The authors consider an autonomous system of three ordinary differential equations: \[ (1)\quad \dot x=-f(z,x),\quad \dot y=g(x),\quad \dot z=h(y), \] where the functions f, g, h verify very simple assumptions: \[ f,\quad g,\quad h\quad are\quad C^ 2,\quad f(0,0)=g(0)=h(0),\quad f_ x\geq 0;\quad f_ z,g',\quad and\quad h'>0. \] This encompasses - after convenient changes of variables - equations in population dynamics previously studied by Levine, MacDonald, Coste et al. There is a unique staionary point (0,0,0) associated with a one-dimensional stable manifold and a two-dimensional unstable one.
The main result is as follows: apart from the solutions which lie on the stable manifold (two of them), all the solutions are unbounded, More precisely, every unbounded solution tends to infinity, by crossing successively and in the same order each one of the six octants which do not contain the stable manifold.
Mostly, this result rests upon a thorough study of the unstable manifold U. The authors are able to prove that U can be globally represented as a graph of a map \(u: {\mathbb{R}}^ 2\Rightarrow {\mathbb{R}}\), \(x=u(y,z).\)
To show that U projects over \({\mathbb{R}}^ 2\) and solutions on U are unbounded the authors use a Lyapunov function on U. They extend this result to all the solutions by noting that the function \((x(t)- u(y(t),z(t)))^ 2\) is decreasing along the solutions. An interesting comparison is made with a class of equations studied by D. Webster, S. Hastings and J. Tyson, J. diff. Equations 25, 39-64 (1977; Zbl 0361.34038): \[ (2)\quad \dot x=-f(z,x);\quad \dot y=g(y,x);\quad \dot z=h(y,z) \] which differs from (1) by the fact that \(g_ y<0\) and \(h_ z<0\). In contrast with (1), the system (2) happens to have periodic solutions. Finally, we note that the equation (1) can be stated in higher dimensions. According to the authors, the problem there is still open but should yield the same result.