Problem 61 - Calculus for Biology and Medicine

Question

The per capita growth rate of a population of cells varies over the course of a day. Assume that time \(t\) is measured in hours and $$ \frac{d N}{d t}=2\left(1-\cos \frac{2 \pi t}{24}\right) N $$ if \(N(0)=5\), find the number of cells after one day (that is, find \(N(24))\)

Step-by-Step Solution

Answer

The number of cells after one day is extremely large: \(5e^{48}\).

Step 1: Understand the Problem

We are given a differential equation \(\frac{dN}{dt} = 2\left(1 - \cos \frac{2\pi t}{24}\right)N\) with an initial condition \(N(0) = 5\). Our task is to find \(N(24)\), or the number of cells after one day.

Step 2: Recognize the Differential Equation Type

This equation is a first-order linear differential equation, where \(\frac{dN}{dt} = g(t)N\), with \(g(t) = 2\left(1 - \cos \frac{2\pi t}{24}\right)\). We can solve it by separating variables or using an integrating factor.

Step 3: Use the Integrating Factor

The general solution of a linear differential equation \(\frac{dN}{dt} + P(t)N = 0\) is \(N(t) = N(0) e^{\int g(t) dt}\). Identify the integrating factor as \(e^{\int 2(1-\cos \frac{2\pi t}{24}) dt}\).

Step 4: Compute the Integral

Calculate the integral \(\int 2(1 - \cos \frac{2\pi t}{24}) \, dt\). First, separate terms into \(\int 2\, dt - \int 2 \cos \frac{2\pi t}{24} \, dt\). This yields \(2t - 48\sin \frac{2\pi t}{24}\).

Step 5: Substitute and Solve for N(t)

Substitute the result into the expression for \(N(t)\): \(N(t) = 5 e^{2t - 48 \sin \frac{2\pi t}{24}}\). Simplify for \(t = 24\): \(N(24) = 5 e^{48}\).

Step 6: Compute Final Value

Calculate \(N(24) = 5 e^{48}\), which is a very large number indicating exponential growth. Without explicit computation of \(e^{48}\), understand that the number of cells is extremely large after a day.

Key Concepts

Per Capita Growth Rate

The term "per capita growth rate" refers to the change in population size relative to the existing number of individuals. It's a measure of how the population's size changes over time.
In mathematical terms, the per capita growth rate is often expressed as \( \frac{1}{N} \frac{dN}{dt} \). This gives us the rate of change per individual in the population, allowing us to understand the growth dynamic on a per-individual basis rather than focusing on absolute numbers.
In the given problem, the rate \( 2(1-\cos \frac{2\pi t}{24}) \) affects how quickly the cell population increases during the day.

• The component \( \cos \frac{2\pi t}{24} \) introduces periodicity, meaning that the growth rate changes at different times of the day.
• Multiplying by \( 2 \) scales this periodic growth, modifying the intensity.

This approach allows for modeling a realistic environment where growth rates aren't static. Such insights are crucial in studies of ecology and other fields that involve dynamic systems.

Population Dynamics

Population dynamics is the study of how populations change over time under the influence of births, deaths, immigration, and emigration. It encompasses the actions and reactions of population components that lead to different growth patterns.
Understanding population dynamics is essential for managing resources, conserving species, and predicting how a population might respond to environmental changes.

• The model given in the exercise introduces time-dependent changes in growth, i.e., the per capita growth rate is a function of time.
• This showcases a dynamic system where parameters like growth are influenced by both internal biological factors and periodic external forces.

By incorporating periodic changes, like the cosine component seen in our differential equation, we simulate scenarios more closely aligned with real-world patterns—such as daily cycles in cellular activities or behaviors linked to environmental changes.

First-order Linear Differential Equation

First-order linear differential equations are equations that involve the first derivative of a function and are linear in the dependent variable. These equations are foundational in modeling natural processes because they often describe systems experiencing change over time.
The given differential equation \( \frac{dN}{dt} = 2(1-\cos \frac{2\pi t}{24})N \) falls under this category. Here's why it's crucial:

• First-order implies the equation involves the first derivative—here, \( \frac{dN}{dt} \).
• Linear means that \( N \) (the number of cells in this context) appears without exponents or other non-linear operations.

To solve this kind of differential equation, we frequently use methods like separation of variables or integrating factors.
In the solution outlined, an integrating factor is used. This approach involves multiplying through by a function derived from \( g(t) \), which transforms our equation so that it can be directly integrated.
This method provides a general solution that reflects how cell populations grow under the specified conditions, and these solutions help predict behaviors of the modeled systems in diverse fields, such as biology, economics, and engineering.