The generation of vascular pattern formation in plants is an interesting process of pattern formation in organisms. It is well known that the plant hormone auxin is involved in plant vascular differentiation and that the PIN1 protein, an auxin efflux carrier, localizes to one side of the cell membrane. Several hypotheses have been proposed to explain the formation of leaf venation. One is the canalization hypothesis that is based on the assumption that a positive feedback regulation exists between the flow of a signal molecule and the capacity of its flow. Here, we attempted to integrate the canalization hypothesis and experimental data. We investigated models of the positive feedback regulation between the auxin flow and PIN1 localization. Model 1, with conserved PIN1 amount in each cell, can generate a branching pattern similar to that of plant leaf venation. We introduced the diffusible enhancer "e" into the model as unknown factor. The obtained patterns show a quasi-periodic distribution of auxin flow paths, when the model dynamics includes domain growth. In order to understand the early initiation process that generates an inhomogeneity from an almost homogeneous distribution, we introduced model 2, a simplified version of model 1. Model 2 can generate inhomogeneity with a parameter dependency similar to that of model 1. To analyse parameter condition required for pattern development, approximated equations are obtained from model 2. The isocline analysis of the equations without spatial structure shows that the inhomogeneous distribution occurs from an almost homogeneous distribution. This parameter condition for generating inhomogeneity is consistent with the results of models 1 and 2.
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