1,721,066 research outputs found
Asymptotic behavior in age-dependent population dynamics with hereditary renewal law.
No full textThis 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
Some considerations on the mathematical approach to nonlinear age dependent population dynamics
No full textAbstractThis paper is devoted to a new approach for the study of nonlinear age dependent population dynamics. It consists to transform the hyperbolic PDE into a functional integral equation and to investigate it. We get global existence, uniqueness, and local stability results. Some applications to the traditional ecological models are made
Stability for second order abstract evolution equations
No full textThe investigates results concerning the stability and asymptotic stability of the null solution of abstract nonlinear second order evolution equations, applying them to semilinear wave equations, strongly dissipative wave equations, an equation for transverse motion of an extensible beam, and to a damped quasilinear wave equation. The main contribution is the removal if the compactness of the orbits required in the LaSalle invariance principl
Decay and stability for nonlinear hyperbolic equations
No full textAbstractThis paper deals with the asymptotic stability of the null solution of a semilinear partial differential equation. The La Salle Invariance Principle has been used to obtain the stability results. The first result is given under quite general hypotheses assuming only the precompactness of the orbits and the local existence. In the second part, under some restrictions, sufficient conditions for precompactness of the orbits and decay of solutions are given. An existence and uniqueness theorem is proved in the Appendix. Some examples are given
Convergence of approximate solutions to scalar conservation laws by degenerate diffusion
No full textThis paper is devoted to establish the existence of weak solutions to the scalar conservation laws (1) ut+f(u)x=0, by compensated compactness theory. He shows that the unique weak entropic solution to (1) can be obtained by replacing the usual viscous approximation by means of the porous media operator, (2) ut+f(u)x=ε(|u|m−1u)xx, m>1, ε>0. The advantage of using the method is that it provides an approximating solution which, in some situations, coincides with the exact solution of (1) outside a compact set (in the space variable, for fixed time), while the perturbation effects of the usual viscosity always lead to undesired modifications of the far fields
Asymptotic behavior of the renewal equation arising in the Gurtin population model.
No full textThis paper is concerned with the asymptotic behaviour in a model for the growth of a spatially diffusing age-structured population. Starting from a renewal equation for the space-dependent birth rate by exploiting Laplace transforms it is possible to deduce that for large time the growth is governed by a (not necessarily simple) dominant eigenvalue
Weak solutions to a nonlinear partial differential equation of mixed type
No full textWe study a convective problem of `mixed type' (i.e., parabolic and hyperbolic) which has a degenerate second-order term. The analysis by the `vanishing viscosity' method is conducted by a careful study of the phase boundaries and of the topological structure of the phase sets. At the end we provide the existence of a discontinuous travelling wav
On the global stability of the logistic age-dependent population growth.
No full textWe study an age-dependent population equation with a nonlinear death rate of "logistic" type. The global asymptotic stability of the null solution is investigated when R(0) less than 1. If R(0) greater than 1 we get the existence of a nontrivial steady state that becomes asymptotically stable itself, while the null solution is unstable. The rate of decay is estimated
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