Lecture 16

Topics

• Ecological Modeling - a short taxonomy

• Darwin's Theory - a format for the the study of population dynamics

• Population growth - a geometric potential

Study Sources

• Ricklefs: Chapter 15 (pp. 298-302) and16 (pp.314-328)

• Study Guide: Biology of Single Populations

The Main Points

• Ecological Modeling is the source of two major subsets of models:
  1. Simulation Models

     These models are very detailed and specific. They are generated from data gathered during very extensive field or laboratory studies. The aim is to be able to make precise predictions of the behavior of selected dependent variables when a system is perturbed. Take a look at the Likens' article listed above in the Study Sources. You will soon appreciate the kind of detail required for these kinds of models.

  2. Strategic Models

     These models are theoretical and abstract (yup. another level of abstraction). They should be thought of as metaphors that may be useful in "sorting things out." Webster's defines metaphor as a "figure of speech in which one thing is spoken of as if it were another. (Ex.:'all the world's a stage')."

     In these models, variables are identified that seem to be important to the dynamics of a system. Assumptions are made about quantitative relationships among the variables and their permissible states and ranges. These assumptions and the relationships become the source of hypotheses which we can take to the field and laboratory for falsification.

     The two major categories of Strategic models are:

     1. Stochastic models are developed by assigning probabilties to the states of the variables, thereby generating a range of predictable variation in the dynamics of the system.

     2. Deterministic models, on the other hand, are very mechanical. In a sense they are simpler and are used to "average out" the stochastic properties of the system in a somewhat mechanical way.

        We will be considering these kinds of models in our look at population dynamics, competition and predation.

• The observation underlying most models of population dynamics is that populations have a potential for geometric growth. A geometic series involves a constant multiplier (like an interest rate) and some variable.

• In population dynamics, the multiplier would be the difference between the birth rate and death rate (r) and the variable would be the number of individuals in a population. Since birth rate and death rates are expressed as a per-individual rate, they may be applied directly to the number of individuals in the population.

• Since we are usually interested in what the population is "doing" at different moments in its history, we most often express the function as a "rate of change of the population size." When we use a graphical model or curve to show the dynamics of this equation, we are interested in the slope of the curve at different moments in time or at different population sizes.

• If we choose to use a large interval of time to examine the rate of increase of the population, we get an approximation of this rate during the interval because we are measuring the slope as a straight line while the actual progress of the population is a curve. You'll have to draw a curve and choose some time intervals to see this.

• What we can do to reduce the degree of approximation is to make the interval infinitely small. Then the slope of the curve at any instant in time is simply the slope of a line drawn tangent to the curve. The slope of such a tangent is called in the calculus, a derivative.

• In the case of a population growth curve, where time (t) is plotted on the X-axis and population size (N) is plotted on the Y-axis, we would speak of this derivative as the "instantaneous rate of change of N with respect to t."