Instructions

• First, do your reading assignment.
• Next, go through Session 1 on the Internet making rough notes and questions you want to discuss.
• Next, try to do the exercises and note any problems you want to discuss.
• Discuss problems or questions with you instructor in person or by e-mail.
• Make a final draft of your notes for this session in your notebook.
• Send a copy of your notebook covering this session to the instructor if you wish or save it until the end of the course.

Reading

• Berryman, A. A. 1997. Chapter 5 in Principles of Population Dynamics.
• Berryman, A. A. and G. T. Ferrell. 1988. The fir engraver beetle in western states. Chapter 26 in Berryman, A. A. (Ed.) Dynamics of forest insect populations. Plenum Press, N.Y.
• Berryman, A. A. 1973. Simulation of intraspecific competition and survival of Scolytus ventralis broods (Coleoptera: Scolytidae). Environmental Entomology 2: 447-459. (optional reading)

Exercises

1. Use the PAS Program PG2 to simulate the dynamics of a population of cellular automata in constant and variable environments. You can also program a computer to do these simulations using the following rules:

• automata with more than 5 neighbors die from overcrowding,
• those with less than 2 neighbors die from isolation,
• those with between 2 and 5 neighbors reproduce offspring into adjacent empty cells, and
• a certain proportion of the offspring are killed randomly.

Notice how the population grows rapidly at first (exponential growth) but then levels off as the lattice becomes fully occupied. Also notice that the growth curve has an S-shaped or sigmoid form. The leveling off of the growth curve around the carrying capacity of the board is due to increasing mortality from overcrowding as the population expands to occupy all the available space (space is the resource). Summarize what you learned in your notebook.

2. Simulate the dynamics of a population according to the nonlinear logistic equation

when the initial density P₀ = 10, the maximum per-capita rate of change is A_P = 0.8, the coefficient of intraspecific competition is C_P = 0.001, the coefficient of curvature is Q_P = 1, and the run length is 20 time steps. Simulation is done by calculating the realized per-capita rate of change

and then substituting this value in the step-ahead forecasting equation to calculate the number of organisms in the following time step

and so on. These calculations can be done by hand, on a standard computer spreadsheet (e.g., Microsoft's Excel), or with the PAS software program P1b. However, you will probably learn more by working out the exercises on a spreadsheet.

• Plot the trajectory (see figure from P1b simulation). Notice how the population grows to equilibrium along a sigmoid or S-shaped trajectory, much like the cellular automata, and that it approaches equilibrium smoothly, without oscillation.
• What is the density of the population at carrying capacity?
• Repeat the above but this time do stochastic simulations (s = 0.2 and 0.4). Notice is that environmental variability can cause a previously stable trajectory to fluctuate irregularly around its equilibrium level and that the amplitude of oscillations depends on the amount of noise.
• Plot the stochastic trajectories in (P,R) phase space.

3. Repeat the above exercise but with A_P = 1.8. Notice that the trajectory damps to a stable equilibrium point but that environmental noise causes it to continue oscillating with amplitude determined by the amount of noise (see figures for s = 0.2 and 0.4).

4. Repeat the above exercise but with A_P = 2.4. Notice that the equilibrium point is now unstable and the trajectory settles into an oscillation with constant amplitude. This is called a limit cycle. The main effect of environmental variability is to cause variations in the amplitude of the oscillations (see figures for s = 0.2 and 0.4).

5. Repeat the above exercise but with A_P = 3.0. Notice that the deterministic population fluctuations now have variable amplitude much like some of the previous ones in the presence of random variability.

The most obvious result of the above analysis is that the dynamics of the population become more complex as the value of the parameter A_P becomes larger. Notice also that the oscillations have a characteristically sharp, or saw-toothed, pattern that repeat itself every two years or so. High-frequency fluctuations with these characteristics are said to be the result of a first order dynamic processes.

What we have done above is what is called a sensitivity analysis of the parameter A_P. In other words, the analysis demonstrates the sensitivity of the model to variations in the maximum per-capita rate of change, A_p, when the other parameters are kept constant. The general result is that the population exhibit the following properties in response to changes in A_P:

1. Smooth growth to equilibrium when .
2. Damped oscillations around equilibrium when .
3. Periodic oscillations that become more and more complex when , eventually leading to aperiodic chaos when . In these cases, the equilibrium point is locally unstable but the oscillations persist indefinitely. Notice that chaotic fluctuations are aperiodic because their amplitude, the difference between peaks and troughs, varies in an irregular manner.

6. Simulate trends caused by exogenous forcing acting on the maximum per-capita rate of change, A_P, and the carrying capacity, K_P, using the linear function

where X_t is the value of the parameter at time t, X₀ is its original value at time t₀, and b is a time-dependent forcing constant.

• Trend in A_p: Set P₀ = 10, A_P = 1 (i.e., X₀ = A_p = 1), C_P = 0.001 , Q_P = 1, b = 0.05, s = 0.2, and the run length 40. (see figure)
• Trend in K_p: Set P₀ = 10, A_P = 1, C_P = 0.001 (i.e., X₀ = K = 1000), Q_P = 1, b = 25, s = 0.2 and the run length 40. (see figure)

Note how gradual environmental forcing, as might be caused by gradual and persistent increases in atmospheric carbon dioxide and other "greenhouse" gasses, could cause increasingly violent oscillations or a gradual change in the average density of the populations.

7. Sudden changes in the exogenous environment can cause discontinuities or shifts in parameter values. For example, a volcanic eruption may change the environment from very favorable to very unfavorable in a matter of hours. Discontinuous changes in parameter values can be simulated by a step function such as

where X_new is the new value of the parameter which replaces the old one X₀ at time t = tstep. Simulate shifts caused by exogenous forcing acting on the maximum per-capita rate of change, A_P, and the carrying capacity, K_P:

• Shift in A_p: Set P₀ = 10, A_P = 1 (i.e., X₀ = A_p = 1), C_P = 0.001, Q_P = 1, A_new = 2, tstep = 20, s = 0.2, and run = 40. (see figure)
• Shift in K_p: Set P₀ = 10, A_P = 1, C_P = 0.001 (i.e., X₀ = K = 1000), Q_P = 1, K_new = 3000, tstep = 20, s = 0.2, run = 40. (see figure)

Examples of the kind of questions that may appear on the next examination are:

1. What is the 3^rd principle of population dynamics?
2. What is intraspecific competition?
3. What is the difference between adapted and incidental competition?
4. Explain how the 3^rd principle affects the shape of the R-function?
5. What is the carrying capacity of the environment?
6. What did the controversy over density-dependence entail and who were the major antagonists?
7. What is meant by enemy-free space?
8. What is the sigmoid behavioral response of a consumer?
9. What characteristics of consumers lead to sigmoid behavioral responses?
10. How does the sigmoid behavioral response of a consumer affect the shape of the prey's R-function?
11. What is phase space?
12. What is a phase portrait?
13. What is the logistic equation?
14. Define the logistic parameters A, K, H, and C.
15. Write an equation for a linear R-function.
16. How do you change a linear into a non-linear R-function?
17. What is sigmoid population growth and what causes it?
18. What is damped stability?
19. What is a periodic fluctuation?
20. What is a chaotic fluctuation?
21. What is an oscillation of period two?
22. What is sensitivity analysis?
23. How does the value of the maximum per-capita rate of change affect the dynamics of populations?
24. How does the level of the carrying capacity affect the dynamics of populations?
25. How does the coefficient of curvature influence the dynamics of populations?
26. How do random variations in the external environment affect the dynamics of populations?
27. How do systematic changes in the external environment affect the dynamics of populations?
28. Derive a mathematical model for the third principle from the point of view of a consumer?
29. What is the basic idea behind Royama's derivation of the third principle?
30. Describe the role of intraspecific competition in the population dynamics of the fir engraver.

e-mail Instructor?

Use BACK key to return to Sessions