Answer Happy

The logistic model often is used to model the initial growth phase of a cancerous tumour or human population growth when resources are limited. The logistic model also plays a role in chemistry; it describes the concentrations of reactants and products in autocatalytic reactions. A constant carrying capacity, K, puts a "cap" on the growth of the population. The logistic growth equation is given by dN dt where N (t) is the population size at time t, r> 0 is the per-capita growth rate, K is a constant carrying capacity, and No is the initial population size at t = 0. a) Separate the variables and show that where CE R is the constant of integration. (Hint: Use partial fractions on b) Use laws of logarithms and show that you obtain a solution of the form N =rN1 ™N (1-2), N(0) = No d) Solve for N, and rearrange to show that In |N| – In |K – N| = rt+C, K N(K – N) - N K-N = where C# 0. c) By letting t = 0 in your result from the previous step, show that the constant of integration C0 is given by No K - No N(t) = = Cert KNO (K- No)ert + No

e) What is the solution N(t) when No = 0? f) What is the solution N(t) when No = K? g) Determine lim N(t) when No # 0. t→∞ h) Set K = 1000 and r = 0.1. Sketch the solutions for No = 0, No limit? = 100, No K, and No = = 1200 on the same axes. Are your graphs consistent with the result of the previous