So I have a system of equations:$$x'(t)=x(t)(4-2x(t)-y(t))\\y'(t)=y(t)(3-x(t)-y(t)) $$
What I understand so far is: if we have x being the population of prey and y is the population of predators. x grows at a rate proportional to x, but decays at a rate proportional to the interaction (xy). y increases at a rate proportional to xy and decreases at a rate due to the size of y (more predators = less food, etc.).
Given that there are no initial values, how can I describe this population system governed by the above equations?
You can define the functions $$ \begin{cases} f_1(x,y)=x(4-2x-y)\\ f_2(x,y)=y(3-x-y) \end{cases} $$ and study them (fixpoints (common zeros) and signs studies and so on). I leave also a plot of the vector field $(x,y)\mapsto (f_1(x,y),f_2(x,y))$. From it, you can see how different solutions will evolve in time. The point $(x,y)=(1,2)$ is marked in red, why?