Is there a simple mathematical way in which the maximum predator and prey populations of the Lotka-Volterra model can be calculated?

I'm currently working on a simulation that applies Euler's method to graph the Lotka-Volterra equations on a scalable graph grid -- unfortunately, due to the way in which I constructed my simulation, it is near impossible for me to dynamically adjust the graph axes. Is there a mathematical way in which these maxima can be calculated -- preferably a formula? A formula + explanation would be really helpful!

The form of the Lotka-Volterra equations that I am using in my program is: $$\frac{dx}{dt} = \alpha x - \beta x y,$$ $$\frac{dy}{dt} = \delta xy - \gamma y,$$where x represents the prey population density and y represents the predator population density (same as the one here).

Thanks!

• What data do you have besides the coefficients $\alpha$, etc.? An initial state $(x(t), y(t) = (x_0, t_0)$? Commented May 27, 2023 at 2:16

The quantity $$V = \delta x - \gamma \log x + \beta y - \alpha \log y$$ is constant, so it can be computed before solving the differential equation.
Also the maximum of $$x$$ is attained when $$y = \frac{\alpha}{\beta}$$, so the maximum value $$x^*$$ satisfies $$V = \delta x^* - \gamma \log x^* + \alpha - \alpha \log (\alpha/\beta)$$ and you can compute $$x^*$$ from that equation numerically.
Similarly the maximum value of $$y$$ occurs when $$x = \frac{\gamma}{\delta}$$ and you can compute the maximum of $$y$$.