Vaccari D.A., Strom P.F., Alleman J.E. Environmental Biology for Engineers and Scientists

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This equatio n has the following analytical solution for N at time t, N(t), based on the

initial population N(0):

NðtÞ¼

1 þ be

r

b ¼

K  Nð0Þ

Nð0Þ

ð14:26Þ

The solution has a sigmoid shape if Nð0Þ < K=2 (Figure 14.15), sometimes referred to

by ecologists as the S-shaped growth curve. This distinguishes the logistic growth curve

from exponential growth, which is called the J-shaped growth curve (Figure 14.15e).

Exponential growth is essentially like logistic growth but with an inﬁnite carrying capa-

city. Curves (b) and (c) show the effect of varying the parameters, and curve (d) shows

what happens if the initial population is greater than the carrying capacity.

Compare the logistic equation with the Monod model for microorganism growth [equa-

tion (11.6), Section 11.7.2]. The Monod model speciﬁcally includes the limiting resource

needed for growth (substrate, S). In the Monod model, exponential growth is possible at

any resource level as long as the resource concentration is held constant. Growth limita-

tion will occur only if the disappearance of the reso urce is modeled separately. In the

logistic equatio n, the organisms in a population are assumed to have to compete with

each other for the resources, so as population increases, the growth rate decrease s. The

total amount of resource available to the population is assumed to be held constant.

100

120

140

160

024681012

(a)

(b)

(c)

(d)

(e)

Figure 14.15 Logistic equation solution (14.26) with several parameter values, compared to

exponential growth equation (14.18). The dashed line is N ¼ 100. (a) logistic equation with

¼ 1:0, K ¼ 100, Nð0Þ¼5:0; ( b) logistic equation with r

¼ 0:75, K ¼ 100, Nð0Þ¼5:0; (c)

logistic equation with r

¼ 1:0, K ¼ 70, Nð0Þ¼5:0; (d) logistic equation with r

¼ 1:0, K ¼ 100,

Nð0Þ¼150; (e) exponential equation with r

¼ 0:7, Nð0Þ¼5:0.

478

ECOLOGY: THE GLOBAL VIEW OF LIFE

The parameters of the logistic equation may be obtained by ﬁtting the equation to

laboratory or ﬁeld data. For example, Pearl and Reed, who originally proposed the logis-

tical equation in 1920, ﬁtted it to U.S. population data from 1790 to 1910, when the popu-

lation was about 92 million. They obtained r

¼ 0:03134 per year and K ¼ 197; 273; 000.

Thus, they projected a doubling of the U.S. population, followed by a leveling off. Notice

that their prediction held for four more decades (Figure 14.16).

In 1950, however, the population started increasing much faster than predicted. The

model fell victim to a basic problem with any model: The conditions on which it is

based may change, resulting in either different parameter values or an altogether different

model. In this case, the growth rate stopped decreasing as the logistic equation would have

predicted. An updated model ﬁtted to the data from 1950 to 1990 predicts almost the

same r

¼ 0:03281 but a much higher K ¼ 324; 796; 000. Thus, the carrying capacity

of the United States seems to have increased, suggesting that the U.S. population will

approach 325 million in the next century if things remain the same. (The population in

2005 is about 296 million.) The increased carrying capacit y may have been due to the

postwar prosperity and exploitation of new resources (raw materials, markets for our

products). This result was a jump in the growth rate of the U.S. population called the

‘‘baby boom.’’

The logistic equation has led to a terminology to describe two basic reproductive stra-

tegies of organisms. An organism is called K-selected if it prospers by making efﬁcient

use of its resources, maximizing its carrying capacity. These organisms are more likely to

be associated with crowded, stable ecosystems. They produce relatively small numbers of

young but may nurture them to ensure their survival. Other organisms, called r-selected,

adopt a strategy of rapid reproduction. These organisms are described as opportunistic.

They can quickly take advantage of stressed conditions. They tend to produce large num-

bers of offspring but have much higher mortality. Weed species often fall into this cate-

gory, as does that urban nuisance, the rat. The r-selected organisms will also have an

advantage in temperate or arctic climates, where the short growing season favors rapidly

growing species.

100

150

200

250

300

350

1750 1800 1850 1900 1950 2000 2050 2100

Year

Population (millions)

Updated

Pearl & Reed

Figure 14.16 U.S. population data with logistic equation ﬁt by Pearl and Reed, and updated

logistic equation ﬁtted to years 1950–1990.

POPULATIONS AND COMMUNITIES 479

Laboratory experiments with single species populations have revealed behaviors that

cannot be described by the models described above. Some species of Daphnia overshoot

their carrying capacity and then oscillate around it. They are able to store food in the form

of fat deposits, which delays the effects of resource limitations. This behavior can be

described by using models with time-delay elements. For example, the logistic equation

may be modiﬁed by using populations taken from \tau time units in the past, Nðt  tÞ:

¼ r

NðtÞ

K  Nðt  tÞ

ð14:27Þ

Each of the foregoing models is deterministic, meaning that for each set of initial con-

ditions, there is a unique prediction. However, the outcome in reality will be affected by

unknown factors which produce random variation. A model that treats the variables as a

random variable is a stochastic model. Each of the preceding models could be made sto-

chastic by adding an inde pendently and randomly distributed ‘‘error’’ at ﬁxed intervals of

time, before continuing the prediction to the next time step. The prediction of a stochastic

model is not just a single value, but a value that has a probability distribution.

None of these models reﬂect interactions with other species. Whatever is the source of

food for the organism will certainly be subject to its own population dynamics, in

response to the predation. In the next several sections we address some of these issues.

Competition The ﬁnal term in equation (14.25) represents the effect of individuals from

a single population competing among themselves. If another organism, species j, is using

the same limi ting resource as the population being modeled, species i, the logistic equa-

tion can be modiﬁed to take this into account:

¼ r

 N

 b

ð14:28Þ

where N

is the population of species i, N

the population of species j, and b

the positive

number coefﬁcient of competition. The coefﬁcient of competition expresses how much

the competing population effectively reduces availability of the resource to population i.

A similar equation would be written for species j by switching the subscripts. If more than

two species are interacting, additional terms of the form of b

would be included. A

differential equation similar to (14.28) must be written for each interacting species, and

the equations solved simultaneously.

A stability analysis of equation (14.28) reveals the conditions under which each species

can survive. For two species we can form the dimensionl ess groups a

¼ b

ðK

Þ and

¼ b

ðK

Þ. If both a

< 1 and a

< 1, the species can coexist, although at a

lower population than each by itself. If both a

 1 and a

 1, only one species can

survive; which one depends on the exact dynamics and the initial conditions. If a

 1

and a

< 1, species 1 will always win out. These dimensionless numbers can be inter-

preted as a measure of the competitiveness of each species. If both numbers are less than

1, competition is weak and the species can coexist. If both are greater than 1, competition

is strong and only one species can survive.

Study of this model has led to the development of the principle of competitive exclu-

sion, which states that species with identical resource requirements cannot coexist. Strong

competition tends to result in the elimination of one species from the ecosystem. Thus,

480 ECOLOGY: THE GLOBAL VIEW OF LIFE

strong competition does not often occur in nature. In cases of apparent strong competition,

it has often been found that the two species may have some signiﬁcant difference. For

example, ﬁve similar species of warblers in New England forests all eat inse cts. However,

they feed at different parts of the forest canopy and in different ways, and they nest at

different times. Strong predation by another species can keep several species sufﬁciently

far below their carrying capacity that competition does not seriously limit the availability

of the mutual resource. Darwin noticed this when he pointed out that grazing increases the

diversity of plants in meadows. When the predation is limited, a single species tends to

dominate. This is what has happened in many ecosystems affected by human activities.

People have tended to remove top carnivores, such as grizzly bears and wolves in western

North America. This actually had a negative impact on most of their prey, by increasing

the role of competition among them. Thus, conservationists have the goal of reintroducing

many of these predators to increase the diversity of the ecosystems.

The principle of competitive exclusion is linked to the concept of niche. The niche can

be described as the resources and other factors favored by a species. Then the principle of

competitive exclusion is equivalent to the statement made above that no two organisms

can occupy the same niche.

Mutualism and Symbiosis Mutualism and symbiosis are simply the opposite of compe-

tition. Instead of a species reducing the carrying capacity of another species, it increases it.

This can be modeled simply by changing the sign of the interaction term in equation (14.28):

¼ r

 N

þ b

ð14:29Þ

If b

< 1, there is a stable steady-state population for both species, which for each

is greater than their respective carrying capacities. However, the model is not realistic for

> 1, in which case the mutual beneﬁt is so great that both populations grow with-

out limit.

Predator–Prey: Lotka–Volterra Equations One of the most interesting types of interac-

tions are those between consumers (predators, parasites, and herbivores) and their hosts.

Lotka and Volterr a derived independently a model which although somewhat simplistic

for modeling real populations, is nevertheless capable of describing some of the important

behaviors found in predator–prey relat ionships. One important behavior is cycles in pre-

dator and prey populations (Figure 14.17). Intensive predation can cause a collapse in

prey population. This is followed by a drop in predator population, due to the drop in

food supply. As a result, the prey population can boom, followed by the predator to com-

plete one cycle.

The Lotka–Volterra predator–prey model has the following form:

¼ Hða bPÞ

¼ PðcH  dÞ

ð14:30Þ

where H is the host or prey population, P is the predator population, and a, b, c, and d are

coefﬁcients. Multiplying through, the host equation is seen to have two terms: The aH

POPULATIONS AND COMMUNITIES 481

term gives it exponential growth when the predator is absent. The bHP term is a nega-

tive interaction due to predation. The predator equation has a positive interaction corre-

sponding to the negative one for the host. The dP term provides an exponential death

rate without which the popul ation could never decrease. There is no exponential growth

term for the predator, and there is no carrying capacity for either equation.

The Lotka–Volterra model has a stable steady-state or equilibrium solution (other than

H ¼ P ¼ 0) when H ¼ d=c and P ¼ a=b. If values other than these are used as an initial

condition, the solution will be a sta ble oscillation. The plot of host and predator popula-

tions vs. time (Figure 14.18a) shows that predator population peaks just after the host

population. Examination of the phase-plane plot (Figure 14.18 b) suggests that predator

population peaks when the host population is at its greatest rate of decline.

More realistic models have been developed. For example, adding a term of the form

eH

to the host equation makes it equivalent to a logistic equation with a negative self-

interaction. Other modiﬁcations place an upper limit on the amount of prey that each

predator can consume. Other factors that affect predator–prey dynamics include the

100

200

300

400

500

600

700

800

900

0 5 10 15 20 25

Generation

Population

Host

Predator

Figure 14.17 Oscillations in predator–prey populations: the predatory wasp Heterospilus

prosopidis and its host the weevil Callosobruchus chinensis. (Data from Utida, 1957.)

0.0

20.0

40.0

60.0

80.0

100.0

120.0

0 0.1 0.2 0.3 0.4 0.5 0.6

Population

0.0

5.0

10.0

15.0

20.0

25.0

30.0

0.0 50.0 100.0 150.0 200.0

(a)

(b)

Figure 14.18 Simulation results of the Lotka–Volterra equations: (a) time-domain plot with

Hð0Þ¼100 and Pð0Þ¼10; (b) phase-plane plot (P vs. H) for various initial conditions, and the

equilibrium point.

482

ECOLOGY: THE GLOBAL VIEW OF LIFE

availability of safe refuges, which limits predation when host population drops, and the

presence of alternative food sources for the predator. Some of these factors make multiple

steady-state values possible. This can lead to epidemics, when some environmental dis-

turbance stimulates the system to jump to a different steady-state condition. Also, preda-

tor–prey populations can stabilize by some sort of accommodation to each other, reducing

the amplitude of oscillations.

Empirical Models Another way to model single populations is to use autoregressive

models of the form

NðiÞ¼f

Nði  1Þþf

Nði  2Þþf

Nði  3Þþ ð14:31Þ

in which N(i) represents the population at equispaced time intervals, and Nði  kÞ is the

population k time steps earlier (lagged by k). The coefﬁcients can be obtained by multi-

linear regression techniques. Because this method is empirical, the coefﬁcients cannot

necessarily be identiﬁed with particular phenomena, such as growth, death, or predation

rates. It is common to perform regressions of the form of equation (14.31) using loga-

rithm-transformed variables instead of raw data. This can make the errors of the regres-

sion more normally distributed, and eliminates the possibility of negative predictions from

the model. In addition, variables other than lagged population, such as the population of

other species, can be used as dependent variables.

Multilinear regression software is capable of determining which coefﬁcients, and

therefore which independent variables, contribute signiﬁcantly to the predictive ability

of the model. Thus, it is possible to tes t which populations affect the dependent population

and whether that effect is positive or negative.

Notice that the models discussed previously, including the exponential growth model

(14.17), the logistic model (14.25), the competition model (14.28), the mutualism mode l

(14.29), and the Lotka–Volterra predator prey model (14.30), can all be expressed as mul-

tivariable polynomials of the following form by combining terms and coefﬁcients:

¼ a

þ a

ð14:32Þ

Effects of even more species can be modeled by adding appro priate terms to the model,

such as a

. By examining the signs of the coefﬁcients of a pair of interacting species

and comparing to the signs in Table 14.5, the type of interaction can be determined from

population dynamics. This information can even help analyze the trophic structure of an

ecosystem.

Terms such as a

test for three-way interactions. Statistical tests can show

whether such effects are signiﬁcant. Otherwise, they can be dropped from the model.

As Krebs has sta ted: ‘‘Spec ies interactions are rarely one-on-one in natural communities,

and the untangling of complex sets of species interactions is an important focus in ecology

today.’’

Stochastic Extinction A population that is in equi librium (birth rate ¼ death rate ¼ b)

can nevertheless experience population ﬂuctuations due to random variations. Thus,

there is a ﬁnite probability that a population could go extinct just by chance. The

POPULATIONS AND COMMUNITIES 483

probability, p(t), of this occurring in a ﬁnite amount of time t can be estimated from b and

from the average population:

pðtÞ¼

1 þ bt



ð14:33Þ

For example, for a population with a birth rate of 0.5 per year, which is reasonable for

terrestrial vertebrates, Table 14.8 shows the probabilities for various population sizes

and periods of time. The table shows that the chance of extinction after a ﬁxed period

increases greatly as the size of the population decreases. For example, a population of

100 has a 14% chance of disappearing over the next century, whereas a population of

10 has an 82% chance.

This effect has been observed in nature in island ecosystems. On average the lost spe-

cies are balanced by immigration or by evolution of new species. However, the role of

humans is disturbing this balance. Urban and agricultural development may destroy

only part of an ecosystem. Yet, even if a portion is preserved out of concern for the wild-

life, the chances are increased that the species contained within will be lost. Simply put-

ting a road through a wilderness divides it into two smaller parts. If movement of

organisms of a particular species is inhibited from crossing, the overall chances of survi-

val will have been decreased.

These concepts have led to the idea of mi nimum viable population, which is a level

sufﬁcient to protect a species not only from stochastic extinction but also from calamities

such as ﬁre or extreme weather. Furthermore, protection of a species requires that a spe-

cies have multiple populations for two reasons: First, if one population is wiped out, it

could be reintroduced from another. Second, having multiple centers of population main-

tains genetic diversity, which may be necessary to develop protection against evolving

diseases.

14.4.2 Species Richness and Diversity: Synoptic Structure of Communities

Now we begin to look at communities instead of the individual populations that constitute

them. Why do similar environmental conditions produce similar sets of species? The

interactions between species are what stabilize a community. The more species that are

present, the more interactions there can be. Removing any single organism eliminates its

interactions, affecting those species with which it interacted. But the more overall species

there are, the easier it will be for an affected organism to ﬁnd a substitute. For example,

TABLE 14.8 Extinction Probability for Birth Rate ¼ 0.5 per Year

Time (yr)

Population 1 10 100 1000

1 0.33 0.83 0.98 0.998

10 10

4

0.16 0.82 0.980

100 10

48

7

0.14 0.819

1000 10

99

79

8

0.135

Source: Ricklefs (1993).

484 ECOLOGY: THE GLOBAL VIEW OF LIFE

suppose that mice are the primary food source for owls in a particular forest, and the mice

are eliminated. If other small mammals, say rabbits, are available, the owls might be able

to substitute them. Therefore, it is thought that species diversity is an indication of com-

munity stability and resistance to disturbance. Th e conce pt of species diversity bringing

stability is used by conservationists to argue for the protection of ecosystems from human

impacts.

There are a numb er of caveats to this idea. Di sturbance, itself, can increase diversity.

Well-developed ecosystems can settle into a lower-diversity state than they had during

their stages of development. Also, it is observed that natural systems do not seem to

have maximized their diversity as measured by the indices described below. This

would require an even distribution of species. Instead, there are always some relatively

rare species and others more dominant. Finally, it is possible that diversity is a conse-

quence of ecosystem stability rather than a cause.

The Number of Species in an Ecosystem The simplest measure of diversity is simply

the number of species, S. It may be difﬁcult to measure this in an ecosystem without

impractically large samples. However, it can be estimated from a limited sample by

assuming a lognormal probability distribution for species abundance. Generally, smaller

geographic areas can be expected to have fewer species. An empirical rule relating S and

area, A,is

S ¼ cA

ð14:34Þ

where c is a parameter that varies with type of ecosystem and z is a parameter that has

consistently been found to fall in the range 0.20 to 0.35 for islands. For large continental

areas, the exponent ranges from 0.15 to 0.24, apparently due to the greater ability to

import species. The number of species in ecosystems tends to increase toward the tropics,

as does species diversity.

The equilibrium theory of biogeography, also calle d island biogeography, proposed

by Robert MacArthur and E. O. Wilson, explains the relationship between area and num-

ber of species. They suggested that S is a result of a balance between immigration and

extinction. Smaller areas would support smaller populations and thus would have higher

rates of stochastic extinction [equation (14.33)]. Islands closer to the mainland have

higher immigration rates and thus higher numbers of species. A similar balance occurs

on the mainland itself, except that instead of immigration providing new species, it occurs

by evolution. Of course, this acts more slowly, so the turnover of species is much slower.

Diversity Indices A number of mathematical diversity indices have been developed to

quantify species diversity better, giving more weight to more dominant species. Measure-

ment of diversity may be subject to the sample-size limitation mentioned above. Also,

measurements of diversity are usually limited to a subset of the community, such as a

single taxa (birds, grasses, aquatic invertebrates, etc.) or a single trophic level. Despite

these measurement problems and the caveats mentioned above with regard to their signif-

icance, diversity indices can be useful for making comparisons or for tracking trends.

Several of the mos t popular indexes can be computed from the proportions, p

, of each

of the S species. One is the Simpson index, D:

D ¼

p

ð14:35Þ

POPULATIONS AND COMMUNITIES 485

The Shannon–Weaver index, H, is based on information theory. It gives more weight

to rare species than does Simpson’s index. It has the advantage of being normally distrib-

uted, making it possible to use statistical tests of signiﬁcance to compare different values.

H ¼ðp

ln p

Þð14:36Þ

Table 14.9 shows examples for several different distributions of ﬁve species. Note that for

H it is necessary to ignore zero values in the calculation (you can’t take the log of zero).

The Shannon–Weaver index is often reported as its exponential form, since e

is equal

to S when the populations are evenly distributed. Both D and e

vary from 1.0 whe n only

one species is represented, to S if the species are evenly distributed. The Shannon–Weaver

index can be used to compute an evenness parameter, Pielou’s index, e. The evenness

index varies from zero for maximally uneven, to 1.0 for p erfectly uniform distribution:

e ¼

ln S

ð14:37Þ

Because e depends explicitly on the number of species, its value is affected by including

species with zero population in the count. Potentially, we could add any number of species

which are absent from the ecosystem. Therefore, there would have to be some indepen-

dent criteria for including such species, such as because they were formerly present in that

ecosystem or because they are typically found in similar ecosystems.

A graphical approach that gives more information is to plot the probability distribution

of the species. This plot is called the dominance-diversity curve. It is made by sorting

the species by p

, and plotting p

vs. rank order. Evenness will be related to the slope of the

curve.

Food web diversity has also been deﬁned as a diversity index computed with abun-

dances classiﬁed by trophic level instead of species.

14.4.3 Development and Succession: Temporal Structure of Communities

Previously, we looked at how to describe changes in individual populations. The basic

population growth models often presuppose that environmental conditions, as reﬂected

in carrying capacity, are constant. Community changes occur in nature as a result of

disturbances that alter individual populations and environmental conditions. The most

TABLE 14.9 Species Diversity Indexes for Hypothetical Communities Having Different

Abundances

Shannon– Pielou’s,

Proportion of Total Population Simpson, D Weaver, H exp H Evenness, e

0.20 0.20 0.20 0.20 0.20 5.00 1.609 5.00 1.000

0.30 0.25 0.20 0.15 0.10 4.44 1.544 4.69 0.960

0.35 0.30 0.20 0.10 0.05 3.77 1.431 4.18 0.889

0.50 0.25 0.15 0.10 0.00 2.90 1.208 3.35 0.751

1.00 0.00 0.00 0.00 0.00 1.00 0.000 1.00 0.000

Source: Based on Ricklefs (1993).

486 ECOLOGY: THE GLOBAL VIEW OF LIFE

commonplace of these are the changes of seasons. More catastrophic changes include

ﬂoods, ﬁres, extreme weather, or volcanic eruptions. The time scales of human distur-

bances can range from practically instantaneous (nuclear explosions) to decades or

more (global warming or soil depletion by agriculture). These disturbances affect the

entire ecosystem, which responds as a unit. Although potentially, we could describe

these ecosystem-wide changes using population models for all the individual species,

we can note some holistic patterns.

A new habitat can be caused by a large event, such as a forest ﬁre or clearcutting of a

forest, or by a small local change such as the falling of a tree or a pile of dung left by a

bear. In each case, the new habitat will be colonized by pioneers who are especially good

at invading. These are often the r-selected organisms. Eventually, they will be replaced by

more efﬁcient K-selected species. In between, a variety of species may come and go in

abundance. The process of sequential population changes initiated by a disturbance is

called succession. The actual sequence of commun ities is called a sere. Barring further

disturbances, succession ends when ecosystems develop a stable condition called a

climax. The climax community is ultimately a community that can succe ed itself. A

climax community may form within a season or it make take decades, as in the case of

forests.

As an example, an abandoned farm ﬁeld may be colonized by grasses, which inhibit

the germi nation of trees. The grasses attract herbivores, which create openings for shrubs

by intense grazing. The shrubs provide shade, which enables pine to germinate and even-

tually to dominate. However, when the cover becomes too dense, the pine seedlings will

not grow, and hardwood trees gain an advantage. Eventually, a climax community is

formed as a hardwood forest. Populations of birds and other animals change as the

food supply changes.

Another example is algal succession in temperate-zone lakes. As the water warms in

the spring, the ﬁrst species that dominate may be the cyanobacter. Some cyanobacter ﬁx

nitrogen and so are limited only by phosphorus. Eventually, they deplete the available

phosphorus, removing it from the water column as they die and settle to the bottom.

Other algae, such as diatoms, may ﬂourish. But these require silicon for their shells,

and when that is depleted, they may make way for green algae.

There are several forces driving succession. A community may change its environ-

ment, making it more suitable for others that succeed it as the shrubs made way for the

pines. Succeeding communities are oft en those that tolerate a lower level of resources, as

the diatoms that followed the cyanobacter. Sometimes the succeeding community must

overcome inhibition by the previous residents, as the shrubs were inhibited by the grasses.

Newly exposed soils may experience a gradual drop in pH from the accumulating pro-

ducts of plant decay.

Overall, succession seems to be related to a balance of colonizing ability of some spe-

cies vs. the competitive ability of others. A particular sequence may not be uniquely deter-

mined by environmental conditions. One could wind up with one of several communities,

depending on contingent factors during succession. However, chance factors seem to

operate most strongly in earlier stages of succession.

Odum (1987) has listed a number of trends identiﬁed with succession in the absence of

further disturbances (Table 14.10). Gross primary productivity tends to form a peak dur-

ing succession and then declines as a community approaches the climax. Succession

occurs in a biological wastewater treatment plant whenever conditions are changed.

Plant startup is a prime example of this. Primary productivity is negligible in this system,

POPULATIONS AND COMMUNITIES 487