A population is modeled by the differential equation (d P)/(d t)=1.2 P(1-(P)/(4200)) (a) For wha

step 1 In order to find the equilibrium solution and increasing, decreasing intervals, the equilibrium

A population is modeled by the differential equation (d...

A population is modeled by the differential equation

$$
\frac{d P}{d t}=1.2 P\left(1-\frac{P}{4200}\right)
$$

(a) For what values of $P$ is the population increasing?
(b) For what values of $P$ is the population decreasing?
(c) What are the equilibrium solutions?

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Key Concepts

Equilibrium Solutions

Equilibrium solutions are the constant solutions where the rate of change is zero. In the context of a differential equation, these correspond to the values of the population where no change occurs over time, marking critical points such as extinction or saturation at the carrying capacity.

Logistic Differential Equation

A logistic differential equation models how a population grows in an environment with limited resources. It typically features a growth term that is proportional to the current population and a limiting factor that decreases the growth as the population approaches a maximum sustainable size, or carrying capacity.

Sign Analysis of the Derivative

Sign analysis involves determining the intervals on which the rate of change (the derivative) is positive or negative. This technique is used to identify when the population is increasing or decreasing by assessing the sign of each multiplicative factor in the differential equation.

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