54 Economic Dynamics express the diffusion process as dN ( t ) dt = bN ( t )(ln m − ln N ( t )) (2.26) or dF ( t ) dt = bF ( t )( − ln F ( t )) Suppressing the time variable for convenience, then the two models are ˙ F = ( a + bF )(1 − F ) and ˙ F = bF ( − ln F ) Pursuing the logistic equation, we can graph ˙ F against F . When F = 0 then ˙ F = a and when ˙ F = 0 then ( a + bF )(1 − F ) = 0 with solutions F 1 =− b / a and F 2 = 1 Since ˙ F denotes the rate of diffusion, then the diffusion rate is at a maximum (penetration is at its maximum rate) when ¨ F = 0, i.e., when d 2 F / dt 2 = 0. Differ-entiating and solving for F , which we denote Fp (for maximum penetration rate), we obtain Fp = 1 2 − a 2 b implying Np = m · Fp = m ± 1 2 − a 2 b ² = m 2 − am 2 b In order to Fnd the time tp when F ( tp ) is at a maximum penetration rate, we must Frst solve for F ( t ). This we indicated above. Since we need to Fnd the value of t satisfying F ( t ) = Fp , then we need to solve 1 − e − ( a + b ) t 1 + ( b / a ) e − ( a + b ) t = 1 2 − a 2 b for t , which we can do using a software package. This gives the time for the maximum penetration of
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This note was uploaded on 12/14/2011 for the course ECO 3023 taught by Professor Dr.gwartney during the Fall '11 term at FSU.