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Calculus and Beyond Homework Help
Population Dynamics: Logistic Model (Differential Equations)
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[QUOTE="KTiaam, post: 5045174, member: 466570"] [h2]Homework Statement [/h2] Population Dynamics: Logistic model. Suppose the environmental carrying capacity of the population is 100000 and the growth rate a t=0 is 5%. . If the population starts at 10000, how long does it take for the population to reach half the carrying capacity? [I]dp/dt[/I] = A [I]P[/I] (P1 – [I]P[/I]), where P1 = 100 using 1000 as the unit of population. Here [I]P[/I]'/[I]P[/I] = 0.05 at t = 0. Use P'/P = 0.05 and the value of P0 given above in the ODE [I]P[/I]' = A [I]P[/I] (P1 – [I]P[/I]) to find A. You have the solution of the ODE… use it to answer the question. [h2]Homework Equations[/h2] [I]dp/dt = [/I]A P p(t) = P[SUB]0[/SUB] e[SUP]kt[/SUP] [I]dp/dt[/I] = A [I]P[/I] (P1 – [I]P[/I])[h2]The Attempt at a Solution[/h2] P[SUB]1[/SUB] = 100 P(0) = 10 From my understanding you want to find P(t) = 50 and t = ? I'm just having a hard time connecting dp/dt = AP and [I]dp/dt[/I] = A [I]P[/I] (P1 – [I]P[/I]) do i find what AP is and then set it equal to A [I]P[/I] (P1 – [I]P[/I])? or do i use p(t) = P[SUB]0[/SUB] e[SUP]kt[/SUP] in some way. It says "[I]you have the solution of the ODE, use it to answer the question[/I]" I am also not understanding what P'/P represents, as it is equal to .05, but that's the answer when i divide 50 by 1000. Any help is appreciated. [/QUOTE]
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Population Dynamics: Logistic Model (Differential Equations)
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