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20 September, 05:04

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.

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  1. 20 September, 05:24
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    (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))
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