This paper treats a model of linear age-dependent population dynamics involving a hereditary birth law. The model is applicable to a bird population in which a maturation period of the eggs is taken into account. The model has the following form: ∂u/∂a+∂u/∂t=−m(a)u(a,t), a∈[0,A], t≥0, u(a,θ)=φ(a,θ), a∈[0,A], θ∈[−r,0], u(0,t)=∫tt−rg(t−s)∫A0b(a)u(a,s)dads, t≥0, where u(a,t) is the density of the population with respect to age a at time t, m(a) is the age-specific mortality rate, b(a) is the age-specific fertility rate, g(s) denotes the proportion of the eggs that will yield living birds s units of time after the eggs are laid, and φ is the initial age distribution in the time period [−r,0]. A generalization of this model to the case in which random diffusion of the population occurs is also treated. The author applies Laplace transform methods, semigroup methods, and spectral theory to study the asymptotic behavior of the solutions. The results demonstrate that the behavior at infinity is qualitatively the same as the behavior at infinity of the model without the gestation delay.

### Asymptotic behavior in age-dependent population dynamics with hereditary renewal law.

#### Abstract

This paper treats a model of linear age-dependent population dynamics involving a hereditary birth law. The model is applicable to a bird population in which a maturation period of the eggs is taken into account. The model has the following form: ∂u/∂a+∂u/∂t=−m(a)u(a,t), a∈[0,A], t≥0, u(a,θ)=φ(a,θ), a∈[0,A], θ∈[−r,0], u(0,t)=∫tt−rg(t−s)∫A0b(a)u(a,s)dads, t≥0, where u(a,t) is the density of the population with respect to age a at time t, m(a) is the age-specific mortality rate, b(a) is the age-specific fertility rate, g(s) denotes the proportion of the eggs that will yield living birds s units of time after the eggs are laid, and φ is the initial age distribution in the time period [−r,0]. A generalization of this model to the case in which random diffusion of the population occurs is also treated. The author applies Laplace transform methods, semigroup methods, and spectral theory to study the asymptotic behavior of the solutions. The results demonstrate that the behavior at infinity is qualitatively the same as the behavior at infinity of the model without the gestation delay.
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Age dependent Population Dynamics; Diffusion; Asymptotic Behaviour
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/20.500.12571/2821`
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