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The stability of a kind of cooperative models incorporating harvesting is considered in this paper. By analyzing the characteristic roots of the models and constructing suitable Lyapunov functions, we prove that nonnegative equilibrium points of the models are globally asymptotically stable. Further, the corresponding nonautonomous cooperative models have a unique asymptotically periodic solution, which is uniformly asymptotically stable. An example is given to illustrate the effectiveness of our results.
This paper investigates the structure of general affine subspaces of L2(Rd) . For a d × d expansive matrix A, it shows that
every affine subspace can be decomposed as an orthogonal sum of spaces each of
which is generated by dilating some shift invariant space in this affine
subspace, and every non-zero and non-reducing affine subspace is the orthogonal
direct sum of a reducing subspace and a purely non-reducing subspace, and every
affine subspace is the orthogonal direct sum of at most three purely
non-reducing subspaces when |detA| =
In this paper, we study the relation of the algebraic
properties of the higher-order Courant bracket and Dorfman bracket on the
direct sum bundle TM⊕∧pT*M？for
an m-dimensional smooth manifold M, and a Lie 2-algebra which is a “categorified” version of a Lie algebra. We prove that the higher-order
Courant algebroids give rise to a semistrict Lie 2-algebra, and we prove that
the higher-order Dorfman algebroids give rise to a hemistrict Lie 2-algebra.
Consequently, there is an isomorphism from the higher-order Courant algebroids
to the higher-order Dorfman algebroids as Lie 2-algebras homomorphism.