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:


 
 

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