AP Biology Exponential and Logistic Growth Problems

INTRODUCTION: The central questions of population ecology relate to the factors thatinfluence the distribution and abundance of animals and plants. An important steptowards answering these questions has been to develop mathematical models thatattempt to predict the growth patterns of populations in different environments.Population ecology modelers construct their models by identifying all relevantvariables (often called parameters) needed to predict changes in population size andthen by attempting to determine the exact manner in which they interact to explainpopulation growth. The models are then tested in the usual manner -- by seeing if thepredictions match the actual behavior of populations. To the extent there is a match, apopulation ecologist becomes more certain that she understands the factors thatactually determine the population's growth.

As an introduction to the field, we will review two general models of populationgrowth:

  • One for a situation where resources are essentially unlimited and therefore individualsdo not compete
  • The other for situations where there are limited resources and therefore members ofthe population must compete for these resources.

THE BASICS. EXPONENTIAL GROWTH: It shouldn't surprise you to know thatpopulation ecology is mathematical in nature. After, models are usually mathematicalabstractions of nature! Let's look over some of the main ideas as painlessly aspossible and then apply them to populations where resources are or are not limiting.Here are some of the main parameters:

  • N - the population size (not necessarily the most important parameter, believe it or not). Usually we specify population size in terms of time starting at some arbitrary point. Thus:
  • N0 - the population size at some arbitrary starting time (time zero)
  • Nt -- the population size at some future time t (where t can have any positive value) forexample, N2 would be the population size two time units (whatever they are -- seconds,years, generations, breeding seasons) in the future.

If we take the ratio of the population size in two adjacent time periods (for instance t =0 and t = 1), we obtain the factor by which the population increases during one unit oftime. We call this factor Y.

Thus equation 1: or restated in an equally useful way:

If a population goes from 100 to 200 in time = 1 unit, then Y is 2; if 100 to 300 then Y is 3, etc. If Y = 4 (a very high number) and the original population is 100, what is it after time 1?After two time units?

Ans: 400, 1600 -- in the second case, the population went from 100 to 400 in time one andthen increased by a factor of four again to 1600.If we are interested in the rate of change of a population's size,, then we canstate that:

Equation 2:

where:

  • B is the birth rate (usually the number of individuals or females born per unit time)
  • D s the death rate (number of individuals or females dying per unit time)
  • I is the immigration rate (individuals or females migrating into the population per time)
  • E is the emigration rate (individuals or females migrating out of the population pertime)

There are additional parameters that can be used to calculate B and D (factors such asage-specific mortality rates and age-specific birth rates).To keep things simple, lets assume that:I = E, and therefore:

Equation 3:

And through some substitutions that I will not explain here you can also say

Growth in an Unlimited Environment: Suppose that organisms find themselves in anideal physical environment where they are not limited by food, space, access to mates,or any other thing that might impede their growth. Examples are species that arerecent invaders of new habitats to which they are well-adapted -- for instance, microbes in a nutrient broth, aphids on an undefended plant, seeds in a freshly tilled, fertilizedand watered plot.For as long as there is no competition for these resources (in ecology we saythat none of the resources are limiting) the population's growth will be entirelydetermined by its intrinsic rate of increase, r0 (see above). Since r0 is largely ameasure of how rapidly organisms of a certain type can turn resources intoreproduction, the higher the r0 value, the faster the population grows. When solved and plotted, it produces what is often called a J-shaped or exponential curve.

However, no population can increase exponentially forever. Sooner or later it will encounter some form of environmental resistance, e.g., individuals will either run outof resources, or encounter a catastrophic change in the environment which causes the population to crash. This latter event is the more usual for a number of species,especially insects and algae. These organisms exhibit exponential growth whenconditions are favorable, but when the environment changes, the population crashesto very low density only to rise again when conditions improve. A good example ofthis type of population growth is seen every spring with algal blooms due to themixing of water in a pond or lake. After the ice melts, the water has a fairly uniformtemperature and this allows nutrients in the sediment at the bottom to mix in thewater at all levels. Algae near the surface use sunlight and these nutrients toreproduce exponentially and soon the pond's surface has a greenish color. As summerapproaches and the sun warms the surface water, a strong temperature barrier iscreated which prevents the nutrients from reaching the surface. Consequently, thepopulation of algae crashes until the next spring. Populations of species which followa series of J-shaped curves as their growth pattern, as with this example, are usuallycontrolled by physical aspects of the environment, e. g. , weather, which changeabruptly. Since the effect of these abiotic factors is independent of the density of thepopulation, this pattern is called density-independent growth. Density-independentgrowth patterns controlled by abiotic factors such as weather characterize speciesvariously referred to as opportunistic species (because they take advantage offavorable conditions to realize their biotic potential), colonizing species (because theyoften disperse to new areas when their density is high), or fugitive species (becausethey cannot persist for long periods in any one area due to competition with morestable or equilibrium species).We will see them discussed below under evolutionary ecology as "r-selected species". Notice that individuals in these "r-selected" speciescompete with each other by being able to disperse better and/or to produce moreoffspring in a shorter period of time.

LOGISTIC GROWTH: Populations in a Limited Environment: More often than not,there is some degree of competition for resources. In fact, in many species and formuch of the time, individuals compete intensely for resources. How can we model thegrowth of such populations?Population ecologists introduce the term carrying capacity (K). It is defined asthe maximum number of individuals of a certain population that can survive in a givenhabitat. It is assumed that any number of individuals beyond K simply cannot survive -- either they die or they are driven off or they emigrate to find a better place to live.Obviously, if the actual population, N, is very close to K, then the population shouldnot be able to grow. On the other hand, if the population is a long way away from K,(i.e., very few individuals compared to the available resources - available open slots formembers of this species) then population growth should be very rapid and in factshould approach exponential growth. Here is one way to treat this mathematically:

where r is growth rate (the maximum rate possible for this species when nothing islimiting).

Notice that the expression (K - N)/ Kdescribes a "braking factor" on the population'sgrowth. The closer N comes to K the smaller that fraction (the more(K - N)/ Kapproaches zero) and the more slowly the population grows. When N = K there is nogrowth, if N > K then the population's growth rate (dN/dt) is negative.

Multiple Choice Questions.

  1. To measure the population density of monarch butterflies occupying a particular park, 100 butterfliesare captured, marked with a small dot on a wing, and then released. The next day, another 100butterflies are captured, including the recapture of 20 marked butterflies. One would estimate thepopulation to be

a. 200.

b. 500.

c. 1,000.

d. 10,000.

e. 900,000.

  1. A population of ground squirrels has an annual per capita birth rate of 0.06 and an annual per capitadeath rate of 0.02. Estimate the number of individuals added to (or lost from) a population of 1,000individuals in one year.
  2. 120 individuals added
  3. 40 individuals added
  4. 20 individuals added
  5. 400 individuals added
  6. 20 individuals lost
  1. A small population of white-footed mice has the same intrinsic rate of increase (r) as a large population. If everything else is equal,
  1. the large population will add more individuals per unit time.
  2. the small population will add more individuals per unit time.
  3. the two populations will add equal numbers of individuals per unit time.
  4. the J-shaped growth curves will look identical.
  5. the growth trajectories of the two populations will proceed in opposite directions.
  1. In the logistic equation , r is a measure of the population's intrinsic rate of increase. It is determined by which of the following?
  2. birth rate and death rates
  3. dispersion
  4. density
  5. carrying capacity
  6. life history
  1. In 2005, the United States had a population of approximately 295,000,000 people. If the birth ratewas 13 births for every 1,000 people, approximately how many births occurred in the United Statesin 2005?
  2. 3,800
  3. 38,000
  4. 380,000
  5. 3,800,000
  6. 38,000,000
  1. As N approaches K for a certain population, which of the following is predicted by the logisticequation?
  2. The growth rate will not change.
  3. The growth rate will approach zero.
  4. The population will show an Allee effect.
  5. The population will increase exponentially.
  6. The carrying capacity of the environment will increase.
  1. Often the growth cycle of one population has an effect on the cycle of another. As moose populations increase, wolf populations also increase. Thus, if we are considering the logistic equation for the wolf population,, which of the factors accounts for the effect on the moose population?
  1. r
  2. N
  3. rN
  4. K
  5. dt
  1. Which of the following might be expected in the logistic model of population growth?
  2. As N approaches K, b increases.
  3. As N approaches K, r increases.
  4. As N approaches K, d increases.
  5. Both A and B are true.
  6. Both B and C are true.
  1. According to the logistic growth equation
  2. the number of individuals added per unit time is greatest when N is close to zero.
  3. the per capita growth rate (r) increases as N approaches K.
  4. population growth is zero when N equals K.
  5. the population grows exponentially when K is small.
  6. the birth rate (b) approaches zero as N approaches K.

Short Answer

  1. Distinguish between exponential and logistic population growth. Give the equations foreach.
  1. At time t1 you capture, mark and release 60 individuals. At time t2you capture 144 individuals; 24 of them are marked (recaptures). What is your estimate of populationsize?
  1. On day 1 you capture, markand release 78 house sparrows. Two days later youcapture both marked and unmarked birds; the number markedis 26. You estimate a population sizeof 195 individuals. How many unmarkedbirds were captured on day 2?
  1. N(t) is population size at time t; time is continuous. Briefly define ; then define. Note that these definitions should apply to any continuous-time model of population growth, not just exponential growth.
  1. A population grows exponentiallyfrom an initial populationsize (i.e., at time 0) of 10 individuals. The intrinsic rate of increaseis 0.1. What is the population sizeat time 10?
  1. Consider a population of two asexually reproducing genotypes (1 and 2). Assume that each genotypic lineage grows exponentially, at constant rates r1and r2, respectively. If r1> r2, what will happen as time grows large?