### Summary

- Top of page
- Summary
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgements
- References
- Supporting Information

**1.** Predation rate (PR) and kill rate are both fundamental statistics for understanding predation. However, relatively little is known about how these statistics relate to one another and how they relate to prey population dynamics. We assess these relationships across three systems where wolf–prey dynamics have been observed for 41 years (Isle Royale), 19 years (Banff) and 12 years (Yellowstone).

**2.** To provide context for this empirical assessment, we developed theoretical predictions of the relationship between kill rate and PR under a broad range of predator–prey models including predator-dependent, ratio-dependent and Lotka-Volterra dynamics.

**3.** The theoretical predictions indicate that kill rate can be related to PR in a variety of diverse ways (e.g. positive, negative, unrelated) that depend on the nature of predator–prey dynamics (e.g. structure of the functional response). These simulations also suggested that the ratio of predator-to-prey is a good predictor of prey growth rate. That result motivated us to assess the empirical relationship between the ratio and prey growth rate for each of the three study sites.

**4.** The empirical relationships indicate that PR is not well predicted by kill rate, but is better predicted by the ratio of predator-to-prey. Kill rate is also a poor predictor of prey growth rate. However, PR and ratio of predator-to-prey each explained significant portions of variation in prey growth rate for two of the three study sites.

**5.** Our analyses offer two general insights. First, Isle Royale, Banff and Yellowstone are similar insomuch as they all include wolves preying on large ungulates. However, they also differ in species diversity of predator and prey communities, exploitation by humans and the role of dispersal. Even with the benefit of our analysis, it remains difficult to judge whether to be more impressed by the similarities or differences. This difficulty nicely illustrates a fundamental property of ecological communities. Second, kill rate is the primary statistic for many traditional models of predation. However, our work suggests that kill rate and PR are similarly important for understanding why predation is such a complex process.

### Introduction

- Top of page
- Summary
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgements
- References
- Supporting Information

Predation and other consumer-resource relationships are among the most fundamental of all ecological relationships and have been the focus of ecology since its inception. The population biology of predation is comprised of two basic elements: the kill rate and the predation rate (PR) (Holling 1959; Taylor 1984). Kill rate (KR) is the number of prey killed per predator per unit time and represents the predator’s supply of food. In this sense, kill rate is the predator population’s perspective of predation. Predation rate is the proportion of the prey population killed by predation and represents the pressure of predation on the prey population. In this sense, PR is the prey population’s perspective of predation. Understanding predation dynamics requires understanding the predator’s rate of food acquisition (KR) and the mortality rate of prey that arises from predation (PR). Despite the fundamental role that KR and PR each play, KR seems to receive more attention (Dale, Adams & Bowyer 1994; Bergstrom & Englund 2004; Nilsen *et al.* 2009). Some studies highlight the difficulties of understanding predation dynamics from assessments that focus on KR and neglect PR (Marshal & Boutin 1999; Jost *et al.* 2005).

The tendency to focus on KR may have arisen largely from the tradition, established by seminal ecologists, to express predator–prey models in terms of per capita kill rate (Lotka 1925; Volterra 1926; Holling 1959; Rosenzweig & MacArthur 1963). That is, prey dynamics are assumed to arise largely from the processes that determine kill rate (i.e. the functional response), and predator dynamics are largely considered some function of the kill rate (i.e. the numerical response (NR), by which we mean the relationship between KR and predator growth [see May 1981; Bayliss & Choquenot 2002]:

- (eqn 1a)

- (eqn 1b)

where *f*(*N*) is some function of prey density and possibly other arguments that represent prey growth in the absence of predation, kr(•) is some function that represents the kill rate, and *g*(•) is the NR, a function whose arguments include the kill rate, and P is predator abundance. In this way, kill rate, especially its relationship to the functional and NR, is conventionally considered the primary determinant of predator–prey dynamics. This convention is likely responsible for apparent confidence about the extent to which empirical assessments of the functional and NRs can, by themselves, explain predator–prey dynamics (e.g. Messier 1994). The appropriateness of treating KR as the fundamental process of predation is further supported insomuch as PR may be expressed in terms of KR. Specifically, PR = (KR × *P*)/*N*, where *P* is the total number of predators and *N* is the total number of prey. This seemingly simple relationship between KR and PR obfuscates what is in reality a far more complex relationship between the two processes.

Here, we first review theoretical models to show how the relationship between KR and PR depends on the nature of predator–prey dynamics, for example, whether dynamics are more influenced by top-down or bottom-up processes. These theoretical results give reason to think KR is a poor predictor of PR. We then test the theoretical predictions with empirical observations from three North American study sites – Isle Royale National Park (IRNP), Yellowstone National Park (YNP) and Banff National Park (BNP) – where long-term observations have been made on wolf (*Canis lupus*) and ungulate dynamics.

For many real populations, estimates of KR or PR are unavailable, but estimates for the ratio of predator-to-prey are available. While this ratio is sometimes considered a useful indicator of predation’s effect on prey population dynamics (Eberhardt 1997), these ratios have been criticized as misleading indicators of predation (Theberge 1990; Abrams 1993; Person, Bowyer & Van Ballenberghe 2005). Here, we show how, for our study sites, these ratios are reasonably good indicators of PR and prey growth rate.

### Results

- Top of page
- Summary
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgements
- References
- Supporting Information

The purpose of the theoretical models is to provide a basis for anticipating and interpreting empirical relationships between KR and PR. Kill rate was a poor predictor of PR for all three study sites (Fig. 3a,b,c). More precisely, KR exhibited no relationship to PR for IRNP (*P* = 0·27, *R*^{2} = 0·17, *n* = 41), a marginally significant, positive relationship to PR for BNP (*P* = 0·13, *R*^{2} = 0·15, *n* = 17), and a marginally significant, positive relationship to PR for YNP (*P* = 0·10, *R*^{2} = 0·27, *n* = 11). However, the ratio of predator-to-prey was a strong indicator of PR for all three study sites (Fig. 3d,e,f; *P* < 0·001, *R*^{2} = 0·63 for IRNP; *P* = 0·009, *R*^{2} = 0·49 for BNP, and *P* < 0·001, *R*^{2} = 0·89 for YNP).

Next, we assessed the extent to which prey growth rate was associated with kill rate and PR. Kill rate was also a poor predictor of prey growth rate (Fig. 4a,b,c). More precisely, KR and prey growth rate were unrelated for YNP (*P* = 0·82) and BNP (*P* = 0·66), and weakly related for IRNP (*P* = 0·05, *R*^{2} = 0·11). The lack of significance for the YNP and BNP relationships may be attributable to their smaller sample sizes.

Predation rate was a good predictor of prey growth rate for IRNP (Fig. 4d, *P* < 0·01, *R*^{2} = 0·67), a weaker predictor for BNP (Fig. 4f, *P* = 0·02, *R*^{2} = 0·31), and bore no statistically significant relationship for YNP (Fig. 4e, *P* = 0·35, *R*^{2} = 0·11). Differences in sample size and observed range of PR may be an important explanation for this pattern among sites.

The ratio of wolf-to-prey abundance performed similarly to PR in terms of its ability to predict prey growth rate (Fig. 4g,h,i). That is, the relationship was strongest for IRNP (*P* < 0·01, *R*^{2} = 0·56), intermediate for BNP (*P* = 0·03, *R*^{2} = 0·23) and weakest for YNP (*P* = 0·33, *R*^{2} = 0·12). The ratio of wolf to prey and PR are also similar in the sense that, for IRNP and BNP, the ratio predicted prey growth rate about as well as PR did.

### Discussion

- Top of page
- Summary
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgements
- References
- Supporting Information

Empirical observations seem to indicate that observed kill rate, which represents the predator population’s perspective on predation, is a poor predictor of PR, which represents the prey population’s perspective on predation (Fig. 3a–c). Our complementary theoretical analyses of a wide variety of predator–prey models give good theoretical reason to expect these empirical results (Figs 1 and 2). Different theoretical models of predator–prey dynamics (prey dependent, ratio dependent, etc.) provided a diverse array of predictions about the predicted relationship between KR and PR (Fig. 1). Given these empirical and theoretical patterns, it is not surprising that KR is also a poor predictor of prey growth rate (Fig. 4a,d,g). Although empirical estimates of KR are critical for understanding the population biology of predators (i.e. the NR, eqn 1b), theoretical and empirical results show how kill rate is not very useful for understanding how prey are affected by predation. That is, kill rate is not a useful indicator of predation pressure. However, the theoretical analyses presented here give reason to think that predator:prey ratios would be a good predictor of PR and prey growth rate (Fig. 3d,e,f), and our empirical analyses suggest that the ratio is the best predictor of these statistics (Fig. 4g,h,i).

The explanation for the contrasting abilities of empirical estimates of KR and the ratio of predator:prey to predict kill rate relates to the fundamental relationship between KR and PR (i.e. PR = KR[*P*/*N*]). That PR is well predicted, in real systems, by *P/N* but not KR (Fig. 3) suggests that much of the variation in PR is attributable to variation in *P/N*, not KR. Logic like this also explains why KR–PR correlations differ among simulated populations with respect to the relative contributions of kill rate and population growth rate to stochasticity (i.e. the relative values of the *s*_{1}, *s*_{2} and *s*_{3} in eqn 4; see Fig. 2a). Moreover, the weak positive KR–PR relationship for BNP (Fig. 3a) suggests KR may be a relatively weak source of stochasticity for the BNP system, compared with YNP and IRNP, where the KR–PR relationships were negative (Fig. 3b,c).

IRNP and YNP were similar insomuch as PR exhibited a strong, positive relationship with wolves-per-prey and weak, negative relationship with kill rate (Fig. 1). Moreover, BNP differed from IRNP and YNP in that PR exhibited a weaker positive relationship with wolves-per-prey, and a positive (though statistically insignificant) relationship with kill rate (Fig. 1a,d). These inter-site differences are consistent with the proposition that the number of wolves-per-prey is a more important (relative to kill rate) source of variation in PR for IRNP and YNP than for BNP.

The relationship between prey growth rate and various predation statistics was also strongest for IRNP and weakest for YNP (Fig. 4). The strength of the IRNP relationships (Fig. 4b,c) may correspond to IRNP being the only site of the three where wolves are the sole predator of moose, and moose represent about 90% of wolf diet. The somewhat weaker relationships observed in BNP (Fig. 4h,i) may correspond to wolves being only one of several predators for the elk in that system and the importance of human harvest on elk and wolves. The weakness of the YNP relationships (Fig. 4d–f) likely corresponds to PR and wolves-per-prey having been low and varied little during the period of observation (cf., the x-axes of Fig. 4b,e,f and compare the axes labels of Fig. 4c,f,i). Compared to the other sites, YNP is likely farthest from its equilibrium because wolves have been present in that system only since the mid-1990s.

IRNP, YNP and BNP are fundamentally similar insomuch as they involve wolves preying on large prey in temperate climates. However, they are also fundamentally different in important respects, including dispersal in and out of the study sites, species diversity of predator and prey communities, and human exploitation. How these differences translate into the differences that we observed in patterns of predation (Figs 3 and 4) is very difficult to know. More generally, it is difficult to judge what is more striking, the similarities in patterns of predation among the three sites, or the differences. This difficulty represents one of the perennial challenges in ecology.

Although predator–prey ratios have been an important basis for predicting the effect predators will have on prey (e.g., Keith 1983; Fuller 1989; Gasaway *et al.* 1992), their use for such purposes has been criticized (Theberge 1990; Abrams 1993; Person, Bowyer & Van Ballenberghe 2005), and basic theoretical considerations offer additional reason to justify the criticism (Theberge 1990; Abrams 1993; Person, Bowyer & Van Ballenberghe 2005). Although our empirical results suggest that predator-to-prey ratios are *relatively* good predictors of prey growth, they also suggest how these ratios may be inadequate for the interest of reliably predicting or controlling predator–prey systems. Even in the best case (IRNP, Fig. 4g), the predator–prey ratio accounts for only 56% of the variance in prey growth rate. Although prey population growth rate is expected to be zero when there are 2·9 wolves per 100 moose on IRNP, the 80% confidence interval for the predator-to-prey ratio corresponding to *r*_{prey} = 0 is [0·8, 4·9]. That is, a prey population that is stationary (in the statistical sense) seems to be associated with a very wide range of predator-to-prey ratio. Knowing the predator-to-prey ratio, even in this best-case scenario, offers only a vague idea about prey growth, probably too vague to be of much value for controlling or precisely predicting predator–prey dynamics. The inability to reliably predict prey growth from predator-to-prey ratio likely arises from factors such as interannual variation in climate, prey age structure, and extent to which predation is compensatory.

Moreover, even knowing that predator–prey ratio is correlated with PR or prey growth rate does not, by itself, indicate that predation is additive, and therefore the ultimate cause of prey dynamics. The correlations we observed here are also consistent with the prospect that some other process (e.g. environmental stochasticity in the form of a severe winter) causes both prey decline and high PR. In other words, a correlation between PR and prey growth rate does not, without additional information, allow one to distinguish between predation being an additive or compensatory source of prey mortality, or allow one to conclude that predation is responsible for lowering prey density. In most cases, data on predator–prey ratio or even PR will be inadequate for reliably inferring whether predation is causing prey population declines. Nevertheless, such inferences are the kinds that are typically used to judge the appropriateness of predator control (Theberge 1990; Abrams & Ginzburg 2000). The limited value of such inferences suggests the need to develop alternative strategies for making such judgments.

The results presented here do not indicate that kill rate is unimportant for understanding predator–prey dynamics. On the contrary, understanding kill rate – that is, its causes and consequences – is important for understanding the energetics and dynamics of predator populations (Scheel 1993; Fuller & Sievert 2001; Vucetich, Peterson & Waite 2004; Packer *et al.* 2005). Nevertheless, our results may feel like a set-back for those hoping that some easily measured predation statistic, such as kill rate, might reliably indicate a prey population’s growth rate or the impact of predation on prey dynamics.

However, our results do represent an opportunity to develop how it is that we commonly think about predation. That is, conventional mathematical models of predation (e.g. eqns 1a,b) are typically expressed in a manner that seem to generate conceptual models (i.e. thoughts we hold in our minds about predation) where kill rate is the central process of predation (Fig. 5a). Our results suggest that a better conceptual model would depict the kill rate and PR, each as a fundamental process (Fig. 5b). Although this second conceptual model is substantially more complex, both arise from the same mathematical models (e.g. eqn 1). The two conceptual models are just different ways of thinking about the same set of equations. The more complex conceptual model should motivate the allocation of greater effort to understand the causes of variation in PR and its consequences. The added complexity of the lower conceptual model also helps one better intuit how complex dynamics can arise from what can otherwise appear to be a relatively simple set of equations.

### Supporting Information

- Top of page
- Summary
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgements
- References
- Supporting Information

**Appendix S1** Details related to estimation of predation rate, analytical results, and simulations.

**Table S1** Model selection results corresponding to Fig. 3.

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