Just a thought... defining "competition"? Let's think about galaxies and animals.  At first, they seem completely different. 
But their behavior seems to be governed by (or at least arise from) the same rules.
Think about competition.  Animals and galaxies can be argued to both compete in very life-like ways...
Image we have a chemist who makes a new kind of blowing bubble for children.  The bubble always floats up.  When the bubbles collide, they either become one bigger bubble, or, if they are a similar size, they both pop.
Now let’s say we have many machines to make these bubbles.  They are lined up in a random array, (with varying spaces between the machines.)
The bubbles, as they rise, will begin to "compete" for the limited resources: space, and the material that makes up the bubble.  In other words some bubbles will gradually become larger and more powerful, while other bubbles will cease to exist.
Now, lets’ say that the chemist has found a way to put a pattern of colors on all the bubbles.  When the bubbles are blown, each one is given a unique pattern. Instead of popping when two bubbles of similar size encounter each other, they change their pattern and become one big bubble.  When a much bigger bubble encounters a much smaller bubble, the smaller bubble simply become part of - and part of the pattern - on the larger bubble.
Viola!  The bubbles can now eat each other!
Or, they can just continue to co-exist if nothing changes size. 
This can serve as an analogy for a number of natural processes I can think of off the top of my head:


*

*Galaxies evolving:  Think of each galaxy as an organization of stars. Galaxies of the same size that run into each other will become disturbed and in the end, form a new organization of stars.  (Two bubbles of same size running into each other.)  In the end, the bubbles will evolve into ‘many’
A much smaller galaxy that runs into a much larger galaxy will simply cease to exist and donate its stars to the much larger organization (Large bubble running into small bubble.)

*Animals living: Animals in the same environment (ie, small animals) will evolve together.  Eventually, many species will change over time to adapt into other species.  For example, the t-rex (a large bubble) will eat smaller bubbles (say, a plant eating dinosaur – which in turn is made out of yet smaller bubbles.  The higher an individual bubble is in the air, the higher it is on the trophic cycle – in terms of how much energy it uses.  Cockroaches, would be countless tiny bubbles – if the big unstable bubbles all get popped by the wind, many of the small ones may live… just like real mass extinctions!)
Or if two animals of the same size encounter each other, one or the other might be destroyed, as they try to fight to defend themselves…
So, what is the difference between the animals eating each other and the galaxies? 
At first, it appears big.  Galaxies gradually just coalesce into bigger and bigger galaxies; animals die and/or evolve, being eaten to become part of bigger animals.  But in reality, the difference is much simpler.
  The animals have a mechanism for keeping smaller bubbles away from larger bubbles or, in other words small bubbles are constantly being made  – the small creatures regenerate themselves when they are eaten!
The galaxies do what they do  - just becoming bigger and bigger –because no more galaxies are being made!  (Also.. perhaps it’s possible they could not ever have had a mechanism for physically avoiding each other…)
So… the only real difference in how galaxies and (statically grouped) animals eating each other is the fact that more energy (from the sun) is being put into the system for the animals, but no more potential energy (ie, hydrogen) is being introduced to the system of the universe.
Now, these similarities are too great to ignore.   
There must be some mathematical way of describing competition – which seems to me there must be some mathematical system for describing both.
I read somewhere (think: the works of Stephen Wolfram, which I’ve never actually inspected directly myself) that complex behavior can arise from a set of very simple laws … is that what this is all about?  The behavior that arises seems very similar, despite the totally different physical nature of galaxies and animals.
(PS:  Einstein wandered that “The most incomprehensible thing about the universe is that it can be comprehended at all…”  If we are just beings evolved to negotiate these simple laws, and the laws governing galaxies and animals are all the same, that would explain why we could comprehend some of it!  Imagine if we were bugs, evolved to live by jumping from bubble to bubble.  We might find it amazing we could somewhat predict where they were going – but could because we would’ve been evolved to jump between them!)
 A: Differential equations and difference equations are used to model population dynamics (see here), which seems to be what you are asking about. Now that is not a definition of "competition", per se, but I think your idea of "competition", as described in your question, is rather too vague to admit a mathematical description. That's not a bad thing, necessarily; it's similar to the notion of a "shape", where there are numerous different areas of math studying various ways of rigorizing this notion, focusing on different aspects of it, no one of them capturing all of the natural-language connotations and ideas of "shape".
By the way, I would advise you read this article before reading anything more by Wolfram.
A: There are many sound, and competing, mathematical approach to competition in terms of species interaction. Lotka-Volterra model is one of these, as there are evolutionary games models, Arditi-Ginzburg ratio dependent model, stochastic models for two species or more (see), ...
I don't know any model truely developing your "bubble" idea, probably because it can't tale into account many important phenomena as speciation (a bubble splitting in two?), population dynamics (bubble increasing and decreasing periodically?), non spacially homogeneous interaction, cooperative behaviour, mimetism, varying efforts, foodweb emergence, ..., while the mathematical models I cited give a really fitting model of the nature
It's maybe a poetic analogy, but seems not a mathematical interesting one.
P.S. Probably many critics cam be done from the astronomical point of view, but I leave them to the field experts...
