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Journal of Mathematical Biology

, Volume 6, Issue 1, pp 1–19 | Cite as

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.

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Copyright information

© Springer-Verlag 1978

Authors and Affiliations

  • Bernard D. Coleman
    • 1
  1. 1.Department of Mathematics and BiologyCarnegie-Mellon UniversityPittsburghUSA

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