The authors consider a class of ecosystems which can be modeled by the Kolmogorov system \[ (1)\quad u'=uf(u,v,w),\quad v'=vg(u,v,w),\quad w'=wh(u,v,w)\quad with \]
\[ u(0)=u_ 0\geq 0,\quad v(0)=v_ 0\geq 0,\quad w(0)=w_ 0\geq 0,\quad('=d/dt), \] where f,g,h are continuously differentiable. The population described by u(t) will always be a prey population, w(t) will always be a predator feeding exclusively on prey within the system [v(t) or u(t) or both], v(t) will be either a predator or a prey or both.
The question addressed is: when do all of the components of the model ecosystem persist? The authors use a stronger definition of persistence than usual, namely, a population \(\rho(t)\) is said to persist if \(\rho(0)>0\) and \(\lim \inf_{t\to \infty}\rho(t)>0\). A system is said to persist if each component population persists. A persistence theorem (Theorem 2.1) is formulated which essentially says that if feasible limit sets on the boundary are unstable, persistence follows. This is applied to the case of two prey and one predator, to two predators and one prey, and to food webs, to obtain persistence. The following references may be of interest: M. W. Hirsch, SIAM J. Math. Anal. 13, 167-179 (1982; Zbl 0494.34017) and The dynamical systems approach to differential equations. Bull. Am. Math. Soc., New Ser. 11, 1-64 (1984).