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The Logistic Equation
Jan. 27th, 2005 10:55 amThat was actually fun!
Background: I spent ten years in college as an aspiring mathematician, undergraduate and graduate. During that time, I carefully avoided taking any course in differential equations. The subject simply did not hold any interest for me.
My first teaching assignment as a new-fledged Ph.D. included... Differential Equations. I was willing to try it, on the Prof Paz Principle: as long as I stay a week ahead of the students, I'm okay. It was, well, a disaster.
Almost two decades pass. The Assistant Chair is drawing up course assignments, and asks me if I'm willing to teach College Algebra again. I ask for alternatives; the only other option, at this point, is DiffEq.
I really don't want to teach College Algebra...
Anyway. Today we were covering population growth. The simplest model of population growth, which is actually the first differential equation students usually encounter, is exponential growth: y'=ry, where r>0 is the growth rate. Solutions of this equation explode as t gets large. This is, of course, completely misleading; somewhere along the line, something - increased predation, disease, shortages of food - kicks things back down.
The next simplest model is "logistic growth"; the equation is y'=r(1-y/K)y, where r is the initial growth rate and K is the "carrying capacity". This equation is not difficult to solve, but the fun part is how much information you can extract about the solutions without solving the equation. I made up a bunch of transparencies (but didn't have time to use all of them) to help them get a visual picture of the behavior of the solutions (First Law! First Law!). (Basically: if the initial population is greater than K, it drops towards K. If it lies between 0 and K, it rises towards K, increasingly quickly until it passes K/2, and afterward with decreasing speed. All of this can be worked out without solving the equation!)
That was fun!
I'm scared.
Background: I spent ten years in college as an aspiring mathematician, undergraduate and graduate. During that time, I carefully avoided taking any course in differential equations. The subject simply did not hold any interest for me.
My first teaching assignment as a new-fledged Ph.D. included... Differential Equations. I was willing to try it, on the Prof Paz Principle: as long as I stay a week ahead of the students, I'm okay. It was, well, a disaster.
Almost two decades pass. The Assistant Chair is drawing up course assignments, and asks me if I'm willing to teach College Algebra again. I ask for alternatives; the only other option, at this point, is DiffEq.
I really don't want to teach College Algebra...
Anyway. Today we were covering population growth. The simplest model of population growth, which is actually the first differential equation students usually encounter, is exponential growth: y'=ry, where r>0 is the growth rate. Solutions of this equation explode as t gets large. This is, of course, completely misleading; somewhere along the line, something - increased predation, disease, shortages of food - kicks things back down.
The next simplest model is "logistic growth"; the equation is y'=r(1-y/K)y, where r is the initial growth rate and K is the "carrying capacity". This equation is not difficult to solve, but the fun part is how much information you can extract about the solutions without solving the equation. I made up a bunch of transparencies (but didn't have time to use all of them) to help them get a visual picture of the behavior of the solutions (First Law! First Law!). (Basically: if the initial population is greater than K, it drops towards K. If it lies between 0 and K, it rises towards K, increasingly quickly until it passes K/2, and afterward with decreasing speed. All of this can be worked out without solving the equation!)
That was fun!
I'm scared.
