Abstract:

In this paper, we study a class of structured bacterial population models under the boundary condition of total transition rule in $L^{1}$ space. The spectral analysis of the transport operators in this model is discussed, and the weak compactness of the positive irreducible $C _0$ semigroup generated by the transport operators is proved. It is concluded that the spectrum of the transport operator consists of only by at most countable isolate eigenvalues with finite algebraic multiplicities, and $-\infty$ is the only possible accumulation point, and the asymptotic behavior of the solution of the model in the topological sense of the uniform operator, so the asynchronous growth characteristics of the bacterial population are given.

Key words: Structured bacterial population, Aggregate transition rule, Transport operator, Weakly compact semigroup, Spectral analysis