A boundedness theorem with application to oscillation of autocatalytic chemical reactions

see section 4 for an account of a few examples. In (∗), q describes the input and possibly output of one agent, r the output (i.e., removal) and possibly input of a second agent, and u and v describe some mechanism of conversion of the first agent to the second agent (e.g., in an enzyme reaction or a population model). ε is a scaling parameter. We are interested in boundedness of the solutions, mainly as a prerequisite for applying the Poincaré–Bendixson theorem to obtain periodic solutions. In section 2, we formulate a set of nine conditions of (∗) under which we prove forward boundedness of all solutions in the nonnegative quadrant (theorem 2.1). The number of conditions admittedly is appalling; but in most applications (with q, r, u, v rational functions) these are easily checked or even trivially fulfilled. On the other hand, some of the conditions are quite natural in the context of the applications, see the comments in section 2. Above all, our result is flexible enough to handle models originating from areas far apart (cf. section 4). Boundedness of solutions in positive time direction and the existence of periodic solutions or even limit cycles are the main features one is looking for in studying 2-dimensional models from a qualitative point of view. Systems of the form (∗) in