1) In this problem we examine the effects of interactions among predators on the stability of a predator-prey system. We will modify the basic Lotka-Volterra model without density dependence in the prey. We will be changing the assumption that the per predator predation rate is independent of the number of predators. We assume that predators help each other, as in the case in which many insects can overwhelm the defenses of a plant (which may be mechanical, such as bark on a tree) only if the density of insects is sufficiently high.
a) Write down a modified version of the predator-prey model where the predation term (in both the prey and the predator equations) has the per predator predation rate an increasing function of the number of predators. The exact form of the predation rate is not critical. Leave the other two terms alone.
b) Draw the prey isocline. (Will it depend on the number of prey?)
c) Draw the predator isocline; a rough sketch will do. (Which way will it ‘tilt' compared to the Lotka-Volterra case?)
d) Use the graphical technique to determine the signs of the terms in the Jacobian (community) matrix.
e) Determine the stability of the equilibrium by looking at the sign of the determinant and the trace.
f) Discuss (verbally) whether this effect of predator interaction makes sense to you - do -insects show outbreaks on plants? What role does the fact that the exact form of the predation rate is not critical here have on your drawing conclusions from this model?
2) Host-parasitoid systems are used to look at biological control, where a parasitoid is introduced to reduce the host population of a pest on a crop. How would the models developed here help to choose a parasitoid to use for biological control, and what qualities should be looked for What are some potential pitfalls? (about 300-600 words - be as specific as possible)
3) This is an open-ended question. How would you determine that disease is regulating a population, rather than just determining that a disease could regulate a population? (about 300-600 words - be as specific as possible)