STUDENTS: Type the answers to the exercises and questions neatly and hand them in or e-mail them to your instructor before the next session.

Exercises

1. Study the dynamics of the logistic individual survival model for a specialized predator without intra-specific competition for fixed resources. Set the parameters: a_h = 1, b_h = 0.003, a_c = 0.5, b_c = 0, c_c = 6, d_c = 4, w_c = 0. You can do this exercise on a spreadsheet or with PAS-P2b, the Two-Species Modeling and Simulation program.
1. Describe the dynamics in a constant and stochastic environment (in stochastic simulations apply the random shocks to the prey R- function only using a standard deviation of s = 0.2).
2. Change the predator maximum rate of change to a_c = 1 and describe the effects on the isocline structure and the deterministic and stochastic dynamics.
3. Change the predator maximum rate of change to a_c = 1.4 and describe the effects on the isocline structure and deterministic and stochastic dynamics.
4. What does this analysis say about the paradoxes of enrichment and biological control? Remember that the carrying capacity for the prey is K_h = a_h / b_h so that one can increase K_h by increasing a_h or decreasing b_h. Reading the paper by Arditi and Berryman (1991) referenced below may help you answer this question.
2. Study the effect of the parameters of the logistic individual survival model on the stability (the degree of fluctuation around equilibrium in deterministic and stochastic environments) and the suppression of the prey population (the degree to which the pest population is reduced from its carrying capacity by the predator). Use the parameters a_h = 1, b_h = 0.003, a_c = 0.5, b_c = 0, c_c = 6, d_c = 4, w_c = 0 for the standard run and then determine how
1. Changes in a_h affect stability and prey suppression?
2. Changes in b_h affect stability and prey suppression?
3. Changes in a_c affect stability and prey suppression?
4. Changes in b_c affect stability and prey suppression?
5. Changes in c_c affect stability and prey suppression?
6. Changes in d_c affect stability and prey suppression?
7. Changes in w_c affect stability and prey suppression?
3. 3. Study the dynamics of the logistic individual survival model for a generalized predator regulated by intra-specific competition for fixed resources (e.g., territories) but uncoupled from the dynamics of its prey. Use the parameters: a_h = 1, b_h = 0.003, a_c = 0.5, b_c = 0.025, c_c = 0, d_c = 4, w_c = 50.
1. Describe the dynamics with different initial densities of each species and in a constant and stochastic environment (apply the random shocks to the prey R- function and use a standard deviation of s = 0.2).
2. Change the predator maximum rate of change parameter to a_c = 0.75 and describe the effects on the isocline structure and deterministic and stochastic dynamics.
3. Change the predator maximum rate of change parameter to a_c = 0.25 and describe the effects on the isocline structure and deterministic and stochastic dynamics.

Reading

1. Arditi, R. and A. A. Berryman. 1991. The paradox of biological control. Trends in Ecology and Evolution, 6: 32.
2. Berryman, A. A. and A. P. Gutierrez. 1999. Dynamics of insect predator-prey interactions. Chapter 12 in Ecological Entomology, C. B. Huffaker and A. P. Gutierrez, Eds., Wiley-Interscience, New York.

Questions

1. Describe the origin and evolution of the Lotka-Volterra model and its derivatives and discuss the strengths and weaknesses of this approach to modeling predator-prey dynamics?
2. Describe the origin and evolution of the energy-flow food chain model and discuss the strengths and weaknesses of this approach to modeling predator-prey dynamics?
3. Describe the origin and evolution of the logistic food chain model and discuss the strengths and weaknesses of this approach to predator-prey dynamics?