<\/p>\n
\u30f4\u30a7\u30a2\u30d5\u30eb\u30b9\u30c8\u306f\uff0c\u4eba\u53e3\u904e\u5bc6\u306e\u8981\u56e0\u3092\u8003\u616e\u306b\u3044\u308c\u3066\uff0c\u6b21\u306e\u3088\u3046\u306a\u4fee\u6b63\u3092\u63d0\u6848\u3057\u305f\u3002
\n\u4eba\u53e3\u306f\u7d99\u7d9a\u3057\u7d9a\u3051\u308b\u9650\u308a\u5897\u52a0\u3059\u308b\u304c\uff0c\u4e0a\u9650 (\u305d\u308c\u3092\u5b9a\u6570 \\(N_{\\max}\\) \u3068\u3057\u3088\u3046) \u304c\u3042\u308b\u3068\u3059\u308b\u3002\u305d\u3057\u3066\uff0c\u4eba\u53e3\u5909\u5316\u306f\uff0c\u6b21\u306e\u5404\u3005\u306b\u6bd4\u4f8b\u3059\u308b\u3068\u4eee\u5b9a\u3059\u308b:
<\/p>\n
- \n
- \u73fe\u5728\u306e\u4eba\u53e3 \\(N\\)<\/li>\n
- \u672a\u4f7f\u7528\u306e\u4eba\u53e3\u8cc7\u6e90\u306b\u5bfe\u3059\u308b\u5272\u5408 \\((1 – N\/N_{\\rm max})\\)<\/li>\n<\/ol>\n
\u3057\u305f\u304c\u3063\u3066\u5fae\u5206\u65b9\u7a0b\u5f0f\u306f
$$
\\frac{dN}{dt} = \\gamma N \\left(1 – \\frac{N}{N_{\\rm max}}\\right)
$$<\/p>\n\u4e21\u8fba\u3092 $N_{\\rm max}$ \u3067\u5272\u308a\uff0c$\\displaystyle y \\equiv N\/N_{\\rm max}$ \u3067\u8868\u3059\u3068<\/p>\n
$$ \\frac{dy}{dt} = \\gamma y (1 – y)$$<\/p>\n
\u3053\u308c\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5909\u6570\u5206\u96e2\u6cd5\u306b\u3088\u3063\u3066\u89e3\u3051\u308b\u3002<\/p>\n
\\begin{eqnarray}
\n\\frac{dy}{y (1-y)} &=& \\gamma dt \\\\
\n\\int\\frac{dy}{y (1-y)} &=& \\gamma \\int dt \\\\
\n\\int\\left\\{\\frac{1}{y} – \\frac{1}{y-1} \\right\\} dy &=& \\gamma \\int dt \\\\
\n\\ln \\left| \\frac{y}{y-1}\\right| &=& \\gamma t + C\\\\
\n\\therefore\\ \\ \\frac{y}{y-1} &=& e^{\\gamma t + C} \\\\
\n&\\equiv& C’ e^{\\gamma t}
\n\\end{eqnarray}<\/p>\n\\(t = t_0\\) \u306b\u304a\u3051\u308b\u4eba\u53e3\u306e\u5024\u3092 \\(N(t_0) \\equiv N_0\\) \u3068\u3059\u308b\u3068\uff0c<\/p>\n
\\begin{eqnarray}
\n\\frac{\\frac{N_0}{N_{\\rm max}}}{\\frac{N_0}{N_{\\rm max}}-1} &\\equiv& C’ e^{\\gamma t_0} \\\\
\n\\therefore\\ \\ C’ &=& \\frac{N_0}{N_0 – N_{\\rm max}} e^{-\\gamma t_0}\\\\ \\ \\\\
\n\\therefore\\ \\ y = \\frac{N}{N_{\\rm max}} &=& \\frac{C’ e^{\\gamma t}}{C’ e^{\\gamma t} -1} \\\\
\n&=& \\frac{\\frac{N_0}{N_0 – N_{\\rm max}} e^{\\gamma(t- t_0)}}{\\frac{N_0}{N_0 – N_{\\rm max}} e^{\\gamma (t-t_0)}-1}\\\\
\n\\therefore\\ \\ N(t) &=& N_{\\rm max} \\frac{N_0 }{N_0 + (N_{\\rm max}-N_0)\\,e^{-\\gamma (t-t_0)}}
\n\\end{eqnarray}<\/p>\n\u4e0b\u56f3\u3092\u307f\u308b\u3068\uff0c\uff08$N_{\\rm max}$\u00a0 \u3068 $\\gamma$ \u3092\u3046\u307e\u304f\u3042\u308f\u305b\u308b\u3053\u3068\u306b\u3088\u3063\u3066\uff09\u30f4\u30a7\u30a2\u30d5\u30eb\u30b9\u30c8\u30e2\u30c7\u30eb\u306e\u307b\u3046\u304c\uff0c\u3053\u306e\u5e74\u4ee3\u306e\u7c73\u56fd\u306e\u4eba\u53e3\u3092\u3088\u304f\u3042\u3089\u308f\u3057\u3066\u3044\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n
![\"\"](\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/ver.svg\")
<\/p>\n<\/p>\n","protected":false},"excerpt":{"rendered":"
\u5fae\u5206\u65b9\u7a0b\u5f0f\u304c\u51fa\u3066\u304f\u308b\u73fe\u5b9f\u4e16\u754c\u306e\u4e00\u5834\u9762\uff1a\u30f4\u30a7\u30a2\u30d5\u30eb\u30b9\u30c8\u306b\u3088\u308b\u4fee\u6b63\u30e2\u30c7\u30eb <\/p>\n
\u30de\u30eb\u30b5\u30b9\u30fb\u30e2\u30c7\u30eb\u306f\uff0c\\(\\gamma > 0\\) \u306e\u5834\u5408\u306b\u306f\u4eba\u53e3\u306e\u969b\u9650\u306a\u3044\u5897\u52a0\u3092\u4e88\u6e2c\u3059\u308b\u3002\u3057\u304b\u3057\uff0c\u5b9f\u969b\u306b\u306f\u98df\u7ce7\u8cc7\u6e90\u306e\u4f9b\u7d66\u4e0d\u8db3\uff0c\u4eba\u53e3\u306e\u904e\u5bc6\uff0c\u305d\u306e\u4ed6\u306e\u74b0\u5883\u7684\u8981\u56e0\u306b\u3088\u308a\uff0c\u3053\u306e\u3088\u3046\u306a\u7121\u5236\u9650\u306a\u5897\u52a0\u306f\u7d9a\u304b\u306a\u3044\u3002<\/p>\n
- \n
- \u3010\u53c2\u8003\u3011 \u300c\u5fae\u5206\u65b9\u7a0b\u5f0f\u3067\u6570\u5b66\u30e2\u30c7\u30eb\u3092\u4f5c\u308d\u3046 \u30d0\u30fc\u30b8\u30a7\u30b9\uff0c\u30dc\u30ea\u30fc\u8457\uff0c\u65e5\u672c\u8a55\u8ad6\u793e\u300d<\/li>\n<\/ul>\n
<\/p>\n
\u30f4\u30a7\u30a2\u30d5\u30eb\u30b9\u30c8\u306f\uff0c\u4eba\u53e3\u904e\u5bc6\u306e\u8981\u56e0\u3092\u8003\u616e\u306b\u3044\u308c\u3066\uff0c\u6b21\u306e\u3088\u3046\u306a\u4fee\u6b63\u3092\u63d0\u6848\u3057\u305f\u3002<\/p>
- \u3010\u53c2\u8003\u3011 \u300c\u5fae\u5206\u65b9\u7a0b\u5f0f\u3067\u6570\u5b66\u30e2\u30c7\u30eb\u3092\u4f5c\u308d\u3046 \u30d0\u30fc\u30b8\u30a7\u30b9\uff0c\u30dc\u30ea\u30fc\u8457\uff0c\u65e5\u672c\u8a55\u8ad6\u793e\u300d<\/li>\n<\/ul>\n