Question

10. The population P(t) of a certain animal species inhabiting in a forest increases according to the logistic equa- tion with growth constant k = 0.2 week and carrying capacity 700. (a) Assuming the initial population is 100, find a formula for the population at time t. (b) How many weeks will the population reach 500?

Answer 1

SON: - Given, growth constant 0. 2 week! carrying capacity a 700 The logistic equation is given by de kp (1- ) L dt where o à population at time t carrying capacity constant of proportionality de dt 0.2p(' ) 700 de 0.2 dt Plic Р 700 Integrating on both sides 0.2 dt dP ol too) The left hand side can be written + 700-P P(700-P) 700 P/I-O Sdp + S 700-7 ap jonat 0.2 et tc 11 Inlol & in 1700-pl

te 700-P. -> 700 where A= 11 Po -0.2t 1 t A e (a) Given Po= 100 700-100 > A = 100 Therefore, population at population at time 't is time 't is a 700 no Platooliran -0.2t It 7e 1 (b) Given PF 500 500 - 700 -0.2+ P 1tq e -0.2 t 3500 e in 1173) > t 0,2 te 14 weeks 1