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Basic Properties of Mathematical Population Models

Mathematical population models are constructed based on plausible explicit and implicit biological assumptions. While it is easy to incorporate explicit assumptions correctly in the models, those implicit ones are often ill treated or forgotten. Indeed, this happens to some well known models in the literature and examples of such will be mentioned and discussed. For a model to be logically credible, we must do our best to ensure that all assumptions are incorporated correctly and consistently. To this end, we exam a simple set of criteria proposed by Arditi and Michalski in 1996. For patchy models, we add an additional criterion to their list. We also add some important criteria in other specific situations and comment on modelling of single species growths. Following criteria of Arditi and Michalski and other well accepted biological assumptions, we introduce some interesting three dimensional predator-dependent (ratio-dependent) population models. We also discuss various aspects of modelling population fluctuations.