# On the growth of populations with narrow spread in reproductive age

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## Summary

General theorems are proven about the existence and asymptotic stability of stationary and temporally periodic age distributions for single species populations which obey the following assumptions: (i) at each instant*t* the rate at which the population loses individuals of age*a* through death and dispersal is given by a function ρ of*a* and the number*x(a, t)* of individuals which have age*a* at time*t*, and (ii) the number*x(0, t)* of individuals born at time*t* is given by a function*F* of the number*x(a* _{ f },*t*) of individuals at a specified reproductive age*a* _{ f }. The theorems given are illustrated for various special cases in which assumptions are made about the forms of loss function ρ and the fecundity function*F*. For example, it is shown that when ρ(*a, x*) has the form ρ=π_{1}(*a)x* + π_{2}(*a)x* ^{2} with π_{1} and π_{2} positive and*F* is monotone increasing and concave with*F*(0)=0, the parameter

*T*>0 there is an asymptotically stable non-zero stationary age distribution, but for

*T*<0 there holds

## Keywords

Periodic Solution Loss Function Narrow Spread Single Species Population Threshold Theorem## Preview

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## References

- 1.Griffel, D. H.: Age-dependent population growth, J. Inst. Maths. Applics.
**17**, 141–152 (1976).MATHMathSciNetGoogle Scholar - 2.Gurtin, M. E., MacCamy, R. C.: Non-linear age-dependent population dynamics, Arch. Rational Mech. Anal.
**54**, 281–300 (1974).MATHMathSciNetCrossRefGoogle Scholar - 3.Hartman, P.: Ordinary Differential Equations. New York: Wiley 1964.Google Scholar
- 4.Hoppensteadt, F.: Mathematical Theories of Populations: Demographics, Genetics, and Epidemics. Philadelphia: S.I.A.M. 1975.Google Scholar
- 5.Li, T.-Y., Yorke, J. A.: Period three implies chaos, Amer. Math. Monthly
**82**, 985–992 (1975).MATHMathSciNetCrossRefGoogle Scholar - 6.Lotka, A. J.: Elements of Physical Biology: Baltimore: Williams and Wilkins 1925.MATHGoogle Scholar
- 7.May, R. M.: Biological populations obeying difference equations: stable points, stable cycles, and chaos. J. Theor. Biol.
**51**, 511–524 (1975).CrossRefGoogle Scholar - 8.May, R. M., Oster, G. F.: Bifurcations and dynamic complexity in simple ecological models, Amer. Naturalist
**110**, 573–597 (1976)CrossRefGoogle Scholar - 9.Maynard Smith, J.: Mathematical Ideas in Biology. Cambridge Cambridge University Press 1968.Google Scholar
- 10.Maynard Smith, J.: Models in Ecology. Cambridge: Cambridge University Press 1974.Google Scholar
- 11.Von Foerster, H.: Some remarks on changing populations. In: The Kinetics of Cellular Proliferation, pp. 382–407. New York: Grune and Stratton 1959Google Scholar