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
Aphids feed on both alfalfa and clover plants. A researcher collected and reared different genotypes of aphids from separate alfalfa and clover fields. He then measured the fecundity of both aphid groups when fed on each of the two host plants. The figure summarizes the performance of aphid groups originating from alfalfa (dashed) or clover (solid).
Which one or more of the following situations does the figure depict?- Which one or more of the following conditions is/are necessary for the evolution of increased nectar production in an insect-pollinated plant via natural selection?
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?A researcher estimates the relationship between reproductive success ($N$, number of offspring) and horn length ($H$, in cm) in a wild goat as
$N = 40 – 2.2H + 0.04H^2$
Horn length typically varies from 10 cm to 50 cm in this species. Which one of the following graphs correctly represents this relationship?
- A new food requesting behaviour has been observed in bonnet macaques in Bandipur National Park. The macaques extend their hand and make a cooing sound only towards humans, which effectively results in food given to them. If this behaviour is to increase in frequency in the population over time by the process of natural selection, which one or more of the options below is/are necessary condition(s)?
