Question

Make a sketch of the population function (as a function of time) that results from the following growth rate functions. Assume the population at time \(t=0\) begins at some positive value.

Short answer

Answer: To find and sketch the population function from a given growth rate function: 1. Understand the relationship between the population function and the growth rate function: \(\frac{dP(t)}{dt} = G(t)\). 2. Integrate the growth rate function: \(P(t) = \int G(t) dt +C\), where \(C\) is the constant of integration. 3. Apply the initial condition to find the value of \(C\). 4. Sketch the population function, P(t), by considering the behavior of the growth rate function and its integral.

Step-by-step solution

Understand the relationship between the population function and the growth rate function

Let \(P(t)\) denote the population at time \(t\), and let \(G(t)\) denote the growth rate function. The growth rate function tells us how the population changes over time, so we can relate the two by a simple equation: \(\frac{dP(t)}{dt} = G(t)\).

Integrate the growth rate function

To get the population function from the growth rate function, we need to integrate \(G(t)\) with respect to time. This will give us an expression for \(P(t)\) that we can sketch: $$ P(t) = \int G(t) dt +C, $$ where \(C\) is the constant of integration. To find \(C\), we can use the given initial condition that the population at time \(t=0\) begins at some positive value. For each given growth rate function, we will need to integrate and find the corresponding population function.

Sketch the population functions

Finally, we need to sketch the population function \(P(t)\) as a function of time. To do so, we will consider the shape of the given growth rate functions and the behavior of their integrals. Since we are sketching the functions, we do not need to find the exact value of \(C\). Instead, we will qualitatively understand the behavior of the population function for each given growth rate function. When working with an actual growth rate function, go through the following steps: 1. Integrate the growth rate function (find \(\int G(t) dt\)) 2. Apply the initial condition (determine the constant of integration, \(C\)) 3. Sketch the population function by considering the behavior of the growth rate function and its integral (use the expression found in step 1) Keep in mind that you need to follow these steps for each given growth rate function in order to sketch the corresponding population function.

Key concepts

Growth Rate Functions

Growth rate functions are pivotal when exploring population growth. They provide the rate at which the population changes over time. In simpler terms, if you imagine population growth as a journey, the growth rate function is like the speedometer, indicating how fast or slow the population is moving at any given moment.

Mathematically, the growth rate of a population is expressed as:

• Let \( P(t) \) be the population at time \( t \).
• The growth rate function is represented as \( G(t) \), meaning it describes how quickly the population size \( P(t) \) is changing.
• The relationship between the population and the growth rate function is given by \( \frac{dP(t)}{dt} = G(t) \).

This relationship forms the basis for predicting future population sizes given a known growth rate, enabling us to map out how populations might expand or contract over time. Each growth rate function offers a unique view of population dynamics due to differences in factors like birth rates, death rates, and environmental conditions.

Integration

Integration is a mathematics tool that helps us transition from knowing a rate of change to understanding the quantity that has changed over time. In the context of population growth, integration allows us to determine the size of a population at any given time, given its growth rate.

To integrate the growth rate function \( G(t) \), we aim to find \( P(t) \), which is the population function:

• The formula we use is \( P(t) = \int G(t) \, dt + C \), where \( C \) is a constant from integration.
• Integration effectively sums up how many new individuals have been added to the population over time, based on the rate given by \( G(t) \).

This process can be likened to accumulating the total number of people added to a city every year. By understanding growth over time through integration, we gather insights into the population's future size, helping us model and plan more effectively.

Initial Conditions

Initial conditions are essential in tailoring a general solution, derived from integration, to a specific scenario. When dealing with population functions, an initial condition helps us find the exact value of that pesky constant \( C \).

Consider an initial condition such as the population at time \( t=0 \) being a known positive value, say \( P_0 \):

• You begin by integrating the growth rate function \( G(t) \), as before, to get \( P(t) = \int G(t) \, dt + C \).
• By substituting the initial condition into this equation, you solve for \( C \). This ensures that the specific population growth model aligns with known facts at a certain starting time.

Such initial conditions ensure that our mathematical model reflects real-world observations. They anchor our equations in reality, providing more accurate, applicable insights into how populations have and will evolve. In practical terms, this means you are accounting for all initial individuals in your population model from the get-go!