Global dynamics of a renewal equation for size-structured population models

We study a size-structured tree growth model from [4–6], described by the nonlinear renewal equation φ(t) = F φₜ, \ φₜ ∈ L¹_ρ(R₋), with reproduction, death, and growth rates β , μ , and g . We prove that, under mild conditions on these rates, the equation generates a semiflow in L¹_ρ(R₋) that is permanent and possesses a compact global attractor A . If β is monotone, A reduces to a single asymptotically stable equilibrium attracting all compact sets with positive initial data. Adapting an approach from [21], originally developed for simpler renewal equations, we investigate stability and persistence in this more complex setting via the one-dimensional recurrence bₙ₊₁ = F bₙ, thereby complementing the functional-analytic framework of [13].