# 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

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