APPENDIX C
Wolves and Caribou in GMU 20: Example of Assessing Predator-Prey Dynamics by Testing the Fit of Different Models to Available Data

Data on wolves and caribou in Alaska are among the most extensive for any large mammal predator-prey interactions. Given this, one might think it easy to determine the impact of wolves on their prey populations. More specifically, game managers should be able to use such data to determine:

  • The extent to which wolves keep prey populations below their carrying capacity.

  • Whether predator control can increase prey populations without extensive killing of predators that must be done year after year.

Well-designed experimental manipulations offer the clearest way of measuring the impact of wolves on their prey, but they are not always feasible. As an alternative to experimentation, one can propose different models or hypotheses to see how well they ''fit" or "explain" the data. To assess how much inference is possible from such model fitting, the committee asked ADFG biologists for their most complete long-term records of wolf and caribou numbers, including data collected during a period of wolf control. Patrick Valkenburg provided population data for GMU20A, including unpublished data. Those data span the years 1970 to 1995 and consist of population counts of caribou and wolves, number of wolves killed, number of caribou killed by hunters, calf:cow ratios for different months, caribou births, and information about weather such as summer temperatures, rain, and snow depths.

One way of asking whether wolves limit caribou populations is to search for correlations between the number of wolves and calf survival (represented by the



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Wolves, Bears, and their Prey in Alaska: Biological and Social Challenges in Wildlife Management APPENDIX C Wolves and Caribou in GMU 20: Example of Assessing Predator-Prey Dynamics by Testing the Fit of Different Models to Available Data Data on wolves and caribou in Alaska are among the most extensive for any large mammal predator-prey interactions. Given this, one might think it easy to determine the impact of wolves on their prey populations. More specifically, game managers should be able to use such data to determine: The extent to which wolves keep prey populations below their carrying capacity. Whether predator control can increase prey populations without extensive killing of predators that must be done year after year. Well-designed experimental manipulations offer the clearest way of measuring the impact of wolves on their prey, but they are not always feasible. As an alternative to experimentation, one can propose different models or hypotheses to see how well they ''fit" or "explain" the data. To assess how much inference is possible from such model fitting, the committee asked ADFG biologists for their most complete long-term records of wolf and caribou numbers, including data collected during a period of wolf control. Patrick Valkenburg provided population data for GMU20A, including unpublished data. Those data span the years 1970 to 1995 and consist of population counts of caribou and wolves, number of wolves killed, number of caribou killed by hunters, calf:cow ratios for different months, caribou births, and information about weather such as summer temperatures, rain, and snow depths. One way of asking whether wolves limit caribou populations is to search for correlations between the number of wolves and calf survival (represented by the

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Wolves, Bears, and their Prey in Alaska: Biological and Social Challenges in Wildlife Management number of calves in the fall divided by the number born in the spring). Figure C.1 shows that calf survival declines as the number of wolves increases. One might be tempted to conclude from such a correlation that in this particular wolf-caribou system wolves control the number of caribou. However, one can find other correlations that support very different conclusions. For example, figure C.1 also shows that the amount of summer rainfall is positively correlated with calf survival. Thus these simple correlations, in which one variable is plotted against another variable, do not tell us which is more important to calf survival: the impact of wolves or the impact of weather. Indeed, to a large extent, the debate about the role of wolves in caribou population dynamics arises from the ambiguity inherent in using simple correlations to determine what factors limit prey populations. One way to weigh the relative importance of different potential limiting factors is to use all the wolf-caribou data in a single analysis, and to ask which possible hypotheses best explain the total pattern of the data, as opposed to a subset of the data used in a single correlation. For example, one can write a population model in which the number of calves killed per unit time is linearly related to the number of calves available and the number of wolves present. Using this quantitative model of wolf predation, one can then ask how much of the variation in caribou numbers can be explained simply by wolf predation. The results of such an analysis are shown in figure C.2. The parameters that describe the graph relating calf survival to wolf numbers were selected to provide the best fit to the data. (In this case, a curve is fitted to the data, and "best fit" is defined as the equation for the curve with the smallest sum of the squared differences between observed data and the predicted data on caribou numbers and calf:cow ratio.) The model, indicated by the solid line in figure C.2, clearly identifies the main trends in the data shown in the open circle. The simple assumption that wolf predation increases linearly with number of caribou is sufficient to account for the major trends in caribou data, without taking any other factors into account. One can then test other factors in the same way. Indeed, none of the weather data could be correlated with the pattern of up and down trends in caribou numbers seen between 1970 and 1995. For an example, see the analysis of snow depth in figure C.3. Although, the model depicted in figures C.2 and C.3 is quite simplistic, it nonetheless suggests a direction of quantitative research that might help wildlife manages learn more from their data than they can by looking for simple correlations between 2 factors at a time. The procedure is straightforward: Propose a specific model for caribou population dynamics. Estimate rates in that model either directly from field data, or by fitting parameters such that the observable variables (such as predator and prey numbers, calf survival, and weather data) are as well-described as possible (by a formal criterion such as least-squares, or maximum likelihood).

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Wolves, Bears, and their Prey in Alaska: Biological and Social Challenges in Wildlife Management FIGURE C.1 Correlations between key caribou demographics (natality and calf survival as measured by calves counted in the fall divided by spring births) and several possible causal factors (such as snow depth, number of wolves, summer temperatures and rainfall, previous summer rainfall). All plots are derived from the same data, albeit with different possible independent variables (causal factors). Evaluate competing hypotheses by comparing their ability to describe trends in caribou numbers. The approach described above differs from the simple correlations depicted in figure C.1 because the changes in population counts over time are analyzed in relation to several factors that are known to be acting concurrently. These time series are ordered and structured. When the analysis is reduced to a single index

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Wolves, Bears, and their Prey in Alaska: Biological and Social Challenges in Wildlife Management FIGURE C.2 The results of a simple model of caribou population trends assuming constant rates of natality, and survival that varies only as a result of wolf predation. Other data used in the model, such as hunting mortality for caribou, were provided by ADFG biologists. The solid line indicates the best fit wolf predation model, that is, the model that is most consistent with the trends observed in the data.

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Wolves, Bears, and their Prey in Alaska: Biological and Social Challenges in Wildlife Management FIGURE C.3 The results of a best fit caribou model in which the only parameter change through time is caribou survival which is assumed to show a straight line correlation with snow depth. Calf survival is assumed to be constant in this model regardless of the number of wolves. The model is not consistent with the patterns seen in the data such as the widely varying calf:cow ratios in October, although it is consistent with varying calf:cow ratios in April.

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Wolves, Bears, and their Prey in Alaska: Biological and Social Challenges in Wildlife Management of survival in a correlation test as in figure C.1 this order and structure is lost, because the information about actual year or sequences in which the caribou data were collected is ignored. It is important to note that this model fitting can still lead to ambiguous answers. In such cases, one can choose management actions that help to discriminate between competing hypotheses. The strategy of adaptive management is to choose actions that increase what is known about the system being managed. This is also best done if there is an actual population model, and not just simple correlations. A key point is that only by writing an explicit model for population change can one adequately measure the impact of predators on prey and interpret the results of different management actions. Otherwise, one is too vulnerable to the illusion that the simple correlations observed between predators and their prey are true indicators of predator-prey dynamics.