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

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