This preview shows page 1. Sign up to view the full content.
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
∝ P (K − P ) (P − T ) = K T · P 1 −
In other words, we have the initial value problem
= −r P 1 −
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
f (P ) = −r P 1 −
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 ...
View Full Document