used analytical approaches and stochastic simulations to test for biases resulting from variance of the matrix elements, while Usher derived analytic solutions for both Leslie matrices and more general models. While there are also methods that attempt to separate sampling variance from the total observed variance in vital rates, , the potential for sampling variance to bias estimates in λ has received limited attention. Several studies have used Jensen's Inequality to predict potential biases in λ resulting from process variance e.g.,, , and methods for dealing with process variance, especially over time, are well-developed. The second source is sampling variance, which is a result of studying a sample rather than the entire population. The first of these is process variance, which results from real variation in the population over space or time –. Variance in vital rates can arise from two sources. The amount and direction of this bias depend both on the strength of the nonlinearity of the relationship between λ and the vital rates, and on the variance of the vital rates themselves.
As for other nonlinear functions describing ecological processes e.g.,, , the mathematical theorem known as Jensen's inequality implies that variance in vital rates–even those that have been accurately measured–will bias estimates of λ. This is because λ is the dominant eigenvalue of the transition matrix, and hence a nonlinear function of the underlying vital rates (i.e., λˆ = f( v 1, v 2,…, v n), where the v i are the n vital rates and f is a nonlinear real-valued function). Nevertheless, even unbiased estimates of vital rates do not ensure unbiased estimates of the population growth rate. Because biased vital rates can lead to inaccurate projections of the population growth rate (i.e., λ), there has been an upsurge in studies exploring alternative sampling designs for demographic studies –. When the number of individuals used to estimate vital rates is low, the resulting vital rates–as well as estimates of variances and covariances among them–can be biased. cogently summarizes, they are data-hungry models requiring detailed estimates of birth, death, reproduction, and other vital rates. However as a recent review by Doak et al. They are flexible, readily applicable to a diversity of life-history strategies, and there is a broad body of literature describing their construction, interpretation, and limitation reviewed in. Matrix models, are an important tool used by ecologists to study the demography of structured populations and for conducting population viability analyses.
The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.Ĭompeting interests: The authors have declared that no competing interests exist. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.įunding: Financial support for collecting demographic data was provided by the National Science Foundation (award numbers DEB-0309819 and INT 98-06351) and the University of Florida. Received: JAccepted: AugPublished: August 28, 2008Ĭopyright: © 2008 Fiske et al. PLoS ONE 3(8):Įditor: Mark Rees, University of Sheffield, United Kingdom A set of options will drop down choose the one you need.Citation: Fiske IJ, Bruna EM, Bolker BM (2008) Effects of Sample Size on Estimates of Population Growth Rates Calculated with Matrix Models. For other items, click the box that has the type of component you need. For the older system, you’ll see a text box. The principles of creating an equation are the same in both systems, but components are in different places. The one that you use to create the equation is the Equation Tools Design tab. In PowerPoint 2010 and later, click the Insert tab, then choose Equation in the Symbols group. You’ll see a window that looks like this. In the Insert Object dialog box, scroll down and select Microsoft Equation 3.0. To access the Equation Editor in PowerPoint 2007 and earlier, choose Insert, Object.
Before that, the Equation Editor was a separate window. The feature is called the Equation Editor and since PowerPoint 2010, it has been incorporated into the ribbon. For years, PowerPoint has had this capability, but not many people know about it. Sometimes, you need to display a complex formula or equation in PowerPoint.
0 kommentar(er)
