![]() ![]() The independent variable is t, not x, and anti-differentiating the right-hand side with respect to x will not make any progress. ![]() ![]() Note that though x appears on the right-hand side, x is being used as a dependent variable. This is an example of differential equation. So the growth rate should be proportional to the size of the population. One can imagine without any limit of resources that the more there is of the colony, the faster it should grow. This model assumes that the derivative of x is some multiple k of x, with k is a positive constant. Suppose we have a population of size x(t), which is a function of time t and want to understand its behavior. We begin with a simpler exponential growth model. There's a point of inflection halfway between, about which the curve has 180-degree rotational symmetry. The resulting curve is increasing, has a sigmoid shape, and sits between two horizontal asymptotes. The logistic model includes a ceiling on the size of the population. In today's video, we discuss the logistic function used to model population dynamics where an inhibition factor is included in the expression describing the rate of growth. ![]()
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