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Population models with state-dependent delays. (English) Zbl 0749.92014
Mathematical population dynamics, Proc. 2nd Int. Conf., New Brunswick/NJ (USA) 1989, Lect. Notes Pure Appl. Math. 131, 165-176 (1991).
[For the entire collection see Zbl 0742.00058.]
We present some population models containing time delays depending on the unknown function, called state-dependent delays. We point out some technical difficulties created by this dependence and show how to circumvent some of them.
In the last section we consider in more details an equation that generalizes a model investigated by K. L. Cooke and J. A. Yorke [Math. Biosci. 16, 75-101 (1973; Zbl 0251.92011)]; this equation has been essentially derived, but not investigated by K. L. Cooke [Differential equations dynamical systems, Proc. internat. sympos. Puerto Rico 1965, 167-183 (1967; Zbl 0189.403)]. We show how the introduction of a variable lifespan in the modeled population, which translates into a variability of the delay in the equation, leads to a destabilization of the stationary solutions. The resulting behavior is investigated numerically.

92D25 Population dynamics (general)
34K25 Asymptotic theory of functional-differential equations
34K20 Stability theory of functional-differential equations