We study a size-structured tree growth model from 4–6, described by the nonlinear renewal equation (t) = F ₜ, \ ₜ L¹_ (R-), with reproduction, death, and growth rates, , and g. We prove that, under mild conditions on these rates, the equation generates a semiflow in L¹_ (R-) that is permanent and possesses a compact global attractor A. If 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 b₍+₁ = F bₙ, thereby complementing the functional-analytic framework of 13.
Herrera et al. (Thu,) studied this question.