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In this paper the dynamics of skew tent maps are classified in terms of two bifurcation parameters. In time series analysis such maps are usually referred to as continuous threshold autoregressive models (TAR(1) models) after Tong (1990). This study contains results simplifying the use of TAR(1) models considerably, e. g. if a periodic attractor exists it is unique. On the other hand we also claim that care must be exercised when threshold autoregressive (TAR) models are used. In fact, they possess a very special type of dynamical pattern with respect to the bifurcation parameters and their transition to chaos is far from standard.

We investigate population models with both continuous and discrete elements. Birth is assumed to occur at discrete instants of time whereas death and competition for resources and space occur continuously during the season. We compare the dynamics of such discrete-continuous hybrid models with the dynamics of purely discrete models where within-season mortality and competition are modelled directly as discrete events. We show that non-monotone discrete single-species maps cannot be derived from unstructured competition processes. This result is well known in the case of fixed reproductive strategies and our results extend this to the case of adjustable reproductive strategies. It is also shown that the most commonly used non-monotone discrete maps can be derived from structured competition processes. 9 1997 Society for Mathematical Biology

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