To solve the two equations in for the values of k and M, begin by solving the first equation for the quantity a = e−50km and the second equation for x2 = e−100km. Upon equating the two resulting expressions for x2 in terms of M, you get an equation that is readily solved for M. With M now known, either of the original equations is readily solved for k. This technique can be used to “fit” the logistic equation to any three population values P0, P1, and P2 corresponding to equally spaced times t0 = 0, t1, and t2 = 2t1.
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