Problem 22 - Calculus for Biology and Medicine

Question

The logistic function defined by: $$N(t)=\frac{1}{a+e^{-t}}, t \geq 0$$ represents the growth of a population. We will derive this function by solving a differential equation model in Chapter \(8 .\) Assume that \(a\) is a positive constant. (a) Determine where \(N(t)\) is increasing and where it is decreasing. (b) Find and classify any local extrema that the function has. (c) Where is the function concave up and where is it concave down? Find all inflection points of \(N(t)\). (d) Find \(\lim _{t \rightarrow \infty} N(t)\) and decide whether \(N(t)\) has a horizontal asymptote. (e) Sketch the graph of \(N(t)\) together with its asymptotes and inflection points (if they exist). (f) Describe in words how the graph of the function changes if \(a\) is increased.

Step-by-Step Solution

Answer

(a) Increasing for all \(t \geq 0\); (b) No local extrema; (c) Concave down for \(t < -\ln(a)\), up for \(t > -\ln(a)\); inflection point at \(t = -\ln(a)\); (d) Horizontal asymptote at \(y = \frac{1}{a}\).

Step 1: Determine where N(t) is increasing or decreasing

To find where the function is increasing or decreasing, we first differentiate the logistic function \(N(t)\) with respect to \(t\): \[ N'(t) = \frac{d}{dt}\left( \frac{1}{a+e^{-t}} \right). \]Using the quotient rule, let \(u = 1\) and \(v = a + e^{-t}\), then \(N'(t) = \frac{v \cdot (0) - u \cdot \left(-e^{-t}\right)}{v^2} = \frac{e^{-t}}{(a+e^{-t})^2}.\)Since both \(e^{-t}\) and \((a+e^{-t})^2\) are positive for \(t \geq 0\), \(N'(t) > 0\). Therefore, \(N(t)\) is increasing for all \(t \geq 0\).

Step 2: Identify any local extrema

As derived above, \(N'(t) > 0\) for all \(t \geq 0\). This indicates that the function is always increasing and thus does not have any local maxima or minima.

Step 3: Determine concavity and inflection points

To find where the function is concave up or down, consider the second derivative \(N''(t)\): \[ N''(t) = \frac{d}{dt} \left( \frac{e^{-t}}{(a+e^{-t})^2} \right). \]Applying the quotient rule again, \[ N''(t) = \frac{(a+e^{-t})^2(-e^{-t}) - e^{-t}(-2)(a+e^{-t})(-e^{-t})}{(a+e^{-t})^4}.\]Simplifying, \[ N''(t) = \frac{-e^{-t}(a+e^{-t}) + 2e^{-2t}(a + e^{-t})}{(a+e^{-t})^4} = \frac{-ae^{-t} + e^{-2t}}{(a+e^{-t})^3}.\]Calculate when \(N''(t) = 0\):\(-ae^{-t} + e^{-2t} = 0\) simplifies to \(e^{-t}(e^{-t} - a) = 0\), which gives \(e^{-t} = a\).Taking natural logs, \(-t = \ln(a)\) or \(t = -\ln(a)\) which is not defined for positive \(a\) and \(t \geq 0\), hence there are no inflection points. Check the sign of \(N''(t)\):For \(t < -\ln(a)\), \(N''(t) < 0\), so concave down. For \(t > -\ln(a)\), \(N''(t) > 0\), so concave up.

Step 4: Find horizontal asymptote

Examine the limit of \(N(t)\) as \(t\) approaches infinity:\[ \lim_{t \to \infty} N(t) = \lim_{t \to \infty} \frac{1}{a+e^{-t}}. \]As \(t \to \infty\), \(e^{-t} \to 0\), thus \(\lim_{t \to \infty} N(t) = \frac{1}{a}.\)This horizontal asymptote exists at \(N(t) = \frac{1}{a}\).

Step 5: Sketch the graph

Plot the logistic function \(N(t)\), marking the horizontal asymptote at \(y = \frac{1}{a}\). Since the function is always increasing from the origin towards the asymptote without any inflection points, the slope is always positive and gradually levels off as it approaches the asymptote.

Step 6: Describe the effect of changing 'a'

If \(a\) increases, the horizontal asymptote \(y = \frac{1}{a}\) decreases, making the curve approach a lower limit. The function will rise from the origin less steeply, meaning the growth rate appears to slow down as \(a\) increases.

Key Concepts

Population Growth

The logistic function is a classic model used to describe population growth. It's an excellent representation of how populations tend to grow in an environment with limited resources. As time passes, resources such as food or space might limit the growth of a population, leading to a slower increase in population size. This adjustment in growth rate is perfectly captured by the logistic function.

• Initially, the population experiences exponential growth as resources are abundant.
• Over time, as the population grows, the availability of resources starts to limit further growth.
• Eventually, the population growth slows, nearing a maximum capacity known as the carrying capacity.

The logistic function used to model this behavior, as given by: \[ N(t) = \frac{1}{a+e^{-t}} \] shows how the population size at any time \( t \) tends to stabilize at a certain value. This limit or horizontal asymptote represents the maximum population that the environment can sustain, thus setting a natural cap on population growth.

Differential Equation

A differential equation is a mathematical expression that describes the relationship between a function and its derivatives. In the context of population growth, differential equations are used to model the rate of change of a population over time. The logistic differential equation is one of the most common models.
The logistic differential equation can be derived from the logistic function by differentiating it with respect to time \( t \): \[ N'(t) = \frac{e^{-t}}{(a+e^{-t})^2} \] This equation captures the rate at which the population grows at any given moment.

Since the initial derivative is positive throughout the domain, it indicates that the population is always growing. There's never a point at which the growth rate is zero, which would signal a stable population size. This is because both the numerator and denominator in the derivative are positive for \( t \geq 0 \).
Understanding the logistic differential equation allows us to predict how a population might change over time based on initial conditions and the value of the constant \( a \).

Concavity

Concavity is a mathematical concept that describes the curvature of a function's graph. It helps to understand how the growth rate of a population, modeled by the logistic function, accelerates or decelerates.

• If the function is concave up, its graph is bowl-shaped, and the population growth is accelerating.
• If the function is concave down, its graph is dome-shaped, and the growth rate is decreasing.

We determine concavity by finding the second derivative of the logistic function: \[ N''(t) = \frac{-ae^{-t} + e^{-2t}}{(a+e^{-t})^3} \]This expression can be used to analyze where the function is concave up or down. It's essential to understand that the sign of the second derivative indicates concavity:

• If \( N''(t) > 0 \), the function is concave up.
• If \( N''(t) < 0 \), the function is concave down.

In this logistic model, since \(-t = \ln(a)\) is not defined for positive \( a \) when \( t \geq 0 \), there are no inflection points. This means there's no switch between concavity. For \( t < -\ln(a) \), the function is concave down; for \( t > -\ln(a) \), it's concave up. However, within our domain (\( t \geq 0 \)), the logistic function is mostly concave up, reflecting a population whose growth rate is gradually slowing as it approaches the carrying capacity.