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BIERZYCHUDEK, PAULETTE. "LOOKING BACKWARDS: ASSESSING THE PROJECTIONS OF A TRANSITION MATRIX MODEL.(Statistical Data Included)." Ecological Applications. Ecological Society of America. 1999. HighBeam Research. 21 Oct. 2018 <https://www.highbeam.com>.
BIERZYCHUDEK, PAULETTE. "LOOKING BACKWARDS: ASSESSING THE PROJECTIONS OF A TRANSITION MATRIX MODEL.(Statistical Data Included)." Ecological Applications. 1999. HighBeam Research. (October 21, 2018). https://www.highbeam.com/doc/1G1-60949669.html
BIERZYCHUDEK, PAULETTE. "LOOKING BACKWARDS: ASSESSING THE PROJECTIONS OF A TRANSITION MATRIX MODEL.(Statistical Data Included)." Ecological Applications. Ecological Society of America. 1999. Retrieved October 21, 2018 from HighBeam Research: https://www.highbeam.com/doc/1G1-60949669.html
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Abstract. Analyses of population projection models are increasingly being used by conservation biologists and land managers to assess the health of sensitive species and to evaluate the likely effects of management strategies, harvesting, grazing, or other manipulations. Here I describe some of the limitations of this approach and illustrate how these limitations may affect its usefulness. I do this by comparing the results of such an analysis, performed in 1979 on two populations of a perennial plant, Arisaema triphyllum, with new information about the size and structure of these same populations gathered in 1994, 15 years later. While one population changed as the model projected it would, the other behaved quite differently from the projection. Instead of increasing in size, this population decreased between 1979 and 1994.
Possible shortcomings in the data and in the model include: too few plants to provide accurate transition probabilities; too few years to capture accurately the complete range of year-to-year environmental variability, and the failure of the most commonly used form of the model to account for density-dependent vital rates. In addition, the asymptotic growth rates ([Lambda]) these models yield may sometimes be irrelevant and even misleading if one's primary interest is in a population's short-term prospects for survival, as is often the case in studies of sensitive species. These shortcomings may apply to many studies involving the use of projection models, and they have important implications for the value of this approach in conservation biology and species management decisions.
Key words: Arisaema; bootstrapping; conservation; demography; management; transition matrix.
INTRODUCTION
Population projection matrix models, also known as transition matrix models, were introduced to biologists by Leslie (1945) and Lefkovitch (1965). They were first applied to data from natural populations of animals in the 1960s (e.g., Pennycuick et al. 1968) and to plant populations in the 1970s (Hartshorn 1975, Werner and Caswell 1977, Caswell and Werner 1978, Enright and Ogden 1979). These studies and others (e.g., Bierzychudek 1982, Meagher 1982, Fiedler 1987, Huenneke and Marks 1987, Lasker 1990, McFadden 1991, Byers and Meagher 1997) used matrix models to assess the fitness consequences of alternative life history strategies. Others have used projection models to document spatial and temporal variation in a species' vital rates (e.g., Gregg 1991, Horvitz and Schemske 1995, Oostermeijer et al. 1996, Kephart and Paladino 1997, Vavrek et al. 1997).
In the last 10 yr, as interest in applied ecology and conservation biology has grown, matrix models have begun to be used in additional ways. Since repeated iterations of a matrix can result in a projection of a population's equilibrium growth rate (under certain assumptions), projection matrices can provide a measure of the health of populations of rare species (Menges 1986, 1990, Waite 1989, Charron and Gagnon 1991, Aplet et al. 1994, Doak et al. 1994, Price and Kelly 1994, Oostermeijer et al. 1996, Kephart and Paladino 1997). Matrix models have been used to assess the likely effects on population growth rates of various management strategies (Crouse et al. 1987, Waite and Hutchings 1991, Aplet et al. 1994, Doak et al. 1994, Smith and Trout 1994, Maschinski et al. 1997, Shea and Kelly 1998) or of natural or anthropogenic disturbances like fire, grazing, or harvesting (de Kroon et al. 1987, Silva et al. 1991, Nault and Gagnon 1993, O'Connor 1993, Pinard 1993, Bullock et al. 1994, Kaye et al. 1994, Moloney et al. 1994, Bastrenta et al. 1995, Olmstead and Alvarez-Buylla 1995, West 1995, Ratsirarson et al. 1996, Batista et al. 1998). Sensitivity or elasticity analyses of these models can identify the life history stages most critical to a rare species' persistence or most in need of additional study (Menges 1986, Crouse et al. 1987, Aplet et al. 1994, Doak et al. 1994, Ginsberg and Milner-Gulland 1994, Schemske et al. 1994, Hitchcock and Gratto-Trevor 1997, Crooks et al. 1998). For this reason, several papers have appeared urging resource managers to employ these models when making management decisions about the conservation of rare species (e.g., Menges 1986, Waite and Hutchings 1991, Schemske et al. 1994).
The enthusiasm with which these models are currently being embraced, however, is sometimes insufficiently tempered by a recognition of their limitations. The population growth rate ([Lambda]) that can be calculated from demographic data using a projection matrix is an asymptotic growth rate, i.e., the growth rate that would result, eventually, if the population's observed vital rates were to be maintained indefinitely. For that reason projection matrix analyses are subject to several important limitations. First, deterministic and stochastic changes in a population's environment make it very unlikely that the vital rates measured for a population will in fact remain constant over time. Caswell (1989) emphasizes that for this reason, matrix analyses should be regarded as projections (what would happen to a population if vital rates remain unchanged) rather than as forecasts (what will happen to a population).
Secondly, applied ecologists and conservation biologists in particular are often forced to be most intensely interested not in a small population's asymptotic, long-term behavior, but rather in whether it can survive into the immediate future--its transient behavior (Menges 1986). A population's short-term, transient behavior may be quite different from its asymptotic behavior. This is because, unlike asymptotic rates, short-term growth rates can be very strongly influenced by a population's current age or size distribution (Caswell and Werner 1978, Caswell 1989, Burgman et al. 1993). A population that is far from its stable size distribution may behave over the short term very differently from the asymptotic projection.
Thirdly, the form of projection matrix model most commonly used assumes that a population's vital rates are independent of population density. In many cases this assumption is appropriate for the size of the population at the time of the study. But any population with a growth rate [is greater than] 1.0 is expected to increase in density and may eventually reach a population size at which density does begin to exert an influence on its birth and death rates.
I present here a vivid example of the ways some of these limitations can exert their influence. The data come from a re-census of two populations of Arisaema triphyllum, jack-in-the-pulpit, that I studied between 1977 and 1979. That study (Bierzychudek 1982) used a transition matrix approach to project asymptotic growth rates for this herbaceous perennial of the forest understory. Here I compare the projections and conclusions of Bierzychudek (1982), as well as the results of a new transient analysis of those same data, with the states of these two populations 15 yr later.
METHODS
Transition matrix analysis
A projection matrix model specifies a matrix of transition probabilities between different size classes, age classes, or stages in a population from time t to time t + 1. These transition probabilities represent observed values of survival, growth, and reproduction (see Bierzychudek 1982, Menges 1986, Groenendael et al. 1988, Caswell 1989, and Horvitz and Schemske 1995 for more detailed explanations). When multiplied by a column vector whose values represent the numbers of individuals in each class at time t, the matrix projects the expected number of individuals in each class at time t + 1. When the matrix is multiplied by the original population vector a sufficient number of times, the population eventually converges to a "stable distribution," at which time each size or age class is changing by the factor k each time period. [Lambda] is the population's asymptotic rate of growth. When [Lambda] exceeds 1.0 the population is projected to increase over time; when [Lambda] is [is less than] 1.0 the population is projected to decline. A population's [Lambda] and its stable distribution are independent of the original column vector and depend only on the values in the matrix. In the terminology of linear algebra, [Lambda] is the dominant eigenvalue of the matrix, and the stable distribution is its right eigenvector.
The analysis in Bierzychudek (1982) was based on 3 yr of demographic data (and thus two different "transition periods") from two populations of A. triphyllum growing near Ithaca, New York: Fall Creek and Brooktondale. In that paper I projected the long-term (asymptotic) population growth rates, [Lambda], of these two populations, creating separate matrices for each of the 1977-1978 and the 1978-1979 transition periods. …
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