Grasshopper Population: 7 Exercises

Introduction

In a text on Population Ecology, Begon (1996, p. 6) describes the life cycle of the field grasshopper (Chorthippus brunneus). The grasshopper is an annual species in which each generation lasts for just one year and there is no overlap between generations. A model of the grasshopper life cycle is shown below:

Time is measured in months, and the DT is 1 month. The model uses three conveyors (see chapter 10) with the following transit times:

egg interval	7 months
youth interval	2 months
adult interval	3 months

The egg loss fraction is set to 92%. The youth loss fraction is a function of the losses expected during the young grasshopper's Instar stages:

Instar I Loss Fraction	28%
Instar II Loss Fraction	24%
Instar III Loss Fraction	24%
Instar IV Loss Fraction	11%

The final molting flow turns a young grasshopper into an adult. Half of the adults are female; the average female lays 7.3 pods; and each pod contains 11 eggs. To simplify the timing, the model assumes that the eggs are laid in the 8th month of the year. A monthly counter (see appendix H) is used to keep track of the months in each new calendar year. The binary variable "ready to lay eggs?" will be 1 in the 8th month; 0 in all other months. This may be achieved with the IF THEN ELSE function:
ready to lay eggs?=
IF monthly counter=8 THEN 1 ELSE 0

1. Gain (with pencil and paper):
Calculate the gain around the positive feedback loop involving adults, eggs, youth and more adults. (To get started, assume that there are 50 adults in the test area. If you calculate that 55 adults would be present one year later, the gain is 10%/year.)

2. How long to Reach 100 Adults?
If the gain is greater than zero, calculate how long it will take for the adult population to reach 100 adults. How long will it take for the population to double again to 200?
(Caution: the doubling time advice in Appendix B applies to exponentially growing systems with continuous compounding.
The grasshopper population grows in discrete steps, so the "continuous compounding" does not apply.) Nevertheless, you should be able to answer this question with pencil and paper.

3. Build the model:
Construct the flow diagram shown above. (The variable names are quite clear, so you should have no trouble writing the equations.) If you are unsure about the algebra for the youth loss fraction, however, you might double check the equation to make sure that the combined effect of the four loss fractions is 63%. And don't forget to set DT to 1month. (This model does not give the expected result if you use a DT of 0.5 or 0.25 months.) Document your work with a printed copy of the Stella diagram.

4. Initialize the model:
Time is in months, with the starting time = 0.The starting time corresponds to January 1st, so you should set the initial number of adult grasshoppers and young grasshoppers to zero. Set the initial value of the overwintering eggs to 300, and be careful to place this initial value on the 4th of the 7th "slats" in the conveyor. Place zero on each of the other slats.

5. Simulate the model:
Run the model over 96 months. Document your work with a time graph of the number of adult grasshoppers.
(Your graph should match the results shown below)

6. Discussion

Does you the simulation result match your estimate from exercise #2?
How large will the simulated population become before it stops growing?

7. Add an Adult Loss Fraction which Increases with larger populations

Expand the model to include a loss of adult grasshoppers. The adult loss fraction will be a function of the number of adult as shown in the chart. If there are less than 100 adults, they suffer no losses. The loss fractions for higher numbers of adults are: 150: 0.2 200: 0.4 250: 0.6 300: 0.8 350: 0.9 400: 0.9 The maximum loss fraction is 0.9 or 90%.

Run the expanded model over a 240 month time period to see if the number of adult grasshoppers will reach equilibrium.
Document your work with a printed copy of your model diagram, the equations and a time graph of the adult population.