Two simple models of almost the same population with very different dynamics (Q1336350)

The authors investigate two related models of self-limiting cell growth, one in which the total number of cells regulates the efficiency of divisions, the other in which the number of dividing cells regulates this efficiency. They show that the former model is always asymptotically stable, while the latter is known to display limit cycles, period doublings, and chaos. They also identify parameter ranges where both models yield equivalent results and discuss the problem of when similar but non-identical biological assumptions lead to the same model behavior. Related problems are well-known from mathematical ecology where, for instance, different discrete versions of the continuous-time logistic equation may have very different dynamical properties for certain parameter ranges [see, e.g., \textit{R. M. May}, Models for single populations. In R. M. May (ed.), Theoretical Ecology, 5-29 (1981)].