^{1}

^{*}

^{2}

^{3}

^{*}

Prominent examples of predator-prey oscillations between prey-specific predators exist, but long-t erm data sets showing these oscillations are uncommon. We explored various mo dels to describe the oscillating behavior of coyote ( Canis latrans ) and black-tailed jackrabbits ( Lepus californicus ) abundances in a sagebrush-steppe community in Curlew Valley, UT over a 31-year period between 1962 and 1993. We tested both continuous and discrete models which accounted for a variety of mechanisms to discriminate the most important factors affecting the time series. Both species displayed cycles in abundance with three distinct peaks at ten-year intervals. The coupled oscillations appear greater in the mid-seventies and a permanent increase in the coyote density seems apparent. Several factors could have influenced this predator-prey system including seasonality, predator satiation, density dependence, social structure among coyotes, and a change in the coyote bounty that took place during the course of data collection. Maximum likelihood estimation was used to obtain parameter values for the models, and Akaike Information Criterion (AIC) values were used to compare models. Coyote social structure and limiting resources in the form of density-dependence and satiation seemed to be important factors affecting population dynamics.

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 [

A well-known example of these trade-offs occurs between the protist ciliate species Paramecium aurelia and Didinium nasutum [

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) [

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 [

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 [

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 [

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 [

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).

Curlew Valley is located in Box Elder County in northeastern Utah on the border of Idaho. This area is dominated by big sagebrush (Artemisia tridentata), rabbitbrush (Chrysothamnus nauseousus or C. viscidiflorus), greasewood (Sarcobatus vermiculatus), as well as agricultural crops including crested wheatgrass (Agropyron cristatum) and alfalfa (Medicago sativa) [

Both coyotes and jackrabbits are considered pest species in Utah, so hunting is unrestricted all year round [

Abundance indices were measured for jackrabbits using flushing transects each spring and fall between fall of 1962 and spring of 1993 in a 640 square kilometer portion of the valley. Data for jackrabbits were complete with the exception of the spring measurements in 1987 and 1988. Coyote indices were estimated by catch-per-unit efforts, scent station visitation, and scat deposition rates. Fall coyote observations were complete from 1963 to 1992. However, spring coyote measurements were only taken from 1974-1986, and from 1989-1992. These values were then normalized and averaged for a single density index value for coyotes each season [

These models assume continuous rates of reproduction and mortality, independent of season since abundance indices are updated constantly. Thus, the effects of a birthing season or harsh winter mortality, for example, are not included; rather, such responses are integrated with decreasing weight over all past population states.

The Lotka-Volterra model was used as the “null” model to test out our hypotheses concerning the various factors influencing the dynamics of this system. This model assumes exponential growth of the prey species, not limited by density-dependence. It also assumes instantaneous conversion of prey into new predators, and does not include satiable predation, density-dependence, or social status among coyotes. The basic model is composed of two equations representing the change in the jackrabbit population, J, and the change in the coyote population, C,

where r is the jackrabbit intrinsic growth rate per year and b is the attack rate of coyotes on jackrabbits per year. The parameter c represents jackrabbit conversion efficiency into new coyotes, measuring the relationship

between new coyotes and the number of prey required to support them. Lastly, coyotes die randomly every year at a constant rate, d. Jackrabbits reproduce at a rate proportional to the current abundance of jackrabbits, while they die of predation at a rate proportional to the abundance of coyotes. Coyote abundance increases simultaneously with the death of jackrabbits, and decreases at a constant rate across all seasons.

The Rosenzweig and MacArthur model incorporates density dependence among the jackrabbits with the addition of a carrying capacity parameter, K. In this model, the per-capita growth rate for jackrabbits would be density-dependent,

where

Handling time (in years) per prey item and the capture rate per year are included in the parameters

where

Additionally, the coyote death rate parameter,

to allow variable death rates in response to changes in coyote bounties and harvest rates that occurred during the course of data collection.

Discrete models reflect seasonal differences occurring in survivorship parameters such as increased mortality in winter. Additionally, it is assumed that reproduction primarily occurs in the spring and thus new individuals are added to the population once annually. Since populations depend only on states in the previous season, seasonality effects are strict and immediate. The Beverton-Holt model was modified to include factors associated with the existence of a hierarchal social structure among the coyotes by specifying a baseline number of territorial coyote pack members, g, which were assumed to always be present. This reflects a division of the Curlew Valley landscape into pack territories. The number of coyotes supported in territories is assumed to be constant, since coyote packs typically cooperate to obtain resources in their own territory and are therefore less sensitive to the fluctuating abundance of jackrabbits.

However, not all coyotes live in packs; young coyotes in particular may become “transient” as pups mature and begin to forage on their own. These transient coyotes, T, were assumed to be most responsive to fluctuations in prey abundance with survivorship of pups dependent on the abundance of jackrabbits during the winter of gestation. Resource availability when coyote pups are born would influence both litter size and pup survival. Hence, jackrabbit abundance when pups are born is likely to affect coyote abundance two years later (

We assumed the transient population,

conversion rate is denoted by c, and the annual survivorship of the transients is denoted by

Jackrabbits in this model were assumed to follow a Beverton-Holt nonlinear response with intrinsic growth rate

The predation and satiation rates are denoted by b and a respectively. This model includes both the effects of social status (i.e. a constant number of territorial coyotes and fluctuating abundance of transients) and satiable predation among coyotes, as well as density-dependent growth for the jackrabbits.

The preceding models were parameterized using only fall indices, but the data set included some years with spring data as well. To take advantage of this, we split the discrete model biannually to test for seasonal differences in survivorship and hopefully to gain leverage on the parameters. Jackrabbits in the spring,

where

Social structure among the coyotes was reflected by a constant number of territorial coyote pack members,

where

This model assumes that mortality primarily occurs in the winter, and new coyote pups are added to the population specifically in the spring since coyotes typically have only one litter of pups per year [

These discrete models were also expanded to reflect possible changes in coyote management by allowing a change in the number of coyote pack members, g, taking place partway through the simulation. Other simulations were also tried which allowed for a change in the transient survivorship rate, f, to occur. This was done in an attempt to reflect how changing mortality might lead to the permanent increase in coyote abundance that started in the early eighties that could have resulted from changing coyote bounty prices.

Parameter estimates for the models were obtained using nonlinear MLE, assuming a Laplace distribution of errors, making it less sensitive to outliers than the normal distribution [

where the error variances are

using

Models were compared using Akaike Information Criterion

where small values indicate a good fit with the data. Consequently, increasing the number of parameters needing to be estimated for the model increases its

We also bootstrapped the data to produce histograms of

Both continuous and discrete models appeared to fit the data fairly well. In fact, the two models with the lowest _{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

Model | Number of parameters | ||
---|---|---|---|

Classic Lotka-Volterra | 8 | 501.5 | 508.05 |

Rosenzweig and MacArthur | 10 | 490.7 | 501.70 |

Rosenzweig and MacArthur w/different coyote death rates | 12 | 505.7 | 523.03 |

Basic Beverton-Holt | 9 | 486.4 | 494.97 |

Basic Beverton-Holt Split | 14 | 535.03 | 561.28 |

Beverton-Holt Split w/different coyote death rates | 16 | 539.3 | 578.16 |

Beverton-Holt Split w/different number of pack members | 16 | 534.1 | 572.96 |

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

Parameters | Description | Values |
---|---|---|

r | Prey growth rate | 4.91 |

b | Attack rate | 0.753 |

c | Conversion rate | 0.0022 |

d | Predator death rate | 0.089 |

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 [

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 (_{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

Parameters | Description | Values |
---|---|---|

r | Prey intrinsic growth rate | 5.18 |

K | Carrying capacity | 251.43 |

a | Satiation rate | 128.22 |

b | Predation rate | 52.49 |

c | Prey conversion rate | 0.014 |

d | Coyote death rate | 0.13 |

mortality rates when bounties decreased (

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 (

Parameters | Description | Values |
---|---|---|

Prey intrinsic growth rate | 1.058 | |

Carrying capacity | 605.23 | |

Satiation rate | 170.55 | |

Predation rate | 29.002 | |

Prey conversion rate | 0.11 | |

Coyote death rate | 0.54 | |

Second coyote death rate | 0.62 | |

Time when switch occurs | 1974 |

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

Parameters | Description | Values |
---|---|---|

Prey intrinsic growth rate | 2.97 | |

Carrying capacity | 278.689 | |

Predation rate | 121.37 | |

Satiation rate | 441.45 | |

Transient survival rate | 0.0763 | |

Conversion rate | 0.0526 | |

No. of coyote pack members | 0.12 |

rate increased to 5.43/jackrabbit, and the carrying capacity remained about the same at 274.72 (

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

Distributions of

Parameters | Description | Values |
---|---|---|

Prey intrinsic growth rate | 5.43 | |

Carrying capacity | 274.72 | |

Predation rate | 75.2 | |

Satiation rate | 299.06 | |

Transient survival rate | 0.21 | |

Conversion rate | 0.104 | |

No. of coyote pack members | 3.21 | |

Jackrabbit winter survival | 0.67 |

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

The model with the lowest

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.