Trajectories with predator-prey population in $xy$-plane I've been solving a predator-prey population DE successfully with programming. But I don't understand exactly what a trajectory in $xy$-plane should look like or if I have done it correctly.
The predator-prey function looks like:
$$\begin{cases}
\dfrac{dx}{dt} = x(1-  y) \\
\dfrac{dy}{dt} = y(0.9 x  - 1)
\end{cases}$$
I have reformulated the DE to the dynamic system: $$C = \ln(y) + \ln(x) - y - 0.9x$$ I "replaced" $C$ to the function and plotted it in the $xy$-plane. But I'm not sure I have come up with a satifying answer to the question.
 A: Your solution seems fine.
The given system is
$$\begin{cases}
\frac{dx}{dt} = x - x y \\
\frac{dy}{dt} = 0.9 x y - y
\end{cases}$$
This is a Lotka-Volterra system with $(\alpha, \beta, \gamma, \delta) = (1, 1, 1, 0.9)$.
The implicit solution to the system has $C = \delta x - \gamma \ln x + \beta y - \alpha \ln y$; substituting the given values, this is $C = 0.9 x - \ln x + y - \ln y$, which is equivalent to what you obtained up to multiplication by $-1$ ($C$ is an arbitrary constant that captures the initial populations).
Edit: Clarification on the "trajectory"
Plotted below is a level curve for the solution in which the initial condition has $x = y = 5$ (if you substitute into the implicit solution, you will find that this corresponds to $C = 9.5 - \ln 25 \approx 6.281$). The prey population is represented by the $x$-coordinate, while the predator population is represented by the $y$-coordinate.

Different initial populations will give you different curves. However, they will all be similar, in that they consist of a closed loop. This loop is traversed counterclockwise. Starting at the bottom right, when prey is plentiful but predators are rare (but non-zero), there are plenty of food resources for predators, and their population will grow at the expense of the prey, following the curve towards the upper left. When the prey population drops below a threshold value, the predator population starts to decline due to starvation, and we move downwards along the curve. When the predator population drops below a certain threshold, the prey population is able to grow again, approximately exponentially until the predator population begins to grow, and we are back at where we started.
