Consider anew the logistic equation,Typically, mathematical ecologists will introduce dime...

Consider anew the logistic equation,

Typically, mathematical ecologists will introduce dimensionless variables to reduce the number of parameters in the logistic equation before proceeding with their analysis.

(a) Show that the substitutions w = αP and s = βt transform equation (1.29) into

(b) Find values of α and β that transform equation (1.30) into

(c) Note that equation (1.31) is a variant of Bernoulli’s equation. Use the technique of Exercise in Section 2.4 to show that equation (1.31) has the solution

(d) Finally, use the change of variables w = αP and s = βt, your parameters α and β found in part (b), and the initial condition P(tQ) = P0 to show that equation (1.32) is equivalent to the solution given in equation (1.15).