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# lecture_9 (dragged) 1 - 2 MA 36600 LECTURE NOTES: MONDAY,...

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Unformatted text preview: 2 MA 36600 LECTURE NOTES: MONDAY, FEBRUARY 2 • There is a threshold T . That is, if P < T then the population will become extinct: P (t) → 0 as t → ∞. We assume that the rate of change of the size of the population is proportional to three factors: (1) The size P . (2) The diﬀerence K − P of the carrying capacity K from the size P . (3) The diﬀerence P − T of the size P from the threshold T . That is, we have the relation ￿ ￿￿ ￿ dP P P ∝ P (K − P ) (P − T ) = K T · P 1 − −1 . dt K T In other words, we have the initial value problem ￿ ￿￿ ￿ dP P P = −r P 1 − 1− , dt K T P (0) = P0 ; where r is a positive constant. We assume that 0 < T < K . This is a logistic equation with threshold. We mention in passing that this equation can be solved explicitly. First, note that the diﬀerential equation is a separable equation. Upon dividing both sides by the cubic polynomial in P , one can compute the antiderivative using partial fraction expansions. We will not concern ourselves will these steps. Instead, we will consider the properties of the solution P = P (t) by focusing on the initial value P (0) = P0 . Denote the cubic polynomial ￿ ￿￿ ￿ P P f (P ) = −r P 1 − 1− . K T This function has the critical points PL = 0, PL = T , and PL = K . Figure 2 contains a plot of f (P ) vs. P . By considering the sign of f ￿ (PL ), we see that ￿ PL = 0 are stable equilibria points; PL = T is an unstable equilibrium point. PL = K Figure 2. Plot of f (P ) = −r P (1 − P/K ) (1 − P/T ) 15 10 Increasing f'(P) > 0 Decreasing f'(P) < 0 Decreasing f'(P) < 0 5 -5 0 5 10 15 -5 -10 To get an idea of how the “limiting solution” PL = lim P (t) t→∞ 20 25 30 ...
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## This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue University-West Lafayette.

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