This paper handles the initial-boundary value problem for a quasilinear parabolic system given in a two-dimensional rectangle which has been presented by Belić, Škarka, Deneubourg and Lax, in order to describe the construction process of parallel honeycombs in a beehive. After constructing the local strict solutions by using the theory of abstract parabolic equations, we define the maximal strict solutions and then investigate their asymptotic behavior. We shall also present some numerical examples which illustrate the honeycomb construction.
