Chapter 9, Problem 37P

The sudden outbreak of an insect population can be modeled by the equation

$\frac{dN}{dt}=RN\left[1-\frac{N}{C}\right]-\frac{r{N}^2}{N_c{}^2-N^2}$

The first term relates to the well-known logistic population growth model where N is the number of insects. R is an intrinsic growth rate, and C is the carrying capacity of the local environment. The second term represents the effects of bird predation. Its effect becomes significant when the population reaches a critical size N_c; r is the maximum value that the second term can reach at large values of N.

Solve the differential equation for $0\le t\le 50$ days and two growth rates. R = 0.55 and R = 0.58 day-1, and with N(0) = 10,000. The other parameters are C = 10⁴, N = 10⁴, r = 10⁴day-1. Make one plot comparing the two solutions and discuss why this model is called an “outbreak” model.