Lecture 17

Topics

• Verhulst (1838, 1845) introduces the "logistic equation

• The nature of r

Study Sources

• Ricklefs: Chapter 16

• Study Guide: Biology of Single Populations

The Main Points

• In Chapter 16, Ricklefs writes a nice treatment of the historical development of the logistic model by Pearl and Reed in 1920. We will be working through that model in the first half of the lecture. If you get a chance to read it before lecture it will help.

• Verhulst (1838, 1845), a Belgian mathematician, had developed a model of population growth that incorporated the idea of a density limit. He reasoned that resources would restrict growth and force the population to reach a stable size at a particular density he labeled "K." At this density the birth rate would equal the death rate.

• Conceptually, this required that r would have to be a "variable." Moreover, this new variable that we'll call "actual r" (r_a) requires that birth rate decrease and deaths increase as the population increases.

• The problem for the model builder is to create a modifier for r_max that would produce a new variable that decreases as N gets larger. More to the point, the new variable r_a should decrease as it approaches a limit (K) imposed by limited resources, such as food and/or space.

• Verhulst's solution was to introduce a "logistic term" into the geometric growth equation. The term, 1 - (N/K), decreases as N approaches K, thereby reducing r_a by an equal amount for each addition of an individual to the population. The equation
  dN/dt = [r_max(1 - N/K)]N

  became known as the "logistic equation." It serves us still as a way to look at the process of population growth.

• The logistic equation, as with any model, has some assumptions. Three of these are rather explicit:
  1. There is no immigration nor emigration to the population.

  2. All individuals are assumed to be equal.

  3. Since this model is an expression of the slope of the logistic growth curve at any instant of time, changes in rate of change (dN/dt) occur instantaneously with changes in N and/or K.

• Two assumptions are implicit in the structure of the equation:
  1. r_a decreases linearly with increases in N.

  2. dN/dt is parabolic function that reaches a maximum at the inflection point of the growth curve and becomes 0 when N reaches K.

• In order to understand these implicit assumptions, make plots of r_a vs N and dN/dt vs N as N increases. To do this, choose values of N in one column and make a table of values for r_a and dN/dt. (you'll have to assume some values for r_max[e.g 0.6] and K[e.g. 150].

• After you have calculated the values, make graphs of r_a vs N and dN/dt vs N. Treat N as the independent variable and plot its values on the X-axis.

• Then describe each of these graphs in words, e.g. "as N increases, r_a ........" etc.

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