Problem 50 - Calculus for Biology and Medicine

Question

In this problem, we will investigate how mutual interference of parasitoids affects their searching efficiency for a host. We assume that \(N\) is the host density and \(P\) is the parasitoid density. A frequently used model for host- parasitoid interactions is the Nicholson-Bailey model (Nicholson, 1933; Nicholson and Bailey, 1935), in which it is assumed that the number of parasitized hosts, denoted by \(N_{a}\), is given by $$ N_{a}=N\left[1-e^{-b P}\right] $$ where \(b\) is the searching efficiency. (a) Show that $$ b=\frac{1}{P} \ln \frac{N}{N-N_{a}} $$ by solving \((10.3)\) for \(b\). (b) Consider $$ b=f\left(P, N, N_{a}\right)=\frac{1}{P} \ln \frac{N}{N-N_{a}} $$ as a function of \(P, N\), and \(N_{a} .\) How is the searching efficiency \(b\) affected when the parasitoid density increases? (c) Assume now that the fraction of parasitized host depends on the host density; that is, assume that $$ N_{a}=g(N) $$ where \(g(N)\) is a nonnegative, differentiable function. The searching efficiency \(b\) can then be written as follows as a function of \(P\) and \(N\) : $$ b=h(P, N)=\frac{1}{P} \ln \frac{N}{N-g(N)} $$ How does the searching efficiency depend on host density when \(g(N)\) is a decreasing function of \(N ?\) (Use the fact that \(g(N)

Step-by-Step Solution

Answer

(a) Showed that formula matches. (b) Efficiency decreases with parasitoid density. (c) Efficiency decreases with host density.

Step 1: Solving for b in the equation for N_a

We start with the equation for parasitized hosts:\[ N_a = N[1-e^{-bP}] \]We need to solve for \( b \). First, isolate the exponential term:\[ e^{-bP} = 1 - \frac{N_a}{N} \]

Step 2: Taking the natural logarithm

To get rid of the exponential, take the natural logarithm on both sides:\[ -bP = \ln\left(1-\frac{N_a}{N}\right) \]

Step 3: Solving for b

Now, we need to solve for \( b \), which gives us:\[ b = -\frac{1}{P} \ln\left(1-\frac{N_a}{N}\right) \]Recognizing that \( \ln(a) = -\ln(1/a) \), we can rewrite it as:\[ b = \frac{1}{P} \ln\left(\frac{N}{N-N_a}\right) \]

Step 4: Investigating dependence on parasitoid density

The expression for searching efficiency is:\[ b=f\left(P, N, N_a\right)=\frac{1}{P} \ln\frac{N}{N-N_{a}} \]As parasitoid density \( P \) increases, the factor \( \frac{1}{P} \) decreases, which leads to a decrease in \( b \). Therefore, searching efficiency decreases as parasitoid density increases.

Step 5: Investigating dependence on host density with g(N)

We have:\[ b=h(P, N)=\frac{1}{P} \ln\frac{N}{N-g(N)} \]If \( g(N) \) is a decreasing function, then as \( N \) increases, \( N-g(N) \) increases. Therefore, \( \frac{N}{N-g(N)} \) becomes smaller, reducing \( b \). So, searching efficiency decreases as host density increases when \( g(N)\) is decreasing.

Key Concepts

host-parasitoid interactions

In the natural world, host-parasitoid interactions represent an intricate dance of survival and predation. A parasitoid is an organism, often an insect, whose larva develops by feeding on a host, ultimately leading to the host's death. This is a unique kind of interaction because the parasitoid relies on the host not just as a food source, but also as an environment for its offspring to develop.

One vital model for understanding these relationships is the Nicholson-Bailey model, which describes the dynamics between host and parasitoid populations over time. This model assumes that parasitoids lay eggs in or on a host and the number of parasitized hosts in a population can be mathematically represented. By converting a specific fraction of the host population into parasitized hosts, researchers can predict how populations might change each generation under various conditions.

This model is crucial for understanding how these interactions might shift due to changes in environmental conditions or population sizes, helping ecologists forecast potential outcomes and design strategies for managing biological control programs.

searching efficiency

Searching efficiency, denoted by the parameter \( b \), is a measure of how effective parasitoids are at finding and parasitizing their hosts. In the Nicholson-Bailey model, searching efficiency is a factor that directly impacts the proportion of the host population that becomes parasitized. The higher the searching efficiency, the more successful the parasitoids are at converting hosts into parasitized individuals.

Mathematically, searching efficiency can be calculated using the formula:

• \( b = \frac{1}{P} \ln\left(\frac{N}{N-N_a}\right) \)

Here, \( N \) represents the host density, \( P \) is the parasitoid density, and \( N_a \) is the number of parasitized hosts.

• As parasitoid density \( P \) increases, the term \( \frac{1}{P} \) diminishes, thus reducing the searching efficiency \( b \).
• This decrease implies that with more parasitoids, the efficiency per individual declines due to mutual interference among parasitoids that competes for the same hosts.

This relationship highlights the delicate balance parasitoids must maintain to optimize their success without overwhelming their environment.

parasitized hosts

Parasitized hosts are an essential outcome of host-parasitoid interactions where a portion of the host population supports the development of parasitoid larvae. Understanding the dynamics of parasitized hosts helps in predicting the impact of parasitoids on host population sizes.

The number of parasitized hosts, denoted \( N_a \), depends on both the interaction dynamics and the efficiency of parasitoids in locating the hosts. According to the Nicholson-Bailey model, knowing how the function \( g(N) \)—which represents the number of parasitized hosts as a function of the total host \( N \)—behaves allows for precise predictions:

When \( g(N) \) decreases as \( N \) increases, it implies fewer hosts are becoming parasitized as the host population grows. This finding is crucial as it suggests that large host populations might overwhelm the parasitoids, reducing their overall impact and efficiency.

By analyzing these interactions, ecologists and entomologists can assess and model the potential effects of introducing or sustaining parasitoid populations in ecosystems for biological control against pest species.