Problem 37 - Calculus for Biology and Medicine

Question

Plants employ two basic reproductive strategies: polycarpy, in which reproduction occurs repeatedly during the lifetime of the organism, and monocarpy, in which the plant flowers and produces seeds only once before dying. (Bamboo, for instance, is a monocarpic plant.) Iwasa et al. (1995) argued that the best strategy for a plant depends on how reproductive success (that is, number of progeny that the plant produces) varies with the investment (that is amount of resource that the plant uses up to reproduce) The optimal strategy is polycarpy if reproductive success increases with the investment at a decreasing rate, [or] monocarpy if the reproductive success increases at an increasing rate. (a) Sketch the graph of reproductive success as a function of reproductive investment for the cases of (i) polycarpy and (ii) monocarpy. (b) Given that the second derivative describes whether a curve bends upward or downward, explain the preceding quote in terms of the second derivative of the reproductive success function.

Step-by-Step Solution

Answer

Polycarpy shows diminishing returns with a negative second derivative, while monocarpy shows increasing returns with a positive second derivative.

Step 1: Understanding the Reproductive Strategies

In polycarpy, plants reproduce multiple times throughout their lifespan. Conversely, monocarpy involves plants flowering and producing seeds once before dying. Reproductive success refers to the number of offspring a plant produces as a result of the resources it invests in reproduction.

Step 2: Analyzing the Graphical Representation

For each strategy, we need to sketch graphs representing how reproductive success varies with reproductive investment. For polycarpy, the graph should show reproductive success increasing at a decreasing rate. Meanwhile, for monocarpy, the graph should display reproductive success increasing at an increasing rate.

Step 3: Sketching the Polycarpy Graph

For polycarpy, sketch a concave down curve where the slope diminishes as investment increases. This represents diminishing returns on investment where each additional unit of investment yields less additional reproductive success.

Step 4: Sketching the Monocarpy Graph

For monocarpy, draw a concave up curve where the slope becomes steeper with increased investment. This reflects increasing returns on investment where each additional unit yields more reproductive success.

Step 5: Explaining in Terms of Second Derivative

The second derivative of the reproductive success function indicates the curvature. If the graph is concave down (as in polycarpy), the second derivative is negative, indicating diminishing returns. If the graph is concave up (as in monocarpy), the second derivative is positive, showing increasing returns.

Key Concepts

Polycarpy and Monocarpy

Plants have developed two main reproductive strategies to ensure their survival and proliferation: polycarpy and monocarpy. Polycarpy is a strategy where a plant reproduces multiple times during its life. This means that throughout various growing seasons, the plant can produce seeds and flowers numerous times.
In contrast, monocarpy can be seen as a more abbreviated strategy. Here, the plant flowers and produces seeds just once before dying. Bamboo is a classic example of a monocarpic plant. This strategy might seem risky since it banks all of the plant's reproductive success on a single event, but it's adapted perfectly to certain environmental conditions.
Choosing between these strategies depends on the conditions in which the plant finds itself. The strategy adopted will depend largely on how reproductive success, in terms of progeny produced, varies with the resources the plant invests in reproduction.

Reproductive Success Function

The reproductive success function is a way to describe how well a plant's reproductive strategy works based on its investment in reproduction. Investment refers to the resources a plant uses to reproduce, such as nutrients and energy.
For both polycarpy and monocarpy, this function looks at how reproductive success changes with increasing investment in reproductive resources.

• Polycarpy: The success function for polycarpic plants shows diminishing returns on investment. Initially, as the plant invests more resources into reproduction, the number of progeny increases significantly. However, over time the additional benefits start to taper off.


• Monocarpy: On the other hand, monocarpic plants depict increasing returns on investment in reproductive success. This means the more resources they invest, the rapidly increasing the number of offspring, maximizing the output in a one-time reproductive event.

Understanding this function helps in analyzing which investment levels benefit reproductive success the most for both strategies.

Second Derivative Analysis

The second derivative offers insight into the curvature of the reproductive success function. It helps answer if reproductive success increases at a decreasing or increasing rate.

• If the second derivative is negative, it suggests a concave down curve—representative of polycarpy. This indicates diminishing returns with additional resource investment, as each unit results in less success.


• Conversely, a positive second derivative implies a concave up curve, characteristic of monocarpy. In this case, the reproductive success accelerates with increased investment, reflecting more offspring per unit of resources.

By using second derivative analysis, we can visualize and understand the efficiency of resource investments in the reproductive strategies of plants, assisting them to optimize their reproductive success based on their environment.