  The amoeboid organism, the plasmodium of Physarum polycephalum, behaves on
the basis of spatio-temporal pattern formation by local
contraction-oscillators. This biological system can be regarded as a
reaction-diffusion system which has spatial interaction by active flow of
protoplasmic sol in the cell. Paying attention to the physiological evidence
that the flow is determined by contraction pattern in the plasmodium, a
reaction-diffusion system having self-determined flow arises. Such a coupling
of reaction-diffusion-advection is a characteristic of the biological system,
and is expected to relate with control mechanism of amoeboid behaviours. Hence,
we have studied effects of the self-determined flow on pattern formation of
simple reaction-diffusion systems. By weakly nonlinear analysis near a trivial
solution, the envelope dynamics follows the complex Ginzburg-Landau type
equation just after bifurcation occurs at finite wave number. The flow term
affects the nonlinear term of the equation through the critical wave number
squared. Contrary to this, wave number isn't explicitly effective with lack of
flow or constant flow. Thus, spatial size of pattern is especially important
for regulating pattern formation in the plasmodium. On the other hand, the flow
term is negligible in the vicinity of bifurcation at infinitely small wave
number, and therefore the pattern formation by simple reaction-diffusion will
also hold. A physiological role of pattern formation as above is discussed.
