- Does the Lotka-Volterra model predict stable or cyclic population variation?
- What determines the amplitude of the cycles predicted by the Lotka-Volterra model?
The Lotka–Volterra equations, also known as the predator–prey equations,
$$
\begin{align}
\frac{du}{dt}&=(a_1-b_1v)u\\
\frac{dv}{dt}&=(-a_2+b_2u)v
\end{align}
$$
The parameters of the model:
$a_1$− growth rate of the prey
$a_2$− death rate of the predator
$b_1$− efficiency of the predator’s ability to capture prey
$b_2$− growth rate of the predator.
And $u(t)$ and $v(t)$ denote the density of the prey and predator populations respectively at time $t$.
I didn't understand what they mean by "predict stable or cycle population variation"? Is it related with Poincaré-Bendixson theorem.
It will be a great help if anyone cite the way to determines the amplitude of the cycles predicted by the Lotka-Volterra model or a dynamical system.
Thanks in advance.
This system has two equilibrium points. The zero equilibrium point $(0,0)$ and the positive equilibrium point $(a_2/b_2,a_1/b_1)$. Trajectories of the system may converge to those points, be repelled from them, there can also be an isolated periodic trajectory (aka limit cycle), or the trajectories can blow-up to infinity. Analyzing those properties will give you the answer to your first question. The local behavior of the system about those equilibrium points can be established by linearizing the system about the equilibrium point of interest and studying the stability of the resulting linear system. This is essentially reducing to an eigenvalue problem.
It is in general difficult to find an explicit solution for the limit cycle. It is also difficult, in general, to obtain an explicit characterization of the amplitude of the limit cycles. What is important to know is that a limit-cycle is a one-dimensional structure. For this specific system, I would recommend to have a look at this paper https://arxiv.org/pdf/2110.11557.pdf.