3.2 Conditional growth: Jumps and Stages

Auto-conditional jumps

Suppose we want to model that the growth rate \(r\) increases after a certain amount has been learned. In general, this is a very common phenomenon, for instance: when becoming proficient at a skill, growth (in proficiency) is at first slow, but then all of a sudden there can be a jump to the appropriate (and sustained) level of proficiency.

  • Take the model you just built as a starting point with \(r = 0.1\) (B5)
    • Type \(0.5\) in C5. This will be the new parameter value for \(r\).
    • Build your new model in column B (leave the original in A).
  • Suppose we want our parameter to change when a growth level of \(0.2\) is reached. We???ll need an IF statement which looks something like this: IF \(L > 0.2\) then use the parameter value in C5, otherwise use the parameter value in B5.
    • Excel has a built in IF function (may be ALS in Dutch).
    • In the first cell in which calculations should start, press \(=\) and then from the formula list choose the IF function, or just type it.
    • Try to figure out what you have to do. In the Logical_test box you should state something which expresses \(L > 0.2\).
    • The other fields tell Excel what to do when this statement is TRUE (then use parameter value in C5) or when it is FALSE (then use paramter value in B5).
    • Note: the word value might be misleading; you can also input new statements.
  • Make a graph in which the old and the new conditional models are represented by lines.
    • Try changing the value of \(r\) in C5 into: \(1, 2, 2.8, 2.9, 3\).

Auto-conditional stages

Another conditional change we might want to explore is that when a certain growth level is reached the carrying capacity K increases, reflecting that new resources have become available to support further growth.

  • Now we want \(K\) to change, so type \(1\) in B7, \(2\) in C7.
  • Build your model in column C. Follow the same steps as above, but now make sure that when \(L > 0.99\), \(K\) changes to the value in C7. Keep \(r = 0.2\) (B5).

  • If all goes well you should see two stages when you create a graph of the timeseries in column C. Change \(K\) in C7 to other values.
    • Try to also change the growth rate r after reaching \(L > 0.99\) by referring to C5. Start with a value of \(0.3\) in C5. Set \(K\) in C7 to \(2\) again.
    • Also try \(1, 2.1, 2.5, 2.6, 3\).

Connected growers

You can now easly model coupled growth processes, in which the values in one series serve as the trigger for for parameter changes in the other process. Try to recreate the Figure of the connected growers printed in the chapter by Van Geert. Demonstrations of dynamic modeling using spreadsheets

See the website by Paul Van Geert, scroll down to see models of:

  • Learning and Teaching
  • Behaviour Modification
  • Connected Growers
  • Interaction during Play