Question

Consider a prolific breed of rabbits whose birth and death rates, β and δ, are each proportional to the rabbit population P = P(t), with β > δ.
Show that:
P(t)= P₀/(1−kP₀t)
with k constant. Note that P(t) → +[infinity] as t→1/(kP₀). This is doomsday.

Answer 1

Answer:

(P(t)) = P₀/(1 - P₀(kt)) was proved below.

Step-by-step explanation:

From the question, since β and δ are both proportional to P, we can deduce the following equation ;

dP/dt = k(M-P)P

dP/dt = (P^(2))(A-B)

If k = (A-B);

dP/dt = (P^(2))k

Thus, we obtain;

dP/(P^(2)) = k dt

((P(t), P₀)∫)dS/(S^(2)) = k∫dt

Thus; [(-1)/P(t)] + (1/P₀) = kt

Simplifying,

1/(P(t)) = (1/P₀) - kt

Multiply each term by (P(t)) to get ;

1 = (P(t))/P₀) - (P(t))(kt)

Multiply each term by (P₀) to give ;

P₀ = (P(t))[1 - P₀(kt)]

Divide both sides by (1-kt),

Thus; (P(t)) = P₀/(1 - P₀(kt))