Problem 23 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Ricker's model for population growth is $$ P^{\prime}=R P e^{-\frac{P}{\alpha}} $$ where \(P\) is population size, \(R\) is the low density growth rate, and \(\alpha\) reflects the carrying capacity of the environment. For what value of \(P\) will the growth rate, \(P^{\prime}\) be greatest?

Step-by-Step Solution

Answer

The growth rate \(P^{\prime}\) is greatest when \(P = \alpha\).

Step 1: Analyze the Function to Maximize

The growth rate given by Ricker's model is defined as \(P^{\prime}=R P e^{-\frac{P}{\alpha}}\). We are asked to find the value of \(P\) that maximizes this growth rate. This means we need to find the critical points by taking the derivative of \(P^{\prime}\) with respect to \(P\).

Step 2: Differentiate the Growth Rate Function

Find the derivative of \(P^{\prime}=R P e^{-\frac{P}{\alpha}} \) with respect to \(P\). Using the product rule for differentiation, where \(u = R P\) and \(v = e^{-\frac{P}{\alpha}}\), we get: \[ \frac{d}{dP}(R P e^{-\frac{P}{\alpha}}) = R e^{-\frac{P}{\alpha}} + R P \left( -\frac{1}{\alpha} e^{-\frac{P}{\alpha}} \right) \]Simplifying, the derivative is: \[ R e^{-\frac{P}{\alpha}} \left( 1 - \frac{P}{\alpha} \right) \]

Step 3: Find Critical Points

To find critical points, set the derivative from Step 2 equal to zero:\[ R e^{-\frac{P}{\alpha}} \left( 1 - \frac{P}{\alpha} \right) = 0 \]Since \(R e^{-\frac{P}{\alpha}}\) is never zero for real \(P\), solve:\[ 1 - \frac{P}{\alpha} = 0 \]Solving yields \(P = \alpha\).

Step 4: Verify Maximum with Second Derivative Test

To confirm that \(P = \alpha\) gives a maximum, examine the second derivative of the growth rate function. The second derivative, using the product rule and chain rule, will involve determining the sign of:\[ \frac{d^2}{dP^2}(R P e^{-\frac{P}{\alpha}}) \]Calculating this shows that at \(P = \alpha\), the second derivative is negative, confirming a local maximum.

Key Concepts

Population Dynamics

Population Dynamics is a fascinating field that explores how populations of living organisms change over time. It considers factors such as birth rates, death rates, immigration, and emigration. These factors collectively determine population sizes and how they fluctuate. In nature, populations are rarely static. They are influenced by a myriad of environmental and biological factors. Population dynamics help us understand these influences, providing insight into the natural world and the sustainability of various ecosystems. Key elements of population dynamics include:

• Growth Rate: The speed at which a population increases or decreases in size. This can be influenced by resources, predators, and environmental pressures.


• Carrying Capacity: The maximum population size an environment can sustain indefinitely. This is influenced by resources like food, water, and space.


• Density-dependent Factors: Factors that change in intensity as the population changes, such as food availability and disease.

Understanding these elements is crucial for managing wildlife, conserving species, and understanding human population trends.

Ricker's Model

Ricker's Model is an important concept in population ecology. Developed by Bill Rickers, it's often used to describe how populations grow and self-regulate in certain environments. The model is particularly effective for species such as fish populations.In mathematical terms, Ricker's Model is expressed as: \[P^{\prime}=R P e^{-\frac{P}{\alpha}}\]Here:

• \(P\) represents the population size.


• \(R\) is the low-density growth rate, showing how quickly the population can grow when density isn't limiting.


• \(\alpha\) is the environment's carrying capacity, representing the maximum population size that the environment can sustain.

Ricker's Model is special because it describes how population growth rates begin to slow as population size approaches the carrying capacity. As the population grows, resources become scarce, causing the growth rate to decline.This model is useful for predicting patterns in population sizes over time, enabling effective resource management and conservation efforts.

Product Rule in Differentiation

The Product Rule in differentiation is a fundamental concept in calculus. It provides a method for finding the derivative of functions that are products of two or more individual functions. If you have two functions, \(u(x)\) and \(v(x)\), their product's derivative is given by:\[ (uv)' = u'v + uv' \]In the context of Ricker's Model, we used the product rule to differentiate the growth rate function \(P^{\prime} = R P e^{-\frac{P}{\alpha}}\). Here:

• Let \(u = R P\) and \(v = e^{-\frac{P}{\alpha}}\).


• Then \(u' = R\) and \(v' = -\frac{1}{\alpha} e^{-\frac{P}{\alpha}}\).

Substituting these into the product rule formula, we obtain:\[ \frac{d}{dP}(R P e^{-\frac{P}{\alpha}}) = R e^{-\frac{P}{\alpha}} + R P \left( -\frac{1}{\alpha} e^{-\frac{P}{\alpha}} \right) \]This differentiation step is crucial for solving problems related to population dynamics models, like Ricker's Model, as it helps us find maximum points such as the one discovered in the exercise.