Differential Inclusions

Dynamical consequences of optimal host feeding on host-parasitoid population dynamics

“…Inclusion of optimal host-feeding behavior of parasitoids lead in this article to nonsmooth differential equations and differential inclusions. A theory for such types of problems has been recently developed (Filippov 1988, Aubin and Cellina 1984, Aubin 1991, Deimling 1992 and it seems that this methodology may prove to be useful when modelling the effects of behavioral trade-offs on population dynamics (Křivan 1996, Křivan 1997. Although we used the simplest possible type of population dynamics in this article, which allowed for a deeper mathematical analysis, the present methodology may be also used for more realistic models, including for example Holing second type functional response function, delays etc.…”

On relations of the adjoint state to the value function for optimal control problems with state constraints

“…By Theorem 3.3 and Filippov's theorem [1,13], there exists a constant M depending only on R such that for some…”

“…Observe next that for all i ≥ 2 we have 1 + α > 0 whenever |α| ≤ 1 i . By the very definition of G for every q ∈ [−2, 0], H i (t, w, x, q, p) = max…”

“…If for some L > 0 and for every trajectory x(·) of (1.1) with x 0 ∈ K, 1] dist(x(t), K), then (1.1)-(1.2) is said to enjoy NFT estimates in · ∞ . For Lipschitz F and bounded K having smooth boundary ∂K, NFT estimates follow from the classical IPC: for all x ∈ ∂K and t ∈ [0, 1], there exists w ∈ F (t, x) satisfying n x , w < 0, where n x is the unit outward normal to K at x (see [4,18]).…”

“…be the set of all real valued nonnegative integrable functions defined of [0,1]. We impose the following regularity assumptions on F…”