Starting with an initial value of P(0)=55, the population of...

Starting with an initial value of $P(0)=55,$ the population of a prairie dog community grows at a rate of $P^{\prime}(t)=20-t / 5$ (in units of prairie dogs/month), for $0 \leq t \leq 200.$ a. What is the population 6 months later? b. Find the population $P(t),$ for $0 \leq t \leq 200.$

Starting with an initial value of $P(0)=55,$ the population of a prairie dog community grows at a rate of $P^{\prime}(t)=20-t / 5$ (in units of prairie dogs/month), for $0 \leq t \leq 200.$
a. What is the population 6 months later?
b. Find the population $P(t),$ for $0 \leq t \leq 200.$

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Key Concepts

Differential Equations

Differential equations describe relationships involving rates of change and the functions themselves. In the context of population models, they are used to represent how the population evolves over time given a specific rate of change, such as birth and death rates in a community.

Rate of Change

The rate of change is a measure of how a quantity, such as a population, increases or decreases over time. It is typically expressed as a derivative, indicating the instantaneous change at any given time. Understanding the rate of change is essential in setting up differential equations for dynamic systems.

Integration and Antiderivatives

Integration is the inverse process of differentiation and is used to find the original function when given its rate of change. In problems involving population growth, the antiderivative allows us to recover the population function from the known rate of change by integrating the differential equation.

Initial Value Problems

An initial value problem involves solving a differential equation with a given starting condition, which determines the particular solution among all possible ones. The initial condition, such as an initial population at a specific time, is used to determine the integration constant, ensuring the model accurately reflects the scenario.

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