Predator-prey interactions among a wide variety of organisms have been studied extensively under various ecological conditions. Studies show that the abundance of prey-specific predators (those which prey upon a single species almost exclusively) is subject to fluctuations dependent on the abundance of the prey species. The most prominent example of such patterns existing in mammalian species is between the lynx (Lynx canadensis) and snowshoe hare (Lepus americanus) using nearly a century’s worth of pelt records from the Hudson Bay Company. These data produced cycles with peak abundance every ten years for both species [1] . Other long-term data sets are rare, and most lab experiments quickly result in the extinction of one or both species [2] . It is also difficult to obtain sufficient lab or field data depicting such oscillatory patterns, especially in vertebrates, since resources are limited and long-term studies can be very expensive.

A well-known example of these trade-offs occurs between the protist ciliate species Paramecium aurelia and Didinium nasutum [3] . Luckinbill was able to produce such patterns with P. aurelia and its predator, D. nasutum, in a homogenous lab environment using methyl cellulose to inhibit the frequency of predator-prey interaction. Previous attempts at producing oscillatory data in a lab environment required the addition of new organisms, mimicking dispersal, or by adding physical complexity to the system which allowed the prey species to find refuge from predation. This type of manipulation simulates a natural heterogeneous environment where prey are able to hide from predators, and are prevented from unlimited growth by natural constraints. Luckinbill was able to create oscillatory data only by limiting the frequency of contact between species, as well as manipulating the food supply for Paramecium. His results suggest that limitations on both the prey’s food availability, as well as the predator’s capture rate, are important factors influencing the existence of coupled oscillations of abundance.

Many mathematical models have been developed to explain this kind of relationship between predators and prey, beginning with Lotka (1925) and Volterra and Brelot (1931) [4] [5] . Their model(s) included a growth rate for the predator depending linearly on prey density and a commensurate death rate for the prey. However, prey conversion into new predators in this model is instantaneous, and the prey species has unrestricted growth. Roszenweig and MacArthur (1963) and Beverton and Holt (1957) later refined these types of models to include resource limitations for one or both species [6] [7] . Saturable predation responses (e.g., the type II functional response described by Holling, 1959) were included so that the frequency of prey consumption was intrinsically limited [8] . Limitations on the growth of the prey species were also included in these models in the form of “carrying capacity,” the number of individuals a certain area can support. Today, it is widely accepted that both density dependence and saturable predation are essential components of predator-prey models.

Harrison (1995) was able to successfully model Luckinbill’s Paramecium and Didinium data by incorporating type II predation and carrying capacity, in addition to a delayed numerical predator response in standardized predator-prey models [9] . Harrison fit several different models to the data, and compared results using summed square error. He also introduced a time-lag in the rate of prey conversion, and this delayed numerical response with saturable predation proved to best fit Luckinbill’s data.

Another study conducted by Dennis et al. (1995) modeled the cannibalistic population interactions of the flour beetle, Tribolium castaneum, using a system of nonlinear difference equations [10] . Notably, a time-series approach using maximum likelihood and conditional least squares estimation was used to determine model parameters. Their experiment included several populations which were subject to identical lab conditions with model coefficients estimated individually for each population. Their analysis failed to reject the hypothesis that all populations had the same parameter values, supporting the use of these methods as an effective way to obtain model coefficients.

Knowlton and Stoddart (1992) modeled a 31-year data set of coyote (Canis latrans) and black-tailed jackrabbit (Lepus californicus) abundances in Curlew Valley, Utah using a seasonal time-step approach [11] . Four equations were developed and parameterized individually representing 1) coyote population change, 2) jackrabbit natality, 3) adult jackrabbit mortality, and 4) juvenile jackrabbit mortality. They were able to successfully reproduce cyclical patterns of abundance in the system, but included two ad hoc constraints (not allowing the jackrabbits to fall below very low levels of abundance, simulating availability of refuges or emigration into the area and placing an upper limit on coyote abundance). Their model reproduced the three distinct cycles evident in the data and results from the study supported the notion that jackrabbits are a majority prey-item for coyotes.

In another study conducted by Bartel and Knowlton (2005), coyotes demonstrated non-trivial functional feed- ing responses in reaction to varying levels of prey abundance [12] . Specifically, coyotes displayed a hyperbolic (type II) feeding response characterized by increasing predation for increasing prey abundance before the relationship asymptotically levels off. Thus, predator satiation is likely to be a factor in this predator-prey system. Pitt et al. (2003) developed an individual-based computer model to test the effects of coyote social structure on population dynamics [13] . This model distinguished between transient coyotes and territorial coyotes since transient coyotes are likely to have higher mortality rates. Social dynamics could affect the success rate of prey capture and handling times because pack coyotes collaborate to take down larger prey. Only territorially dominant females were allowed to reproduce in the simulation. However, this model assumed a homogeneous food supply, unlikely in a natural environment. Still, results suggested that territoriality and social structure limited reproduction, which could influence predator-prey dynamics.

Multiple models have been developed to describe predator-prey relationships, but rarely have individual influencing factors been assessed using competing models on the same system. Here we used several predator- prey models to describe Knowlton’s unique data set to discriminate which mechanisms have the greatest effect on abundance, testing effects of satiable predation, density dependence and consequences of social hierarchy at the population level. Continuous and discrete models were examined to evaluate the effects of seasonality and changing anthropogenic influences (e.g., changes in coyote bounties) on the populations. Parameter estimates were obtained using nonlinear maximum likelihood estimation (MLE), and models were compared using the Akaike Information Criterion (AIC).

Both continuous and discrete models appeared to fit the data fairly well. In fact, the two models with the lowest values were the annual discrete Beverton-Holt model without seasonality, and the continuous Rosenzweig and MacArthur model (Table 1). Models that included density-dependence and satiable predation produced much better fits than the basic Lotka-Volterra model. The discrete Beverton-Holt models, split to account for spring/fall seasonality produced the largest AIC_c values, but this can partially be explained by the increase in the number of model coefficients.

The Lotka-Volterra model, our “null” model, turned out to give the least appealing visual fit to the data (Fig- ure 4). Problematically, coyote abundance remained about the same over the course of the simulation. The model correctly predicted the time of the third peak in jackrabbit abundance, but did not predict the second peak as a maximum (the peak in jackrabbit abundance around 1980 is higher than in 1970 and 1990) and over-predicted the first peak. However, it was not the worst fit of the models. The intrinsic growth rate of the jackrabbits was estimated to be 4.91/year and the capture rate, 0.753/coyote-year (Table 2). This was consistent with existing research on this species having high rates of both growth and mortality [11] . The prey conversion rate affecting the coyotes was estimated to be 0.0022 new coyotes/jackrabbit and their death rate 0.089/year.

This model did not correctly capture whatever influenced coyote fluctuations since abundances were predicted to be nearly constant over all years.

Adding a carrying capacity for the jackrabbits (although there is currently no direct evidence that jackrabbits are in any way resource-limited in Curlew Valley) and predator satiation for the coyotes greatly improved the fit in the Rosenzweig and MacArthur model, which had the second-lowest value. The first peak in jackrabbit abundance was clearly predicted accurately, and the other peaks were predicted quite close to the observed values as well (Figure 5). The jackrabbit growth rate, valued at 5.18/year, remained consistent with expectations (Table 3). The index value for carrying capacity was estimated to be approximately 251 individuals.

Predictions were again too stable for coyote abundance, producing only two minor maxima shortly after the rise in jackrabbit abundance in the early seventies and again in the early eighties. However, the prey conversion rate, estimated to be 0.014 new coyotes/jackrabbit, was more consistent with existing theory regarding energy transfer between trophic levels [22] . The coyote handling time per prey item and attack rate were estimated to be 78.3 minutes and 0.41/coyote-year, which both seem like realistic values. The coyote death rate also appeared reasonable at 0.13, suggesting approximately ten percent of coyotes die each year.

The same model was tried again allowing for a change to occur in the coyote death rate partway through the simulation to reflect a decrease in the coyote bounty that happened in the eighties. This produced a good visual fit to the data (Figure 6), but the addition of two new parameters (the new death rate and time of change) increased the AIC_c value to 523. Oddly enough, this significantly changed the parameter estimates for the jackrabbit growth rate (1.058) and their carrying capacity (605.23), but similar values were obtained for the satiation (170.55) and predation (29) rates. However, parameter estimates for the death rates were contrary to our expectations since the death rate increased slightly in 1974 from 0.54 to 0.62 whereas one would expect lower

mortality rates when bounties decreased (Table 4). Nonetheless, this model adequately predicted the oscillations of jackrabbit abundance and produces two distinct maxima in coyote abundance.

The Beverton-Holt model had the lowest AIC_c value (495) and included the effects of density-dependence, satiable predation, delayed predator response, and social structure among the coyotes. This discrete model predicted jackrabbit abundance fairly accurately, even capturing a higher abundance in 1980 than in any other year (Figure 7). The intrinsic growth rate for the jackrabbits was estimated to be 2.97/jackrabbit/year, and the carrying capacity 278.689 (Table 5). Satiation and predation rates were estimated to be 441.45 and 121.37, respec- tively. Transient survival, valued at 0.0763/year, was expectedly low. The number of coyote pack members was

approximately 0.12, and the prey conversion rate 0.0526/jackrabbit (recall that these parameter estimates reflect indices values, rather than actual population numbers). This was the only model in which the graph displayed three maxima of coyote abundance. However, the coyote population appeared to crash in 1990 when observed values were high.

Splitting the Beverton-Holt model by season failed to improve the fit. In fact, these models had the highest values (561.28 - 572.9), which were computed using only fall values for model comparison. Three distinct peaks in jackrabbit abundance were evident in both seasons, and coyote abundance seemed to be more accurately predicted by this model in the early nineties than in any other model (Figure 8). The jackrabbit growth

rate increased to 5.43/jackrabbit, and the carrying capacity remained about the same at 274.72 (Table 6). Both the predation and satiation rates were reduced by almost half because this model updates indices biannually instead of annually. Jackrabbit winter survival was estimated to be 0.67, and the prey conversion rate remained consistent at 0.104/jackrabbit. However, the number of coyote pack members in this model was estimated to be 3.21, significantly higher than the value obtained from the annual model. Coyote transient survivorship also changed considerably, increasing to 0.21.

This model was also adjusted, much like the Rosenzweig and MacArthur model, to allow for either a change in the number of coyote pack members, or for a change in the survivorship of transients to occur. This was done to account for changes that could have taken place in coyote harvest rates due to a decrease in the price of the coyote bounty. However, these adjustments produced unrealistic parameter estimates and neither simulation visually improved the fit.

Distributions of values for each model were computed by randomly eliminating three years from the data and fitting each model with the new data for 1000 different samples. values for converging solutions were then graphed with lines indicating the values for the original model fits consisting of all data years (Figure 9). These distributions depict the probability that one model could perform better than another. For example, the probability that seasonality would improve the Beverton-Holt model of the time series is. Similarly, it is shown that satiation and resource limitations are important factors affecting this system since both the Beverton-Holt model and the Rosenzweig and MacArthur model most probably outperform the Lotka-Vol- terra model.

A variety of both continuous and discrete models competed to describe the details of a time series linking coyote and jackrabbit abundance indices in northern Utah/southern Idaho; the best models proved to be the simplest. The Lotka-Volterra model (1-2) was used as the null model and did not include any factors tested in the other models. Models which included the effects of density-dependence on jackrabbits and satiable predation performed better than the others. The model with the lowest value included the effects of social status and delayed predator response, suggesting these may be significant factors affecting this predator-prey system at the population level. The models that accounted for seasonal differences in critical parameters had much higher values, indicating that seasonality is not a significant factor affecting predator-prey oscillations.

The model with the lowest value, the basic Beverton-Holt model (5-7), included social status among coyotes and an effective carrying capacity of packs in Curlew Valley (i.e. a fixed number of territorial packs and a fluctuating number of transients). This supports the hypothesis that social structure could be a contributing factor to predator-prey dynamics in this system. Social status among canids may affect their response to fluctuating prey densities since coyote pack members are less likely to be prey-selective. Pack association allows coyotes to hunt collaboratively, giving them the ability to take down larger prey which could sustain the pack for a longer period of time than smaller prey. Thus, we can assume that a transient coyote’s diet mainly consists of small prey and in this case, almost exclusively black-tailed jackrabbits [11] .

Additionally, the basic Beverton-Holt model included a delayed predator response since incoming transients were dependent on the abundance of jackrabbits at the time of pup gestation (an index offset of two, reflecting dependence of pup survival on winter jackrabbit abundance during gestation, a year of maturation, and subsequent delay until effects of new transients are reflected in fall abundances). This is not a novel idea since predator abundance in these cyclical predator-prey systems often lags several years behind the prey species; Harrison (1995) included a delayed numerical response when modeling Luckinbill’s data. Thus, the inclusion of this type of time-lag of prey conversion into new predators could enhance other predator-prey models as well.

This paper illustrates how multiple, competing models can be used to evaluate the importance of possible mechanisms contributing to complex ecological interactions. Rather than trying to nest models, which intrinsically links some mechanisms to others, we constructed specific models for specific clusters of mechanisms. Models competed via model-independent information theoretic criteria, allowing us to discriminate among mechanisms, even using the non-reproducible time series for coyote-jackrabbit oscillations in the Curlew Valley. Similar approaches can help wildlife managers to evaluate population models and contributing mechanisms without as much need for difficult observations and manipulations, which would otherwise be required to reproduce long- term oscillations between predator and prey species.

We thank the Editor and the referee for their comments.