1. From: McCune, B. & J. B. Grace. 2002. Analysis of Ecological Communities. MjM Software Design, Gleneden Beach, Oregon http://www.pcord.comhttp://www.pcord.com Tables, Figures, and Equations

2. Figure 5.1. Hypothetical species abundance in response to an environmental gradient. Lettered curves represent different species. Figure adapted from Whittaker (1954).

3. Figure 5.2. Hypothetical linear responses of species abundance to an environmental gradient. Lettered lines represent different species.

4. Three major problems with community data 1. Species responses have the zero truncation problem.

6. Three major problems with community data 1.Species responses have the zero truncation problem. 2.Curves are “solid” due to the action of many other factors.

7. Figure 5.4. A “solid” response curve. Points represent a species abundances.

8. Three major problems with community data 1.Species responses have the zero truncation problem. 2.Curves are “solid” due to the action of many other factors. 3.Response surfaces can be complex: polymodal, asymmetric, or discontinuous

9. Figure 5.5. Scatterplot of abundance, measured as cover, of four species in relation to a gradient. Each point is a plot. Least-squares linear regression lines and fitted envelope lines are shown.

10. Figure 5.6. Frequency distribution of abundance for a typical tree species in a forest in southern Indiana. The abundance data are basal areas in about 90 plots. Note that most plots do not contain the species, but a few plots have large amounts of the species.

11. Figure 5.7. Bivariate species distributions from species responses to environmental gradients. Left column: two positively associated species. Right column: two negatively associated species. (A)  Positively associated species following Gaussian ideal curves. (B)  Negatively associated species following Gaussian ideal curves. (C)  Bivariate distribution of species abundances corresponding to A. (D)  Bivariate distribution of species abundances corresponding to B. (E)  Positively associated species with “solid” responses to the environmental gradient. (F)  Negatively associated species with “solid” responses to the environmental gradient. (G)  Bivariate distribution of species abundances corresponding to E. (H)  Bivariate distribution of species abundances corresponding to F.

12. Figure 5.8. The dust bunny distribution in ecological community data, with three levels of abstraction. Background: a dust bunny is the accumulation of fluff, lint, and dirt particles in the corner of a room. Middle: sample units in a 3-D species space, the three species forming a series of unimodal distributions along a single environmental gradient. Each axis represents abundance of one of the three species; each ball represents a sample unit. The vertical axis and the axis coming forward represent the two species peaking on the extremes of the gradient. The species peaking in the middle of the gradient is represented by the horizontal axis. Foreground: The environmental gradient forms a strongly nonlinear shape in species space. The species represented by the vertical axis dominates one end of the environmental gradient, the species shown by the horizontal axis dominates the middle, and the species represented by the axis coming forward dominates the other end of the environmental gradient. Successful representation of the environmental gradient requires a technique that can recover the underlying 1-D gradient from its contorted path through species space.

13. Figure 5.9. Comparison of the normal and dust bunny distributions. Upper left.  the bivariate normal distribution forms an elliptical cloud most dense near the center and tapering toward the edges. The more strongly correlated the variables, the more elongate the cloud. Upper right.  the bivariate dust bunny distribution has most points lying near one of the two axes. A distribution like this results from two overlapping “solid” Gaussian curves (Fig. 5.4). Lower left.  the multivariate normal distribution (in this case 3-D) forms a hyperellipsoid most dense in the center. Elongation of the cloud is described by correlation between the variables. Lower right.  the multivariate dust bunny has most points lying along the inner corners of the space. Two positively associated species would have many points lying along the wall of the hypercube defined by their two axes.

14. Figure 5.10. Plotting abundance of one species against another reveals the bivariate dust bunny distribution. Note the dense array of points near the origin and along the two axes. This bivariate distribution is typical of community data. Note the extreme departure from bivariate normality.

15. Figure 5.11. Nature abhors a vacuum. A sample unit with all species removed is usually soon colonized. The vector shows a trajectory through species space. The sample unit moves away from the origin (an empty sample unit) as it is colonized. In this case, species B and a bit of species C colonized the sample unit. As in this example, successional trajectories tend to follow the corners of species space.

16. Figure 5.12. The consequence for the correlation coefficient of adding (0,0) values between species.

17. Box 5.1. Basic properties of ecological community data. 1. Presence or abundance (cover, density, frequency, biomass, etc.) is used as a measure of species performance in a sample unit. 2. Key questions depend on how abundances of species relate (a) to each other and (b) to environmental or habitat characteristics. 3. Species performance over long environmental gradients tends to be hump-shaped or more complex, sometimes with more than one hump. 4. The zero-truncation problem limits species abundance as a measure of favorability of a habitat. When a species is absent we have no information on how unfavorable the environment is for that species.

18. Basic properties, cont... 5. Species performance data along environmental gradients form “solid” curves because species fail for many reasons other than the measured environmental factors. 6. Abundance data usually follow the “dust bunny” distribution, whether univariate or multivariate; the data rarely follow normal or lognormal distributions. 7. Relationships among species are typically nonlinear.