A model of random walk with memory enhancement and decay was presented on the basis of the characteristics of the biological intelligent walks. In this model, the movement of the walker is determined by the difference between the remaining information at the jumping-out site and jumping-in site. The amount of the memory information s(i)(t) at a site i is enhanced with the increment of visiting times to that site, and decays with time t by the rate e(-beta(t)), where beta is the memory decay exponent. When beta=0, there exists a transition from Brownian motion (BM) to the compact growth of walking trajectory with the density of information energy u increasing. But for beta>0, this transition does not appear and the walk with memory enhancement and decay can be considered as the BM of the mass center of the cluster composed of remembered sites in the late stage.

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A cellular automation model of the "game of Life" on a two-dimensional small-world network is presented in order to count in long-range interactions among living individuals in social or biological systems. The density of the life and its fluctuation are calculated, respectively. The present model exhibits a nonequilibrium phase transition from an "inactive-sparse" state to an "active-dense" one at a certain intermediate value of the network disorder. Employing finite-size scaling analysis, we estimate the location of the critical point with p(c)( infinity ) approximately 0.3685. The transition is of the "second-order" type with power-law diverging length. We obtain the critical exponents 1/nu approximately 1.70, beta approximately 0.50, and beta/nu approximately 0.85. The calculated results indicate that the present model may belong to the universality class of directed percolation.