Let's look at an example. Suppose a yeast colony starts off with a population of a 100 cells. Suppose that it's growing most rapidly 10 hours later, at which time 500 cells are observed. Assuming a logistic model, let's find (a), an upper bound for the colony size, and (b), how long it takes for the colony to reach 90 percent of its maximum. Let t be the number of hours, and x be the size of the colony, which is a function of t. Since we're assuming logistic growth, then x must take this particular form for some positive constants M, capital K and little k. The population is growing most rapidly at t equals 10 hours, at which time 500 cells are present. But the moment of most rapid growth occurs when x is M on two, which is 500. Hence, the maximum M equals 1,000 answering part (a). We can revise our formula and then think about tackling part (b). We want time t at which x reaches 90 percent of a 1,000, which is 900. So, we just have to unravel the formula for t. This is a straightforward algebraic manipulation. You can pause the video if you like and check this derivation. We have an answer for t, but we haven't finished because it involves capital K and little k. We know that the initial population was a 100 cells, so that a 100 equals x evaluated at 0 which is 1,000 on 1 plus capital K. And we find that capital K equals nine. It remains to find little k. We know that the population was 500 when t equals 10. So, we can feed that information into the formula also using the fact that capital K equals nine. In a couple of steps, we get that 9 e to the negative 10 k equals one, and you can check that k equals ln of 9 over 10. We have all of the ingredients to evaluate t. We seem to have a lucky cancellation and the whole expression quickly evaluates to 20. The model predicts that it takes 20 hours for the colony to grow to 90 percent of the maximum. Is this just a coincidence that we produced such a simple whole number? It's always good practice to draw a diagram, and check if your answer makes sense. Here, t labels the horizontal axis, and x the vertical axis. The initial yeast colony had size 100, which becomes the vertical intercept. After 10 hours, the population reached 500, so we now have two points on the curve. Because the rate of growth is greatest when the population reached 500, we can double that to get a 1,000, representing the limiting maximum predicted by the model, and this gives us a horizontal asymptote. We thus get a smooth sigmoid curve having an inflection with coordinates (10,500). We can mark off the value 900 on the x axis, which is 90 percent of the maximum, move across to the curve, and down to the t axis to get the answer to part b of our problem, which we worked out before to be t equal to 20. Now the answer becomes transparent. The intercept on the x axis, and the point with coordinates (20,900) exactly match up by a 180 degree rotation about the inflection point. If we'd thought of this, then we could have solved the original problem without using any formulae at all. Here's an application to sociology. Suppose that a rumour has been spread that calculus is fun in a small town with a population of 1,000 people. Initially, 100 people know about it. After 10 days, 500 people know about it. How many days does it take for 900 people to know about it? We know the answer immediately. It must be 20 days, assuming that the spread of the rumour is modeled by the logistic equation. We know this answer because the data exactly matches the problem that we solved earlier regarding the yeast culture. Even though the physical manifestations are completely different, the underlying mathematics is the same. Is it reasonable for sociologists to employ the logistic model? Let's have a look. Here's the logistic differential equation where x represents the number of people that know about the rumour after t days. We can rewrite the right-hand side as k times M times x on M times one minus x on M. You can think of the product k times M as a single constant out the front. The factor x on M represents the proportion of the population that knows about the rumour. The second factor represents the remaining proportion of the population that is not yet aware of the rumour. This now makes sense. Initially, only a few people know about the rumour, and it spreads like wildfire with approximately exponential growth. But after a while a lot of people get to know about the rumour. When exactly half of the population knows about it, then the growth rate reaches a maximum, but after that it starts to decline. It gets increasingly difficult to find someone that doesn't know about the rumour, and the growth rate dwindles to a trickle. In today's video, we modified the usual exponential growth model to form the logistic equation used to model population dynamics, by incorporating an inhibition factor as well as a growth factor in the equation. The solution is called a logistic function. We derived the general formula and described its most important features, which include a limiting ceiling on the size of the population, which is approached as the time variable gets arbitrarily large, and a sigmoid shape with a 180 degree rotational symmetry about its point of inflection. We applied the logistic model to predict behavior of a growing yeast population, and also to predict the spread of rumours. Please read the notes and when you're ready, please attempt the exercises. Thank you very much for watching, and I look forward to seeing you again soon.
