In this paper, we consider certain analytical features of a stochastic model that can explain competition among species and simultaneous predation on the competing species from a geometric perspective. This allows us to build a systematic description of models admitting singular Lagrangians. The model equations are shown to admit a Jacobi Last Multiplier which allows us to construct an appropriate Lagrangian. Due to the singular nature of the Lagrangian, the Hamiltonian formalism may be shown to exist in a submanifold of the cotangent space under certain minimal regularity conditions. In this communication, the Hamiltonian description of the model is constructed via the introduction of Dirac brackets and explicit results for the “Kill the winner” model and its reductions are presented.

在本文中，我们考虑了随机模型的某些分析特征，该模型可以从几何角度解释物种之间的竞争和对竞争物种的同时捕食。这使我们能够对承认奇异拉格朗日量的模型进行系统描述。模型方程显示允许 Jacobi Last Multiplier，它允许我们构造适当的 Lagrangian。由于拉格朗日的奇异性质，哈密顿形式主义可能被证明存在于某些最小规则条件下的余切空间的子流形中。在本次交流中，模型的哈密顿描述是通过引入狄拉克括号构建的，并给出了“杀死赢家”模型的显式结果及其简化。