数理解析研究所講究録第1940巻

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$1940 2015 101-109 101 Formation mechanism and dynamics of localized bioconvection by photosensitive microorganisms $A$, $A$ Erika Shoji, Nobuhiko $Suematsu^{A}$, Shunsuke Izumi, Hiraku Nishimori,$

1 Formation mechanism and dynamics of localized bioconvection by photosensitive microorganisms $A$, $A$ Erika Shoji, Nobuhiko $Suematsu^{A}$, Shunsuke Izumi, Hiraku Nishimori, Akinori Awazu, Makoto Iima Hiroshima University, Meiji $University^{A}$ 1 Introduction (gravitaxis)[l] gyrotaxis $[$2, $3]$ (phototaxis)[3, 4, 5] (Euglena gracilis) 10 $\mu$m $50$ $100\mu m$ $100W/m^{2}$ ( ) $200W/m^{2}$ ()

2 102 (step-down photophobic response) (step-up photophobic response) [6] ( 1) 1: ( ) [7] [10] 1 traveling wave 2

$103 ( ) 2 [8] $I$ $T(I)$ [11] ( :1 $\mu$l $:30mm$) ( ):5.$

3 103 ( ) 2 [8] $I$ $T(I)$ [11] ( :1 $\mu$l $:30mm$) ( ):5.0 $=$ $\cross$ $\Delta S$ $10^{5}cells/ml)$ $10mm$ 30 $T(=30)$ $(n_{+})$ $(n_{-})$ $\triangle S$ $(n_{+}-n_{-})/(t\triangle S)$ ( $=$ / ) 2 $(n_{+}-n_{-})/t\triangle S$ $T(I)$ $400lux$ $T(I)$ $400lux$ $400lux$ 3 2 Vincent and Hill [8] (4 )

4 $\approx\triangleleft\infty$ 104 $\vee-+$ $ (n $ light intensity(lx) 2: $T(I)$ 3.1 $($ $:1\mu$ l $:30mm)$ $($ $:2.5\cross 10^{5}cell_{\mathcal{S}}/ml)$ $(x=0)$ ( 1) ( 2) $x=0$ $( x <\Delta)$ 30 1( ) 2( ) ( 1 n1 2 $n_{2}$ ) $J$ $J=J_{p}+D \frac{dn}{dx}$ $J=0$ $[-\triangle, \triangle]$ $J$ $0= \int_{-\delta}^{\delta}jdx = \int_{-\delta}^{\delta}\sqrt{}dxp-d(n(\triangle)-n(-\triangle))$ (1) $= \int_{-\delta}^{\delta}j_{p}dx-d(n_{2}-n_{1})$ (2) $ x <\Delta$ $\sqrt{}p$ $\sqrt-p$ $0=2\triangle\overline{J_{p}}-D\triangle n, \Delta n=n_{2}-n_{1}$ (3) $\overline{j_{p}}$ $\triangle n$ $\triangle n$ $\Delta I=I_{1}-I_{2}$ 3.2 $3$ $4$, 680 $\overline{i}=160$ $\Delta I$ $\triangle n$ (4 $160lux$ $680lux$

5 $\vee\overline{\underline{ 6}}I$ n}{\overline{n}}$ 105 ) $\triangle I$ $\frac{\delta $\overline{\sqrt{}p}$ $\overline{n}\nabla I\ovalbox{\tt\small REJECT}$ $I=160 I=6S0$ - $\backslash$ $\wedge-$ difference of light intensity(lx) difference of light intensity(lx) 3: $160lux$ 4: $680lux$ 4 [8] ( $5$ 6) ( 10 ) $\frac{\partial n}{\partial t}+\frac{\partial J}{\partial x}=0$ (4) (3 ) J $=$ ( ) $+$ ( ) $J=D(f(I)n \frac{\partial I}{\partial x}-\frac{\partial n}{\partial x})$ (5)

$I}{\partial x}$ 106 5: 6: $J=0$ $\frac{n_{x}}{n}=f(i)\frac{\partial I}{\partial x}$ (6)$ $5\cross 10^{5}$ cells/ml) 5( $\triangle$x 2)mm 5mm $1$mm $+$ 2 ( ) $\triangle x$ $\Delta$

6 I}{\partial x}$ 106 5: 6: $J=0$ $\frac{n_{x}}{n}=f(i)\frac{\partial I}{\partial x}$ (6) $\frac{\partial $=\gamma$( ) $f(i)=_{n}^{n_{a}} \frac{1}{\gamma}-$ $n(x)$ $f(i)$ ( ) 4.1 $x$ ( $0.5\cross 10^{5}$ cells/ml) 5( $\triangle$x 2)mm 5mm $1$mm $+$ 2 ( ) $\triangle x$ $\Delta x=5mm$ $1$mm $\triangle x$ ,8 $n_{g}(i)$ $n(i)=a\exp(-b(x-c)^{2})+d$ (7)

7 $\frac{>}{\overline{q^{)}q-\infty}})$ $a\overline{e_{\underline{\#}}\xi}$ $\frac{\geq}{\overline{\frac{\infty}{\approx Q)}}}\backslash$ $\overline{\underline{g_{3}\xi}}$ 107 $a= ,$ $b= \cross 10^{-5},$ $c= ,$ $d= $ $n_{w}$ ng $f(i)= \frac{1}{\gamma}\frac{-2ab(i-c)\exp(-b(i-c)^{2})}{a\exp(-b(i-c)^{2}+d)}$ (8) 9 $305lux$ $305lux$ $1000lux$ $0$ tted function $n_{g}$ fitted function $n_{w}$ $ $ $0150.2$ $/\}^{:^{=}}f_{\phi}^{fi}$ $\#_{\ovalbox{\tt\small REJECT}\cdot\cdots n\ldots \mathbb{a}\ldots\#}\backslash.\ldots\#\cdots\varphi$ $\backslash..\},..,\cdots\{\cdots\cdots\ldots\cdots i\cdots\cdots\iota\ldots\ldots\iota\ldots..\sim 4$ 0. $05$ 0. $05$ $0$ $0$ $ $ light intensity (1x) light intensity (1x) 7: 8: 5 2 $T(I)$ $400lux$ 3 2 I$ $\triangle n$ 4 $\triangle

8 108 $ $ light intensity (lux) 9: $f(i)$ $300lux$ $1000lux$ $0$ ( ) CREST(PJ ) [1] Y.Mogani, A.Yamane, A.Gino, amd S.A.Baba, J.Exp.Biol.207,3349(2004). [2] A.Kage, C.Hosoya, S.A.Baba, and M.Mogami, J.Exp.Biol.216,4457(2013). [3] C.R.Williams and M.A.Bees, J.Exp.Biol.214,2398(2001). [4] S.Ghorai, M.K.Panda, and N.A.Hill,Phys.Fluids 22,071901(2010). [5] S.Ghorai and M.K.Panda,Eur.J.Mech. $B$ 41,81(2013). [6] (1989). [7] N. J. Suematsu, A. Awazu, S. Izumi, S. Noda, S. Nakata and H. Nishimori,J.Phys. Soc. Jpn (2011).

9 109 [8] R.V.Vincent, and N.A.Hill, J. Fluid Mech.300(1996). [9] N.A.Hill, T.J.Pedley, Fluid Dyn. ${\rm Res}$., 37(2005). [10] E. Shoji, H. Nishimori, A. Awazu, S. Izumi, M. Iima, J.Phys. Soc. Jpn (2014). [11] D.P.H\"ader, Arch. Macrobiol. 147(1987).

Title 微生物の局在対流形成機構に関する光走性の数理モデル ( 流体と気体の数学解析 ) Author(s) 飯間, 信 Citation 数理解析研究所講究録 (2016), 1985: Issue Date URL

Title 微生物の局在対流形成機構に関する光走性の数理モデル ( 流体と気体の数学解析 ) Author(s) 飯間, 信 Citation 数理解析研究所講究録 (2016), 1985: 138-143 Issue Date 2016-04 URL http://hdl.handle.net/2433/224511 Right Type Departmental Bulletin Paper

数理解析研究所講究録 第1940巻

数理解析研究所講究録第1940巻