Logistic growth: The logistic function
Definition and basic properties
The logistic function, also called sigmoid function, is with and . The graph of this function in the case is outlined below: the graph is an increasing sigmoid, i.e., an S-shaped curve: the growth is first increasing, and then progressively decreases and disappears over time so that the limit will be approached. It holds that and approaches with increasing time . The parameter is called the carying capacity in the logistic growth model and the intrinsic growth factor.
Consider how the graph looks like in case.
A special case of the logistic function is This function displays real values in between 0 and 1 and is widely used in neural network theory in order to limit response values.
The derivative of the logistic function can be computed with the rules of differentiating and satisfies the relationship This is the differential equation of logistic growth.
Sometimes it is written as We speak of the logistic growth model, also known as the Verhulst model.
For the math enthusiast, we prove that the derivative of the logistic function satisfies the relationship
Uding the rules of differentiation we can determine the derivative : The derivative is because therefore applies
The general solution of the differential equation with constants and is equal to for some constant .
The constant is often determined by the initial value . Only in the case of , i.e., when is between 0 and , there is a logistic function with a sigmoid as function graph.
A concrete example of a logistic differential equation is Three solution curves are drawn in the slope field below. You get such curves by 'going with the flow', i.e., by following the lineal elements of the slope field in small time steps. For example, the green S-shaped curve belongs to the solution with initial value . The other two curves, belonging to initial values outside the interval , do not have the shape of a sigmoid. What the diagram also illustrates is that in this example the equilibrium is attracting: any solution curve which comes in the vicinity of this straight line approaches this equilibrium over time. The equilibrium is the opposite: this equilibrium is repelling; every solution curve that deviates even a little bit will at some point go away from this equilibrium.
Solve the differential equation in explicit form with the initial value .
Use numbers rounded to two decimals or fractions in your answer.
The differential equation is So we are dealing with a special case of the logistic equation with The solution of the general logistic equation is for some constant . In this case we have a solution We can calculate by solving the equation .
Because we must solve This can be done as follows: The graph for is
