Next: Equilibria, Stability, and Phase Space
Return to: Unbounded Growth
Up: Outline

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

Question 4

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.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

Next: Equilibria, Stability, and Phase Space
Return to: Unbounded Growth
Up: Outline

The Geometry Center Calculus Development Team

[HOME] The Geometry Center Home Page

Comments to:
Created: May 15 1996 --- Last modified: May 15 1996
Copyright © 1996 by The Geometry Center All rights reserved.