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Volume 11,     Number 2,     Summer 2003

 

WHERE TO PUT DELAYS IN POPULATION MODELS, IN PARTICULAR IN THE NEUTRAL CASE
K. P. HADELER AND G. BOCHAROV

Abstract. Hutchinson's model or the delayed logistic equation explains oscillations in populations by delayed response to exhaustion of nutrients. The exploration of periodic solutions and global behavior of nonlinear delay equations has started from this model. Another delay model for oscillations in populations is the blowfly equation which has been thought to be better suited to explain experiments. Also delay differential equations of neutral type have been proposed as population models. The Gurtin-MacCamy system of a partial differential equation with boundary condition models populations structured by age with birth and death rates depending on age and on total population size. If written as a renewal equation then piecewise constant coefficients, even with delta peaks, are admissible. It is shown that these "realistic caricatures" of age structure models are equivalent to delay equations, in general of neutral type. In the standard retarded case (as opposed to the neutral case) the resulting systems have the form of the blowfly equation. Hence the latter seems indeed better justified than the delayed logistic equation. In the case of neutral equations the age structure approach presented here is a rigorous derivation of such population models. Also in the retarded case the view on oscillations is drastically changed since Hutchinson's model and the blowfly model have distinct stability properties.

 

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