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Limits on Growth

No population grows without bounds, so we need to modify our population model to predict the fact that many populations have a so-called limiting population that is determined by the carrying capacity of their environment.

The easiest way to model a limiting population is to introduce a new term into our population model. This term is called an overcrowding term and the coefficient of this term is called the coefficient of overcrowding. The simplest overcrowding term is proportional to the square of the current population. In other words:

dP/dt = k P - A P².

Assuming that A>0, the negative sign in the second term indicates that this term decreases the population. This population model is called the logistic model.

If a differential equation has a solution that does not change over time (that is, a solution of the form P(t)=C for some constant C), then we say that the differential equation has a equilibrium solution.

Group Discussion

How does a population change at an equilibrium solution? In other words, at an equilibrium solution, is dP/dt<0, dP/dt>0, dP/dt=0, or can we tell?

Question 3

Using the Population Simulator, graphically produce several solutions to the logistic model for a variety of initial populations. Determine the limiting population size when the initial population is large and when the initial population is small for
- k=0.2; A=0.05
- k=0.2; A=0.005
- k=0.1; A=0.005
Do you see a pattern? Conjecture what the carrying capacity is for a net birth rate of k and a coefficient of overcrowding given by A.
The growth curve of a population growing according to logistic growth is typically characterized by three phases: an initial establishment phase in which growth is slow, a rapid expansion phase in which the population grows relatively quickly, and a a long entrenchment stage in which the population is close to its limiting population due to intra-species competition.
Sketch one of the curves above and identify each phase on your plot. How do the values of k and A affect the time it takes for the population to progress through each phase?
Do populations that are decreasing in number according to a logistic model go through the same stages? If not, what stages do they go through? Identify the stages on your plot and explain how k and A affects each phase.

Question 4

The answer to the group discussion indicates that we can find equilibrium solutions by solving for values of P such that dP/dt=0. Explicitly solve for the two equilibria solutions for the logistic model with net birth rate k and coefficient of overcrowding A. Compare this theoretical result with your numerical estimates of the carrying capacity for the values of k and A given in the previous question.
Let's name the two equilibria C and E, with C < E. According to the logistic equation, in terms of the quantities given in the group discussion, how will the population change if
- P > E?
- C < P < E?
- Can the population ever satisfy P < C? Why or why not?

Invasion of the White Pine

The Bufo marinus data we worked with in the previous section fit the exponential model well. In this section we will examine data that indicates the prevalence of white pine (Pinus strobus) in the vicinity of the Lake of the Clouds, a lake in the Boundary Waters Canoe Area of northeastern Minnesota.

The lake is deep (31 meters), calm, sheltered from wind, and devoid of inflowing streams. Consequently, the lake's bottom is covered with layers of annual sedimentary deposits. Each layer contains a sampling of pollen, and by counting the pollen belonging to each species of tree and herb, it is possible to estimate the ratio of one plant species to another.

White pine became extinct in northern Minnesota during the last period of glaciation, although it remained in southern climates such as Virginia. Once the glaciers began to retreat, the white pine began to expand northward again; it reappeared in northern Minnesota about 9400 years ago (H. E. Wright, "The roles of pine and spruce in the forest history of Minnesota and adjacent areas", Ecology, 49, 937-55, 1968).

A. J. Craig (Absolute pollen analysis of laminated sediments: a pollen diagram from northeastern Minnesota, M.S. thesis, University of Minnesota, 1970) counted pollen in a phenomenal 9400 sedimentary laminae from a core at Lake of the Clouds. According to his data, as white pine invaded the region surrounding the lake, it competed with entrenched populations of jack pine (Pinus banksiana) and red pine (Pinus resinosa), which occupy essentially the same coarse, sandy soil as P. strobus. The combined pine tree pollen accounted for about 60-70% of the pollen during the period of interest; other plant species remained essentially constant during the time period (with the exception of spruce (Picea) which decreased).

Craig's data is condensed and analyzed by W. A. Watts ("Rates of change and stability in vegetation in the perspective of long periods of time", Quaternary Plant Ecology, H.J.B. Birks and R.G. West, eds, Blackwell Scientific, 1973), and we have reproduced portions of this data below. Note that in the second column, time is measured in units of thousands of years.

       Years since
Years     9131      P.bank/P.resin  P.strobus
 Ago   (Thousands)  percentage      percentage
-----  ----------   --------------  ----------
9131	0.0		53.4		 3.2
8872	0.259		65.5		 0.0
8491	0.640		61.8		 3.7

8121	1.010		55.2		 3.4
7721	1.410		60.4		 1.7
7362	1.769		59.4		 1.8
7005	2.126		50.6		10.6
6699	2.432		51.6		 7.0

6444	2.687		40.0		21.2
5983	3.148		29.7		34.2
5513	3.618		25.0		40.4
5022	4.109		32.5		29.8
4518   	4.613		22.7		46.2
4102   	5.029		31.6		33.0
3624   	5.507		32.5		37.6
3168   	5.963		27.1		39.5

Table 2: Percentages of pollen for red/jack pine and white pine for sedimentary layers at Lake of the Clouds, MN.

By scanning the data, it is clear that percentages of red and jack pine decreased during the time period indicated, whereas white pine pollen increased. If we assume that these pollen counts are representative of the relative populations of these species, then we have a basis for examining the population growth of P. strobus and the simultaneous decline of P. banksiana and P. resinosa.

Question 5

For the P. strobus data set, use the Population Simulator with the option to "Plot data for trees" to model the growth of the white pine population near Lake of the Clouds. (The green data points indicate red/jack pine pollen; the blue data points indicate the invading white pine.) In particular,
- Find and record an initial population, a net birth rate, and a coefficient of overcrowding that qualitatively matches the growth curve for this species of pine.
- Identify the three phases of logistic growth for this species of pine (if applicable). How long was the expansion phase?
- Estimate the equilibrium value for this species.
Repeat the above analysis for the P. banksiana and P. resinosa data set (green data points). Again, identify phases of growth, if applicable.
Recall that for the exponential growth model, we chose our parameters to try to minimize the sum of the squares of the residuals between the data and a solution curve. Explain how one might attempt a similar procedure to find parameter values that best fit this data set.
Because of your success in the mathematical modeling of population data, you are invited to give an address at the next International Congress of Mathematical Ecologists. You decide that the title of your talk will be "Post-Glacial Populations of Minnesota Pines: Are the Growth Curves Logistic?" Write an abstract (one page or less) describing the main points of your talk. In particular, argue why you do or do not believe that the pine data we are considering comes from a population that obeys a logistic growth law.

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