Logistic Models with Differential Equations

Logistic Model

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Logistic Differential Equation

The logistic differential equation equation image indicator $LaTeX: \frac{dy}{dt}=ky\left(1-\frac{y}{L}\right)$ , where k and L are positive constants, is used to model populations that grow logistically, i.e., growth that occurs when some upper limit L exists beyond which no additional growth can occur. Notice that if y is small compared to L, then y/L is close to 0 and dy/dt ≈ ky. If y approaches its carrying capacity L, then equation image indicator $LaTeX: \frac{y}{L}\rightarrow1$ and $LaTeX: \frac{dy}{dt}\rightarrow0$ . If the population y lies between 0 and the carrying capacity L, then dy/dt > 0, and the population increases. If y exceeds the carrying capacity L, then dy/dt < 0, and the population decreases.

General and Particular Solutions of Logistic Differential Equations

The logistic differential equation equation image indicator $LaTeX: \frac{dy}{dt}=ky\left(1-\frac{y}{L}\right)$ is separable and a general solution can be found as illustrated below.

$solving a logistic differential equation$

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Applications and Modeling with Differential Equations Practice

Logistic Models with Differential Equations: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of logitics models with differential equations.

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