Problem 91 - Brief Applied Calculus

Question

Beverton-Holt Recruitment Curve 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 organism's environment. Show that \(\frac{d y}{d x}>0\), and interpret this as a statement about the parents and the offspring.

Step-by-Step Solution

Answer

The child density \( y \) rises as the parent density \( x \) increases, indicating a positive relationship between parental density and offspring density.

Step 1: Differentiate the Beverton-Holt Recruitment Curve

To find \( \frac{d y}{d x} \), differentiate the given function:\[y = \frac{R x}{1 + \left(\frac{R-1}{K}\right) x}\]We use the quotient rule, which states that the derivative of \( \frac{u}{v} \) is \( \frac{u'v - uv'}{v^2} \), where \( u = R x \) and \( v = 1 + \left(\frac{R-1}{K}\right) x \).

Step 2: Find Derivatives of u and v

Calculate the derivatives: \( u' = \frac{d}{dx}(R x) = R \) and \( v' = \frac{d}{dx}\left(1 + \left(\frac{R-1}{K}\right) x\right) = \frac{R-1}{K} \).

Step 3: Apply the Quotient Rule

Substitute the derivatives and the expressions for \( u \) and \( v \) into the quotient rule formula: \[\frac{d y}{d x} = \frac{(R)\left(1 + \left(\frac{R-1}{K}\right) x\right) - (R x)\left(\frac{R-1}{K}\right)}{\left(1 + \left(\frac{R-1}{K}\right) x\right)^2}\]

Step 4: Simplify the Expression

Simplify the numerator:\[R + \frac{R(R-1)}{K} x - \frac{R^2-1}{K} x = R + \frac{R(K+1)-R}{K} x - \frac{R x (R-1)}{K}\]This reduces to:\(R + \left( R - \frac{R-1}{K} R x\right)\)Since the numerator simplifies positive, we know \( \frac{d y}{d x} > 0 \).

Step 5: Interpret the Result

The result \( \frac{d y}{d x} > 0 \) indicates that the density of surviving offspring \( y \) increases with an increase in the density of parents \( x \), given a fixed reproductive rate and carrying capacity. This illustrates that more parents tend to produce more offspring within the environmental constraints.

Key Concepts

Density-Dependent Mortality

Density-dependent mortality describes how the survival rate of a population changes with the density of the population itself. As population density increases, resources such as food, space, and mates become limited. This limitation leads to higher competition, which can cause an increase in mortality rates.

In the context of the Beverton-Holt model, density-dependent mortality helps explain the decrease in offspring survival as the number of parents (\(x\)) increases. Unlike constant mortality, density-dependent mortality reflects more realistic ecological scenarios where survival depends on environmental factors and population size.

This concept is important because it highlights the dynamic relationship between population size and environmental capacity, illustrating how populations self-regulate through natural mechanisms. It tells us why, sometimes, despite having high reproductive rates, species may not grow indefinitely as their environment can only support a limited number.

Carrying Capacity

Carrying capacity (\(K\)) is a fundamental concept in ecology that defines the maximum number of individuals an environment can sustain indefinitely without degrading the resource base. In the Beverton-Holt recruitment curve, it acts as a balancing force that limits the population growth.

As populations near their carrying capacity, density-dependent factors start to play a significant role, reducing the net growth rate of the population. In essence, \(K\) represents the "saturation point" beyond which the population cannot viably grow without leading to resource exhaustion or increased mortality.

This concept helps students understand why unlimited growth is unsustainable in biological systems and emphasizes the importance of environmental constraints in population dynamics. The recruitment curve itself balances the reproductive potential (\(R\)) against the carrying capacity to provide a realistic model of population growth.

Quotient Rule

The quotient rule is a method from calculus used to find the derivative of a function that is the ratio of two differentiable functions. The formula used in the quotient rule is: \[\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}\]

In the Beverton-Holt recruitment curve, the quotient rule is applied to find the rate of change of surviving offspring density (\(y\)) with respect to parent density (\(x\)).

• Step 1: Identify the numerator and denominator. Here, \(u = R x\) and \(v = 1 + \left(\frac{R-1}{K}\right) x\).
• Step 2: Compute the derivatives \(u' = R\) and \(v' = \frac{R-1}{K}\).
• Step 3: Substitute in the quotient rule formula to obtain the derivative \(\frac{dy}{dx}\).

By using the quotient rule, students can understand how changes in parent density affect offspring density, providing insight into population dynamics.

Net Reproductive Rate

The net reproductive rate (\(R\)) is a critical metric in population ecology, representing the average number of offspring that can be expected to survive per parent. In the context of the Beverton-Holt recruitment curve, it reflects the potential for population growth.

A net reproductive rate greater than 1 (\(R > 1\)) indicates that each generation is producing more offspring than the number of parents, suggesting potential for population growth. Conversely, a rate less than 1 would imply a declining population under stable environmental conditions.

The net reproductive rate is fundamental because it provides a measure of the intrinsic growth capability of a population, independent of environmental constraints. When paired with concepts like carrying capacity, it helps students understand the balance required for sustainable population dynamics.