Singular Lagrangians and the dynamics of the "Kill the Winner" model and its descendants
We consider certain analytical features of the "Kill the Winner" model which is a stochastic model that can explain among other things competition among species and simultaneous predation on the competing species. The model equations are shown to admit a Jacobi Last Multiplier which in turns allows for the construction of a Lagrangian. The Lagrangian is of singular nature so that construction of the Hamiltonian via a Legendre transformation is not possible. A Hamiltonian description of the model therefore requires the introduction of Dirac brackets. Explicit results are presented for the model and its reductions.