Cycle lengths and phase portrait characteristics as probes for predator-prey interactions: comparing simulations and observed data.(Report)

Introduction

Predator-prey systems may show regular cyclic behavior over time. We here hypothesize that cycle lengths and quantitative phase portrait characteristics of a predator-prey system can be used to predict important characteristics of both prey and predator populations, such as cycle amplitude and the tightness of their coupling. Explaining the cyclicity of populations has been an area of interest in ecology ever since Elton (1924) first recorded the phenomenon. The cycles can be externally driven, or be caused by internal factors related to the interaction between the prey and the predator. In rodent-specialist predator interactions, it has been thoroughly demonstrated that the causes of at least some of the cycles are the interactions between the species, e.g., for voles in Fennoscandia (Hanski et al. 1991; Turchin et al. 2000) and in Hokkaido, Japan (Saitoh et al. 1998; Stenseth et al. 2003). Recently, cycles of voles and lemmings in Scandinavia have shown a tendency for dampening of their cycles (Hdrnfeldt 2004; Hdrnfeldt et al. 2005), and climate effects may be a possible explanation for this dampening.

The cyclic nature of predator-prey interactions are easily seen in simple mathematical models of prey-predation, such as the Lotka-Volterra model (Lotka 1925; Volterra 1926) and the Rosenzweig-MacArthur model (Rosenzweig and MacArthur 1963), as well as in more detailed models. We compare cycle characteristics extracted from the simulations of the Lotka-Volterra and Rosenzweig-MacArthur models with characteristic parameters of field data, here the fur return data from Canada, to evaluate the ability of the model to capture the relationship between parameters that expresses cyclic properties of the time series. In addition we run simulations of a specialist and generalist predator model (Hanski et al. 1991). This model adds more realism, compared with the other two models, and we expect that it may be better at predicting the dynamics of the mink-muskrat system. We examine three characteristics of the population cycles: the amplitude of the single-species cycles, the time lag between the cycle peaks of the interacting species, and the pattern of the trajectory of species pairs plotted in phase space. For a predator-prey pair, it has been shown that the trajectories rotate counter-clockwise when the phase space has the prey on the x axis and the predator on the y axis (Seip 1997).

The analyses of the American mink (Neovison vison (Schreber, 1777)) and muskrat (Ondatra zibethicus (L., 1766)) fur return time series from the Hudson's Bay Company in Canada have made a significant contribution to the study of predator-prey interactions. Mink prey species richness is greater in the southeastern part of Canada (C.J. Shier and M.S. Boyce), (2) and previous autocorrelation analyses of the spatiotemporal dynamics of the mink and muskrat fur return data have shown that the coupling between the two species is strongest in the northwestern part of Canada (Erb et al. 2001; Haydon et al. 2001; Viljugrein et al. 2001). A thorough evaluation of the geographic variability in four phases of mink-muskrat phase plots from the Hudson's Bay Company fur returns will be published elsewhere (Holmengen et al. 2009).

Cycle length and phase portrait characteristics are easily obtained probes of population dynamics. We here relate cycle lengths to other probes of population interactions. There are two major objectives of this study. Firstly, we hypothesized that cycle amplitude will increase with cycle length for both species, because populations that reach very low numbers would need more time to reach the cycle maximum than populations that have higher minimum abundances (presupposing that the low minimum abundance would not be compensated by, e.g., higher growth rates). We should be able to find an optimum time lag between the predator and the prey cycles that corresponds to the tightest coupling between the two species. The time lag and the rotations of the trajectories in the phase portraits should also depend on the cycle lengths of the species, since it is the relationship between the two species that determine the strength of the interaction. Secondly, we examine if the Lotka-Volterra model, the Rosenzweig-MacArthur model, or the more detailed model by Hanski et al. (1991) is capable of predicting the observed relationships among the species characteristics in the observed mink-muskrat data. One possible effect of climate changes and other human disturbances is a damping of the variation in population size of species, because the climate zones may be shifted northwards (Williams et al. 2007), and it is thus desirable to find the factors that cause damping. If simple models are capable of capturing the main trends in the relationships between predator and prey, they could be a useful tool in predicting future development.

Materials and methods

Test data

The need of coupled time series of predator-prey systems has led researchers to study simultaneous harvest data. The records of the Hudson's Bay Company in Manitoba includes long time series of fur return data for many predatory and some prey species. For the analyses performed here, fur return data of mink and muskrat from 75 different stations across Canada for the time period spanning 1925-1949 were used (25 years, one observation per year). These data have been shown to reflect variations in relative abundance of mink and muskrat, and not just information about the trapping efforts (Chitty and Chitty 1941, see also Viljugrein et al. 2001). Mink is a predator of the muskrat (Hamilton 1936; Errington 1943; O'Neil 1949), and throughout Canada, both species are widely distributed. In this study, we re-examine the Hudson's Bay Company fur return data. As the analyses undertaken here presuppose cyclic behavior of the populations, time series from two fur posts were omitted because of their lack of cyclicity.

The Hudson's Bay Company data from different stations are numerically very different (range of maximum fur return numbers between stations--mink: 142--3 157 individuals; muskrat: 307--117 483 individuals), and normally such time series should be log-transformed and standardized, as comparing times series from different areas is otherwise difficult. However, in part of our analyses, one major point is to examine the amplitude in different areas; thus, for this analysis, time series were not standardized, as valuable information would be lost in this process. The time lag between prey and predator is defined as the mean difference between one predator peak and the previous prey peak, a definition which results in the lag always being positive. For those time series that by this definition yielded a time lag higher than 3 years, a subjective evaluation of the time series was considered necessary. In the few situations where the mink populations had peaks close to, and before, the muskrat populations, the populations were considered to have negative …