Problem 20 - Calculus for Biology and Medicine

Question

Assume that the population growth is described by the Beverton-Holt recruitment curve with parameters \(R_{0}\) and a. Find the population sizes for \(t=1,2, \ldots, 5\) and find \(\lim _{t \rightarrow \infty} N_{t}\) for the given initial value \(N_{0} .\) \(R_{0}=3, a=1 / 10, N_{0}=3\)

Step-by-Step Solution

Answer

The population sizes are \(6.923, 12.275, 16.532, 18.696, 19.547\) and \(\lim_{t \to \infty} N_t = 20\).

Step 1: Introduction to Beverton-Holt Formula

The Beverton-Holt recruitment model is used to calculate population sizes over time. The formula to find the population size at any time \(t\) is given by \(N_{t+1} = \frac{R_{0}N_{t}}{1 + aN_{t}}\), where \(R_0\) is the maximum recruitment rate, and \(a\) is the density-dependent parameter.

Step 2: Calculate Population for t=1

Using the formula \(N_{1} = \frac{R_{0}N_{0}}{1 + aN_{0}}\), substitute \(R_{0} = 3\), \(a = \frac{1}{10}\), and \(N_{0} = 3\). This results in \(N_{1} = \frac{3 \times 3}{1 + \frac{1}{10} \times 3} = \frac{9}{1.3} \approx 6.923\).

Step 3: Calculate Population for t=2

Next, use the result from \(N_1\) to find \(N_{2}\). Substitute into the formula: \(N_{2} = \frac{R_{0}N_{1}}{1 + aN_{1}} = \frac{3 \times 6.923}{1 + \frac{1}{10} \times 6.923} \approx \frac{20.769}{1.6923} \approx 12.275\).

Step 4: Calculate Population for t=3

Use \(N_{2}\) to find \(N_{3}\). Substitute into the formula: \(N_{3} = \frac{R_{0}N_{2}}{1 + aN_{2}} = \frac{3 \times 12.275}{1 + \frac{1}{10} \times 12.275} \approx \frac{36.825}{2.2275} \approx 16.532\).

Step 5: Calculate Population for t=4

Use \(N_{3}\) to find \(N_{4}\). Substitute: \(N_{4} = \frac{R_{0}N_{3}}{1 + aN_{3}} = \frac{3 \times 16.532}{1 + \frac{1}{10} \times 16.532} \approx \frac{49.596}{2.6532} \approx 18.696\).

Step 6: Calculate Population for t=5

Finally, use \(N_{4}\) to find \(N_{5}\). Substitute: \(N_{5} = \frac{R_{0}N_{4}}{1 + aN_{4}} = \frac{3 \times 18.696}{1 + \frac{1}{10} \times 18.696} \approx \frac{56.088}{2.8696} \approx 19.547\).

Step 7: Calculate Limit as t Approaches Infinity

The equilibrium solution is found by setting \(N_{t} = N_{t+1}\) in the Beverton-Holt equation: \(N_{\infty} = \frac{R_{0}N_{\infty}}{1 + aN_{\infty}}\). This leads to \(N_{\infty}(1 + aN_{\infty}) = R_0 N_{\infty}\). Solving for \(N_{\infty}\), we have \(aN_{\infty}^2 + (1-R_{0})N_{\infty} = 0\). Simplifying, \(N_{\infty} = \frac{R_{0} - 1}{a} = \frac{3 - 1}{0.1} = 20\).

Key Concepts

Understanding Population Dynamics

Population dynamics refers to the ways in which a population's size or composition changes over time. Understanding these changes is crucial because it helps us predict the future of the population. In biology, this encompasses everything from the birth and death rates to immigration and emigration.

The Beverton-Holt model captures this dynamic by modeling the changes in population size across different generations. Each step uses the current population size to calculate the next, considering factors like the maximum recruitment rate and density dependence. This iterative process provides a series of population sizes that help predict future growth and sustainability. As we've seen, this model can predict population sizes well into the future, showing whether the population tends towards stability or continual growth.

Density Dependence in Population Growth

Density dependence is a key concept in ecology that describes how the growth rate of a population is affected by the population's density. In simple terms, when a population is small, resources are abundant, and intraspecific competition is low, so the population can grow rapidly.

However, as population density increases, individuals compete more fiercely for limited resources such as food or space. The Beverton-Holt model integrates density dependence through the parameter \(a\). As the population size \(N_t\) increases, the term \(aN_t\) in the denominator of the formula \(N_{t+1} = \frac{R_{0}N_{t}}{1 + aN_{t}}\) becomes significant, thereby slowing the growth.

This model elegantly demonstrates that, though a population with low density can grow rapidly, a higher population density leads to reduced growth rates, eventually resulting in equilibrium.

Exploring Recruitment Models

Recruitment models, like the Beverton-Holt model, are essential tools in ecology used to predict the number of new individuals joining a population. They take into account various factors such as birth rates, survival rates, and the effects of population density to predict how many newborns reach a certain age or contribute to the next generation.

The Beverton-Holt model, specifically, assumes a maximum recruitment rate, \(R_0\), which represents ideal conditions without any density effects. As population size increases, density-dependent factors come into play, limiting recruitment to sustainable levels. This approach helps in understanding the balance between birth rates and the factors inhibiting them.

• **Maximum recruitment rate** - This is how many individuals would be added if there were unlimited resources.
• **Limitations due to density** - As the population grows, recruitment slows because of competition for resources.

Understanding recruitment models is vital for determining sustainable population levels and managing wildlife conservation efforts.