mathematic in biology

periodic solutions of the volterra's model for the interaction between the biological species.
We first study the volterra's model for k=1 by the liapunov's method.
a family of periodic solutions is obtained in the form of convergent series easly computed in terms of elementary periodic functions, comparison with the results of the numerical integration gives a good coincidence of the two first approximations of the solutions in rather large neighbourhood of the point d'equilibrium.
For the 2k-species set we can prove the existence of k families of periodic selection depending on two parameters in the previous form, this existence is proved in nearly all the cases if there is a certain equilibrium between the disappearance of the species we also study the evolution of periodic solutions (k=1) if the coefficients of the equations are adiabetic variable and periodic.
the thorem of V.Arnold and adiabatic invariants is applied, it proves that for most initial data, the movement is quasi periodic.