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:
$$\frac{dx}{dt} = \alpha x - \varphi x,$$
$$\frac{dy}{dt} = \beta y - \varphi y,$$ where $\varphi$ is the average species fitness.
Which one of the following options correctly represents the expression for $\varphi$?
Species Fitness φ Derivation from Rate Equations
The problem provides two species, X and Y, with abundances $x$ and $y$ respectively, and growth rates $\alpha$ and $\beta$. The sum of their abundances is constant, $x + y = 1$. The rate equations governing their population dynamics are given as:
$ \frac{dx}{dt} = \alpha x - \varphi x $
$ \frac{dy}{dt} = \beta y - \varphi y $
Here, $\varphi$ represents the average species fitness. Our goal is to find the expression for $\varphi$.
Deriving Average Species Fitness φ
The rate equations can be rewritten by factoring out the abundance:
$ \frac{dx}{dt} = (\alpha - \varphi) x $
$ \frac{dy}{dt} = (\beta - \varphi) y $
The total change in abundance over time is the sum of the individual changes:
$ \frac{d(x+y)}{dt} = \frac{dx}{dt} + \frac{dy}{dt} $
Given that the sum of abundances is constant ($x+y=1$), its derivative with respect to time is zero:
$ \frac{d(x+y)}{dt} = 0 $
Substitute the rate equations into the total change equation:
$ 0 = (\alpha x - \varphi x) + (\beta y - \varphi y) $
Rearrange the terms to group $\varphi$:
$ 0 = \alpha x + \beta y - \varphi x - \varphi y $
$ 0 = \alpha x + \beta y - \varphi (x + y) $
Since $x + y = 1$, we substitute this value:
$ 0 = \alpha x + \beta y - \varphi (1) $
Now, solve for $\varphi$:
$ \varphi = \alpha x + \beta y $
This expression correctly represents the average species fitness based on the given rate equations and the constraint $x+y=1$. This matches Option 2.
Important Questions from Natural selection
- Which one or more of the following is/are NOT essential for evolution by natural selection to take place in a population?
- Two bacterial variants are growing together in the same flask. At any relative frequency of the two variants, the population growth rate of the rarer variant is higher. The above is an example of ________.
The figure illustrates the soil zinc tolerance of the grass species Anthoxanthum along a transect from inside a mine to the middle of a pasture outside the mine.
Which one or more of the following processes explain(s) the observed pattern of zinc tolerance in this grass species?From an original population \(P\) of a butterfly species, two experimental populations \(X\) and \(Y\) were established. In \(X\), males and females were maintained in standard conditions, and females were allowed to mate and lay eggs. Only eggs from females laying small clutches (i.e., \(S\) eggs or fewer) were allowed to hatch and the rest were not utilized. In \(Y\), males and females were maintained in standard conditions and females were allowed to mate and lay eggs. From each female, \(S\) eggs were randomly selected and allowed to hatch, and the rest were not utilized. After 20 generations of these experimental conditions, relative to the original population \(P\), and assuming that clutch size is under genetic control, we expect clutch size to be ______________________in \(X\) and ___________________ in \(Y\).
- In which of the following four plots, showing reproductive fitness versus a trait, is the strength of selection MAXIMUM?
