Not sure whether this is what you meant, but there is a difference between a situation when resources are abundant and the reproduction is an exponential function of the speed of reproduction, and when resources become scarce and reproduction is only one important parameter along with survival and interaction with competitors.
To continue with your example, imagine that Y has faster doubling rate than X (assuming abundant resources), but X can disassemble Y to create its own copy while Y can’t do the same to X. So there will be first a period when Y exponentially outgrows X, followed by a period where Y greadually disappears.
If you want to model this by matrices or something similar, you need to somehow include this aspect.
Also, the reality will be more complicated, because the values of X and Y and their interaction may depend on local environment. So it is possible that X eliminates Y in warm waters, but Y survives around the poles. Then it is possible that X evolves into intelligent species that causes global warming… okay, this is probably outside the scope of the original question.
Not sure whether this is what you meant, but there is a difference between a situation when resources are abundant and the reproduction is an exponential function of the speed of reproduction, and when resources become scarce and reproduction is only one important parameter along with survival and interaction with competitors.
To continue with your example, imagine that Y has faster doubling rate than X (assuming abundant resources), but X can disassemble Y to create its own copy while Y can’t do the same to X. So there will be first a period when Y exponentially outgrows X, followed by a period where Y greadually disappears.
If you want to model this by matrices or something similar, you need to somehow include this aspect.
Also, the reality will be more complicated, because the values of X and Y and their interaction may depend on local environment. So it is possible that X eliminates Y in warm waters, but Y survives around the poles. Then it is possible that X evolves into intelligent species that causes global warming… okay, this is probably outside the scope of the original question.