered to be small compared with opportunity costs (Cooper and Frederick 2007a). However, after autotomy, energetic costs might increase due to inefficient locomotion or decrease due to the lower body mass to be moved. Both effects might occur simultaneously with opposing, but very small, net effect on flight initiation distance.
Autotomy, lethality, and optimal escape
Autotomy clearly has multiple effects relevant to escape behavior. We next use the extended optimal escape model incorporating the effect lethality to interpret and predict effects of this complex defense. If nothing else were changed as a result of autotomy, its effect would be to increase the coefficient L for lethality in Equation 3 and, thus, increase optimal flight initiation distance as shown in Figure 1. However, the multiple consequences of autotomy outlined above may augment or counteract the effect of increased lethality. Changes in running speed and ability to obtain benefits in some circumstances lead to greater optimal flight initiation distance in ways readily interpretable in Equation 3. Decrease in running speed entails increase in predation risk at a given distance, which corresponds to a decrease in c and acts in concert with increased lethality to increase flight initiation distance.
Another possibly major contributor to increase in optimal flight initiation distance is decrease in benefits obtainable during encounters. If autotomized individuals forage less efficiently, B* may decrease and n » 1 may increase or n in the range 0 < n < 1 may decrease, lowering B(d). For the linear benefit function shown in Figure 1, (n = 1), B(d) is reduced for all d when B* is reduced. In social contexts, B(d) after autotomy may at worst be zero for all d, maximizing the optimal flight initiation distance. The effects of autotomy on ability to obtain benefits may have strong but contextually limited effects on optimal flight initiation distance. For example, when no food is visible, the curve relating benefits to distance between predator and prey may be nearly identical for intact and autotomized ambush foragers. In general, decrease in ability to obtain benefits, if it were the only effect of autotomy, would be to increase the optimal flight initiation distance, as shown in Figure 3 when the only effect of autotomy is diminution of ability to gain benefits.
Other factors that may lead to shorter optimal flight initiation distance after autotomy include lowered fitness and, for at least a few species, increase in running speed (Daniels 1983). The decline in expected fitness after autotomy corresponds to a lowered value of expected fitness at the outset of later encounters, F0. Shorter flight initiation distance is predicted by the decrease in F0, as shown in Figure 4 in which only F0 changes after autotomy. In species that run faster after autotomy, risk of capture decreases, which is modeled by an increase in c, decreasing the optimal flight initiation distance as shown in Figure 2.
Use of economy of autotomy (shedding less than the complete tail) may have little or no influence on qualitative effects of autotomy on factors affecting flight initiation distance. We predict that the major consequence of economy of autotomy is to lessen the magnitude of the effects of autotomy on the variables modeled and on flight initiation distance.
Given that some effects of autotomy favor increase and others decrease in flight initiation distance, the overall effect of autotomy may be that flight initiation distance increases, is unaffected, or even decreases. Because autotomy has multiple costs and benefits, its effect flight initiation distance is difficult to predict for a particular species. The balance between effects of factors that lead to predictions of increased optimal flight initiation distance (greater lethality, decreased running speed, and lesser ability to obtain benefits) and factors leading to predictions of decreased flight initiation distance (lower initial fitness and faster running speed) determines whether the predicted optimal flight initiation is greater before or after autotomy.
The extended optimal escape model including the effect of lethality offers a clear explanation for variable effects of autotomy on escape decisions. It applies when prey use the same escape tactics after autotomy as when intact. However, some prey alter escape tactics after autotomy. Some prey after autotomy may become less active to enhance crypsis by remaining immobile or stay closer to refuges (Cooper 2003; Fleming et al. 2007; Bateman and Fleming 2009). Escape tactics may change not only due to autotomy but also during periods of reduced mobility while gravid, injured, or molting, and for other reasons. In such cases, the extended optimal escape model does not apply.
Other factors having multiple effects on escape parameters
The optimal escape model incorporating lethality is useful for conceptualizing and predicting effects of several other factors that influence multiple parameters. These include age, sex, gravidity, injury, disease, and parasitism. In some species, juveniles have lower fitness than adults due to low probability of surviving to reproduce, as well as slower running speed and reduced ability to defend themselves (greater lethality). Flight initiation distance may be shorter for juveniles than adults if the effect of lower fitness is stronger than the combined effects of lower speed and greater lethality; it may be longer if the greater predation risk due to lower speed and greater lethality predominates.
Gravid females are slower runners (e.g., Shine 1980; Bauwens and Thoen 1981; Cooper et al. 1990, 2009) and often may have higher initial fitness than nongravid females, suggesting that flight initiation distance should be longer for gravid than nongravid females provided that gravid and nongravid females use the same escape tactics. Being gravid might also reduce ability to obtain benefits in some species, which would augment effects of reduced speed and higher initial fitness. For injured, parasitized, or diseased organisms, lower initial fitness may predict shorter flight initiation distance, but reduced ability to obtain benefits and impairment of escape ability and defensive capability if caught may predict longer flight initiation distance due to increased risk of being caught and greater lethality.
Applicability and limitations
The original version of optimal escape theory successfully predicts effects of many factors that affect predation risk and costs of escaping (reviewed by Stankowich and Blumstein 2005). As discussed above, naive predictions based on increased predation risk after autotomy have not always been supported because autotomy affects multiple factors that affect predation risk. The extended optimal escape model presented here improves the original model by adding effects of the degree or predator lethality, which must be an important consideration for assessing overall predation risk and for making escape decisions, on flight initiation distance. The extended model also applies to sublethal predation for factors including autotomy, other injury, envenomation, and others.
A major advantage of the model is that it clarifies how changes in factors that have opposing effects on parameters of the optimal escape model may lead to increase, no change, or decrease in the predicted fight initiation distance. This has not been possible using earlier theory. Optimal escape theory provides a conceptual framework that accommodates varying effects of single-state variables on flight initiation distance, greatly improves our understanding of cases in which multiple parameters have reinforcing effects, and permits qualitative predictions when the relative magnitudes of opposing effects can be estimated. The model can account for many observed effects of factors such as autotomy that influence multiple parameters affecting escape decisions, including combinations of lethality, other risk factors, and escape costs. It facilitates understanding of effects of age, sex, reproductive condition, and health that are likely to impact flight initiation distance via a balance between effects on predation risk and escape costs.
Although we have limited the extension of the optimality model to flight initiation distance, similar cost–benefit logic applies to some other escape variables and to refuge use. For prey that do not enter refuges, the distance fled before stopping once escape begins has been successfully predicted by the optimality model when the predator ceases approaching as soon as escape begins (Snell et al. 1988; Martín and López 2003, Cooper, 2009). For prey that use refuges, the probability of entering refuge also is predictable (Martín and López 2003; Cooper and Whiting 2007; Cooper, 2009). Finally, effects of lethality on refuge occupation can readily be incorporated in optimality models of hiding time (=emergence time, the interval between entering and emerging from a refuge; Cooper and Frederick 2007b) by multiplying e-ct by L in the risk on emergence function.