Hutson V, Law R
J Math Biol. 1985;21(3):285-98. doi: 10.1007/BF00276227.
We address the question of the long term coexistence of three interacting species whose dynamics are governed by the ordinary differential equations xi = xifi(i = 1,2,3). In order for any theory in this area to be useful in practice, it must utilize as little information as possible concerning the forms of the fi, in view of the great difficulty of determining these experimentally. Here we obtain, under rather general conditions on the equations, a criterion for judging whether the species will coexist in a biologically realistic manner. This criterion depends only on the behaviour near the one or two species equilibria of the two dimensional subsystems, the behaviour there being relatively easy to examine experimentally. We show that with the exception of one class of cases, which is a generalization of a classical example of May and Leonard [21], invasibility at each such equilibrium suitably interpreted is both necessary and sufficient for a strong form of coexistence to hold. In the exceptional case, a single additional condition at the equilibria is enough to ensure coexistence.
我们研究了由常微分方程(x_i = x_if_i)（(i = 1,2,3)）描述其动力学的三种相互作用物种长期共存的问题。鉴于通过实验确定(f_i)的形式存在很大困难，为使该领域的任何理论在实际中有用，它必须尽可能少地利用关于(f_i)形式的信息。在此，我们在方程的相当一般条件下，得到了一个判断物种是否以生物学上现实的方式共存的标准。该标准仅取决于二维子系统在一物种或两物种平衡点附近的行为，那里的行为相对容易通过实验进行研究。我们表明，除了一类情况（它是梅和伦纳德[21]的一个经典例子的推广）外，在每个这样的平衡点处经过适当解释的可入侵性对于一种强形式的共存成立既是必要的也是充分的。在例外情况下，在平衡点处一个额外的条件就足以确保共存。
