S.O.S. Mathematics CyberBoard

Consider the generalised logistic equation

dN/dt = (rN^(lamda)) (1-(N/K)^μ) (1)

where r,K, (lamda) and μ are positive constants. (Logistic equation ⇐⇒ (lamda) = 1, μ = 1)

For a small initial condition N(0) = NS ≪ K, equation (1) can be approximated by

dN/dt = rN^(lamda) (2)

(i) Solve equation (2) for the cases (a) lamda = 1 and (b) lamda (does not equal) 1.
(ii) In the case (lamda) ∈ (0, 1), what does the model predict to be possible when Ns = 0?
(iii) When will the approximate solutions of equation (1) found in part (i) become poor?
Show for case the lamda = 1 that this will occur when

t ≈ (1/r)ln(K/Ns)