I've got the following simple system of ODEs:
x': x*(1-x-y)
y': y*(1-a* y-b* x).
Plotting the system in a x' y' plot is not fully satisfying to me as what I want to achieve is that the sum of x and y, has a constant value, as I am simulating a space which is fully occupied by the species x and y, e.g. with no empty space in between. Thus, if my total space has a value of 10, and y gets a value of 8, x must automatically get a value of 2.
So, for example I could substitute x for 10-y and get:
y': y*(1-a*y - b *(10-y)).
And then I could look at the y versus t plot and by that I could also derive the x-value as x = 10 - y.
As I am quite new to ODEs, my one and only question actually is, is this a correct way of doing or is there another way? To me it sounds logical on the one hand, but on the other hand, I will obviously lose the ODE for x', as this ODE will now simply become a 'filler-up' to get to the maximum value. Or maybe there is also a way to look at the maximum value by conserving the system? Thanks in advance!