Asymptotic behavior of age-structured and delayed Lotka-Volterra models

Abstract

In this work we investigate some asymptotic properties of an age-structured Lotka-Volterra model, where a specific choice of the functional parameters allows us to formulate it as a delayed problem, for which we prove the existence of a unique coexistence equilibrium and characterize the existence of a periodic solution. We also exhibit a Lyapunov functional that enables us to reduce the attractive set to either the nontrivial equilibrium or to a periodic solution. We then prove the asymptotic stability of the nontrivial equilibrium where, depending on the existence of the periodic trajectory, we make explicit the basin of attraction of the equilibrium. Finally, we prove that these results can be extended to the initial PDE problem.

Global stability in a competitive infection-age structured model

“…As above, the Lasalle invariance principle implies that t −→ L x (Φ t (v)) is constant for every v ∈ ω(z). Using (40), we obtain:…”

“…Note that similar functionals are used for delayed equations (see e.g. [40] and the references therein). The latter attractiveness combined with the stability then yield the global asymptotic stability of the corresponding equilibrium.…”

“…As written in Figure 1, the set of equilibria {E * α , α ∈ [1, 2]} is proved to be globally attractive in S x ∩ S y . Finally, following [16] and [40], we handle the stability of the disease-free equilibrium E 0 in the case max{R x 0 , R y 0 } = 1, by making use of the Lyapunov functionals.…”