# Interpreting differential equation models

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)?

## 1 Answer

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:

1. $x^2$ represents the growth of the prey population in absence of predators if it has free acces to resources.
2. $-x^3$ represents the effects of overpopulation.
3. $1$ is again a threshold. If the population grouths above $1$, then it declines due to overpopulation. If it is below $1$, it grows.
• are you basing your interpretation on the solutions of these equations?
– abel
Commented Mar 9, 2015 at 11:31
• No, I do not know the solutions. My interpretation comes from the equations. Commented Mar 9, 2015 at 11:44
• The term $-ay$ may act as a threshold, post hoc, but its interpretation is definitely different than what you say: actually, $-a$ is the intrinsic growth rate of the $y$-population, that is, the Malthusian parameter describing the extinction of the $y$-population (predators) in the absence of the $x$-population (preys).
– Did
Commented Jun 27, 2015 at 9:06
• Thanks for the explanation. i needed that, too ;) Commented Jul 27, 2017 at 13:42
• Oh one question: I plucked in several combinations of values for predators and prey... e.g. 2 prey, 1 pred., 10 prey, 1 pred. and so on... there is no way in this equation that population will grow, because x² will always be smaller than x³ for whole numbers in x*(x(1-x)). so is this the fault of the euquation? Thanks for answering! :) Commented Jul 27, 2017 at 13:52