Problem 19 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

For what population size is the growth rate \(P^{\prime}\) of the logistic population function the greatest? The equation $$ P^{\prime}(t)=r P(t)\left(1-\frac{P(t)}{M}\right) $$ provides an answer. Observe that \(y=r p(1-p / M)=r p-(r / M) p^{2}\) is a quadratic whose graph is a parabola. The answer to this question is important, for the population size for which \(P^{\prime}\) is greatest is that population that wildlife managers may wish to maintain to provide maximum growth.

Step-by-Step Solution

Answer

Maximum growth occurs at population size \( P = \frac{M}{2} \).

Step 1: Identify the equation and variables

The growth rate of the logistic function is given by the equation \( P'(t) = rP(t) \left( 1 - \frac{P(t)}{M} \right) \), where \( r \) is the intrinsic growth rate and \( M \) is the carrying capacity. We need to find the population size \( P \) where \( P' \) is maximized.

Step 2: Simplify the function

Let's rewrite the equation as a quadratic function in terms of \( P \): \( y = rP - \frac{r}{M}P^2 \). Here, \( y \) is considered as the growth rate, and the equation is a parabola opening downwards with respect to \( P \).

Step 3: Determine the vertex of the parabola

The growth rate \( y = rP - \frac{r}{M}P^2 \) represents a quadratic equation of the form \( y = ax^2 + bx + c \) where \( a = -\frac{r}{M} \) and \( b = r \). The vertex of a parabola \( y = ax^2 + bx + c \) occurs at \( P = -\frac{b}{2a} \).

Step 4: Calculate the vertex for maximum growth rate

Substitute \( a = -\frac{r}{M} \) and \( b = r \) into the vertex formula: \( P = -\frac{r}{2(-\frac{r}{M})} = \frac{M}{2} \). Here, \( P = \frac{M}{2} \) represents the population size that results in the maximum growth rate.

Key Concepts

Population Dynamics

Population dynamics is a fascinating field that examines how populations of organisms change over time. In the context of population growth, scientists use models to predict how a population size will evolve. These models take into account birth rates, death rates, immigration, and emigration factors, among others.

One of the key aspects is understanding the logistic growth model. Unlike exponential growth, which assumes unlimited resources, logistic growth considers a carrying capacity.

• Carrying Capacity (M): The maximum population size that an environment can sustain indefinitely given the food, habitat, water, and other necessities available in the environment.
• Intrinsic Growth Rate (r): This is the rate at which the population would grow if there were no limits on its growth.

The logistic equation helps wildlife managers determine the optimal population size for conservation efforts to maintain ecological balance. In this model, the growth slows as the population approaches the carrying capacity, making it crucial for understanding and managing real-world populations.

Quadratic Functions

Quadratic functions are mathematical expressions that form a parabola when graphed. In the context of the logistic growth equation, it helps us determine the population size where growth is maximized. The equation we work with is a specific form of a quadratic function:

• The logistic growth rate, given by: \[y = rP - \frac{r}{M}P^2\]creates a downward opening parabola.
• Standard form of a quadratic equation is: \(y = ax^2 + bx + c\)where in our case, \(a = -\frac{r}{M}\) and \(b = r\), effectively having no \(c\) term.

The vertex of the parabola indicates the maximum or minimum point, depending on the direction it opens. For our equation, knowing it opens downwards means the vertex represents the point of maximum growth rate, which is calculated using \(P = -\frac{b}{2a}\). This understanding is fundamental in determining critical points in population dynamics.

Mathematical Modeling

Mathematical modeling is an essential tool in understanding real-world phenomena through mathematical concepts. In the case of logistic growth and population dynamics, mathematical models allow us to project population changes over time. These models use equations to simplify complex interactions in an environment.

Using the logistic growth model, we derive the behavior of a population based on simple interactions: birth rates and competition for resources. The ability to represent these dynamics through a model aids in several ways:

• Predictive power: Enables researchers and wildlife managers to anticipate future population sizes and understand potential impacts on ecosystems.
• Optimization: Helps in making informed decisions regarding conservation efforts, ensuring sustainability of a population around its optimal size.

By employing mathematical tools, we can simplify, analyze, and understand intricate details of population behavior, leading to more effective environmental strategies and policies.