Abstract
The phenomenon of replacement migration into declining population prompts development of multicomponent models in population dynamics. We propose a simple model of population including resident and migrant components with migration flow as an external input. The main assumption is that offspring that are born to migrants will have the same vital rates as the resident population. The proposed model is based on partial differential equation to take into account the age structure of the population. The formulae for exact solutions are derived. We focus on the case when native population declines in the absence of migration. Assuming a sufficiently large constant growth rate of migration we obtain asymptotic solutions as t → ∞. Using the asymptotic solutions, we have calculated “critical” value of growth rate of migrant inflow that is the value that provides, as time tends to infinity, the equal number of residents and migrants in the population. We provide numerical illustrations using demographic data for Germany in 2010.
Highlights
Models of population dynamics play major roles in many areas of science such as human demography, ecology, and biology (Stott, 2011) and are integral to population management (Menges, 1990; Fujiwara and Caswell, 2001)
The discretization of age distribution in the boundary condition leads to errors, since the true projected value, depends on the way in which the population is distributed within the age intervals
We suggest a simple two–component population model including immigration as external input that is continuous in age and in time, and derive formulae to effectively investigate and calculate the solution to the corresponding mathematical problem
Summary
Models of population dynamics play major roles in many areas of science such as human demography, ecology, and biology (Stott, 2011) and are integral to population management (Menges, 1990; Fujiwara and Caswell, 2001). We study a linear model of McKendrick–von Foerster type for the time development of an age structured two component female population including internal migration. Where (x, t) ∈ Ω = {(x, y) ∈ R2 | x ∈ (0, X), t > 0}, which is a differential form of the conservation law of the number of individuals in the continuous model of the age structure of a population that consists of two subpopulations with external immigration into the second subpopulation. Μ1, μ2, p are continous nonnegative functions, and X is a positive number It is a dynamical model of a female population consisting of residents and migrants; x denotes the age of individuals, t – the time, N1 – the age density of residents, N2 – the age density of migrants, p – the migrant inflow age density, μ1, μ2 – the age specific mortality rates, and X – the maximum lifespan of an individual.
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