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.
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Notes
In (Pielou 1977, p. 38 ) the author speaks of a ‘concave’ curve, but the Fig. 2.7 in the same paper shows clearly a convex curve.
Throughout this paper, a prime refers to differentiation with respect to \(U\), and a dot to differentiation with respect to the time variable \(t\).
Think of a government that changes the voting age at a given moment (which is a very rapid change). If the voting age is decreased, a class of juveniles become instantaneously adults, which seems acceptable. But if the voting age in increased, then a certain class of adults lose their voting right, i.e., become instantaneously juveniles.
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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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Barbarossa, M.V., Hadeler, K.P. & Kuttler, C. State-dependent neutral delay equations from population dynamics. J. Math. Biol. 69, 1027–1056 (2014). https://doi.org/10.1007/s00285-014-0821-8
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DOI: https://doi.org/10.1007/s00285-014-0821-8
Keywords
- Neutral delay equation
- State-dependent delay
- Blowfly equation
- Age structure
- Quasi-linear
- Population dynamics