Iterative Methods for Generalized von Foerster Equations with Functional Dependence

Abstract

We investigate when a natural iterative method converges to the exact solution of a differential-functional von Foerster-type equation which describes a single population depending on its past time and state densities, and on its total size. On the right-hand side, we assume either Perron comparison conditions or some monotonicity.

Stability of finite difference schemes for generalized von Foerster equations with renewal

“…The existence of solutions for generalized von Foerster equations with renewal conditions and with the functional dependence is proved in [13], which continues the sequence of results [3,4] and [10,12], focused on integral fixed-point equations, generated by differential-functional problems. As a main tool in the existence theory there are constructed integral fixed-point equations and functional spaces, invariant …”

Differential functional von Foerster equations with renewal

“…In the present paper we generalize the L ∞ ∩ L 1 -convergence results of [9] to the case of renewal boundary conditions with natural assumptions on the flow of bicharacteristics. An associate result to [9] on fast convergent quasi-linearization methods has been published in [8].…”

“…Nonlocal terms always cause huge problems. Satisfactory conditions for convergence of iterative methods were provided in [9], where (for the sake of simplicity) some boundary data were prescribed. This forced additional restrictions on the (tangential!)…”

“…An associate result to [9] on fast convergent quasi-linearization methods has been published in [8]. The renewal condition of the form u(t, 0) = ∞ 0 k(t, x, y) u(t, x)dx is interpreted as giving birth to young individuals at the age 0 by mature individuals (0 < x < +∞).…”

“…This approach can be extended to many species models. There is also a difference between the renewal case and [9]: the renewal involves such coupling of the unknown function that excludes monotone iterations.…”