{"id":1520,"date":"2016-05-11T15:30:29","date_gmt":"2016-05-11T06:30:29","guid":{"rendered":"http:\/\/www.bi.appi.keio.ac.jp\/?p=1520"},"modified":"2016-05-14T14:35:23","modified_gmt":"2016-05-14T05:35:23","slug":"%e9%a4%8c%e3%81%a8%e6%8d%95%e9%a3%9f%e8%80%85%e3%81%ae%e9%96%a2%e4%bf%82","status":"publish","type":"post","link":"https:\/\/www.bi.appi.keio.ac.jp\/?p=1520","title":{"rendered":"\u990c\u3068\u6355\u98df\u8005\u306e\u95a2\u4fc2"},"content":{"rendered":"<p>\u7269\u7406\u60c5\u5831\u6570\u5b66B\u3067\uff0c\u884c\u5217\u306e\u7d2f\u4e57\u306e\u5fdc\u7528\u4f8b\u3068\u3057\u3066\uff0c\u990c\u3068\u6355\u98df\u8005\u306e\u95a2\u4fc2\u3092\u8868\u3059\u884c\u5217\u5dee\u5206\u65b9\u7a0b\u5f0f\u3092\u8aac\u660e\u3057\u3066\u3044\u307e\u3059\uff0e<\/p>\n<p>\u300c\u7b2c[latex]k[\/latex]\u5e74\u306e\u30a4\u30ef\u30b7\u306e\u6570\u3092[latex]x_1(k)[\/latex]\uff0c\u30af\u30b8\u30e9\u306e\u6570\u3092[latex]x_2(k)[\/latex]\u3068\u3059\u308b\uff0e\u30af\u30b8\u30e9\u304c\u3044\u306a\u3044\u3068\u304d\u306e1\u5e74\u5f8c\u306e\u30a4\u30ef\u30b7\u306e\u6570\u3092[latex]\\alpha x_1(k)[\/latex]\u3068\u3059\u308b\uff0e\u3064\u307e\u308a\uff0c\u7b2c[latex]k[\/latex]\u5e74\u306e\u30a4\u30ef\u30b7\u306e\u6570\u306b\u6bd4\u4f8b\u3057\u3066\u5909\u5316\u3059\u308b\uff0e\u30af\u30b8\u30e9\u306b1\u5e74\u9593\u306b\u98df\u3079\u3089\u308c\u308b\u30a4\u30ef\u30b7\u306e\u6570\u3092[latex]\\beta x_2(k)[\/latex]\u3068\u3059\u308b\uff0e\u3064\u307e\u308a\uff0c\u30af\u30b8\u30e9\u306e\u6570\u306b\u6bd4\u4f8b\u3059\u308b\u30a4\u30ef\u30b7\u304c\u98df\u3079\u3089\u308c\u308b\uff0e\u30a4\u30ef\u30b7\u304c\u3044\u306a\u3044\u3068\u304d\u306e1\u5e74\u5f8c\u306e\u30af\u30b8\u30e9\u306e\u6570\u3092[latex]\\gamma x_2(k)[\/latex]\u3068\u3059\u308b\uff0e\u307e\u305f\uff0c\u30a4\u30ef\u30b7\u3092\u98df\u3079\u30661\u5e74\u9593\u306b\u5897\u52a0\u3059\u308b\u30af\u30b8\u30e9\u306e\u6570\u3092[latex]\\delta x_1(k)[\/latex]\u3068\u3059\u308b\uff0e\u300d<\/p>\n<p>\u4e0a\u8a18\u306e\u95a2\u4fc2\u3092\u884c\u5217\u5dee\u5206\u65b9\u7a0b\u5f0f\u3067\u8868\u3059\u3068<\/p>\n<p>[latex]<\/p>\n<p>\\left[\\begin{array}{c}x_1(k+1) \\\\x_2(k+2)\\end{array}\\right]<\/p>\n<p>=<\/p>\n<p>\\left[\\begin{array}{rr}\\alpha &amp; -\\beta \\\\ \\delta &amp; \\gamma\\end{array}\\right]<\/p>\n<p>\\left[\\begin{array}{c}x_1(k) \\\\x_2(k)\\end{array}\\right]<\/p>\n<p>[\/latex]<\/p>\n<p>\u306b\u306a\u308a\u307e\u3059\uff0e\u4e00\u822c\u89e3\u306f<\/p>\n<p>[latex]\\boldsymbol{x}(k) = \\left[\\begin{array}{rr}\\alpha &amp; -\\beta \\\\ \\delta &amp; \\gamma\\end{array}\\right]^k\\boldsymbol{x}(0)[\/latex]<\/p>\n<p>\u306b\u306a\u308a\u307e\u3059\uff0e<\/p>\n<p>\u3053\u306e\u884c\u5217\u5dee\u5206\u65b9\u7a0b\u5f0f\u3092\u8b1b\u7fa9\u3067\u306f\u624b\u8a08\u7b97\u3067\u89e3\u3051\u308b\u3088\u3046\u306b\uff0c[latex]\\alpha=1.5[\/latex]\uff0c[latex]\\beta=0.25[\/latex]\uff0c[latex]\\gamma=1.1[\/latex]\uff0c[latex]\\delta=0.07[\/latex]\u3068\u3057\u3066\u89e3\u304d\u307e\u3059\uff0e\u3057\u304b\u3057\uff0c\u3053\u306e\u884c\u5217\u306e\u56fa\u6709\u5024\u306f1\u4ee5\u4e0a\u306e\u5b9f\u6570\u304c2\u3064\u3067\u3059\u306e\u3067\uff0c\u990c\u3068\u6355\u98df\u8005\u306e\u6570\u306e\u5909\u5316\u306b\u671f\u5f85\u3055\u308c\u308b\u632f\u52d5\u7684\u306a\u632f\u308b\u821e\u3044\u306b\u306f\u306a\u308a\u307e\u305b\u3093\uff0e\u521d\u671f\u5024\u306e\u30af\u30b8\u30e9\u306e\u6570\u304c\u591a\u3044\u3068\uff0c\u4e0b\u56f3\u306e\u3088\u3046\u306b\u30a4\u30ef\u30b7\u306e\u6570\u306f\u6e1b\u5c11\u306b\u8ee2\u3058\u307e\u3059\uff0e<\/p>\n<figure id=\"attachment_1580\" aria-describedby=\"caption-attachment-1580\" style=\"width: 320px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.bi.appi.keio.ac.jp\/?attachment_id=1580\" rel=\"attachment wp-att-1580\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1580\" src=\"https:\/\/www.bi.appi.keio.ac.jp\/wordpress\/wp-content\/uploads\/2016\/05\/Graph0.png\" alt=\"Graph0\" width=\"320\" height=\"281\" \/><\/a><figcaption id=\"caption-attachment-1580\" class=\"wp-caption-text\">\u990c\u3068\u6355\u98df\u8005\u306e\u6570\uff08x1(0)=100\uff0cx2(0)=150\uff09<\/figcaption><\/figure>\n<p>\u990c\u3068\u6355\u98df\u8005\u306e\u95a2\u4fc2\u306e\u30e2\u30c7\u30eb\u3068\u3057\u3066\u306f\uff0c\u30ed\u30c8\u30ab\u30fb\u30dc\u30eb\u30c6\u30e9\u306e\u30e2\u30c7\u30eb\u304c\u77e5\u3089\u308c\u3066\u3044\u307e\u3059\uff0e\u3053\u306e\u30e2\u30c7\u30eb\u3067\u306f\uff0c<\/p>\n<p>[latex]<\/p>\n<p>\\displaystyle\\frac{dx_1(t)}{dt} = \\alpha x_1(t) &#8211; \\beta x_1(t)x_2(t) \\\\<\/p>\n<p>\\displaystyle\\frac{dx_2(t)}{dt} = \\delta x_1(t)x_2(t) &#8211; \\gamma x_2(t)<\/p>\n<p>[\/latex]<\/p>\n<p>\u3067\u8868\u3055\u308c\u307e\u3059\uff0e\u975e\u7dda\u5f62\u5fae\u5206\u65b9\u7a0b\u5f0f\u3067\u3059\u304b\u3089\uff0c[latex]x_1(t)[\/latex]\u3068[latex]x_2(t)[\/latex]\u3092\u89e3\u6790\u7684\u306b\u6c42\u3081\u308b\u3053\u3068\u306f\u3067\u304d\u307e\u305b\u3093\uff0e\u9069\u5f53\u306a\u5024\u3092\u4ee3\u5165\u3057\u3066\u6570\u5024\u89e3\u3092\u6c42\u3081\u308b\u3068\uff0c[latex]x_1(t)[\/latex]\u3068[latex]x_2(t)[\/latex]\u304c\u632f\u52d5\u7684\u306a\u632f\u308b\u821e\u3044\u3092\u3059\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\uff0e<\/p>\n<p>\u307e\u305f\uff0c\u5e73\u8861\u70b9\u306f[latex]dx_1(t)\/dt=0[\/latex]\u3068[latex]dx_2(t)\/dt=0[\/latex]\u304b\u3089\uff0c\u539f\u70b9\u3068[latex]x_1=\\gamma\/\\delta[\/latex]\uff0c[latex]x_2=\\alpha\/\\beta[\/latex]\u306b\u3042\u308a\u307e\u3059\uff0e\u5e73\u8861\u70b9\u8fd1\u508d\u3067\u306f\uff0c[latex]x_1(t)x_2(t)[\/latex]\u306e\u9805\u304c\u5c0f\u3055\u3044\u306e\u3067\u7121\u8996\u3067\u304d\u308b\u3068\u4eee\u5b9a\u3059\u308b\u3068\uff0c[latex](0, 0)[\/latex]\u306e\u5e73\u8861\u70b9\u8fd1\u508d\u3067\u306f\uff0c<\/p>\n<p>[latex]<\/p>\n<p>\\displaystyle\\frac{d}{dt}\\left[\\begin{array}{c}x_1(t) \\\\x_2(t)\\end{array}\\right]<\/p>\n<p>=<\/p>\n<p>\\left[\\begin{array}{rr}\\alpha &amp; 0 \\\\<\/p>\n<p>0 &amp; -\\gamma\\end{array}\\right]<\/p>\n<p>\\left[\\begin{array}{c}x_1(t) \\\\x_2(t)\\end{array}\\right]<\/p>\n<p>[\/latex]<\/p>\n<p>\u3068\u306a\u308a\uff0c\u884c\u5217\u306e\u56fa\u6709\u5024\u306f\u5b9f\u6570\u30671\u3064\u306f\u6b63\u3067\u3082\u30461\u3064\u306f\u8ca0\u3067\u3059\uff0e\u3057\u305f\u304c\u3063\u3066\uff0c[latex](0, 0)[\/latex]\u306f\u978d\u70b9\u3067\u3059\uff0e<\/p>\n<p>\u307e\u305f\uff0c\u3082\u30461\u3064\u306e\u5e73\u8861\u70b9[latex](\\gamma\/\\delta, \\alpha\/\\beta)[\/latex]\u306e\u8fd1\u508d\u3067\u306f\uff0c[latex]x_1(t)[\/latex]\u3092[latex]x_1+\\gamma\/\\delta[\/latex]\uff0c[latex]x_2(t)[\/latex]\u3092[latex]x_2+\\alpha\/\\beta[\/latex]\u3068\u304a\u3044\u3066\uff0c[latex]x_1(t)x_2(t)[\/latex]\u306e\u9805\u304c\u5c0f\u3055\u3044\u306e\u3067\u7121\u8996\u3067\u304d\u308b\u3068\u4eee\u5b9a\u3059\u308b\u3068<\/p>\n<p>[latex]<\/p>\n<p>\\displaystyle\\frac{d}{dt}\\left[\\begin{array}{c}x_1(t) \\\\x_2(t)\\end{array}\\right]<\/p>\n<p>=<\/p>\n<p>\\left[\\begin{array}{rr}0 &amp; -\\frac{\\beta\\gamma}{\\delta} \\\\<\/p>\n<p>\\frac{\\alpha\\delta}{\\beta} &amp; 0\\end{array}\\right]<\/p>\n<p>\\left[\\begin{array}{c}x_1(t) \\\\x_2(t)\\end{array}\\right]<\/p>\n<p>[\/latex]<\/p>\n<p>\u3068\u8868\u3055\u308c\u307e\u3059\uff0e\u3057\u305f\u304c\u3063\u3066\uff0c\u56fa\u6709\u5024\u304c\u7d14\u865a\u6570\u3068\u306a\u308a\uff0c\u632f\u52d5\u3059\u308b\u89e3\u306b\u306a\u308a\u307e\u3059\uff0e<\/p>\n<p>[latex]\\alpha=10[\/latex]\uff0c[latex]\\beta=0.1[\/latex]\uff0c[latex]\\gamma=10[\/latex]\uff0c[latex]\\delta=0.1[\/latex]\u3068\u3057\uff0c[latex]x_1(0)=10[\/latex]\u3068[latex]x_2(0)=100[\/latex] \u3067\u6570\u5024\u89e3\u3092\u6c42\u3081\u308b\u3068\u4e0b\u56f3\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\uff0e\u4e0a\u306e\u30d1\u30cd\u30eb\u306f[latex]x_1(t)[\/latex]\u3068[latex]x_2(t)[\/latex]\u306e\u6642\u9593\u767a\u5c55\u3067\u3059\uff0e \u4e0b\u306e\u30d1\u30cd\u30eb\u306f[latex]x_1(t)[\/latex]\u3068[latex]x_2(t)[\/latex]\u306e\u95a2\u4fc2\u3067\u3059\uff0e<\/p>\n<figure id=\"attachment_1598\" aria-describedby=\"caption-attachment-1598\" style=\"width: 320px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.bi.appi.keio.ac.jp\/?attachment_id=1598\" rel=\"attachment wp-att-1586\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1598\" src=\"https:\/\/www.bi.appi.keio.ac.jp\/wordpress\/wp-content\/uploads\/2016\/05\/lv1-2.png\" width=\"320\" height=\"171\" \/><\/a><figcaption id=\"caption-attachment-1598\" class=\"wp-caption-text\">x1(t)\u3068x2(t)\u306e\u6642\u9593\u767a\u5c55<\/figcaption><\/figure>\n<figure id=\"attachment_1587\" aria-describedby=\"caption-attachment-1587\" style=\"width: 320px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.bi.appi.keio.ac.jp\/?attachment_id=1587\" rel=\"attachment wp-att-1587\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1587\" src=\"https:\/\/www.bi.appi.keio.ac.jp\/wordpress\/wp-content\/uploads\/2016\/05\/lv2-1.png\" alt=\"lv2\" width=\"320\" height=\"286\" \/><\/a><figcaption id=\"caption-attachment-1587\" class=\"wp-caption-text\">x1(t)\u3068x2(t)\u306e\u95a2\u4fc2\uff08t=0\u3067\uff0810, 100\uff09\u3068x1=0\u306e\u30cc\u30eb\u30af\u30e9\u30a4\u30f3\u306b\u8fd1\u304f\uff0cx2\u304c\u6025\u901f\u306b\u6e1b\u5c11\u3059\u308b\u304c\uff0c\u539f\u70b9\u304c\u978d\u70b9\u3067\u3042\u308b\u306e\u3067\u3042\u308b\u7a0b\u5ea6\u6e1b\u5c11\u3057\u305f\u3068\u3053\u308d\u3067x1\u304c\u6025\u901f\u306b\u5897\u52a0\u3059\u308b\uff0e\uff09<\/figcaption><\/figure>\n<p>\u521d\u671f\u5024\u3092\u5e73\u8861\u70b9\uff08(100, 100)\uff09\u306b\u8fd1\u3044\u70b9(99, 99)\u3068\u3059\u308b\u3068\u4e0b\u56f3\u306e\u3088\u3046\u306b\u5186\u306b\u8fd1\u3044\u8ecc\u9053\u3092\u63cf\u304d\u307e\u3059\uff0e<\/p>\n<figure id=\"attachment_1589\" aria-describedby=\"caption-attachment-1589\" style=\"width: 320px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.bi.appi.keio.ac.jp\/?attachment_id=1589\" rel=\"attachment wp-att-1589\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1589\" src=\"https:\/\/www.bi.appi.keio.ac.jp\/wordpress\/wp-content\/uploads\/2016\/05\/lv3.png\" alt=\"lv3\" width=\"320\" height=\"286\" \/><\/a><figcaption id=\"caption-attachment-1589\" class=\"wp-caption-text\">\u521d\u671f\u5024\u3092(99, 99)\u3068\u5e73\u8861\u70b9\uff08100, 100\uff09\u306b\u8fd1\u3044\u5024\u306b\u3057\u305f\u306e\u3067\u5186\u306b\u8fd1\u3044\u8ecc\u9053\u3092\u63cf\u304f<\/figcaption><\/figure>\n<p>MATLAB\u306e\u30b3\u30fc\u30c9\u3067\u3059\uff0e<\/p>\n<ul>\n<li>lv.m<\/li>\n<\/ul>\n<pre>tspan = [0:0.001:5];\r\n[t y] = ode45(@LotkaVolterra,tspan, [99 99]);\r\nfigure(1);\r\nplot(t, y(:,1), 'b', t, y(:,2), 'r');\r\nfigure(2);\r\nplot(y(:,1), y(:,2));<\/pre>\n<ul>\n<li>LotkaVolterra.m<\/li>\n<\/ul>\n<pre>function dx = LotkaVolterra(t, x)\r\nalpha = 10;\r\nbeta = 0.1;\r\ngamma = 10;\r\ndelta = 0.1;\r\ndx = zeros(2,1);\r\ndx(1) = alpha * x(1) - beta * x(1) * x(2);\r\ndx(2) = delta * x(1) * x(2) - gamma * x(2);\r\nend<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>\u7269\u7406\u60c5\u5831\u6570\u5b66B\u3067\uff0c\u884c\u5217\u306e\u7d2f\u4e57\u306e\u5fdc\u7528\u4f8b\u3068\u3057\u3066\uff0c\u990c\u3068\u6355\u98df\u8005\u306e\u95a2\u4fc2\u3092\u8868\u3059\u884c\u5217\u5dee\u5206\u65b9\u7a0b\u5f0f\u3092\u8aac\u660e\u3057\u3066\u3044\u307e\u3059\uff0e \u300c\u7b2c[latex]k[\/latex]\u5e74\u306e\u30a4\u30ef\u30b7\u306e\u6570\u3092[latex]x_1(k)[\/latex]\uff0c\u30af\u30b8\u30e9\u306e\u6570\u3092[latex &hellip; <a href=\"https:\/\/www.bi.appi.keio.ac.jp\/?p=1520\" class=\"more-link\"><span class=\"screen-reader-text\">&#8220;\u990c\u3068\u6355\u98df\u8005\u306e\u95a2\u4fc2&#8221; \u306e<\/span>\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[9],"tags":[],"class_list":["post-1520","post","type-post","status-publish","format-standard","hentry","category-9"],"_links":{"self":[{"href":"https:\/\/www.bi.appi.keio.ac.jp\/index.php?rest_route=\/wp\/v2\/posts\/1520","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.bi.appi.keio.ac.jp\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.bi.appi.keio.ac.jp\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.bi.appi.keio.ac.jp\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.bi.appi.keio.ac.jp\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1520"}],"version-history":[{"count":62,"href":"https:\/\/www.bi.appi.keio.ac.jp\/index.php?rest_route=\/wp\/v2\/posts\/1520\/revisions"}],"predecessor-version":[{"id":1600,"href":"https:\/\/www.bi.appi.keio.ac.jp\/index.php?rest_route=\/wp\/v2\/posts\/1520\/revisions\/1600"}],"wp:attachment":[{"href":"https:\/\/www.bi.appi.keio.ac.jp\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1520"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.bi.appi.keio.ac.jp\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1520"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.bi.appi.keio.ac.jp\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1520"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}