THE LOTKA-VOLTERRA MODEL
Results

Influence of dt

In theory, when you plot y as a function of x, we are supposed to obtain closed lines as trajectories (solutions are periodic).
But if we plot the solution obtained with the set of parameters (0.04, 0.005, 0.1, 0.1, 150, 4, dt, nite), for dt=0.1 and nite=10000, we can observe that we don't have closed trajectories.

An hypothesis to explian this result is that dt is not small enough (moreover, the numerical scheme is not very precise). With the same set of parameters, we plot:

• y(x) for dt=0.01 and nite=35000
• x(t) and y(t) for dt=0.1, nite=3500 and dt=0.01, nite=35000

It is easy to see that the trajectory plotted for dt=0.01 is periodic whereas the other one isn't (despite the fact that the simulation time is the same). So it seems that in this case, it is the influence of the time step which leads to bad results. We can also see on the temporal plots that the two curves get separed little by little.
For the following simulations,  we will use a time step in order to have better results. As in this study, we only need qualitative results, it is not too im;portant, but in the case in which we would have needed very precise results, a better numerical scheme (Runge-Kutta for example) should hava been used, as well as simulations with a very small time step.

Global observations

We made some simulations with realistic values. The values of the parameters are not set for the sharks and sardines case (sorry, I didn't find those values!) but are characteristic of a natural system.
We choose:

a=0.04
b=0.0005
c=0.2
d=0.1
dt=0.01
nite=30000
We plotted on the same figures different simulations made with different initial conditions (y0=80 and x0=3000,2000,1000 and 80) and the same set of parameters (the one defined previously).

y as a function of x

x and y as functions of t

• We can notice that a dynamic trajectory starting at any point in phase-space will form a closed orbit whose amplitude is determined by the starting point (plotted in magenta). We can say that this model has no asymptotic stability, it doesn't converge to an attractor (it never forgets its initial conditions).
• Those simulations also confirm the theoritical study of the stability of the critical points. It is clear that (0,0) is an unstable critical point and we also find the fact that the second critical point is a center (the "center" of the trajectories). It is neither attractive, nor repulsive (marginal state for the stability) and the trajectories oscillate around it.
• The parameters we chose leads to those results. We can see that the more we decrease the initial value of the sardine population, the more it gets bigger and bigger, whereas the population of sharks remain quite stable (not stable, but at least in the same order of values). As a consequence, the period of the trajectories increase.