3-species Lotka-Volterra food chain in Hamiltonian formalism

[mastrofoffi]

Hello, I have been assigned to write a report on a topic of my choice for my Computational Physics class, and I chose to focus on the symplectic integration of Hamiltonian systems, in particular the Lotka-Volterra model.
A 3-species model($\gamma$ eats $\beta$, $\beta$ eats $\alpha$) is not, unlike the classic Prey-Predator model, directly representable by an Hamiltonian since we have 3 coordinates, so I splitted my set of differential equations in a system of 2 coupled Hamiltonians, so from this
$$
\begin{cases}
\dfrac{dx}{dt} = x(A - By)\\
\dfrac{dy}{dt} = y(-C + Dx - Ez)\\
\dfrac{dz}{dt} = z(-F + Gy)\\
\end{cases}
$$
I obtained this:
\begin{cases}
H_1 = Be^y - Ay - Cx + De^x - Exe^z\\
H_2 = Ge^y - Fy + Cz + Ee^z - Dze^x\\
\end{cases}

where $x=ln\alpha, y=ln\beta, z=ln\gamma$; in $H_1$ i consider $z$ to be instantaneously constant, and in $H_2$ I do the same for $x$; all the interaction parameters are positive.

So in $H_1$ my hamiltonian variables are p=x q=y, while in the other they are p=y q=z.

I used a symplectic Euler method to integrate these equations(at every instant I first integrate H1 and then H2 with the newly obtained values for x and y; I also tried to invert the order of integration and there are no significative changes) and I manage to get some fairly nice orbits in 3d space

Here i used parameters A=B=C=D=E=F=G=1 and same starting conditions for every species (1, 2, 3, 4); the orbit gets larger as the starting number of individuals increases.

Now I tried to interpret these closed orbits as oscillations around a stable equilibrium point(as in the 2 species prey-predator case), and I made many different assumptions to try to find where this equilibrium points were located, but had no luck.
First of all I assumed that motions sharing the same "total Hamiltonian"(i know that the sum of these two hamiltonians is not the hamiltonian of the whole system, but I tried anyway) $H = H1+H2$ should be lying on the same surface, since H is a first integral of the motion, but to now I couldn't manage to verify nor to exclude this (I also tried to ask my Analysis professor, but she said she couldn't really help me with this, even though she said the assumption might be reasonable).
I also tried a more geometrical approach, seeing as the orbits seem to open up in some conical shape around a vertex, which should be a stable eq point, but had no luck with this either.
Again, solving the set of diff. equations, I found out that all the equilibrium points for this system should be on a straight line over plane y=1: $z = x - 1$ for my set of parameters
Even with this I could not associate any orbit with a specific point on the line.

I feel like I'm either lacking some fundamental mathematical knowledge to approach the problem of finding the barycenters of these orbits or I'm missing something very important.
I actually believe there may be some fallacies in my reasoning, especially in the Hamiltonian approach I used, which, even being correct(and I'm not even sure about this) from a purely computational standpoint, feels a bit weird from a physical point of view.

Any suggestion or help will be greatly appreciated

 

[eljose79]

.

Thank you for sharing your interesting research topic with us. As a fellow scientist, I would like to offer some suggestions and insights that may help you in your report on the symplectic integration of Hamiltonian systems, specifically in the context of the Lotka-Volterra model.

Firstly, I would like to commend you on your approach of splitting the set of differential equations into two coupled Hamiltonians for the 3-species model. This is a valid and effective way to handle the complexity of the model and make it more amenable to Hamiltonian analysis. However, I would like to point out that the Lotka-Volterra model is not strictly a Hamiltonian system, as it does not satisfy the necessary conditions for Hamiltonian dynamics. This is because the Hamiltonian for the system is not a constant of motion and does not satisfy the Poisson bracket condition. This is why you have encountered difficulties in finding the equilibrium points and interpreting the orbits as oscillations around them.

That being said, your approach of using the total Hamiltonian H = H1 + H2 as a first integral of motion is a valid one, and it is worth exploring further. In fact, it can be shown that the level sets of the total Hamiltonian correspond to the orbits of the system, and the equilibrium points can be found at the intersections of these level sets. However, as you have rightly pointed out, this approach may not be physically meaningful since the total Hamiltonian is not a constant of motion. This is where the concept of "adiabatic invariants" comes into play. Adiabatic invariants are quantities that are approximately conserved when the system evolves slowly, and they can be used to approximate the motion of the system. In your case, the total Hamiltonian can be considered as an adiabatic invariant, and the equilibrium points can be approximated by the intersections of the level sets of the total Hamiltonian. This approach has been used in the study of non-Hamiltonian systems, such as the Lotka-Volterra model, and has yielded good results.

Another approach that you can consider is the use of geometric methods, such as the Poincaré map, to analyze the behavior of the system. The Poincaré map is a powerful tool in the study of dynamical systems and can provide insights into the stability and bifurcations of the system. In your case, you can construct a P