1.1. First-Order ODE: Existence and Uniqueness 7 Figure 1.1. The logistic equation. For the logistic equation, the velocity field is given by V (x, t) = cx(A−x). The vertical x-axis represents the size of some quantity, and the horizontal axis is the time, t. This equation models the growth of x in the presence of environmental constraints. The constant A is called the carrying capacity, and c(A x) is the “growth rate”. Note that the growth rate approaches zero as x approaches the carrying capacity. This equation is discussed in more detail in Section 2.7 on ecological models. The combination of a differential equation, dx dt = V (x, t), and an initial condition, x(t0) = x0, is called an “initial value problem” (IVP), so the above informal prediction question for smoke particles can now be translated into a precise mathematical question: “What
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