(Algebraic, exponential, and explosive growth) We saw, in Section 2.3.3, that the populati...

(Algebraic, exponential, and explosive growth) We saw, in Section 2.3.3, that the population model

gives exponential growth, whereby N → ∞ as t → ∞. More generally, consider the model

where p is a positive constant. Solve (12.2) and show that if 0 < p < 1 then the solution exhibits algebraic growth [i.e., N(t) ~ αtβ as t → ∞]. Show that as p → 0 the exponent β tends to unity, and as p → 1 the exponent β tends to infinity. (Of course, when p = 1 we then have exponential growth, as mentioned above, so we can think - crudely - of exponential growth as a limiting case of algebraic growth, in the limit as the exponent β becomes infinite. Thus, exponential growth is powerful indeed.) If p is increased beyond 1 then we expect the growth to be even more spectacular. Show that if p > 1 then the solution exhibits explosive growth, explosive in the sense that N → ∞ in finite time, as t→T, where

N0 denotes the initial value N(0).Observe that T diminishes as p increases.