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.

(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))