How does the rate of environmental change affect density-dependent population dynamics?

Natural populations experience variable environments. Anthropogenically driven environmental change, in particular, is expected to impose trends on key demographic parameters such as reproduction and survival. Theoretical studies of how such environmental changes affect populations have highlighted dynamical phenomena including bifurcation-related tipping points – typically identified by comparing different, but constant, environmental states – and long transients – that can arise after sudden environmental perturbations. However, real-world environmental trends are neither instantaneous nor slow enough to justify treating the environment as constant, motivating recent interest in r-tipping points – regime shifts induced by \emph{r}, the rate of environmental change in demographic parameters. Most existing work examines this phenomenon in complex ecological models and for specific values of \emph{r}. Here, we develop tools to help ecologists investigate how populations and communities respond to environmental trends across a continuum of \emph{r} values. Using a simple density-dependent model, we identify four qualitatively distinct responses to a trend as a function of \emph{r} – patterns that traditional methods fail to reveal – and we visualize them using an \emph{r-bifurcation diagram} introduced here. We also describe and mathematically explain the emergence of abrupt regime shifts linked to delayed bifurcations, revealed by a novel \emph{superimposition diagram}. These findings are robust across modelling frameworks and ecological contexts, providing new insights into interactions between short- and long-term environmental change processes.