Problem 67 - Calculus for Biology and Medicine

Question

(Adapted from Moss, 1980) \(\quad\) Hall (1964) investigated the change in population size of the zooplankton species Daphnia galeata mendota in Base Line Lake, Michigan. The population size \(N(t)\) at time \(t\) was modeled by the equation $$ N(t)=N_{0} e^{r t} $$ where \(N_{0}\) denotes the population size at time \(0 .\) The constant \(r\) is called the intrinsic rate of growth. (a) Plot \(N(t)\) as a function of \(t\) if \(N_{0}=100\) and \(r=2\). Compare your graph against the graph of \(N(t)\) when \(N_{0}=100\) and \(r=3\). Which population grows faster? (b) The constant \(r\) is an important quantity because it describes how quickly the population changes. Suppose that you determine the size of the population at the beginning and at the end of a period of length 1, and you find that at the beginning there were 200 individuals and after one unit of time there were 250 individuals. Determine \(r\). [Hint: Consider the ratio \(N(t+1) / N(t) .]\)

Step-by-Step Solution

Answer

The population with \(r = 3\) grows faster. For part (b), the intrinsic rate \(r \approx 0.223\).

Step 1: Identifying Parameters and Equation

The growth of the population is modeled by the equation \( N(t) = N_0 e^{rt} \). We need to graph this equation for different values of \(r\) and compare their growth rates. For \(N(t)\), when \(N_0 = 100, r=2\), and \(N_0 = 100, r=3\).

Step 2: Graphing for r = 2

Set \(N_0 = 100\) and \(r = 2\). The equation becomes \( N(t) = 100 e^{2t} \). Plot \(N(t)\) against \(t\) using software or graphing tools. This exponential function will result in a curve that rises sharply as \(t\) increases.

Step 3: Graphing for r = 3

Set \(N_0 = 100\) and \(r = 3\). The equation becomes \( N(t) = 100 e^{3t} \). Plot this equation on the same graph to visualise how the increase in \(r\) affects growth. The graph should rise even more sharply compared to \(r=2\).

Step 4: Comparing Graphs

Examine both plots on the graph. Notice that the graph with \(r=3\) rises faster than the graph with \(r=2\). Thus, the population grows faster when \(r=3\) compared to when \(r=2\).

Step 5: Calculate r for Part b

Given the initial population \(N(0) = 200\) and after one unit of time \(N(1) = 250\), use the relationship \(N(t+1)/N(t) = e^r\). Substitute \(N(1) = 250\) and \(N(0) = 200\) into the formula \(\frac{N(1)}{N(0)} = e^r\) to get \(\frac{250}{200} = e^r\).

Step 6: Solve for r

Simplify the previous equation: \(\frac{250}{200} = \frac{5}{4}\). So \(e^r = \frac{5}{4}\). Take the natural logarithm: \(r = \ln(\frac{5}{4})\). Use a calculator to find \(r \approx 0.223\).

Key Concepts

Population Dynamics

Population dynamics is the study of how and why populations change over time. It involves examining changes in population size, structure, and distribution in response to various ecological processes.
In the context of the exercise with the zooplankton species *Daphnia galeata mendota*, population dynamics focuses on how these organisms multiply and grow in a specific environment, such as a lake. Understanding these changes is crucial for ecologists as it helps them to make predictions about future population sizes and understand the potential effects of environmental changes.

Key aspects to consider in population dynamics include:

• **Population Size**: The number of individuals present in the population at any given time.
• **Birth Rate**: The rate at which new individuals are added to the population through reproduction.
• **Death Rate**: The rate at which individuals leave the population due to death.
• **Migration**: The movement of individuals into or out of the population.'

These factors interact with each other in complex ways, influencing the overall trajectory of population growth or decline over time.

Intrinsic Rate of Growth

The intrinsic rate of growth, denoted as "r", is a fundamental concept in ecology and is critical in understanding how populations grow. It represents the rate at which a population increases in size if there are no constraints, such as limited resources or environmental factors.
This rate is "intrinsic" because it assumes ideal conditions where every individual has the maximum potential to reproduce.

In mathematical terms, with exponential growth modeled by the formula:

\[ N(t) = N_0 e^{rt} \]
"r" allows us to determine how quickly the population grows:

• If **r > 0**, the population grows exponentially. The larger the value of r, the faster the growth.
• If **r = 0**, the population size remains constant over time.
• If **r < 0**, the population declines, potentially leading to extinction if the trend continues.

Understanding the intrinsic rate of growth helps in predicting future changes in population size, especially in contexts like conservation biology where maintaining a healthy population is crucial.

Mathematical Modeling

Mathematical modeling in ecology is a powerful tool used to simplify and understand the complex interactions within biological systems. By representing biological processes with mathematical formulas, ecologists can simulate and predict how populations behave under various conditions.
In our exercise, the population of *Daphnia galeata mendota* was modeled with the equation \( N(t) = N_0 e^{rt} \). This is a classic example of a mathematical model used in population dynamics to illustrate exponential growth.

Benefits of using mathematical models include:

• **Prediction:** Models allow researchers to predict future population sizes based on current data.
• **Experimentation:** Virtual experiments can be conducted to test different scenarios without impacting actual populations.
• **Understanding Dynamics:** Complex systems can be broken down into understandable components.

However, these models are simplifications and rely on assumptions that may not hold true in real-world situations, such as constant growth rate "r" without environmental constraints. Despite this, they offer invaluable insights and are integral to many ecological studies.