We investigate the existence of a two-dimensional invariant manifold that attracts all nonzero orbits in 3 species Lotka-Volterra systems with identical linear growth rates. This manifold, which we call the balance simplex, is the common boundary of the basin of repulsion of the origin and the basin of repulsion of infinity. The balance simplex is linked to ecological models where there is ‘growth when rare’ and competition for finite resources. By including alternative food sources for predators we cater for predator-prey type models. In the case that the model is competitive, the balance simplex coincides with the carrying simplex which is an unordered manifold (no two points may be ordered componentwise), but for non-competitive models the balance simplex need not be unordered. The balance simplex of our models contains all limit sets and is the graph of a piecewise analytic function over the unit probability simplex.

我们研究了二维不变流形的存在，该流形吸引了具有相同线性增长率的3种Lotka-Volterra系统中的所有非零轨道。这个歧管，我们称为平衡单纯形，是原点排斥盆和无穷斥盆的共同边界。平衡单纯形法与生态模型联系在一起，在生态模型中，“稀有时会增长”并争夺有限资源。通过包括捕食者的替代食物来源，我们迎合了捕食者-猎物类型的模型。在模型具有竞争性的情况下，平衡单形与无形流形的携带单形重合（两个点不能按分量顺序排序），但是对于非竞争性模型，平衡单形不必是无序的。我们模型的平衡单纯形包含所有极限集，并且是单位概率单纯形上分段分析函数的图形。