Problem 90 - Brief Applied Calculus

Question

Some organisms exhibit a density-dependent mortality from one generation to the next. Let \(R>1\) be the net reproductive rate (that is, the number of surviving offspring per parent), let \(x>0\) be the density of parents and \(y\) be the density of surviving offspring. The Beverton-Holt recruitment curve is $$ y=\frac{R x}{1+\left(\frac{R-1}{K}\right) x} $$ where \(K>0\) is the carrying capacity of the environment. Notice that if \(x=K\), then \(y=K\). Show that if \(x>K\), then \(K

Step-by-Step Solution

Answer

If the initial population exceeds carrying capacity, it decreases over generations but remains above \(K\) initially before stabilizing.

Step 1: Understanding the Given Formula

We begin with the Beverton-Holt recruitment curve where the density of surviving offspring \(y\) is given by \[y = \frac{R x}{1 + \left(\frac{R-1}{K}\right) x}.\] This equation describes how population size is regulated by both the reproductive rate and the carrying capacity \(K\).

Step 2: Substitute and Simplify for x > K

Assume \(x > K\). Substitute into the equation and analyze: \[ y = \frac{R x}{1 + \left(\frac{R-1}{K}\right) x}. \] Substituting \(x = K\) gives \(y = K\). For \(x > K\), we need to compare \(y\) with \(K\) and \(x\).

Step 3: Show y > K for x > K

To show \(y > K\), observe: \[ y = \frac{R x}{1 + \left(\frac{R-1}{K}\right) x} > K. \] For \(y\) to be greater than \(K\), the numerator \(R x\) must grow faster relative to the denominator \(1 + \left(\frac{R-1}{K}\right) x\) after substitution. Since \(x > K\) and \(R > 1\), and the function is increasing, we have \(y > K\).

Step 4: Show y K

Next, confirm \(y < x\) for \(x > K\): \[ y = \frac{R x}{1 + \left(\frac{R-1}{K}\right) x} < x. \] Rearranging, we want: \[ R x < x + \left(\frac{R-1}{K}\right) x^2. \] Dividing by \(x\), since \(x > K\), leaves \(R < 1 + \left(\frac{R-1}{K}\right)x\), which holds as \(R > 1\) and \(x > K\).

Step 5: Conclusion on Population Size

Thus, we confirmed that for \(x > K\), the resulting surviving population size \(y\) satisfies \(K < y < x\). This indicates that while the population initially exceeds the carrying capacity, it will decrease over successive generations as the environment cannot sustain it at \(x\). The population size stabilizes over generations to react towards \(K\), the carrying capacity.

Key Concepts

Density-Dependent Mortality

Density-dependent mortality is a concept that refers to the impact of population density on mortality rates within an ecological community. As the density of a population increases, the mortality rate may also increase due to factors such as increased competition for resources, spread of disease, or predation. This concept is crucial in understanding how populations are naturally regulated by their environment.

In the context of the Beverton-Holt model, density-dependent mortality affects how many offspring survive to adulthood. The model assists in predicting population dynamics by considering how density may impact survival rates. In essence:

• Higher densities can lead to increased competition for food and space, making it difficult for all individuals to survive.
• As density increases, mortality rates may also increase, preventing the population from growing indefinitely.
• This regulation ensures a balance between the population size and the resources available in the environment.

Net Reproductive Rate

The net reproductive rate, denoted as \( R \) in the Beverton-Holt model, is a key concept that describes the average number of offspring that survive per parent. It is crucial in determining whether a population will grow, shrink, or remain stable.

Assuming \( R > 1 \) means that each parent produces more than one surviving offspring, indicating a potential for population growth. Here is why the net reproductive rate is important:

• If \( R = 1 \), the population remains stable since each parent replaces itself with exactly one offspring.
• If \( R > 1 \), the population has the potential to grow because more offspring are surviving than what parents simply replace.
• If \( R < 1 \), the population is likely to decrease over time since not enough offspring survive to replace the parents.

Thus, \( R \) helps us understand how reproductive dynamics relate to population size changes over generations. In the formula, \( R \) directly impacts the numerator, indicating how many offspring are expected under various conditions.

Carrying Capacity

The carrying capacity, denoted as \( K \) in the model, represents the maximum population size that the environment can sustainably support over a long period of time. It is a central concept in the Beverton-Holt model, as it provides a limit to population growth, balancing out the net reproductive rate and the density-dependent mortality.

• When population size \( x \) reaches \( K \), the population is at its equilibrium, meaning the environment can just about support the numbers without the additional strain on resources.
• If \( x < K \), the environment has the capacity to support more individuals, and the population can grow towards \( K \).
• When \( x > K \), the population exceeds the carrying capacity. This excess leads to reduced resources and an increase in mortality rate, encouraging a stabilization of the population size back to \( K \).

In simple terms, carrying capacity reflects how many individuals an environment can support before experiencing negative effects, such as resource depletion or increased mortality. Understanding it helps us predict how populations interact with their environment and what may happen when they exceed these limits.