Question

There are two species, X and Y, with abundances \(x\) and \(y\), respectively. Species X has growth rate \(\alpha\), and species Y has growth rate \(\beta\). Assume that the sum of the species abundances is constant over time, i.e., \(x + y = 1\). Let \(x\) and \(y\) follow the rate equations: \[ \frac{dx}{dt} = \alpha x - \varphi x, \quad \frac{dy}{dt} = \beta y - \varphi y, \] where \(\varphi\) is the average species fitness. Which one of the following options correctly represents the expression for \(\varphi\)?

• A.
• B.
• C.
• D.

Hint:

To calculate average fitness in population models: 1. Use weighted contributions based on growth rates and proportions.
2. Verify that the total population remains constant, ensuring \(x + y = 1\).
3. Avoid introducing unnecessary terms or operations.

Answer 1

The Correct Option is B

Step 1: Understand the definition of average species fitness (\(\varphi\)). The average fitness, \(\varphi\), is determined by the weighted contributions of each species' growth rate to the total population, based on their relative abundances. Since \(x + y = 1\), the abundances \(x\) and \(y\) can be treated as the respective proportions of the two species in the population. Step 2: Derive the expression for \(\varphi\). The average fitness is given by the sum of the contributions of both species: \[ \varphi = \alpha x + \beta y, \] where: - \(\alpha x\) represents the contribution of species X (growth rate \(\alpha\) multiplied by its proportion \(x\)), and - \(\beta y\) represents the contribution of species Y (growth rate \(\beta\) multiplied by its proportion \(y\)). Step 3: Evaluate the options. Option (A): Incorrect. This expression introduces terms that do not correspond to the definition of average fitness. Option (B): Correct. This matches the derived expression for average fitness, \(\varphi = \alpha x + \beta y\). Option (C): Incorrect. The denominator \(x^2 + y^2\) is unnecessary and does not fit the definition of average fitness. Option (D): Incorrect. The reciprocal of \(\alpha x + \beta y\) is not relevant to the calculation of \(\varphi\).