91Ó°ÊÓ

As part of a study of the effects of timber management strategies (Ecological Applications [2003]: \(1110-1123\) ) investigators used satellite imagery to study abundance of the lichen Lobaria oregano at different elevations. Abundance of a species was classified as "common" if there were more than 10 individuals in a plot of land. In the table below, approximate proportions of plots in which Lobaria oregano were common are given. $$ \begin{array}{llllllll} \hline \text { Elevation }(\mathrm{m}) & 400 & 600 & 800 & 1000 & 1200 & 1400 & 1600 \\ \hline \text { Prop. of plots } \\ \text { with Lichen } & & & & & & & \\ (>10 / \text { plot }) & 0.99 & 0.96 & 0.75 & 0.29 & 0.077 & 0.035 & 0.01 \\ & & & & & & \\ \hline \end{array} $$ a. As elevation increases, does Lobaria oregano become more common or less common? What aspect(s) of the table support your answer? b. Using the techniques introduced in this section, calculate \(y^{\prime}=\ln \left(\frac{p}{1-p}\right)\) for each of the elevations and fit the line \(y^{\prime}=a+b(\) Elevation \() .\) What is the equation of the best-fit line? c. Using the best-fit line from Part (b), estimate the proportion of plots of land on which Lobaria oregano are classified as "common" at an elevation of \(900 \mathrm{~m}\).

Step by step solution

01

- Interpret the Table and Identify Trend

The data in the table shows that as elevation increases, the proportion of plots in which Lobaria oregano is common decreases. This is evident because as one goes from 400m to 1600m, the proportion decreases from 0.99 to 0.01.

02

- Calculate y-prime for Proportions

Calculate \(y^{\prime}\) value for each elevation using the formula \(y^{\prime} = \ln\left(\frac{p}{1-p}\right)\), where \(p\) is the proportion. The calculated value of \(y^{\prime}\) would be fit against the elevation to get the best-fit line.

03

- Fit Best Line

With the calculated values of \(y^{\prime}\) and the given elevations, use the least-squares method or a suitable technique to fit the best line with equation of the form \(y^{\prime}=a+b(\) Elevation \(). Assume the line as \(y^{\prime} = a + b * x\), where x is the elevation.

04

- Predict Proportion from Fitted Line

Substitute the elevation \(x = 900\) into the fitted line equation. This would give the \(y^{\prime}\) at 900m. Next, rearrange logarithmic expression to find the corresponding proportion using the formula \(p = \frac{e^{y'}}{1+e^{y'}}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Species Abundance Distribution

Understanding how species are distributed within an ecosystem provides insights into biodiversity, population dynamics, and conservation strategies. Species abundance distribution (SAD) refers to the pattern of distribution of individuals among species in a community. Simplistically, it helps us understand how many species are common or rare within a certain area.

In the context of the exercise, the SAD indicates that Lobaria oregana becomes less common as the elevation increases. The table provided in the exercise records the abundance of this lichen in various plots at different elevations. By categorizing the abundance as 'common' if there are more than 10 individuals per plot, researchers can make species abundance comparisons across the elevation gradient.

The SAD can be plotted as a frequency distribution, which often reveals that few species are very common, while most are rare—a pattern described by models like the lognormal distribution or the broken stick model in ecology.

Satellite Imagery in Ecological Studies

Satellite imagery has become indispensable in modern ecological research, providing extensive spatial coverage and the ability to monitor changes over time. It allows ecologists to analyze habitats, land cover, and even species distributions from a vantage point that is unachievable through ground surveys alone.

In our exercise, satellite imagery was used to assess the abundance of Lobaria oregana across various elevations. Such imagery can capture multiple plots of land simultaneously, providing data for species abundance distribution that would be labor-intensive to collect otherwise.

Advantages of Using Satellite Imagery

• Large-scale monitoring of ecological changes
• Cost-effective in comparison to extensive ground surveys
• Access to remote or difficult-to-reach areas
• Provides a historical archive of ecological information
• Holistic view enabling multi-scale analysis

Through these images, researchers can detect variations in the landscape and identify patterns that inform about the species' habitat preferences and threats.

Logistic Regression

Logistic regression is a statistical method used when the dependent variable is categorical—often binary. It estimates the probability that the dependent variable belongs to a particular category. This technique is particularly suitable for situations where there is a non-linear relationship, like the one in the exercise involving species abundance and elevation.

When applying logistic regression to ecological data, we often work with transformed data. The transformation, as used in the exercise, employs the logit function, which is the natural logarithm of the odds ratio: \( y' = \ln\left(\frac{p}{1-p}\right) \), where \( p \) is the probability of an event (in this case, the lichen being classified as 'common').

The output of the logistic regression is an equation that relates the log odds of the dependent variable to one or more independent variables, allowing for predictions about the probability of occurrence based on factors like elevation.

Proportion and Elevation Relationship

The relationship between the proportion of a species in a given area and the elevation can be critical in understanding ecological patterns. In the exercise, we see that the proportion of plots where the lichen is classified as 'common' decreases with increasing elevation. This relationship can be indicative of the ecological limits or preferences of a species.

By applying logistic regression, we can model this relationship and predict the abundance of a species at different elevations. The best-fit line from such a regression provides a mathematical interpretation of this relationship. From the equation of the line, \( y' = a + b\times\text{Elevation} \), one can predict the probability of finding the species in common abundance by inputting the elevation of interest. This information is valuable for ecologists in predicting habitat suitability across elevation gradients and is vital for management and conservation efforts.