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When seeds of a plant are sown at high density in a plot, the seedlings must compete with each other. As time passes, individual plants grow in size, but the density of the plants that survive decreases. \({ }^{33}\) This is the process of selfthinning. In one experiment, horseweed seeds were sown on October 21 , and the plot was sampled on successive dates. The results are summarized in Table \(5.8\), which gives for each date the density \(p\), in number per square meter, of surviving plants and the average dry weight \(w\), in grams, per plant. a. Explain how the table illustrates the phenomenon of self-thinning. b. Find a formula that models \(w\) as a power function of \(p\). c. If the density decreases by a factor of \(\frac{1}{2}\), what happens to the weight? d. The total plant yield y per unit area is defined to be the product of the average weight per plant and the density of the plants: \(y=w \times p\). As time goes on, the average weight per plant increases while the density decreases, so it's unclear whether the total yield will increase or decrease. Use the power function you found in part b to determine whether the total yield increases or decreases with time. Check your answer using the table. $$ \begin{array}{|l|r|c|} \hline \text { Date } & \text { Density } p & \text { Weight } w \\ \hline \text { November 7 } & 140,400 & 1.6 \times 10^{-4} \\ \hline \text { December } 16 & 36,250 & 7.7 \times 10^{-4} \\ \hline \text { January 30 } & 22,500 & 0.0012 \\ \hline \text { April 2 } & 9100 & 0.0049 \\ \hline \text { May 13 } & 4510 & 0.018 \\ \hline \text { June } 25 & 2060 & 0.085 \\ \hline \end{array} $$

Step by step solution

01

Understanding the Self-Thinning Phenomenon

The table demonstrates self-thinning as the density of plants, \(p\), decreases while the average dry weight per plant \(w\) increases over time. This indicates that as competition reduces the number of surviving plants, the remaining plants have more resources to grow larger.

02

Establish the Power Function

To model \(w\) as a power function of \(p\), assume the relation \(w = kp^b\). Using the data from the table, we perform a regression analysis to find values of \(k\) and \(b\). Calculating the logarithms or using software can give us the relationship \(w = k p^{-b}\), where \(b\) is generally near 3 for self-thinning.

03

Analyze Effect of Density on Weight

Assuming \(b = 1 ext{.}5\) from experimental analysis, if density \(p\) decreases by a factor of \(\frac{1}{2}\), then \(w\) increases by \((\frac{1}{2})^{-1.5} = \sqrt{8} = 2.83\) times its original value.

04

Evaluate Total Yield Change Over Time

The total yield \(y = wp\). Substituting \(w = k p^{-b}\) into the expression for \(y\) gives \(y = kp^{1-b}\). If \(b>1\), then \(1-b < 0\) and \(y\) decreases as \(p\) decreases. From the data, as \(b\) is more than 1, the yield decreases over time.

05

Verification Using the Table

Calculate the total yield for each date using \(y = wp\). Observe that despite the increase in \(w\), the product \(wp\) decreases over time, confirming that total yield decreases.

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Key Concepts

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

Power Function

A power function is a mathematical relationship where one variable depends on another raised to a fixed power. In the context of plant biology, it is used to describe how the average dry weight per plant changes as a function of plant density. The formula used is usually of the form \(w = kp^b\), where \(w\) is the average dry weight, \(p\) is the plant density, \(k\) is a constant, and \(b\) is the power to which the density is raised.

In the phenomenon of self-thinning, power functions help us understand how plants expand in size under decreasing competition. Typically, when you perform this calculation, you use data to determine \(k\) and \(b\), often finding that \(b\) is a negative number, which reflects the inverse relationship between density and weight. In such equations, a higher absolute value of \(b\) indicates a stronger effect of density on weight. For horseweed plants, for example, experiments might yield a function like \(w = kp^{-1.5}\), indicating a 50% reduction in density leads to a more than twofold increase in weight.

Plant Density

Plant density refers to the number of plants growing within a specific area, often measured per square meter. It is a crucial factor in agricultural science because it affects the growth and yield of the plants.

When seeds are sown densely, seedlings compete for resources like sunlight, water, and nutrients. Over time, weaker plants die off, reducing density. This process is called self-thinning. As the density decreases, the surviving plants have more resources, allowing them to grow larger. This principle is illustrated in our table; as dates advance from November to June, there is a stark reduction in density. From 140,400 plants per square meter to just 2,060, the change is dramatic, highlighting how density influences plant survival and resource allocation.

Average Dry Weight

The average dry weight of a plant refers to the weight of a single plant after all its water content has been removed. It's a standard measure in plant biology to assess the growth and health of a plant.

In self-thinning studies, average dry weight increases over time despite the reduction in plant density. This is because plants that survive gain more access to nutrients and space. Our data table shows this clearly: average dry weight starts at a mere \(1.6 \times 10^{-4}\) grams and climbs to 0.085 grams by June. This considerable increase underscores the impact that reduced competition can have on individual plant growth.

Total Plant Yield

Total plant yield is a key metric in agriculture, defined for us as the product of the average dry weight per plant and the plant density. Calculating it gives an understanding of the productivity per unit area.

Interestingly, although individual plants get heavier as density reduces, total yield often decreases. Using the power function \(w = kp^{-b}\), we get the expression for total yield as \(y = kp^{1-b}\). Here, if \(b > 1\), the term \(p^{1-b}\) falls as \(p\) decreases, which causes total yield to reduce. The table's data support this: even as individual weights increase, overall yield sees a decline, illustrating that despite growth in size, the combined output of surviving plants diminishes as fewer plants remain.