How do you overlay trajectory graphs on vector fields?

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Hi, I'm doing a programming problem where I have to graph a vector field, then overlay trajectory graphs with different initial values on it.

So far this is what I have:

var('R, S')
plot_vector_field([3*R - R^2 - 2*R*S, 2*S - S^2 - R*S], (R,-50, 50), (S, -50, 50))
R = sol[:,0]
S = sol[:,1]
list_plot(zip(R,S), plotjoined=True)

The last line always produces a separate graph. I've tried using the "+" operator to "add" it to the vector field graph, but that gives me an error instead. I also don't know how to repeat the process (i.e. overlay additional trajectories) without having to make a new graph for each. Truth be told I'm only a beginning coder and have some trouble synthesizing different coding aspects together.

Any help would be greatly appreciated!