Journal of Mathematical Biology

, Volume 6, Issue 1, pp 1–19

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

I. General theory and examples
• Bernard D. Coleman
Article

## 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 instantt the rate at which the population loses individuals of agea through death and dispersal is given by a function ρ ofa and the numberx(a, t) of individuals which have agea at timet, and (ii) the numberx(0, t) of individuals born at timet is given by a functionF of the numberx(a f ,t) of individuals at a specified reproductive agea 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 functionF. For example, it is shown that when ρ(a, x) has the form ρ=π1(a)x + π2(a)x 2 with π1 and π2 positive andF is monotone increasing and concave withF(0)=0, the parameter

$$T = ln\frac{d}{{dx}}F(0) - \int_0^{a_f } {\pi _1 (a) da}$$
determines whether the population is “endangered” in the following sense: forT>0 there is an asymptotically stable non-zero stationary age distribution, but forT<0 there holds
$$\begin{array}{*{20}c} {\lim } \\ {t \to \infty } \\ \end{array} \begin{array}{*{20}c} {\sup } \\ {0 \leqslant a \leqslant a_f } \\ \end{array} x(a,t) = 0.$$

## Keywords

Periodic Solution Loss Function Narrow Spread Single Species Population Threshold Theorem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Griffel, D. H.: Age-dependent population growth, J. Inst. Maths. Applics.17, 141–152 (1976).
2. 2.
Gurtin, M. E., MacCamy, R. C.: Non-linear age-dependent population dynamics, Arch. Rational Mech. Anal.54, 281–300 (1974).
3. 3.
Hartman, P.: Ordinary Differential Equations. New York: Wiley 1964.Google Scholar
4. 4.
5. 5.
Li, T.-Y., Yorke, J. A.: Period three implies chaos, Amer. Math. Monthly82, 985–992 (1975).
6. 6.
Lotka, A. J.: Elements of Physical Biology: Baltimore: Williams and Wilkins 1925.
7. 7.
May, R. M.: Biological populations obeying difference equations: stable points, stable cycles, and chaos. J. Theor. Biol.51, 511–524 (1975).
8. 8.
May, R. M., Oster, G. F.: Bifurcations and dynamic complexity in simple ecological models, Amer. Naturalist110, 573–597 (1976)
9. 9.
Maynard Smith, J.: Mathematical Ideas in Biology. Cambridge Cambridge University Press 1968.Google Scholar
10. 10.
Maynard Smith, J.: Models in Ecology. Cambridge: Cambridge University Press 1974.Google Scholar
11. 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