Problem 14 - Calculus for Biology and Medicine

Question

In the negative binomial model, the fraction of hosts escaping parasitism is given by $$ f(P)=\left(1+\frac{a P}{k}\right)^{-k} $$ (a) Graph \(f(P)\) as a function of \(P\) for \(a=0.1\) and \(a=0.01\) when \(k=0.75\) (b) For \(k=0.75\) and a given value of \(P\), how are the chances of escaping parasitism affected by increasing \(a\) ?

Step-by-Step Solution

Answer

A decrease in \(a\) increases the chances of escaping parasitism as it leads to higher \(f(P)\) values.

Step 1: Understand the Function

The function given is \(f(P) = \left(1 + \frac{aP}{k}\right)^{-k}\). This represents the fraction of hosts escaping parasitism as a function of \(P\), which is the number of parasites. The parameters \(a\) and \(k\) are constants where \(a\) is the searching efficiency of parasites, and \(k\) is a dispersion parameter.

Step 2: Set Parameters for Plotting

For the first part of the exercise, two scenarios are given: \(a = 0.1\) and \(a = 0.01\), with a common dispersion parameter \(k = 0.75\). We will graph \(f(P)\) over a sensible range of \(P\), for example, from \(P = 0\) to \(P = 10\).

Step 3: Calculate and Graph for a = 0.1

Substitute \(a = 0.1\) and \(k = 0.75\) into the function: \(f(P) = \left(1 + \frac{0.1P}{0.75}\right)^{-0.75}\). Compute \(f(P)\) for \(P\) values ranging from 0 to 10, and graph these values to visualize the function's behavior for \(a = 0.1\).

Step 4: Calculate and Graph for a = 0.01

Substitute \(a = 0.01\) into the function while keeping \(k = 0.75\): \(f(P) = \left(1 + \frac{0.01P}{0.75}\right)^{-0.75}\). Again, compute \(f(P)\) for the same range of \(P\) values, and create a second graph to compare the behavior with the scenario where \(a = 0.1\).

Step 5: Analyze the Effect of Changing a

Observe the graphical results. For both \(a = 0.1\) and \(a = 0.01\), as \(P\) increases, \(f(P)\) decreases. A smaller \(a\) (i.e., \(a = 0.01\)) results in higher values of \(f(P)\) compared to \(a = 0.1\) for the same \(P\). This indicates that lower searching efficiency \(a\) leads to a higher chance of escaping parasitism.

Step 6: Conclusion on the Effects of a

In conclusion, as \(a\) increases, the function \(f(P)\) decreases more rapidly with respect to \(P\), implying that higher searching efficiency \(a\) reduces the chances of escaping parasitism for the hosts.

Key Concepts

Parasitism

Parasitism is a relationship between two organisms where one organism, the parasite, benefits at the expense of the other, the host. In the context of the negative binomial model, parasitism is analyzed by examining how likely hosts are to escape being parasitized by a certain number of parasites. The function given, \( f(P) = \left(1 + \frac{aP}{k}\right)^{-k} \), allows us to calculate the likelihood of escape based on different values of \( P \), the number of parasites. By graphing \( f(P) \), we can understand the behavior of the escape probability as the number of parasites increases. One can observe that as \( P \) increases, \( f(P) \) decreases, indicating that with more parasites, the likelihood of escaping parasitism reduces. This function thus provides a framework for predicting and understanding the dynamics of parasitism under different conditions.

Searching Efficiency

Searching efficiency, denoted as \( a \), refers to how effectively a parasite can locate its host. High searching efficiency means that a parasite is very good at finding hosts, whereas low searching efficiency indicates the opposite. In our model, the parameter \( a \) plays a crucial role in determining \( f(P) \), the fraction of hosts escaping parasitism.By examining different \( a \) values, such as \( a = 0.1 \) and \( a = 0.01 \), we can discern their impact on the escape likelihood. When \( a \) equals 0.1, the parasites are more efficient in finding hosts, resulting in a steeper decrease in \( f(P) \) as \( P \) increases. In contrast, when \( a \) is only 0.01, the decrease is more gradual, indicating that lower searching efficiency allows more hosts to escape parasitism. This demonstrates the inverse relationship: higher searching efficiency typically leads to fewer hosts escaping parasitism.

Dispersion Parameter

The dispersion parameter \( k \) in the negative binomial model helps quantify the variance in the number of parasites per host. It's an important factor when assessing the distribution and clustering of parasites among hosts. A smaller \( k \) indicates a more aggregated distribution, meaning that most hosts have few parasites, but a few have many. Conversely, a larger \( k \) suggests a more uniform distribution.Within the exercise, \( k = 0.75 \) is used as the baseline for analysis. This specific value helps us gauge how clustered parasites might be in their interaction with hosts. Understanding \( k \) allows researchers and students to better predict how changes in other parameters such as searching efficiency \( a \) might impact the probability of escape, providing a comprehensive insight into the behavior of parasitism across different scenarios. Ultimately, the dispersion parameter is vital for interpreting results and making predictions about the distribution of parasites, impacting effective decision-making in ecological and zoological studies.