As N increases, resources become limited and intraspecific competition can become important.
As N increases, density-dependent factors negatively affect mortality , survival , and/or birth rates, which lead to a decrease in population growth rate.
Population growth rates are not constant.
In the beginning, population growth is nearly exponential, with increases close to rmax.
There is a constant linear decrease in the growth rate (r) as population size increases.
Population size plateaus and fluctuates around some mean.
The resulting logistic growth curve is S-shaped (sigmoid curve) and has 3 important characteristics:
The curve has an upper asymptote called the carrying capacity (K). This is the maximum population size at which dN/dt = 0.
Deceleration in population growth is smooth as it approaches K; thus, when the curve is cut in half, the upper and lower halves are mirror images. The point where you could cut the curve in half is called the inflection point, and it occurs at K/2.
The maximum number of individuals being "added" to the population per unit of time (dN/dt) occurs at N = K/2. Theoretically, this point on the growth curve is the population level at which one could manage for Maximum Sustained Yield (MSY).
Population starts with a stable-age distribution.
Density is measured in appropriate units (i.e., individuals are equal versus age-specific differences).
There is a real attribute of the population corresponding to r.
The relationship between density and rate of increase per individual is linear.
The relationship between density and the rate of increase operates instantaneously without any time lags.
Carrying capacity is constant.
The population is large.
The logistic equation provides a relatively good fit to many case histories of population growth observed in the lab and field; however, it is just a model and contains several oversimplifications. Nevertheless, the logistic model provides a good introduction to more complex subjects such as density-dependence and mean population level. |
lamda = 1 - B(N-Neq)
where N = current population size.
Neq = equilibrium population size (at Neq, lambda = 1).B = rate of change in lambda.
Nt+1 = Nt(lambda) = Nt [1-B(Nt-Neq)]
or
Nt+1 = Nt + rmax (1 - Nt/K) Nt
L = BNeq
If L < 1.0, then population will approach Neq smoothly.
If 1.0 < L < 2.0, then oscillations of decreasing amplitude around Neq.
If 2.0< L < 2.57, then stable, cyclic oscillations around Neq .
If L > 2.57, then population will fluctuate chaotically.
Biological Significance: if a population has even a moderate net-reproductive rate, and if it has a density-dependent reaction which even moderately overcompensates, then far from being stable, it may fluctuate in numbers without any extrinsic factor acting (Begon and Mortimer 1986:49). |
Note: This population behavior is very similar to the behavior of a famous model used extensively in fisheries population studies, the Ricker model. This model was explored in some depth by Robert May (1974) in a famous paper in Science (186:645-647). He pointed out that as the populations maximum rate of increase went above 2 the population behavior first became cyclic, showing different length periods, then above 2.69 it showed chaotic population changes. The Ricker model takes this form:
Nt+1 = Nt exp[r(1-N/K)]
You can explore this model in the lab on the Populus program.
Assume that lambda at time t is dependent on the population size at time t. However, lambda at time t may in fact be dependent on the population size at some previous time denoted Nt-L, where L = the time lag. Lambda can be calculated as:
lambdat = [1-B(Nt-L - Neq)]
and the population at time t+1 is:
Nt+1 = Nt [1 - B(Nt-L - Neq)] .
or
where dN/dt = rate of population increase per unit of time.
K = the maximum number of individuals that the given environment can support, i.e., the upper limit of population growth.
Basically, this equation states that:
rate of increase of population per unit of time | = | innate capacity for increase | x | Population size | x | Unutilized opportunity for population growth |
where "a" is a constant =
Note: the basic logistic equation, with no time lags, results in a population always approaching K with exponential damping .
where L = lag time
a converging oscillation toward equilibrium,
a stable oscillation about the equilibrium level, or
a divergent oscillation that is unstable and leads to extinction of the population.
Note: Time lags, high reproductive rates, and highly overcompensating density-dependence (either alone or in combination) are capable of provoking all types of fluctuations in population density, without invoking any extrinsic factor (Begon and Mortimer 1986:50). |
Begon M., and M. Mortimer. 1986. Population ecology: A unified study of animals and plants. Blackwell Scientific Publ., Boston. 220pp.
Elseth, G. D., and K. D. Baumgardner. 1981. Population biology. D. Van Nostrand Co., New York. 623pp.
Krebs, C. J. 1972. Ecology: The experimental analysis of distribution and abundance. Harper & Row, Publ., New York. 694pp.
Ricklefs, R. E. 1979. Ecology, second edition. Chiron Press, New York. 966pp.
Wilson, E. O., and W. H. Bossert. 1971. A primer of population biology. Sinauer Assoc., Inc., Sunderland, Mass. 192pp.
Revised: 22 October 2002