Question

The general logistic growth equation is $$ f(t)=\frac{C}{1+A e^{-k t}} $$ (a) Let \(A=50\) and \(k=0.1\). Graph the logistic curves with \(C=100, C=500\), and \(C=1000\) on a single set of axes. Include a legend. What does \(C\) represent? (b) Now, let \(A=50\) and \(C=100\). Graph the curves with \(k=0.1, k=0.4,\) and \(k=1\). How does the parameter \(k\) affect the shape of the curve? (c) Notice that in each case, the curve has a single inflection point. Find its coordinates, in terms of the parameters \(A, C,\) and \(k,\) using symbolic operations.

Short answer

(a) C is the carrying capacity. (b) Larger \( k \) causes faster growth. (c) Inflection at \( t = \frac{1}{k} \ln(A) \).

Step-by-step solution

Graph with varying C

To graph the logistic curves for part (a), we use the specific values \( A=50 \) and \( k=0.1 \), and vary \( C \) among 100, 500, and 1000. Each curve will maintain the same basic shape but will approach different horizontal asymptotes, determined by \( C \). The logistic function can be plotted using these values to compare them visually. The parameter \( C \) in a logistic curve often represents the carrying capacity, which is the maximum value that the function approaches as \( t \to \infty \). This means that \( C \) is indeed the upper horizontal asymptote of the growth function.

Graph with varying k

For part (b), use the fixed values \( A=50 \) and \( C=100 \), and vary \( k \) among 0.1, 0.4, and 1. The curve changes shape depending on \( k \). A smaller \( k \) value results in a slower growth rate and a more stretched curve. Conversely, a larger \( k \) will steepen the curve, indicating faster growth towards the maximum value, \( C \). These changes should be graphed to see the impact of each \( k \) on curve steepness.

Determine the inflection point

The inflection point of a logistic function occurs where the second derivative equals zero. First, differentiate the logistic function:1. First derivative with respect to \( t \): \[ f'(t) = \frac{d}{dt} \left( \frac{C}{1+Ae^{-kt}} \right) \] By applying the chain rule and quotient rule, we get: \[ f'(t) = \frac{CAe^{-kt}(-k)}{(1+Ae^{-kt})^2} \] Simplified as: \[ f'(t) = \frac{-kCAe^{-kt}}{(1+Ae^{-kt})^2} \]2. Second derivative with respect to \( t \): Differentiate \( f'(t) \) again, using the quotient rule: \[ f''(t) = \frac{-kC(Ae^{-kt})(-k)(1+Ae^{-kt})^2 - (-kCAe^{-kt})(2)(Ae^{-kt})(-k)}{(1+Ae^{-kt})^4} \] Simplification leads to solving \( f''(t) = 0 \), and we find the inflection point at \( t = \frac{1}{k} \ln(A) \).

Key concepts

Carrying Capacity

In the logistic growth equation, carrying capacity is a fundamental concept. Represented by the parameter \( C \), it defines the maximum population size the environment can sustainably support. This means, in the context of a population model, it is the largest value that \( f(t) \), the population, approaches over time. As \( t \) tends to infinity, all growth plates account for environmental constraints, stabilizing near this limit, \( C \). The graph of a logistic function reveals this as a horizontal asymptote.

• For different values of \( C \), the curve will reach different levels.
• The approach speed to this level will depend on the combined influence of other parameters, like \( A \) and \( k \).

Hence, understanding \( C \) as carrying capacity allows us to grasp how it impacts the limiting behavior of populations in a given environment.

Inflection Point

Every logistic growth curve contains an inflection point. This is the spot on the curve where it changes from being concave up to concave down or vice versa. In simpler terms, it's where the increase of the curve shifts pace. Mathematically, the inflection point is where the second derivative of the function is zero. For the logistic growth equation, this occurs at:\[t = \frac{1}{k} \ln(A) \]Understanding this critical point provides insight into when the growth rate transitions from accelerating to decelerating:

• The location of the inflection point can affect the shape of the curve significantly
• Knowing how to calculate and interpret it helps predict population dynamics and control resource usage effectively

This understanding is crucial when assessing population health in biological or social sciences.

Differentiation

Differentiation is a mathematical tool used to understand the rate of change in variables. In the context of the logistic growth equation, differentiation allows us to explore how and at what rate the population (\( f(t) \)) changes over time.By calculating the first derivative \( f'(t) \), we gain initial insight into the growth rate:\[f'(t) = \frac{-kCAe^{-kt}}{(1+Ae^{-kt})^2}\]

• The derivative tells us how fast the population is growing at any given time \( t \)
• When \( f'(t) = 0 \), the population reaches its peak growth rate

Second derivatives give even more depth:

• Second derivatives show us the acceleration of the rate of growth
• Solving \( f''(t) = 0 \), identified by \( t = \frac{1}{k} \ln(A) \), helps find the inflection point

This analysis of derivatives forms a basis for understanding complex dynamic systems.

Graphical Representation

Creating a graphical representation of the logistic growth equation can help us visualize complex population dynamics through a clear graphical format.Graphs of the logistic function typically maintain an S-shape or sigmoid curve:

• The steepness of the curve on a graph reflects the rate of population growth
• Different parameters like \( C, A, \) and \( k \) affect the slope and eventual leveling of the curve

When \( A \), \( k \), and \( C \) are modified, their impact can distinctly be visualized:

• Varying \( C \) allows us to see different carrying capacities
• Adjusting \( k \) shifts the growth speed, giving the curve a stretched or compressed appearance

Graphical analysis improves our comprehension by presenting a clear visual understanding of how populations evolve over time and under varying conditions.