Appl Math Optim 51:35–59 (2005)
2004 Springer Science+Business Media, Inc.
A Quasilinear Hierarchical Size-Structured Model:
Well-Posedness and Approximation
Azmy S. Ackleh, Keng Deng, and Shuhua Hu
Department of Mathematics, University of Louisiana at Lafayette,
Lafayette, LA 70504, USA
Abstract. A ﬁnite difference approximation to a hierarchical size-structured model
with nonlinear growth, mortality and reproduction rates is developed. Existence-
uniqueness of the weak solution to the model is established and convergence of
the ﬁnite-difference approximation is proved. Simulations indicate that the mono-
tonicity assumption on the growth rate is crucial for the global existence of weak
solutions. Numerical results testing the efﬁciency of this method in approximating
the long-time behavior of the model are presented.
Key Words. Hierarchical size-structured model, Existence-uniqueness, Finite-
AMS Classiﬁcation. 35L60, 65M06, 65M12, 92D25.
In this paper we consider the following initial-boundary value problem which models
the evolution of a hierarchical size-structured population:
+ (g(x , Q(x, t))u)
+ m(x, Q(x, t))u = 0,(x, t) ∈ (0, L] × (0, T ],
g(0, Q(0, t))u(0, t) = C(t ) +
β(x, Q(x, t ))u(x, t) dx, t ∈ (0, T ],
u(x, 0) = u
(x), x ∈ [0, L].
The work of ASA and SH was supported in part by the National Science Foundation under Grant
No. DMS-0311969. The work of KD was supported in part by the National Science Foundation under Grant