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We consider a system of nonlinear delay differential equations that describes the growth of the mature population of a species with age-structure living over three patches. We analyze existence of nonnegative homogeneous equilibria and their stability and discuss possible Hopf bifurcation from these equilibria. More precisely, by employing both the standard Hopf bifurcation theory and the symmetric bifurcation theory for functional differential equations, we obtain very rich dynamics for the system, includResearch of P. Weng was supported by ing bistable equilibria, transient oscillations, synchronous periodic solutions, phase-locked periodic solutions, mirror-reflecting waves and standing waves.

Principal Component Analysis(PCA) is intrinsically a ridge regression problem in statistical view. By imposing l1 constraint on the regression coefficients, we have Sparse Principal Component Analysis(SPCA) which is easier to interpret and better for generalization. But traditional SPCA is difficult to be used on 2-d face data for its high dimensionality of covariance matrix because of the matrix-to-vector transformation, especially when the number of dimensionality and training samples are all in large scale. In this paper,we proposed a bi-directional Two-dimensional Sparse Principal Component Analysis(2dSPCA) to overcome the above shortcoming of SPCA. 2dSPCA is directly calculated by elastic net regularization on image covariance matrix without vectorization. Sparsity of projection vectors makes the results more interpretable,also helps us find the important local areas of face image for face recognition,for example, the areas around the corner of eye,nose and mouth include significantly discriminative information. Experiments on some benchmark face databases show that 2dSPCA achieves comparable or higher performance in face recognition compared with 2dSPCA. We also propose a 2dSPCA+LDA algorithm to improve the effectiveness of face recognition.Index Terms-Two-dimensional sparse principal component analysis, feature extraction, elastic net, face recognition

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