WHAT IS ECOLOGICAL STABILITY? In 2019 I posed this question informally to colleagues, using Twitter, a professional workshop that I lead, and a conference. Respondents on Twitter consisted mostly of ecological scientists, but the workshop included paleontologists, biologists, physicists, applied mathematicians, and an array of social scientists, including sociologists, anthropologists, economists, archaeologists, political scientists, historians and others. And this happened…

Previous posts in this series:

The logistic equation, covered in the previous post, is a differential equation, where time is divided up into infinitesimal bits to model the growth and size of X (“infinitesimal” knots your stomach? I cannot recommend Steven Strogatz’s “Infinite Powers” enough!). We can also simulate the logistic model in discrete time to get a better feeling for it, where X(t + 1) is population size in the next “time step” or generation. This approach is instructive because anyone can play with the calculations using a calculator or spreadsheet! Here is an example of a discrete version of logistic growth, the Ricker difference equation (Ricker, 1954).

EQ. 1: (FUTURE POPULATION SIZE) = (CURRENT POPULATION SIZE) x (EXPONENTIAL REPRODUCTION LIMITED BY CARRYING CAPACITY AS IN THE LOGISTIC MODEL)

$X_{t+1} = X_{t}e^{R\left (1-\frac{X_{t}}{K}\right )}$

r has been replaced by R, the main difference between the logistic equation and Eq. 1 being that, because we no longer measure time as continuous but instead step discretely from one generation to the next, we measure the intrinsic rate of increase as the “net population replacement rate”. The population again, after an initial interval of near-exponential growth, settles down to a fixed value at K (Fig. 1). Both the continuous and discrete logistic patterns of population growth are, by all definitions, stable populations in equilibrium. They are stable because, at least within the scope of the models, once a population attains its carrying capacity there is no more variation of population size. This brings us to our first definition of stability.

Stability: An absence of change.

In the following sections we will cover a real-world example of logistic growth, and then go through the derivation of the logistic function itself.

An Example of Logistic Growth

The state of Washington in the United States employed a program of harbor seal (Phoca vitulina) culling during the first half of the twentieth century. The seals were considered to be direct competitors to commercial and sport fishermen. The state sponsored monetary bounties for the killing of seals until 1960, by which time seal populations must have been reduced significantly below historical levels. Additional relief arrived for the seals in 1972 with passage of the United States Marine Mammal Protection Act. Monitoring of seal populations along the coast, estuaries and inlets of Washington, primarily by the Washington Department of Fish and Wildlife, and the National Marine Mammal Laboratory provided a time series of seal population size, spanning the beginning of recovery in the 1970’s to the end of the century (Jeffries et al., 2003). Population sizes from one region of the coastal stock, the “Coastal Estuaries”, show a logistic pattern of growth (Fig. 2). The function fitted to the data (using a nonlinear least squares regression) is y = 7511.541/[1 + exp[−0.265(x − 1980.63)]] (r-squared = 0.98; p < 0.0001; note that “r-squared” is the coefficient of correlation, not our intrinsic rate of increase). Given an initial population size of X(0) = 1,694 in year 1975, the function yields estimates of r = 0.265 and K = 7,511. This excellent example of logistic growth in the wild, or recovery in this case, was unfortunately brought to us courtesy of the ill-informed belief that the success of human commercial pursuits necessitate, or even benefit from, the destruction of wild species.

Deriving the logistic equation
Equation 1 in the previous post is the logistic growth rate of the population, but it is not the logistic function itself. That function is obtained by integrating the growth rate dX/dt, and the process is instructive because, as illustrated in later sections, our ability to do so with more complicated dynamic equations is quite limited.

The logistic growth rate is first re-written to eliminate the X/K ratio (makes it easier to proceed)
$\frac{dX}{dt} = rX\left ( 1-\frac{X}{K}\right )$
$\Rightarrow K\frac{dX}{dt} = rX\left ( K-X\right )$
and then re-arranged to separate variables,
$\frac{K\, dX}{X\left ( K-X\right)} = r\,dt$
The logistic function is derived by integrating both sides, but doing so with the left hand side (LHS) requires simplification using partial fractions (some of you might remember those from high school math; or not).
$\frac{K}{X\left ( K-X\right)} = \frac{A}{X} + \frac{B}{K-X}$
$\Rightarrow K = X\left ( K-X\right ) \left [ \frac{A}{X} + \frac{B}{K-X} \right ]$
$\Rightarrow K = A(K-X) + BX$
$\Rightarrow K = AK - X(B-A)$
The solutions to the final equation are A=1 and B-A=0, yielding B=1. Therefore
$\frac{K}{X\left ( K-X\right)} = \frac{1}{X} + \frac{1}{K-X}$
Now if we wish to integrate our logistic differential equation,
$\int \frac{K\, dX}{X\left ( K-X\right)} = \int r\,dt$,
we can substitute our partial fractions solution and proceed as follows.
$\Rightarrow \int \left ( \frac{1}{X} + \frac{1}{K-X}\right ) dX = \int r\,dt$
$\Rightarrow \int \frac{1}{X}\, dX + \int \frac{1}{K-X}\, dX = \int r\,dt$
And if you recall our integration of the Malthusian Equation, the solution is
$\Rightarrow \ln{\vert X\vert} - \ln{\vert K-X\vert} = rt + C$
$\Rightarrow \frac{K-X}{X} = e^{-rt-C}$
Let $A=e^{-C}$, a constant. Then
$\frac{K}{X} - 1 = Ae^{-rt}$
$\Rightarrow \frac{K}{X} = Ae^{-rt} + 1$
$\Rightarrow X(t) = \frac{K}{Ae^{-rt}+1}$
which is the equation for logistic population growth! Whew.

References
Jeffries, S., Huber, H., Calambokidis, J., and Laake, J. (2003). Trends and status of harbor seals in Washington State: 1978-1999. The Journal of Wildlife Management, 67:207–218.
Ricker, W. E. (1954). Stock and recruitment. Journal of the Fisheries Board of Canada, 11:559–623.