Chapter 1

A community measure that takes both species abundance anc species richness into account is the Shannon diversity index \(H\). To calculate \(H\), the proportion \(p_{i}\) of species \(i\) in the community used. Assume that the community consists of \(S\) species. Then $$ H=-\left(p_{1} \ln p_{1}+p_{2} \ln p_{2}+\cdots+p_{S} \ln p_{S}\right) $$ (a) Assume that \(S=5\) and that all species are equally abundant that is, \(p_{1}=p_{2}=\cdots=p_{5}\). Compute \(H\). (b) Assume that \(S=10\) and that all species are equally abundant that is, \(p_{1}=p_{2}=\cdots=p_{10}\). Compute \(H\). (c) A measure of equitability (or evenness) of the specie distribution can be measured by dividing the diversity index \(H\) by \(\ln S\). Compute \(H / \ln S\) for \(S=5\) and \(S=10\). (d) Show that, in general, if there are \(N\) species and all specie are equally abundant, then $$ \frac{H}{\ln S}=1 $$

Simplify each expression and write it in the standard form \(a+b i\). \((2-3 i)(5+2 i)\)

The absorption of light in a uniform water column follows an exponential law; that is, the intensity \(I(z)\) at depth \(z\) is $$ I(z)=I(0) e^{-\alpha z} $$ where \(I(0)\) is the intensity at the surface (i.e., when \(z=0)\) and \(\alpha\) is the vertical attenuation coefficient. (We assume here that \(\alpha\) is constant. In reality, \(\alpha\) depends on the wavelength of the light penetrating the surface.) (a) Suppose that \(10 \%\) of the light is absorbed in the uppermost meter. Find \(\alpha\). What are the units of \(\alpha\) ? (b) What percentage of the remaining intensity at \(1 \mathrm{~m}\) is absorbed in the second meter? What percentage of the remaining intensity at \(2 \mathrm{~m}\) is absorbed in the third meter? (c) What percentage of the initial intensity remains at \(1 \mathrm{~m}\), at 2 \(\mathrm{m}\), and at \(3 \mathrm{~m} ?\) (d) Plot the light intensity as a percentage of the surface intensity on both a linear plot and a log-linear plot. (e) Relate the slope of the curve on the log-linear plot to the attenuation coefficient \(\alpha\). (f) The level at which \(1 \%\) of the surface intensity remains is of biological significance. Approximately, it is the level where algal growth ceases. The zone above this level is called the euphotic zone. Express the depth of the euphotic zone as a function of \(\alpha\). (g) Compare a very clear lake with a milky glacier stream. Is the attenuation coefficient \(\alpha\) for the clear lake greater or smaller than the attenuation coefficient \(\alpha\) for the milky stream?

In Problems \(91-96\), for each given pair of functions, use a graphing calculator to compare the functions. Describe what you see. \(y=\sin x\) and \(y=2 \sin x\)

Simplify each expression and write it in the standard form \(a+b i\). \((6-i)(6+i)\)

When plants are grown at high densities, we often observe that the number of plants decreases as plant weights increase (due to plant growth). If we plot the logarithm of the total aboveground dry-weight biomass per plant, \(\log w\), against the logarithm of the density of survivors, \(\log d\) (base 10 ), a straight line with slope \(-3 / 2\) results. Find the equation that relates \(w\) and \(d\), assuming that \(w=1 \mathrm{~g}\) when \(d=10^{3} \mathrm{~m}^{-2}\) In Problems 93-98, find each functional relationship on the basis of the given graph.

For each given pair of functions, use a graphing calculator to compare the functions. Describe what you see. \(y=\sin x\) and \(y=\sin (2 x)\)

Simplify each expression and write it in the standard form \(a+b i\). \((-4-3 i)(4+2 i)\)

For each given pair of functions, use a graphing calculator to compare the functions. Describe what you see. \(y=\cos x\) and \(y=2 \cos x\)

Let z \(=3-2 i\), \(u=-4+3 i, v=3+5 i\), and \(w=1-i .\) Compute the following expressions: \(\bar{z}\)

For each given pair of functions, use a graphing calculator to compare the functions. Describe what you see. \(y=\cos x\) and \(y=\cos (2 x)\)

Let z \(=3-2 i\), \(u=-4+3 i, v=3+5 i\), and \(w=1-i .\) Compute the following expressions: \(z+u\)

For each given pair of functions, use a graphing calculator to compare the functions. Describe what you see. \(y=\tan x\) and \(y=2 \tan x\)

Let z \(=3-2 i\), \(u=-4+3 i, v=3+5 i\), and \(w=1-i .\) Compute the following expressions: \(\overline{z+v}\)

For each given pair of functions, use a graphing calculator to compare the functions. Describe what you see. \(y=\tan x\) and \(y=\tan (2 x)\)

Let z \(=3-2 i\), \(u=-4+3 i, v=3+5 i\), and \(w=1-i .\) Compute the following expressions: \(\overline{v-w}\)

Let $$ f(x)=3 \sin (4 x), \quad x \in \mathbf{R} $$ Find the amplitude and the period of \(f(x)\).

Let z \(=3-2 i\), \(u=-4+3 i, v=3+5 i\), and \(w=1-i .\) Compute the following expressions: \(\overline{v w}\)

Let $$ f(x)=-2 \sin \left(\frac{x}{2}\right), \quad x \in \mathbf{R} $$ Find the amplitude and the period of \(f(x)\).

Let z \(=3-2 i\), \(u=-4+3 i, v=3+5 i\), and \(w=1-i .\) Compute the following expressions: \(\overline{u z}\)

Let $$ f(x)=4 \sin (2 \pi x), \quad x \in \mathbf{R} $$ Find the amplitude and the period of \(f(x)\).

If \(z=a+b i\), find \(z+\bar{z}\) and \(z-\bar{z}\).

Logistic Transformation Suppose that $$ f(x)=\frac{1}{1+e^{-(b+m x)}} $$ A function of the form (1.8) is called a logistic function. The logistic function was introduced by the Dutch mathematical biologist Verhulst around 1840 to describe the growth of populations with limited food resources. Show that $$ \ln \frac{f(x)}{1-f(x)}=b+m x $$ This transformation is called the logistic transformation. It is a standard transformation for linearizing functions of the form (1.8).

Let $$ f(x)=-\frac{3}{2} \sin \left(\frac{\pi}{3} x\right), \quad x \in \mathbf{R} $$ Find the amplitude and the period of \(f(x)\).

If \(z=a+b i\), find \(\bar{z}\). Use your answer to compute \(\overline{(\bar{z})}\), and compare your answer with \(z\).

Not every study of species richness as a function of productivity produces a hump-shaped curve. Owen (1988) studied rodent assemblages in Texas and found that the number of species was a decreasing function of productivity. Sketch a graph that would describe this situation.

Let $$ f(x)=4 \cos \left(\frac{x}{4}\right), \quad x \in \mathbf{R} $$ Find the amplitude and the period of \(f(x)\).

Solve each quadratic equation in the complex number system. \(2 x^{2}-3 x+2=0\)

Species diversity in a community may be controlled by disturbance frequency. The intermediate disturbance hypothesis states that species diversity is greatest at intermediate disturbance levels. Sketch a graph of species diversity as a function of disturbance level that illustrates this hypothesis.

Let $$ f(x)=7 \cos (2 x), \quad x \in \mathbf{R} $$ Find the amplitude and the period of \(f(x)\).

Solve each quadratic equation in the complex number system. \(3 x^{2}-2 x+1=0\)

Preston (1962) investigated the dependence of number of bird species on island area in the West Indian islands. He found that the number of bird species increased at a decelerating rate as island area increased. Sketch this relationship.

Let $$ f(x)=-3 \cos \left(\frac{\pi x}{5}\right), \quad x \in \mathbf{R} $$ Find the amplitude and the period of \(f(x)\).

Solve each quadratic equation in the complex number system. \(-x^{2}+x+2=0\)

Phytoplankton converts carbon dioxide to organic compounds during photosynthesis. This process requires sunlight. It has been observed that the rate of photosynthesis is a function of light intensity: The rate of photosynthesis increases approximately linearly with light intensity at low intensities, saturates at intermediate levels, and decreases slightly at high intensities. Sketch a graph of the rate of photosynthesis as a function of light intensity.

Let $$ f(x)=-\frac{2}{3} \cos \left(\frac{3 x}{\pi}\right), \quad x \in \mathbf{R} $$ Find the amplitude and the period of \(f(x)\).

Solve each quadratic equation in the complex number system. \(-2 x^{2}+x+3=0\)

Brown lemming densities in the tundra areas of North America and Eurasia show cyclic behavior: Every three to four years, lemming densities build up very rapidly, and they typically crash the next year. Sketch a graph that describes this situation.

Use the fact that \(\sec x=\frac{1}{\cos x}\) to explain why the maximum domain of \(y=\sec x\) consists of all real numbers except odd integer multiples of \(\pi / 2\).

Solve each quadratic equation in the complex number system. \(4 x^{2}-3 x+1=0\)

Nitrogen productivity can be defined as the amount of dry matter produced per unit of nitrogen per unit of time. Experimental studies suggest that nitrogen productivity increases as a function of light intensity at a decelerating rate. Sketch a graph of nitrogen productivity as a function of light intensity.

Use the fact that \(\csc x=\frac{1}{\sin x}\) to explain why the maximum domain of \(y=\csc x\) consists of all real numbers except integer multiples of \(\pi\).

Solve each quadratic equation in the complex number system. \(-2 x^{2}+4 x-3=0\)

A study of Borchert's (1994) investigated the relationship between stem water storage and wood density in a number of tree species in Costa Rica. The study showed that water storage is inversely related to wood density; that is, higher wood density corresponds to lower water content. Sketch a graph of water content as a function of wood density that illustrates this situation.

First determine whether the solutions of each quadratic equation are real or complex without solving the equation. Then solve the equation. \(3 x^{2}-4 x-7=0\)

Species richness can be a hump-shaped function of productivity. In the same coordinate system, sketch two hump-shaped graphs of species richness as a function of productivity, one in which the maximum occurs at low productivity and one in which the maximum occurs at high productivity.

First determine whether the solutions of each quadratic equation are real or complex without solving the equation. Then solve the equation. \(3 x^{2}-4 x+7=0\)

The size distribution of zooplankton in a lake is typically a hump-shaped curve; that is, if the frequency (in percent) of zooplankton is plotted against the body length of zooplankton, curve that first increases and then decreases results. Brooks and Dodson (1965) studied the effects of introducing a planktivorous fish in a lake. They found that the composition of zooplankton after the fish was introduced shifted to smaller individuals. In the same coordinate system, sketch the size distribution of zooplankton before and after the introduction of the planktivorous fish.

First determine whether the solutions of each quadratic equation are real or complex without solving the equation. Then solve the equation. \(-x^{2}+2 x-1=0\)

Daphnia is a genus of zooplankton that comprises a number of species. The body growth rate of Daphnia depends on food concentration. A minimum food concentration is required for growth: Below this level, the growth rate is negative; above, it is positive. In a study by Gliwicz (1990), it was found that growth rate is an increasing function of food concentration and that the minimum food concentration required for growth decreases with increasing size of the animal. Sketch two graphs in the same coordinate system, one for a large and one for a small Daphnia species, that illustrates this situation.