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4 MA 36600 MIDTERM #1 REVIEW is an equilibrium solution. In general, an equilibrium solution is a constant which is a solution, so it satisfies the equation f ( y L ) = 0 . Any constant y L such that f ( y L ) = 0 is called a critical point for the function f ( y ). A critical point y L is a stable equilibrium solution if f ( y L ) < 0; and y L is an unstable equilibrium solution if f ( y L ) > 0. Say that we have a population which has a size P = P ( t ) at time t . Assume that the rate of change of the size is proportional to both the size of the population and the di ff erence of the size from some environmental carrying capacity K . That is dP dt P ( K P ) = dP dt = r P 1 P K , P (0) = P 0 ; for some positive constant r called the intrinsic growth rate . This is known as the logistic equation . The solution is P ( t ) = P 0 K P 0 + ( K P 0 ) e rt . More generally, say that the rate of change of the size of the population is proportional to three factors: the size P , the di ff erence K P of a carrying capacity K from the size P , and
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