{"id":29600,"date":"2022-09-15T15:31:05","date_gmt":"2022-09-15T07:31:05","guid":{"rendered":"https:\/\/ocw.nycu.edu.tw\/?post_type=course_page&#038;p=29600"},"modified":"2022-09-15T16:05:34","modified_gmt":"2022-09-15T08:05:34","slug":"%e5%96%ae%e8%aa%bf%e5%8b%95%e6%85%8b%e7%b3%bb%e7%b5%b1%e5%8f%8a%e5%85%b6%e6%87%89%e7%94%a8-monotone-dynamical-systems-with-applications-in-biology-%e6%87%89%e7%94%a8%e6%95%b8%e5%ad%b8%e7%b3%bb","status":"publish","type":"course_page","link":"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course\/college-of-science\/am\/%e5%96%ae%e8%aa%bf%e5%8b%95%e6%85%8b%e7%b3%bb%e7%b5%b1%e5%8f%8a%e5%85%b6%e6%87%89%e7%94%a8-monotone-dynamical-systems-with-applications-in-biology-%e6%87%89%e7%94%a8%e6%95%b8%e5%ad%b8%e7%b3%bb","title":{"rendered":"\u55ae\u8abf\u52d5\u614b\u7cfb\u7d71\u53ca\u5176\u61c9\u7528 Monotone Dynamical Systems with Applications in Biology | \u61c9\u7528\u6578\u5b78\u7cfb \u8a31\u4e16\u58c1\u8001\u5e2b"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"29600\" class=\"elementor elementor-29600\">\n\t\t\t\t\t\t\t\t\t<section class=\"penci-section penci-disSticky penci-structure-10 elementor-section elementor-top-section elementor-element elementor-element-90a0113 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"90a0113\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"penci-ercol-100 penci-ercol-order-1 penci-sticky-ct    elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-78bf394\" data-id=\"78bf394\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-9a5e0e8 elementor-widget elementor-widget-breadcrumbs\" data-id=\"9a5e0e8\" data-element_type=\"widget\" data-widget_type=\"breadcrumbs.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<link rel=\"stylesheet\" href=\"https:\/\/ocw.nycu.edu.tw\/wp-content\/plugins\/elementor-pro\/assets\/css\/widget-theme-elements.min.css\"><p id=\"breadcrumbs\"><span><span><a href=\"https:\/\/ocw.nycu.edu.tw\/\">Homepage<\/a><\/span><\/span><\/p>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"penci-section penci-disSticky penci-structure-20 elementor-section elementor-top-section elementor-element elementor-element-0025485 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"0025485\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"penci-ercol-50 penci-ercol-order-1 penci-sticky-ct    elementor-column elementor-col-50 elementor-top-column elementor-element elementor-element-0446ed9\" data-id=\"0446ed9\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2f5840a elementor-widget elementor-widget-penci-info-box\" data-id=\"2f5840a\" data-element_type=\"widget\" data-widget_type=\"penci-info-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div id=\"penci_info_box_84218\" class=\"penci-block-vc penci-info-box penci-ibox-float-left penci-view-default penci-shape-circle\">\r\n\t\t\t<div class=\"penci-ibox-inner\">\r\n\t\t\t\t\t\t\t\t<div class=\"penci-ibox-icon penci-ibox-icon--icon penci-icon penci-tibox-text\"><span class=\"penci-ibox-icon-fa\"><\/span><\/div>\t\t\t\t<div class=\"penci-ibox-content-wrap\">\r\n\t\t\t\t\t<div class=\"penci-ibox-stit\">Monotone Dynamical Systems with Applications in Biology<\/div>\t\t\t\t\t<h3 class=\"penci-ibox-title\">\u55ae\u8abf\u52d5\u614b\u7cfb\u7d71\u53ca\u5176\u61c9\u7528 <\/h3>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\r\n\t\t\t<\/div>\r\n\t\t<\/div>\r\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-c6381f4 elementor-widget elementor-widget-eael-adv-tabs\" data-id=\"c6381f4\" data-element_type=\"widget\" data-widget_type=\"eael-adv-tabs.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t        <div id=\"eael-advance-tabs-c6381f4\" class=\"eael-advance-tabs eael-tabs-horizontal eael-tab-auto-active\" data-tabid=\"c6381f4\">\n            <div class=\"eael-tabs-nav\">\n                <ul class=\"eael-tab-inline-icon\">\n                                            <li id=\"home\" class=\"active-default eael-tab-item-trigger\" aria-selected=\"true\" data-tab=\"1\" role=\"tab\" tabindex=\"0\" aria-controls=\"home-tab\" aria-expanded=\"false\">\n                            \n                                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4.4-3.6 8-8 8H168c-4.4 0-8-3.6-8-8v-16c0-4.4 3.6-8 8-8h144c4.4 0 8 3.6 8 8v16z\"><\/path><\/svg>                                                            \n                                                            <span class=\"eael-tab-title  title-after-icon\">\u8ab2\u7a0b\u7db1\u8981<\/span>                            \n                                                    <\/li>\n                                    <\/ul>\n            <\/div>\n            \n            <div class=\"eael-tabs-content\">\n\t\t        \n                    <div id=\"home-tab\" class=\"clearfix eael-tab-content-item active-default\" data-title-link=\"home-tab\">\n\t\t\t\t        \t\t\t\t\t        <p><style type=\"text\/css\">\n.\u3002 {<br \/>\tfont-size: 18px;<br \/>}<br \/>.\u3002 {<br \/>\tfont-size: 18px;<br \/>}<br \/>.\u3002 {<br \/>\tfont-size: 18px;<br \/>}<br \/><\/style><\/p><p style=\"margin-bottom: 10px; line-height: 28px; color: #101010; font-family: \u5fae\u8edf\u6b63\u9ed1\u9ad4, Arial; font-size: 16px;\">\u672c\u8ab2\u7a0b\u662f\u7531\u00a0<a href=\"https:\/\/www.math.nycu.edu.tw\/\" target=\"_blank\" rel=\"noopener\">\u570b\u7acb\u967d\u660e\u4ea4\u901a\u5927\u5b78\u61c9\u7528\u6578\u5b78\u7cfb<\/a>\u63d0\u4f9b\u3002<\/p><p>(1) Review on General Dynamical System, topological property of w-limit set, alpha limit set, 2-d Poincare Bendixson Theorem , Stable Manifold Theorem, Hopf Bifurcation, Chaotic Dynamical Systems.<br \/><br \/><\/p><p>(2) Monotone Dynamical System<br \/>Definitions, Monotone, Strongly monotone, Strongly order preserving (SOP)<br \/>Hirsch\u2019s Convergence Theorem, No attracting periodic orbit in monotone system. Non-ordering of limit sets, limit set dichotomy , generic quasi-convergence.<br \/><br \/><\/p><p>(3) Competitive and Cooperative Differential Equations<br \/>Kamke Condition, Positively invariant sets and Monotone Solutions, Non-oscillation principle (Ito\u2019s Lemma), Topological Equivalence of flow of competitive or cooperative n-d systems to a flow on (n-1)-d system of Lipschitz differential equations, Poincare-Bendixson Theorem of 3-d competitive systems, Alternative Cones, The Field-Noyes Model.<br \/><br \/><\/p><p>(4) Irreducible Cooperative Systems<br \/>Strong Monotonicity, A Biochemical Control Circuit, Stability and the Perron-Frobenius Theorem, Competition and Migration, Smale\u2019s Construction<br \/><br \/><\/p><p>(5) Quasimonotone Systems of Parabilic Equations<br \/>Parabolic Systems: The Basic Setup, Maximum Principles, Positively Invariant Sets , Comparison and Monotonicity, The Strong Order Preserving Property, The Biochemical Control Circuit with Diffusion.<br \/><br \/><\/p><p>(6) Two species competition in the flow reactor model in a river system.<br \/><br \/><\/p><p>(7) Sublinearity of Monotone Dynamical System, Butler McGhee Lemma, Uniform Persistence Theory.<br \/><br \/><\/p><p>(8) Jiang\u2019s global stability theorem for cooperative systems<br \/><br \/><\/p><p>(9) PDE, Reaction-Diffusion system, Principle eigenvalue<br \/><br \/><\/p><p>(10) Krein-Rutman Theorem , Systems with coupled PDE and ODE, Generalized Krein-Rutman\u2019s Theorem.<br \/><br \/><\/p><p>(11) Abstract theory of two species competition in ordered Banach Spaces, Dancer-Hess Lemma.<br \/>Slower Diffuser vs Fast Diffuser, Two species Competition in an unstirred chemostat.<br \/><br \/><\/p><p>(12) Poincare Bendixson Theorem for Monotone Cyclic Feedback System.<br \/><br \/><\/p><p><span style=\"color: #3366ff; font-size: 16px;\">\u53c3\u8003\u66f8\u76ee\uff1a<\/span><\/p><p class=\"text\">Hal L. Smith.\u00a0<em>Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems (Mathematical Surveys &amp; Monographs),<\/em>Amer Mathematical Society, 1995.<br \/>SZE-BI HSU,\u00a0<em>Ordinary Differential Equations with Applications\u00a0(Applied Mathematics),\u00a0<\/em>2nd Ed.\u00a0World Scientific Publishing Company, 2013.<br \/>Xiao-Qiang Zhao,\u00a0Dynamical Systems in Population Biology, 2nd Ed,\u00a0Springer, 2017.<\/p><p><span style=\"color: #ff00ff;\">\u70ba\u6c42\u5b78\u7fd2\u6210\u6548\u5b8c\u7f8e\uff0c\u8acb\u8cfc\u8cb7\u8ab2\u672c\uff01<br \/><br \/><\/span><\/p><table style=\"color: #101010; font-family: \u5fae\u8edf\u6b63\u9ed1\u9ad4, Arial; font-size: 16px;\" border=\"1\"><tbody><tr><th style=\"text-align: center;\" nowrap=\"nowrap\" bgcolor=\"#FFF3DB\" width=\"100\"><span style=\"color: #313131; font-family: PT Serif, serif;\"><span style=\"font-weight: 400; white-space: normal;\">\u6388\u8ab2\u6559\u5e2b<\/span><\/span><\/th><td bgcolor=\"#EEEEEE\" width=\"518\"><span style=\"color: #313131; font-family: PT Serif, serif;\"><span style=\"color: #313131; font-family: PT Serif, serif;\"><span style=\"color: #313131; font-family: PT Serif, serif;\"><span style=\"color: #313131; font-family: PT Serif, serif;\">\u00a0\u61c9\u7528\u6578\u5b78\u7cfb \u8a31\u4e16\u58c1\u8001\u5e2b<\/span><\/span><\/span><\/span><\/td><\/tr><tr><th style=\"text-align: center;\" nowrap=\"nowrap\" bgcolor=\"#FFF3DB\"><span style=\"color: #313131; font-family: PT Serif, serif;\"><span style=\"font-weight: 400; white-space: normal;\">\u8ab2\u7a0b\u5b78\u5206<\/span><\/span><\/th><td class=\"text\"><span style=\"color: #313131; font-family: PT Serif, serif;\"><span style=\"color: #313131; font-family: PT Serif, serif;\"><span style=\"color: #313131; font-family: PT Serif, serif;\"><span style=\"color: #313131; font-family: PT Serif, serif;\">\u00a02\u5b78\u5206<\/span><\/span><\/span><\/span><\/td><\/tr><tr><th class=\"text\" style=\"text-align: center;\" nowrap=\"nowrap\" bgcolor=\"#FFF3DB\"><span style=\"color: #313131; font-family: PT Serif, serif;\"><span style=\"font-weight: 400; white-space: normal;\">\u6388\u8ab2\u5e74\u5ea6<\/span><\/span><\/th><td class=\"text\" bgcolor=\"#EEEEEE\"><span style=\"color: #313131; font-family: PT Serif, serif;\"><span style=\"color: #313131; font-family: PT Serif, serif;\"><span style=\"color: #313131; font-family: PT Serif, serif;\"><span style=\"color: #313131; font-family: PT Serif, serif;\">\u00a0107\u5b78\u5e74\u5ea6<\/span><\/span><\/span><\/span><\/td><\/tr><tr><th style=\"text-align: center;\" nowrap=\"nowrap\" bgcolor=\"#FFF3DB\"><span style=\"color: #313131; font-family: PT Serif, serif;\"><span style=\"font-weight: 400; white-space: normal;\">\u6388\u8ab2\u5c0d\u8c61<\/span><\/span><\/th><td class=\"text\"><span style=\"color: #313131; font-family: PT Serif, serif;\"><span style=\"color: #313131; font-family: PT Serif, serif;\"><span style=\"color: #313131; font-family: PT Serif, serif;\"><span style=\"color: #313131; font-family: PT Serif, serif;\">\u00a0\u78a9\u58eb\u751f<\/span><\/span><\/span><\/span><\/td><\/tr><tr><th style=\"text-align: center;\" nowrap=\"nowrap\" bgcolor=\"#FFF3DB\"><span style=\"color: #313131; font-family: PT Serif, serif;\"><span style=\"font-weight: 400; white-space: normal;\">\u9810\u5099\u77e5\u8b58<\/span><\/span><\/th><td class=\"text\" bgcolor=\"#EEEEEE\"><span style=\"color: #313131; font-family: PT Serif, serif;\"><span style=\"color: #313131; font-family: PT Serif, serif;\"><span style=\"color: #313131; font-family: PT Serif, serif;\"><span style=\"color: #313131; font-family: PT Serif, serif;\">\u00a0\u7121<\/span><\/span><\/span><\/span><\/td><\/tr><tr><th style=\"text-align: center;\" nowrap=\"nowrap\" bgcolor=\"#FFF3DB\"><span style=\"color: #313131; font-family: PT Serif, serif;\"><span style=\"font-weight: 400; white-space: normal;\">\u8ab2\u7a0b\u63d0\u4f9b<\/span><\/span><\/th><td class=\"text\"><span style=\"color: #313131; font-family: PT Serif, serif;\"><span style=\"color: #313131; font-family: PT Serif, serif;\"><span style=\"color: #313131; font-family: PT Serif, serif;\"><span style=\"color: #313131; font-family: PT Serif, serif;\">\u00a0<a href=\"#video\">\u8ab2\u7a0b\u5f71\u97f3<\/a>\u00a0 \u00a0<a href=\"#syllabus\">\u8ab2\u7a0b\u7db1\u8981<\/a>\u00a0<\/span><\/span><\/span><\/span><\/td><\/tr><\/tbody><\/table>\t\t\t\t                            <\/div>\n\t\t        \n                    <div id=\"video-tab\" class=\"clearfix eael-tab-content-item inactive\" data-title-link=\"video-tab\">\n\t\t\t\t        \t\t\t\t\t        <p>\n<table id=\"tablepress-74\" class=\"tablepress tablepress-id-74\">\n<thead>\n<tr class=\"row-1 odd\">\n\t<th class=\"column-1\">\u9031\u6b21<\/th><th class=\"column-2\">\u8ab2\u7a0b\u5167\u5bb9<\/th><th class=\"column-3\">\u8ab2\u7a0b\u5f71\u97f3<\/th>\n<\/tr>\n<\/thead>\n<tbody class=\"row-hover\">\n<tr class=\"row-2 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Review on General Dynamical System<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=29642\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-3 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Monotone Dynamical System<br \/>\nDefinitions<br \/>\nMonotone<br \/>\nStrongly monotone<br \/>\nStrongly order preserving (SOP)<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=34198\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-4 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Hirsch\u2019s Convergence Theorem<br \/>\nNo attracting periodic orbit in monotone system <\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=34204\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-5 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Non-ordering of limit sets<br \/>\nLimit set dichotomy<br \/>\nGeneric quasi-convergence <\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=34210\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-6 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Competitive and Cooperative Differential Equations<br \/>\nKamke Condition<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=34219\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-7 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Positively invariant sets and Monotone Solutions<br \/>\nNon-oscillation principle  <\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=34225\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-8 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Topological Equivalence of flow of competitive or cooperative n-d systems to a flow on (n-1)-d system of Lipschitz differential equations<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=34231\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-9 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Poincare-Bendixson Theorem of 3-d competitive systems<br \/>\nAlternative Cones<br \/>\nThe Field-Noyes Model <\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=34237\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-10 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Strong Monotonicity<br \/>\nA Biochemical Control Circuit<br \/>\nStability and the Perron-Frobenius Theorem <\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=34243\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-11 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Irreducible Cooperative Systems<br \/>\nCompetition and Migration<br \/>\nSmale\u2019s Construction<\/td><td class=\"column-3\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?post_type=course_page&amp;p=34249\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-12 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Quasimonotone Systems of Parabilic Equations<br \/>\nThe Basic Setup, Maximum Principles<br \/>\nPositively Invariant Sets<br \/>\nComparison and Monotonicity<br \/>\nThe Strong Order Preserving Property <\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=34255\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-13 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Irreducible Cooperative Systems<br \/>\nA Biochemical Control Circuit<br \/>\nTwo Patch Competitive Models<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=34274\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-14 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Quasimonotone Systems of Parabilic Equations<br \/>\nThe Basic Setup<br \/>\nStrong Parabilic Maximum Principles<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=34282\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-15 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Comparison Principle<br \/>\nKrein-Rutman Theorem<br \/>\nWeak Maximum Principle<br \/>\nPositively Invariant Sets<br \/>\nComparison and Monotonicity<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=34297\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<tr class=\"row-16 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Krein-Rutman Theorem<br \/>\nReaction-Diffusion system<br \/>\nPrinciple eigenvalue<br \/>\nTwo species competition in the flow reactor model <\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=34303\">\u7dda\u4e0a\u89c0\u770b<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<!-- #tablepress-74 from cache --><\/p>\t\t\t\t                            <\/div>\n\t\t        \n                    <div id=\"syllabus-tab\" class=\"clearfix eael-tab-content-item inactive\" data-title-link=\"syllabus-tab\">\n\t\t\t\t        \t\t\t\t\t        <p class=\"title\"><strong>\u8ab2\u7a0b\u76ee\u6a19<\/strong><\/p><p class=\"text\">(1) Review on General Dynamical System, topological property of w-limit set, alpha limit set, 2-d Poincare Bendixson Theorem , Stable Manifold Theorem, Hopf Bifurcation, Chaotic Dynamical Systems. <br \/>(2) Monotone Dynamical System Definitions, Monotone, Strongly monotone, Strongly order preserving (SOP) Hirsch\u2019s Convergence Theorem, No attracting periodic orbit in monotone system. Non-ordering of limit sets, limit set dichotomy , generic quasi-convergence. <br \/>(3) Competitive and Cooperative Differential Equations Kamke Condition, Positively invariant sets and Monotone Solutions, Non-oscillation principle (Ito\u2019s Lemma), Topological Equivalence of flow of competitive or cooperative n-d systems to a flow on (n-1)-d system of Lipschitz differential equations, Poincare-Bendixson Theorem of 3-d competitive systems, Alternative Cones, The Field-Noyes Model. <br \/>(4) Irreducible Cooperative Systems Strong Monotonicity, A Biochemical Control Circuit, Stability and the Perron-Frobenius Theorem, Competition and Migration, Smale\u2019s Construction <br \/>(5) Quasimonotone Systems of Parabilic Equations Parabolic Systems: The Basic Setup, Maximum Principles, Positively Invariant Sets , Comparison and Monotonicity, The Strong Order Preserving Property, The Biochemical Control Circuit with Diffusion. <br \/>(6) Two species competition in the flow reactor model in a river system. <br \/>(7) Sublinearity of Monotone Dynamical System, Butler McGhee Lemma, Uniform Persistence Theory. <br \/>(8) Jiang\u2019s global stability theorem for cooperative systems <br \/>(9) PDE, Reaction-Diffusion system, Principle eigenvalue <br \/>(10) Krein-Rutman Theorem , Systems with coupled PDE and ODE, Generalized Krein-Rutman\u2019s Theorem. <br \/>(11) Abstract theory of two species competition in ordered Banach Spaces, Dancer-Hess Lemma. Slower Diffuser vs Fast Diffuser, Two species Competition in an unstirred chemostat. <br \/>(12) Poincare Bendixson Theorem for Monotone Cyclic Feedback System.<\/p><p>\u00a0<\/p><p class=\"title\"><strong>\u53c3\u8003\u66f8\u76ee<\/strong><\/p><p class=\"text\">Hal L. Smith.\u00a0<em>Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems (Mathematical Surveys &amp; Monographs),<\/em>Amer Mathematical Society, 1995.<br \/>SZE-BI HSU,\u00a0<em>Ordinary Differential Equations with Applications\u00a0(Applied Mathematics),\u00a0<\/em>2nd Ed.\u00a0World Scientific Publishing Company, 2013.<br \/>Xiao-Qiang Zhao,\u00a0Dynamical Systems in Population Biology, 2nd Ed,\u00a0Springer, 2017.<\/p>\t\t\t\t                            <\/div>\n\t\t                    <\/div>\n        <\/div>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-d7723a8 elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"d7723a8\" data-element_type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.13.3 - 28-05-2023 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