{"id":32802,"date":"2022-03-15T13:00:55","date_gmt":"2022-03-15T13:00:55","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=32802"},"modified":"2025-04-01T18:37:01","modified_gmt":"2025-04-01T18:37:01","slug":"%e5%b8%b8%e5%be%ae%e5%88%86%e6%96%b9%e7%a8%8b%e5%af%bc%e8%ae%ba","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/zh\/%e5%b8%b8%e5%be%ae%e5%88%86%e6%96%b9%e7%a8%8b%e5%af%bc%e8%ae%ba\/","title":{"rendered":"\u5e38\u5fae\u5206\u65b9\u7a0b\u5bfc\u8bba"},"content":{"rendered":"<style>\np, ul, ol{\ntext-align: justify;\n}\nh1{\ntext-align:center;\ntext-transform: uppercase;\n}\nh2{\ntext-align:center;\ntext-transform: uppercase;\nfont-size:24pt;\n}\nh3 { \n    text-align: center;\n    text-transform: uppercase;\n    font-size: 24px !important;\n}\n<\/style>\n<h1>\u5e38\u5fae\u5206\u65b9\u7a0b\u5bfc\u8bba<\/h1>\n<p style=\"text-align:center;\"><em>\u672c\u8bfe\u7a0b\u8be6\u7ec6\u63a2\u8ba8\u4e86\u63a7\u5236\u8fd9\u4e9b\u65b9\u7a0b\u53ca\u5176\u5728\u591a\u4e2a\u9886\u57df\u4e2d\u5e94\u7528\u7684\u57fa\u672c\u601d\u60f3\u3002\u4ece\u5206\u6790\u6211\u4eec\u5468\u56f4\u4e16\u754c\u4e2d\u6301\u7eed\u53d8\u5316\u7684\u672c\u8d28\u5f00\u59cb\uff0c\u4ecb\u7ecd\u4e86\u51fd\u6570\u3001\u5bfc\u6570\u7b49\u57fa\u672c\u6982\u5ff5\uff0c\u4ee5\u53ca\u5b83\u4eec\u4e0e\u8fde\u7eed\u548c\u79bb\u6563\u53d8\u5316\u7684\u5173\u7cfb\u3002\u5f15\u5165\u504f\u5fae\u5206\u65b9\u7a0b\uff08PDE\uff09\u4e0e\u5e38\u5fae\u5206\u65b9\u7a0b\uff08ODE\uff09\u4e4b\u95f4\u7684\u533a\u522b\uff0c\u8bfe\u7a0b\u91cd\u70b9\u653e\u5728\u5e38\u5fae\u5206\u65b9\u7a0b\u7684\u7814\u7a76\u4e0a\u3002\u901a\u8fc7\u5b9e\u9645\u4f8b\u5b50\u8fdb\u884c\u8bf4\u660e\uff0c\u5982\u5496\u5561\u51b7\u5374\u3001\u725b\u987f\u5b9a\u5f8b\u548c\u4eba\u53e3\u6a21\u578b\u3002\u5b66\u751f\u5c06\u6709\u673a\u4f1a\u719f\u6089\u63a7\u5236\u81ea\u7136\u4e0e\u7269\u7406\u73b0\u8c61\u7684\u5fae\u5206\u65b9\u7a0b\uff0c\u4e86\u89e3\u5982\u4f55\u7528\u6570\u5b66\u65b9\u5f0f\u8868\u793a\u8fd9\u4e9b\u73b0\u8c61\uff0c\u5e76\u638c\u63e1\u7814\u7a76\u5176\u89e3\u6cd5\u7684\u4e00\u4e9b\u6280\u672f\u3002\u8fd9\u4e9b\u521d\u6b65\u77e5\u8bc6\u5c06\u6210\u4e3a\u4eca\u540e\u5728\u79d1\u5b66\u4e0e\u5de5\u7a0b\u4e2d\u6df1\u5165\u7814\u7a76\u5fae\u5206\u65b9\u7a0b\u53ca\u5176\u5e94\u7528\u7684\u57fa\u7840\u3002<\/em><\/p>\n<p style=\"text-align:center;\"><strong><u>\u5b66\u4e60\u76ee\u6807<\/u>\uff1a<\/strong><br \/>\u5b8c\u6210\u672c\u8bfe\u7a0b\u540e\uff0c\u5b66\u751f\u5c06\u80fd\u591f\uff1a<\/p>\n<ol>\n<li><strong>\u7406\u89e3<\/strong> \u4e0e\u5fae\u5206\u65b9\u7a0b\u76f8\u5173\u7684\u57fa\u672c\u6982\u5ff5\uff0c\u4f8b\u5982\u53d8\u5316\u7684\u672c\u8d28\u3001\u51fd\u6570\u3001\u5bfc\u6570\uff0c\u4ee5\u53ca\u504f\u5fae\u5206\u65b9\u7a0b\uff08PDE\uff09\u4e0e\u5e38\u5fae\u5206\u65b9\u7a0b\uff08ODE\uff09\u4e4b\u95f4\u7684\u5dee\u5f02\u3002<\/li>\n<\/ul>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/bYwm6NAEvVA\" title=\"YouTube \u89c6\u9891\u64ad\u653e\u5668\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><br \/>\n<center><\/p>\n<p style=\"text-align:center;\"><strong>\u76ee\u5f55<\/strong><br \/>\n<a href=\"#LasEcuacionesDiferencialesYLaNaturalezaDeLasCosas\"><strong>\u5fae\u5206\u65b9\u7a0b\u4e0e\u4e8b\u7269\u7684\u672c\u8d28<\/strong><\/a><br \/>\n<a href=\"#ElCambioIncesante\">\u6301\u7eed\u53d8\u5316<\/a><br \/>\n<a href=\"#FuncionesDerivadasYSusCambios\">\u51fd\u6570\u3001\u5bfc\u6570\u4e0e\u5176\u53d8\u5316<\/a><br \/>\n<a href=\"#EDOyEDP\">\u5e38\u5fae\u5206\u65b9\u7a0b\u4e0e\u504f\u5fae\u5206\u65b9\u7a0b<\/a><br \/>\n<a href=\"#EjemplosDeEcuacionesDiferencialesOrdinarias\"><strong>\u5e38\u5fae\u5206\u65b9\u7a0b\u5b9e\u4f8b<\/strong><\/a><br \/>\n<a href=\"#ElEnfriamientoDeUnaTazaDeCafe\">\u5496\u5561\u676f\u7684\u51b7\u5374<\/a><br \/>\n<a href=\"#LasLeyesDeNewton\">\u725b\u987f\u5b9a\u5f8b<\/a><br \/>\n<a href=\"#ModeloDePoblaciones\">\u4eba\u53e3\u6a21\u578b<\/a>\n<\/p>\n<p><\/center><\/p>\n<p><a name=\"LasEcuacionesDiferencialesYLaNaturalezaDeLasCosas\"><\/a><br \/>\n<center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/KgUDA2Q1qaA\" title=\"YouTube \u89c6\u9891\u64ad\u653e\u5668\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n<h2>\u5fae\u5206\u65b9\u7a0b\u4e0e\u4e8b\u7269\u7684\u672c\u8d28<\/h2>\n<p><a name=\"ElCambioIncesante\"><\/a><\/p>\n<h3>\u6301\u7eed\u53d8\u5316<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=KgUDA2Q1qaA&#038;t=133s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u5728\u81ea\u7136\u754c\u4e2d\uff0c\u4e00\u5207\u90fd\u5728\u4e0d\u65ad\u53d8\u5316\u3002<\/span><\/strong><\/a> \u5373\u4f7f\u662f\u90a3\u4e9b\u770b\u8d77\u6765\u6c38\u8fdc\u4e0d\u4f1a\u6539\u53d8\u7684\u4e8b\u7269\uff0c\u6bd4\u5982\u592a\u9633\u7684\u5149\u8f89\uff0c\u5982\u679c\u5728\u5408\u9002\u7684\u65f6\u95f4\u5c3a\u5ea6\u4e0a\u89c2\u5bdf\uff0c\u5b83\u4e5f\u4f1a\u53d8\u5316\u3002\u4e00\u5207\u90fd\u5728\u53d8\u5316\uff1a\u661f\u661f\u7684\u4eae\u5ea6\u3001\u4e00\u676f\u5496\u5561\u7684\u6e29\u5ea6\u3001\u7269\u4f53\u7684\u4f4d\u7f6e\uff0c\u4ee5\u53ca\u4eba\u53e3\u7684\u6570\u91cf\u90fd\u662f\u4f8b\u5b50\uff0c\u800c\u8fd9\u4e9b\u53d8\u5316\u7387\u901a\u5e38\u4e0e\u6b63\u5728\u53d1\u751f\u53d8\u5316\u7684\u72b6\u6001\u6709\u5173\u3002<\/p>\n<p>\u4e00\u79cd\u76f4\u89c2\u7406\u89e3\u53d8\u5316\u7684\u65b9\u5f0f\u662f\u89c2\u5bdf\u4e8b\u7269\u968f\u65f6\u95f4\u5982\u4f55\u53d8\u5316\u3002\u76f8\u5bf9\u4e8e\u65f6\u95f4\u53d1\u751f\u7684\u53d8\u5316\u88ab\u79f0\u4e3a\u6f14\u5316\uff0c\u800c\u6211\u4eec\u80fd\u89c2\u5bdf\u5230\u7684\u4e00\u5207\u90fd\u5728\u6301\u7eed\u6f14\u5316\u3002\u4f46\u6f14\u5316\u4e0d\u662f\u552f\u4e00\u7684\u53d8\u5316\u5f62\u5f0f\uff1b\u4f8b\u5982\uff0c\u867d\u7136\u6211\u4eec\u76f8\u5bf9\u4e8e\u6d77\u5e73\u9762\u7684\u9ad8\u5ea6\u53ef\u80fd\u4f1a\u968f\u7740\u65f6\u95f4\u800c\u53d8\u5316\uff0c\u4f46\u66f4\u6709\u53ef\u80fd\u662f\u6839\u636e\u6211\u4eec\u7684\u4f4d\u7f6e\uff08\u6216\u5730\u7406\u5750\u6807\uff09\u800c\u53d8\u5316\u3002<\/p>\n<p><a name=\"FuncionesDerivadasYSusCambios\"><\/a><\/p>\n<h3>\u51fd\u6570\u3001\u5bfc\u6570\u4e0e\u5176\u53d8\u5316<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=KgUDA2Q1qaA&#038;t=301s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u66f4\u4e00\u822c\u5730\u8bf4\uff0c<\/span><\/strong><\/a> \u4e00\u4e2a\u591a\u53d8\u91cf\u51fd\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x_1,x_2, \\cdots, x_n)<\/span><\/span> \u82e5\u5176\u4e2d\u67d0\u4e2a\u53d8\u91cf\u53d1\u751f\u53d8\u5316\uff0c\u8be5\u51fd\u6570\u4e5f\u4f1a\u53d1\u751f\u53d8\u5316\uff0c\u8fd9\u79cd\u53d8\u5316\u53ef\u4ee5\u662f\u8fde\u7eed\u7684\uff0c\u4e5f\u53ef\u4ee5\u662f\u79bb\u6563\u7684\u3002\u5bf9\u4e8e\u591a\u53d8\u91cf\u51fd\u6570\uff0c\u8fde\u7eed\u53d8\u5316\u53ef\u4ee5\u901a\u8fc7<strong>\u504f\u5bfc\u6570<\/strong>\u6765\u7814\u7a76\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{\\partial f(x_1, \\cdots, x_n)}{\\partial x_1} = \\lim_{\\Delta x_1 \\to 0} \\frac{ f(x_1 + \\Delta x_1, \\cdots, x_n) -  f(x_1, \\cdots, x_n)}{\\Delta x_1} <\/span>\n<p>\u5982\u679c\u51fd\u6570\u53ea\u6709\u4e00\u4e2a\u53d8\u91cf\uff0c\u5219\u4f7f\u7528<strong>\u666e\u901a\u5bfc\u6570<\/strong>\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{df(x)}{dx} = \\lim_{\\Delta x \\to 0} \\frac{ f(x + \\Delta x) -  f(x)}{\\Delta x} <\/span>\n<p>\u5982\u679c\u53d8\u5316\u662f\u79bb\u6563\u7684\u800c\u975e\u8fde\u7eed\u7684\uff0c\u5219\u7701\u7565\u5bfc\u6570\u4e2d\u51fa\u73b0\u7684\u6781\u9650\u8fd0\u7b97\u3002<\/p>\n<p><a name=\"EDOyEDP\"><\/a><\/p>\n<h3>\u5e38\u5fae\u5206\u65b9\u7a0b\u4e0e\u504f\u5fae\u5206\u65b9\u7a0b<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=KgUDA2Q1qaA&#038;t=624s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u5305\u542b\u4e00\u4e2a\u51fd\u6570\u53ca\u5176\u5bfc\u6570\u7684\u65b9\u7a0b<\/span><\/strong><\/a> \u88ab\u79f0\u4e3a<strong>\u5fae\u5206\u65b9\u7a0b<\/strong>\u3002\u5982\u679c\u6d89\u53ca\u7684\u662f\u504f\u5bfc\u6570\u6216\u666e\u901a\u5bfc\u6570\uff0c\u5206\u522b\u79f0\u4e3a<strong>\u504f\u5fae\u5206\u65b9\u7a0b\uff08PDE\uff09<\/strong>\u6216<strong>\u5e38\u5fae\u5206\u65b9\u7a0b\uff08ODE\uff09<\/strong>\u3002\u672c\u8bfe\u7a0b\u5f53\u524d\u805a\u7126\u4e8e\u5e38\u5fae\u5206\u65b9\u7a0b\u7684\u7814\u7a76\uff0c\u5e76\u5c06\u56de\u987e\u51e0\u4e2a\u76f8\u5173\u7684\u5b9e\u4f8b\u3002<\/p>\n<p><a name=\"EjemplosDeEcuacionesDiferencialesOrdinarias\"><\/a><\/p>\n<h2>\u5e38\u5fae\u5206\u65b9\u7a0b\u5b9e\u4f8b<\/h2>\n<p><a name=\"ElEnfriamientoDeUnaTazaDeCafe\"><\/a><\/p>\n<h3>\u5496\u5561\u676f\u7684\u51b7\u5374<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=KgUDA2Q1qaA&#038;t=680s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u5496\u5561\u51b7\u5374\u7684\u901f\u7387\u4e0e\u5176\u4e0e\u73af\u5883\u4e4b\u95f4\u7684\u6e29\u5dee\u6210\u6b63\u6bd4\u3002<\/span><\/strong><\/a> \u5982\u679c\u7a7a\u6c14\u7684\u6e29\u5ea6 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T_a<\/span><\/span> \u662f\u6052\u5b9a\u7684\uff0c\u800c\u5496\u5561\u7684\u6e29\u5ea6\u662f\u65f6\u95f4\u7684\u51fd\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T_c=T_c(t),<\/span><\/span>\uff0c\u6211\u4eec\u53ef\u4ee5\u5efa\u7acb\u4e00\u4e2a\u5fae\u5206\u65b9\u7a0b\u6765\u63cf\u8ff0\u5496\u5561\u5728\u4efb\u4e00\u65f6\u523b\u7684\u6e29\u5ea6\u3002\u521d\u59cb\u65b9\u7a0b\u4e3a\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{dT_c(t)}{dt} = -\\alpha^2(T_c(t) - T_a) <\/span>\n<p>\u5176\u4e2d <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> \u662f\u6bd4\u4f8b\u5e38\u6570\uff0c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T_a \\lt T_c(t)<\/span><\/span> \u4e14\u8d1f\u53f7\u8868\u793a\u5496\u5561\u6e29\u5ea6\u6b63\u5728\u964d\u4f4e\u3002\u7a0d\u540e\u6211\u4eec\u4f1a\u770b\u5230\uff0c\u8fd9\u4e2a\u65b9\u7a0b\u7684\u89e3\u5f62\u5f0f\u4e3a\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T_c(t) = T_a + Be^{-\\alpha^2 t}<\/span>\n<p>\u5176\u4e2d <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">B<\/span><\/span> \u662f\u4e00\u4e2a\u9700\u8981\u786e\u5b9a\u7684\u5e38\u6570\u3002<\/p>\n<p><a name=\"LasLeyesDeNewton\"><\/a><\/p>\n<h3>\u725b\u987f\u5b9a\u5f8b<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=KgUDA2Q1qaA&#038;t=885s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u725b\u987f\u7b2c\u4e8c\u5b9a\u5f8b\u672c\u8d28\u4e0a\u662f\u4e00\u4e2a\u5e38\u5fae\u5206\u65b9\u7a0b\uff0c<\/span><\/strong><\/a> \u56e0\u4e3a\u5728\u8868\u8fbe\u5f0f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F=ma<\/span><\/span>\uff08\u529b\u7b49\u4e8e\u8d28\u91cf\u4e58\u4ee5\u52a0\u901f\u5ea6\uff09\u4e2d\uff0c\u52a0\u901f\u5ea6 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a=d^2x(t)\/dt^2,<\/span><\/span> \u662f\u7269\u4f53\u4f4d\u7f6e\u5173\u4e8e\u65f6\u95f4\u7684\u4e8c\u9636\u5bfc\u6570\u3002\u901a\u8fc7\u8be5\u5b9a\u5f8b\uff0c\u6211\u4eec\u53ef\u4ee5\u627e\u5230\u63cf\u8ff0\u7269\u4f53\u8fd0\u52a8\u7684\u5173\u7cfb\u5f0f\uff0c\u8fd9\u4e9b\u5173\u7cfb\u5f0f\u672c\u8d28\u4e0a\u5c31\u662f\u5fae\u5206\u65b9\u7a0b\u3002\u4e00\u4e2a\u7b80\u5355\u7684\u4f8b\u5b50\u662f\u5f39\u7c27\u7814\u7a76\uff1a\u82e5\u6709\u4e00\u7aef\u56fa\u5b9a\u5728\u5899\u4e0a\u7684\u5f39\u7c27\uff0c\u53e6\u4e00\u7aef\u8fde\u63a5\u8d28\u91cf\u5757\uff0c\u7cfb\u7edf\u5904\u4e8e\u5e73\u8861\u4f4d\u7f6e\u65f6\uff0c\u6211\u4eec\u5c06\u8d28\u91cf\u5757\u79fb\u52a8\u8ddd\u79bb <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span>\uff0c\u6839\u636e\u80e1\u514b\u5b9a\u5f8b\uff0c\u8be5\u8d28\u91cf\u5757\u5c06\u53d7\u5230\u56de\u590d\u529b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F=-kx<\/span><\/span>\u3002\u7136\u540e\uff0c\u6839\u636e\u725b\u987f\u7b2c\u4e8c\u5b9a\u5f8b\uff0c\u6709\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle -kx(t) = m\\frac{d^2x(t)}{dt^2} <\/span>\n<p>\u4e4b\u540e\u6211\u4eec\u5c06\u770b\u5230\uff0c\u5176\u89e3\u4e3a\u5982\u4e0b\u5f62\u5f0f\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle x(t) = A\\sin\\left(\\sqrt{\\frac{k}{m}}t + \\phi \\right)<\/span>\n<p>\u5176\u4e2d <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> \u548c <span class=\"katex-eq\" data-katex-display=\"false\">\\phi<\/span> \u662f\u7531<strong>\u95ee\u9898\u7684\u521d\u59cb\u6761\u4ef6<\/strong>\u51b3\u5b9a\u7684\u5e38\u6570\u3002<\/p>\n<p><a name=\"ModeloDePoblaciones\"><\/a><\/p>\n<h3>\u4eba\u53e3\u6a21\u578b<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=KgUDA2Q1qaA&#038;t=1184s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u6bcf\u4f4d\u4e2a\u4f53\u7684\u4eba\u53e3\u589e\u957f\u7387<\/span><\/strong><\/a> \u7b49\u4e8e\u51fa\u751f\u7387\u4e0e\u6b7b\u4ea1\u7387\u4e4b\u5dee\uff0c\u5373\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{1}{x(t)} \\frac{dx(t)}{dt} = N - M<\/span>\n<p>\u5982\u679c\u51fa\u751f\u7387 <span class=\"katex-eq\" data-katex-display=\"false\">N<\/span> \u968f\u65f6\u95f4\u4fdd\u6301\u6052\u5b9a\uff0c\u800c\u6b7b\u4ea1\u7387\u4e0e\u4eba\u53e3\u6210\u6b63\u6bd4\uff0c\u5373 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">M=\\alpha^2 x(t),<\/span><\/span>\uff0c\u90a3\u4e48\u4e0a\u5f0f\u53ef\u8f6c\u5316\u4e3a\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{dx(t)}{dt} = x(t) (N - \\alpha^2 x(t))<\/span>\n<p>\u8fd9\u88ab\u79f0\u4e3a<strong>\u201c\u4eba\u53e3\u7684\u903b\u8f91\u65b9\u7a0b\u201d<\/strong>\u3002\u57fa\u4e8e\u8be5\u65b9\u7a0b\uff0c\u53ef\u4ee5\u63a8\u5e7f\u4e3a\u591a\u4e2a\u79cd\u7fa4 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_1(t), x_2(t), \\cdots, x_n(t)<\/span><\/span> \u76f8\u4e92\u7ade\u4e89\u7684\u6a21\u578b\uff0c\u8868\u793a\u5982\u4e0b\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{dx_i(t)}{dt} = x_i(t) \\left(N_i - \\displaystyle \\sum_{j=1}^n\\alpha^2_{ij} x_j(t)  \\right)<\/span>\n<p>\u5176\u4e2d <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">i\\in\\{1,\\cdots, n\\}<\/span><\/span>\u3002\u8fd9\u5c31\u662f\u8457\u540d\u7684<strong>\u6d1b\u7279\u5361-\u6c83\u5c14\u6cf0\u62c9\u65b9\u7a0b\u7ec4\uff08Lotka-Volterra \u65b9\u7a0b\uff09<\/strong>\u3002<\/p>\n<h2>\u7ed3\u8bba<\/h2>\n<p>\u5728\u672c\u6b21\u5e38\u5fae\u5206\u65b9\u7a0b\u5bfc\u8bba\u4e2d\uff0c\u6211\u4eec\u63a2\u8ba8\u4e86\u6570\u5b66\u5982\u4f55\u7cbe\u786e\u800c\u4f18\u96c5\u5730\u63cf\u7ed8\u81ea\u7136\u754c\u4e2d\u7684\u53d8\u5316\u3002\u4ece\u4e00\u676f\u5496\u5561\u7684\u51b7\u5374\uff0c\u5230\u5f39\u7c27\u7684\u8fd0\u52a8\uff0c\u518d\u5230\u4eba\u53e3\u7684\u589e\u957f\uff0c\u5e38\u5fae\u5206\u65b9\u7a0b\u8ba9\u6211\u4eec\u80fd\u5c06\u590d\u6742\u7684\u52a8\u6001\u8fc7\u7a0b\u8f6c\u5316\u4e3a\u53ef\u7406\u89e3\u3001\u53ef\u5206\u6790\u7684\u6570\u5b66\u5173\u7cfb\u3002<\/p>\n<p>\u7406\u89e3\u8fd9\u4e9b\u65b9\u7a0b\u7684\u7ed3\u6784\u548c\u542b\u4e49\uff0c\u4e3a\u8fdb\u5165\u7269\u7406\u5b66\u3001\u751f\u7269\u5b66\u3001\u7ecf\u6d4e\u5b66\u548c\u5de5\u7a0b\u5b66\u7b49\u591a\u4e2a\u5b66\u79d1\u63d0\u4f9b\u4e86\u91cd\u8981\u57fa\u7840\u3002\u672c\u8bfe\u5960\u5b9a\u4e86\u7ee7\u7eed\u6df1\u5165\u5b66\u4e60\u7684\u6982\u5ff5\u6846\u67b6\uff0c\u5728\u540e\u7eed\u8bfe\u7a0b\u4e2d\u6211\u4eec\u5c06\u6df1\u5165\u89e3\u51b3\u6280\u672f\u3001\u5b9a\u6027\u5206\u6790\u53ca\u6570\u503c\u65b9\u6cd5\u7b49\u5185\u5bb9\u3002\u7136\u800c\u6700\u91cd\u8981\u7684\u662f\uff0c\u6211\u4eec\u521d\u6b65\u5efa\u7acb\u4e86\u4e00\u4e2a\u76f4\u89c9\uff1a\u53d8\u5316\u7684\u8bed\u8a00\u2014\u2014\u5fae\u5206\u65b9\u7a0b\u2014\u2014\u4f7f\u6211\u4eec\u80fd\u591f\u63cf\u8ff0\u3001\u7406\u89e3\u5e76\u9884\u6d4b\u52a8\u6001\u7cfb\u7edf\u7684\u884c\u4e3a\u3002<\/p>\n<p>\u5728\u63a5\u4e0b\u6765\u7684\u8bfe\u7a0b\u4e2d\uff0c\u6211\u4eec\u5c06\u7ee7\u7eed\u5f00\u53d1\u66f4\u5f3a\u5927\u7684\u5de5\u5177\uff0c\u5e76\u5c06\u5176\u5e94\u7528\u4e8e\u65b0\u7684\u80cc\u666f\u4e2d\u3002\u5fae\u5206\u65b9\u7a0b\u4e0d\u4ec5\u4e3a\u6211\u4eec\u63d0\u4f9b\u4e86\u89e3\u6790\u73b0\u5b9e\u7684\u65b9\u6cd5\uff0c\u4e5f\u8ba9\u6211\u4eec\u80fd\u591f\u8bbe\u60f3\u5728\u4e0d\u540c\u6761\u4ef6\u4e0b\u7cfb\u7edf\u53ef\u80fd\u5982\u4f55\u6f14\u5316\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u5e38\u5fae\u5206\u65b9\u7a0b\u5bfc\u8bba 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\u5b66\u4e60\u76ee\u6807\uff1a\u5b8c\u6210\u672c\u8bfe\u7a0b\u540e\uff0c\u5b66\u751f\u5c06\u80fd\u591f\uff1a \u7406\u89e3 \u4e0e\u5fae\u5206\u65b9\u7a0b\u76f8\u5173\u7684\u57fa\u672c\u6982\u5ff5\uff0c\u4f8b\u5982\u53d8\u5316\u7684\u672c\u8d28\u3001\u51fd\u6570\u3001\u5bfc\u6570\uff0c\u4ee5\u53ca\u504f\u5fae\u5206\u65b9\u7a0b\uff08PDE\uff09\u4e0e\u5e38\u5fae\u5206\u65b9\u7a0b\uff08ODE\uff09\u4e4b\u95f4\u7684\u5dee\u5f02\u3002 \u76ee\u5f55 \u5fae\u5206\u65b9\u7a0b\u4e0e\u4e8b\u7269\u7684\u672c\u8d28 \u6301\u7eed\u53d8\u5316 \u51fd\u6570\u3001\u5bfc\u6570\u4e0e\u5176\u53d8\u5316 \u5e38\u5fae\u5206\u65b9\u7a0b\u4e0e\u504f\u5fae\u5206\u65b9\u7a0b \u5e38\u5fae\u5206\u65b9\u7a0b\u5b9e\u4f8b \u5496\u5561\u676f\u7684\u51b7\u5374 \u725b\u987f\u5b9a\u5f8b \u4eba\u53e3\u6a21\u578b \u5fae\u5206\u65b9\u7a0b\u4e0e\u4e8b\u7269\u7684\u672c\u8d28 \u6301\u7eed\u53d8\u5316 \u5728\u81ea\u7136\u754c\u4e2d\uff0c\u4e00\u5207\u90fd\u5728\u4e0d\u65ad\u53d8\u5316\u3002 \u5373\u4f7f\u662f\u90a3\u4e9b\u770b\u8d77\u6765\u6c38\u8fdc\u4e0d\u4f1a\u6539\u53d8\u7684\u4e8b\u7269\uff0c\u6bd4\u5982\u592a\u9633\u7684\u5149\u8f89\uff0c\u5982\u679c\u5728\u5408\u9002\u7684\u65f6\u95f4\u5c3a\u5ea6\u4e0a\u89c2\u5bdf\uff0c\u5b83\u4e5f\u4f1a\u53d8\u5316\u3002\u4e00\u5207\u90fd\u5728\u53d8\u5316\uff1a\u661f\u661f\u7684\u4eae\u5ea6\u3001\u4e00\u676f\u5496\u5561\u7684\u6e29\u5ea6\u3001\u7269\u4f53\u7684\u4f4d\u7f6e\uff0c\u4ee5\u53ca\u4eba\u53e3\u7684\u6570\u91cf\u90fd\u662f\u4f8b\u5b50\uff0c\u800c\u8fd9\u4e9b\u53d8\u5316\u7387\u901a\u5e38\u4e0e\u6b63\u5728\u53d1\u751f\u53d8\u5316\u7684\u72b6\u6001\u6709\u5173\u3002 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