In this problem we examine a continuous plant-herbivore model. We shall define q as the chemical state of the plant. Low values of q mean that the plant is toxic; higher values mean that the herbivores derive some nutritious value from it. Consider a situation in which plant quality is enhanced when the vegetation is subjected to a low to moderate level of herbivory, and declines when herbivory is extensive. Assume that herbivores whose density is I are small insects (such as scale bugs) that attach themselves to one plant for long periods of time. Further assume that their growth rate depends on the quality of the vegetation they consume. Typical equations that have been suggested for such a system are
dq/dt = K1-K2ql (I-Io),
dI/dt = K3l (1 - K4l/q)
(a) Explain the equations, and suggest possible meanings for K1, K2, Io, K3, and K4.
(b) Show that the equations can be written in the following dimensionless form:
dq/dt = 1- Kql (I-1), dI/dt= al(1-I/q).
Determine K and a in terms of original parameters.
(c) Find qualitative solutions using phase-plane methods. Is there a steady state? What are its stability properties?
(d) Interpret your solutions in part (c).