For my question, I'm given a situation where I let $x(t)$ and $y(t)$ represent the population sizes of two discrete species. The scenario I'm attempting to model goes something like this:
When both populations are isolated from each other, the groups grow, but there are limited resources. Also, the relationship between these two species is cooperative.
With that description in mind, my task is to come up with with a system of differential equations that models this. I've worked with species that don't benefit from each other through interactions, like foxes and rabbits, but this situation is a bit different. If I'm interpreting this correctly, the species will both benefit through interactions and during isolation, correct? I have an idea of what my system of differential equations should look like. So, I have
$$\frac{dy}{dt} = \fbox{numbers involving variables x and y} + kxy$$ $$\frac{dx}{dt} = \fbox{numbers involving variables of x and y} + kxy$$
Here, I let $k \in \mathbb{R^+}$ except the number $0$. In terms of the species that I can use to model my system, what about clown fish and sea anemone? Thoughts and inquiries are welcome here.
This is a spin on the classical Lotka - Volterra system.
My first impulse would be to try a model such as the following: $$ x'=x(a+by) \\ y'=y(c+dx) $$
Where $a,b,c,d$ are positive terms.
$a$ and $c$ represent the reproduction rate when the populations are isolated, while $b$ and $d$ represent the cooperation between the two species (which is proportional to the number of interactions, itself proportional to the population of both species, thus the $b$,$d$ terms must go together with the product of the variables $xy$ ).
$b$, $d$ could be the same as in your question, though it does not really simplify the system too much and in general the benefits do not have to be symmetric for both species.
For the species choice, clown fish and anemone look good :)