Consider the model $$\dot{x}=x[x(1-x)-y], \qquad \dot{y}=y(x-a)$$ where $x \geq0$ represents the population of prey and $y\geq0$ represents the population of a predator, with $a\geq0$ as a control parameter.
Describe in words what the various terms in the model may represent. Make a story about it.
This question has come up in a differential equations course and I have not done modelling, so this is completely out of my depth. I certainly don't understand how I can "make a story" about the equations.
Obviously the $\dot{x}$ and $\dot{y}$ terms represent the growth or decline of populations of prey and predator respectively. I assume the $-yx$ term in the first equation represents the population decline in prey due to being eaten by the predators, and similarly the $yx$ term in the second equation represents the growth in population of predator due to consumption of prey. If the $a$ term is a control parameter, I'm not really sure what I can say about it. Finally, the $x^2(1-x)$ term must represent some sort of growth for the prey independent of the existence of the predators, but I'm not quite sure what or how.
Is this interpretation along the right lines? Are there any other ways to interpret this appropriately? Do I have to make any sort of observations about the rate of growth/decline in each population respective to each other (ie. if the prey will die out before the predator etc)?
Your interpretation of the $x\,y$ terms is correct.
$a$ is a threshold. If the prey population is above $a$, then $\dot y>0$ and the predator population grows. If the prey population is below $a$, then $\dot y<0$ and the predator population decreases (not enough food.)
The term $x^2(1-x)$ has two components: