# State-dependent neutral delay equations from population dynamics

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## Abstract

A novel class of state-dependent delay equations is derived from the balance laws of age-structured population dynamics, assuming that birth rates and death rates, as functions of age, are piece-wise constant and that the length of the juvenile phase depends on the total adult population size. The resulting class of equations includes also neutral delay equations. All these equations are very different from the standard delay equations with state-dependent delay since the balance laws require non-linear correction factors. These equations can be written as systems for two variables consisting of an ordinary differential equation (ODE) and a generalized shift, a form suitable for numerical calculations. It is shown that the neutral equation (and the corresponding ODE—shift system) is a limiting case of a system of two standard delay equations.

## Keywords

Neutral delay equation State-dependent delay Blowfly equation Age structure Quasi-linear Population dynamics## Mathematics Subject Classification

34K40 34K17 92D25 34K20## Notes

### Acknowledgments

MVB was supported by the ERC Starting Grant No. 259559 as well as by the European Union and the State of Hungary, co-financed by the European Social Fund in the framework of TÁMOP-4.2.4. A/2-11-1-2012-0001 National Excellence Program.

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