In some circumstances, individuals reproduce better when the population size is large, and fail to reproduce when the population size is small (the Allee effect introduced in Exercise 46). Suppose that per capita production is an increasing linear function with λ(0) = -2 and a slope of 0.01. Find λ(b) and the differential equation for b. Is b(t) increasing when b = 100? Is b(t) increasing when b = 300?
The simple model of bacterial growth assumes that per capita production does not depend on population size.
The above problems help you derive models of the form
db/dt = λ(b)b
where the per capita production λ. is a function of the population size b.