Asymptotic Behavior of Solutions for Forest Kinematic Model

Abstract

Abstract. We continue a study of the mathematical model of forest ecosystem which has been introduced by Kuznetsov et al. [3]. In this paper, we will introduce three kinds of o-limit sets, namely, oðU 0 Þ H L 2 -oðU 0 Þ H w à -oðU 0 Þ, for each point U 0 of the dynamical system which was constructed in our preceding paper [1]. Using a Lyapunov function, we will then investigate basic properties of these o-limit sets.Especially it shall be shown that L 2 -oðU 0 Þ consists of stationary solutions alone.

A Forest Model with Memory

“…In the case of no memory, the dynamical systems for (1.1) were constructed by [1,8]. Asymptotic behavior of solutions were studied by [2,9]. Furthermore, homogeneous and inhomogeneous stationary solutions were studied by [3,10].…”

“…For investigating the asymptotic behavior of global solutions of (1.1), we have to use much more sophisticated techniques than those in [2,9]. This will be studied in forthcoming papers.…”

Stationary Solutions to Forest Kinematic Model

“…While the first and second equations describe the growth of young and old trees, respectively; 0 < δ 1 is an establishment rate of seeds, f > 0 is an aging rate and h > 0 is a mortality of old trees. And γ (v) > 0 is a mortality of young trees which is allowed to depend on the old-tree density v and is expected to hit a minimum at a certain optimal value of v. In the preceding two papers [7,8], the authors have already studied this system analytically. In [7], they constructed a dynamical system (S(t), K, X) determined from the initial-boundary value problem (1.1).…”

“…Moreover, even if the initial functions u 0 , v 0 and w 0 are smooth, its solution (u, v, w) can tend to a discontinuous stationary solution as t → ∞ (see [8,Section 6] and [13]). That is, the continuous function space C( ) is not suitable.…”

“…The non-linear semigroup S(t) acts on K for 0 ≤ t < ∞. In [8], the authors found a Lyapunov function and investigated asymptotic behaviour of trajectories S(t)U 0 , U 0 ∈ K. Since some S(t)U 0 can converge to a discontinuous stationary solution even if the initial value U 0 ∈ K consists of smooth functions and since if so the trajectory S(t)U 0 has an empty ω-limit set in X, the dynamical system (S(t), K, X) never enjoys any compact attractor in general. Due to this reason we introduced three kinds of ω-limit sets for…”