Problem 49 - Calculus for Biology and Medicine

Question

The functional responses of some predators are sigmoidal; that is, the number of prey attacked per predator as a function of prey density has a sigmoidal shape. If we denote the prey density by \(N\), the predator density by \(P\), the time available for searching for prey by \(T\), and the handling time of each prey item per predator by \(T_{h}\), then the number of prey encounters per predator as a function of \(N, T\), and \(T_{h}\) can be expressed as $$ f\left(N, T, T_{h}\right)=\frac{b^{2} N^{2} T}{1+c N+b T_{h} N^{2}} $$ where \(b\) and \(c\) are positive constants. (a) Investigate how an increase in the prey density \(N\) affects the function \(f\left(N, T, T_{h}\right)\) (b) Investigate how an increase in the time \(T\) available for search affects the function \(f\left(N, T, T_{h}\right)\). (c) Investigate how an increase in the handling time \(T_{h}\) affects the function \(f\left(N, T, T_{h}\right)\) (d) Graph \(f\left(N, T, T_{h}\right)\) as a function of \(N\) when \(T=2.4\) hours, \(T_{h}=0.2\) hours, \(b=0.8\), and \(c=0.5\)

Step-by-Step Solution

Answer

(a) Initially increases and then decreases. (b) Increased search time increases encounters. (c) Increased handling time reduces encounters. (d) Displays a sigmoidal curve.

Step 1: Analyze Effect of Prey Density (N)

Investigate the derivative of the function with respect to prey density \(N\). Compute \(\frac{\partial }{\partial N} f(N, T, T_h)\). Using the quotient rule, we find:\[\frac{\partial f}{\partial N} = \frac{2b^2NT(1+cN+bT_hN^2) - b^2N^2T(2c+bT_h2N)}{(1+cN+bT_hN^2)^2}\]The function \(f\) will increase initially as \(N\) increases but will eventually decrease when the denominator grows faster than the numerator, leading to a sigmoidal shape.

Step 2: Explore Effect of Search Time (T)

Find how increasing the available search time \(T\) affects the function by calculating \(\frac{\partial }{\partial T} f(N, T, T_h)\). The derivative is:\[\frac{\partial f}{\partial T} = \frac{b^2N^2}{1+cN+bT_hN^2}\]Since \(\frac{\partial f}{\partial T} \) is positive and independent of \(T\), an increase in \(T\) leads to an increase in \(f\). Thus, function response is positively affected by increasing search time.

Step 3: Evaluate Effect of Handling Time (T_h)

Investigate the derivative concerning the handling time \(T_h\) by computing \(\frac{\partial }{\partial T_h} f(N, T, T_h)\). The derivative is:\[\frac{\partial f}{\partial T_h} = \frac{-b^3N^4T}{(1+cN+bT_hN^2)^2}\]Since this derivative is negative, increasing \(T_h\) decreases the value of \(f\), indicating that longer handling times reduce the number of prey encounters per predator.

Step 4: Graph Function of Prey Density (N)

Graph the function \(f(N, 2.4, 0.2)\) using given values: \(T=2.4\), \(T_h=0.2\), \(b=0.8\), and \(c=0.5\). The expression becomes:\[ f(N, 2.4, 0.2) = \frac{0.8^2N^2 \cdot 2.4}{1+0.5N+0.8\cdot 0.2N^2}\]Create a graph for \(N\) in a reasonable range. The graph will show a sigmoidal curve; initially rising with \(N\) and plateauing as \(N\) increases further.

Key Concepts

Prey Density

In the context of a predator-prey relationship, prey density refers to the number of prey individuals available in a given area. It is denoted by \( N \) and plays a crucial role in the dynamics between predators and prey. As prey density increases, predators have more opportunities to encounter and capture prey. This relationship can be represented using a complex function that captures the dynamics based on various parameters, such as search time and handling time.

The provided function \( f(N, T, T_h) = \frac{b^{2} N^{2} T}{1+c N+b T_{h} N^{2}} \) highlights how prey density \( N \) influences predator behavior. To understand how changes in prey density affect this function, we use the calculus-derived derivative \( \frac{\partial f}{\partial N} \). This derivative shows us that the function initially increases with \( N \), as more prey is beneficial to predators, but at some point, it begins to decrease because the resources (such as time) become limiting factors. This creates a sigmoidal or "S"-shaped curve, indicating that benefits rise quickly, then level off, and can even decline if overcrowding occurs.

• Initially, function \( f \) benefits from increased \( N \) as encounters rise.
• After a threshold, increased prey density may not lead to more prey captures.
• The shape of the curve is influenced significantly by constants \( b \) and \( c \).

Derivative Calculation

Derivative calculation is a method used to find the rate at which a function changes as its input changes. It's a fundamental concept in calculus that helps analyze complex functions. In predator-prey models, derivatives help understand how variables like prey density \( N \), search time \( T \), and handling time \( T_h \) affect the number of prey encounters.

When calculating the derivative of the function \( f \) concerning a specific variable, like prey density \( N \), you're assessing how a small change in \( N \) alters \( f \). This is crucial because it shows sensitivity in predator-prey dynamics. For example, the calculus-derived derivative \( \frac{\partial f}{\partial N} = \frac{2b^{2}NT(1+cN+bT_hN^2) - b^{2}N^{2}T(2c+bT_h2N)}{(1+cN+bT_hN^2)^2} \) provides insights:

• The numerator represents the potential benefits of adding more prey.
• The denominator grows quicker than the numerator over time, showing limits on benefits.
• This highlights how derivatives can map responses to different ecological scenarios.

Understanding derivatives' roles in these models is essential for predicting how ecosystems might react to changes, helping ecologists make more accurate predictions.

Predator-Prey Model

The predator-prey model is an essential framework in ecology used to describe the interactions between predators and their prey. This model provides a quantitative way to explore these complex interactions, often depicted as equations that describe how predator and prey dynamics change over time.

In our specific context, the model is represented by the function \( f(N, T, T_h) \), which calculates prey encounters as a function of prey density, search time \( T \), and handling time \( T_h \). By utilizing derivatives, the model helps predict how changes in these variables influence the predator-prey relationship:

• Increasing \( N \) leads to an initial rise in encounters until resource limitations create a plateau (a sigmoidal response).
• Additional \( T \) (search time) typically results in more encounters as time allows for more prey searching.
• To contrast, increasing \( T_h \) (handling time) reduces encounters, as more time is spent dealing with each caught prey.

Using such a model, ecologists can simulate different scenarios within ecosystems, predicting outcomes like population stability or oscillations between predator and prey numbers. The predator-prey model is invaluable for understanding real-world dynamics, guiding conservation efforts and ecological research.