Competition Model?

Question

imagine a small pond that is mature enough to support wild life. we desire to stock the pond with game fish, say trout ad bass. let (x(t) denote the population f the trout at any time t, and let y(t) denote the bass population.

Assumptions: initially assume that the environment can support an unlimited number of trout and bass.

also assume that trout and bass compete for living sace and a common food supply.

a) obtain a competition model for this problem

b) perfom a graphical analysis of your model

c) how would you modify the model to have a more realistic one

Answer 1

I don't understand the first part of this question, in which you seem to be asked to assume that there is no environmental limitation on the number of trout and bass in the pond, yet the trout and bass are competing for living space and food.  If there are really no limitations on resources, then it makes no sense to me to model either inter- or intra-species competion.  The populations of both species will simply grow exponentially with time (assuming there are at least two fish of the opposite s*x initially!), at a rate given by the difference between each specie's birth and death rates.

A more realistic model would assume competion between members of the same species, as well as competition between the different species.  This leads to the well-known Lotka-Volterra equations that describe this situation.  I won't reproduce the derivation here, but will point you to a couple of good sources that should allow you to answer questions a-c above.  If you search for "Lotka-Volterra" using your favorite search engine, you'll come up with many more pages that may be of help.