If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

AP®︎/College Calculus BC

Course: AP®︎/College Calculus BC>Unit 7

Lesson 9: Logistic models with differential equations

Logistic equations (Part 2)

Finding the general solution of the general logistic equation dN/dt=rN(1-N/K). The solution is kind of hairy, but it's worth bearing with us!

Want to join the conversation?

• I Integrated and found N/(K-N)= Ce^rt, Is it any bit different from Sal's result? or did I made a mistake in calculation, because I checked it thrice.
Edit: Got the answer, no big deal here.
• You did it correctly, but instead of leaving the factor of 1/K on top of the second term before integrating, you simplified the term completely. This doesn't come to the same result after solving for N. I have no idea what is going wrong.
• What values of K, No and r I should take to perform the graph?
• `r` should be a positive number greater than 1, (since we're modelling population growth); you can choose any number greater than 1, the bigger the number the faster the population will grow.

`K` is the limit of growth, so you also need a positive number, the larger this number is the more space the population has to grow.

`N₀` is the starting population, so it has to be greater than 0, but less than the population limit `K`.

Within those parameter any numbers should give you a representation for the population growth.
• How do I know the value of "r"?
• As said in : I have plotted the logistic growth model using Euler's method and the logistic function found in this video (N(t)= (N_0*K)/((K-N_0)e^(-rt)+N_0)).
Here's the graph: https://www.desmos.com/calculator/aagshxl1mg
• For all we know that N = "population number/number of humans " is a discrete function, that is, it takes on discrete values like 1,2,3,... not 1.23 or 3.14 etc. Because number of people = 3.14 doesn't make sense. Since discrete functions aren't continuous hence non-differentiable, so how can one calculate dN/dt and build a differential equation for it? It makes no sense or at least to me.
• This is only a model of population, that is, N(t) is the statistically expected value at t. The model is continuous, while the real world, αs you point out, is not.
• Do you have to take the reciprocal of both sides? Can't you just take (N/1)-(K)= Ce^rt and solve to get N=Ce^rt + K? Because 1/ (C_3)= C_2 and 1/ e^-rt is e^rt and 1/ (1/k) is just K?
• I'm assuming you are referring to ? No you can't do that. I don't know how you got to the step where (N/1)-(K)
If you could clarify from what position of the video you started and the steps taken, it would be very helpful.
• When Sal finds the solution to 1/N and finds the reciprocal again, why doesnt Ce^-rt + 1/k just become Ce^rt + k? Why is it 1/Ce^ -rt + 1/k?
• In order to find the reciprocal of 1/N, one must take the reciprocal of the entire equation, not just one of the variables. Therefore we place the entire right side of the equation into the denominator, and as an unfortunate byproduct, we are stuck with 1/k in the denominator as well. If you multiply the right side by `k/k`, though, you could get rid of the `1/k` term, leaving you with: `N = k/(Cke^(-rt) + 1)`, but in my opinion that is less of a "clean" answer.
• How can I plot this logistic differential model equation on Wolfram Alpha?
• Why did the absolute value signs disappear in the integrals of ln(N) and ln(1-n/k)?
• No, all the solutions of the logistic function approach `K` asymptotically without ever reaching it, much less overshooting it (which you would need to have oscillations).