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# lecture_8 - MA 36600 LECTURE NOTES FRIDAY JANUARY 30...

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MA 36600 LECTURE NOTES: FRIDAY, JANUARY 30 Population Dynamics Logistic Equations. We discuss an application of autonomous equations by considering population growth. 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 difference of the size from some maximal sustainable size K . That is dP dt P ( K - P ) = K · P 1 - P K . Hence there exists a positive constant r such that dP dt = r P 1 - P K . This is known as the logistic equation . The equilibrium solutions P = P L can be found by computing the critical points of the function f ( P ) = r P 1 - P K . Clearly these are P L = 0 and P L = K . Sometimes we refer to K as the saturation level or the environmental carrying capacity . Today we will study the solution to this problem. Solution to Initial Value Problem. We solve this problem explicitly. To this end, it will be useful to consider the “normalized” function and the “normalized” initial condition y ( t ) = P ( t ) K and y 0 = P 0 K . We substitute P = K y into the differential equation above: dP dt = r P 1 - P K d dt [ K y ] = r K y 1 - K y K K dy dt = r K y (1 - y ) = dy dt = r y (1 - y ) .

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lecture_8 - MA 36600 LECTURE NOTES FRIDAY JANUARY 30...

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