Observations on evolutionary models with (or without) time lag, and on problematical paradigms

Models of phenomena that evolve with time, notably those in bioscience, may be more realistic as scientific models if they incorporate a memory or time-lag rather than assume instantaneous effects. The differences and similarities between the two types of model, having regard to their ‘faithfulness’ as models, the sensitivity, stability, and qualitative behaviour of their solutions, are often discussed through canonical models or suggested paradigms using various ‘mathematical toolkits’. We offer insight (largely self-contained, but supplemented by a bibliography) into mathematical models with time-lag and we consider how appropriate or limited are various exemplars when used as paradigms. We also mention possibilities for extending the application of the mathematical tools at our disposal.