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).


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

1 Answer 1


Read the Wikipedia article carefully.

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 $y$.


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