An introduction to delay differential equations with applications to the life sciences:
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
New York, NY [u.a.]
Springer
2011
|
| Schriftenreihe: | Texts in applied mathematics
57 |
| Schlagwörter: | |
| Links: | http://deposit.dnb.de/cgi-bin/dokserv?id=3518636&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020766525&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Beschreibung: | Literaturverz. S. 167 - 170 |
| Umfang: | XI, 172 S. graph. Darst. 240 mm x 160 mm |
| ISBN: | 9781441976451 9781461426974 |
Internformat
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Datensatz im Suchindex
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| adam_text | Titel: An introduction to delay differential equations with applications to the life sciences
Autor: Smith, Hal L.
Jahr: 2011
Contents
1 Introduction................................................... 1
1.1 Examples of Delay Differential Equations...................... 1
1.2 Some Terminology......................................... 9
1.3 Solving Delay Equations Using a Computer.................... 11
2 Delayed Negative Feedback: A Warm-Up......................... 13
2.1 Preliminaries.............................................. 13
2.2 The Simplest Delay Equation................................. 16
2.3 Oscillation of Solutions..................................... 20
2.4 Solutions Backward in Time................................. 22
3 Existence of Solutions .......................................... 25
3.1 The Method of Steps for Discrete Delay Equations............... 25
3.2 Positivity of Solutions....................................... 27
3.3 A More General Existence Result............................. 29
3.4 Continuation of Solutions................................... 36
3.5 Remarks on Backward Continuation........................... 37
3.6 Stability Definitions ........................................ 38
4 Linear Systems and Linearization................................ 41
4.1 Autonomous Linear Systems................................. 41
4.2 Laplace Transform and Variation of Constants Formula........... 43
4.3 The Characteristic Equation.................................. 45
4.4 Small Delays Are Harmless.................................. 48
4.5 The Scalar Equation x1(t) = Ax(t) + Bx{t - r)................... 49
4.6 Principle of Linearized Stability.............................. 54
4.7 Absolute Stability.......................................... 56
5 Semidynamical Systems and Delay Equations.............-----.... 61
5.1 The Dynamical Systems Viewpoint............................ 61
5.2 Semiflows and Omega Limit Sets............................. 64
Contents
5.3 SemiDynamical Systems Induced by Delay Equations............ 65
5.4 Monotone Dynamics........................................ 70
5.5 Delayed Logistic Equation................................... 73
5.6 Delayed Microbial Growth Model............................. 76
5.7 Liapunov Functions......................................... 78
5.7.1 Logistic Equation with Instantaneous and Delayed Density
Dependence......................................... 80
Hopf Bifurcation............................................... 87
6.1 A Canonical Example....................................... 87
6.2 Hopf Bifurcation Theorem................................... 89
6.3 Delayed Negative Feedback.................................. 91
6.3.1 Computation of the Hopf Bifurcation.................... 92
6.3.2 Series Expansion of Hopf Solution...................... 94
6.3.3 The Logistic Equation................................ 97
6.4 A Second-Order Delayed Feedback System..................... 99
6.4.1 Delayed Feedback Dominates Instantaneous Feedback.....101
6.4.2 Instantaneous Feedback Dominates Delayed Feedback.....104
6.4.3 Stabilizing the Straight-Up Steady State of the Pendulum... 106
6.5 Gene Regulation by End-Product Repression ...................Ill
6.6 A Poincaré-Bendixson Theorem for Delay Equations.............115
Distributed Delay Equations and the Linear Chain Trick...........119
7.1 Infinite Delays of Gamma Type...............................119
7.1.1 Characteristic Equation and Stability....................120
7.1.2 The Linear Chain Trick...............................123
7.2 A Model of HIV Transmission ...............................126
7.3 An ODE Approximation to a Delay Equation...................129
Phage and Bacteria in a Chemostat..............................131
8.1 Model....................................................131
8.2 Positivity and Boundedness of Solutions.......................133
8.3 Basic Reproductive Number for Phage.........................134
8.4 Persistence of Host and Phage Extinction.......................135
8.5 The Coexistence Equilibrium.................................137
8.6 Another Formulation of the Model............................141
Results from Real and Complex Analysis.........................149
A.I Analytic Functions.........................................149
A.2 Implicit Function Theorem for Complex Variables...............151
A.3 Rouché s Theorem.........................................152
A.4 Ascoli-Arzela Theorem.....................................153
A.5 Fluctuation Lemma.........................................154
A.6 General Implicit Function Theorem...........................155
A.7 GronwalPs Inequality.......................................155
Contents xi
 Hopf Bifurcation for Delayed Negative Feedback..................157
B. 1 Basic Setup and Preliminaries................................157
B.2 The Solution ..............................................160
B.2.1 Solve for q..........................................161
B.2.2 Solve for ì and ä....................................163
References.........................................................167
Index.............................................................171
|
| any_adam_object | 1 |
| author | Smith, Hal L. 1947- |
| author_GND | (DE-588)111530326 |
| author_facet | Smith, Hal L. 1947- |
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| classification_rvk | SK 520 SK 950 |
| classification_tum | BIO 105f MAT 348f |
| ctrlnum | (OCoLC)696034580 (DE-599)DNB1005136343 |
| discipline | Biologie Mathematik |
| format | Book |
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| id | DE-604.BV036850548 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T14:43:14Z |
| institution | BVB |
| isbn | 9781441976451 9781461426974 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-020766525 |
| oclc_num | 696034580 |
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| physical | XI, 172 S. graph. Darst. 240 mm x 160 mm |
| publishDate | 2011 |
| publishDateSearch | 2011 |
| publishDateSort | 2011 |
| publisher | Springer |
| record_format | marc |
| series | Texts in applied mathematics |
| series2 | Texts in applied mathematics |
| spellingShingle | Smith, Hal L. 1947- An introduction to delay differential equations with applications to the life sciences Texts in applied mathematics Biomathematik (DE-588)4139408-2 gnd Differentialgleichung mit nacheilendem Argument (DE-588)4199298-2 gnd |
| subject_GND | (DE-588)4139408-2 (DE-588)4199298-2 |
| title | An introduction to delay differential equations with applications to the life sciences |
| title_auth | An introduction to delay differential equations with applications to the life sciences |
| title_exact_search | An introduction to delay differential equations with applications to the life sciences |
| title_full | An introduction to delay differential equations with applications to the life sciences Hal Smith |
| title_fullStr | An introduction to delay differential equations with applications to the life sciences Hal Smith |
| title_full_unstemmed | An introduction to delay differential equations with applications to the life sciences Hal Smith |
| title_short | An introduction to delay differential equations with applications to the life sciences |
| title_sort | an introduction to delay differential equations with applications to the life sciences |
| topic | Biomathematik (DE-588)4139408-2 gnd Differentialgleichung mit nacheilendem Argument (DE-588)4199298-2 gnd |
| topic_facet | Biomathematik Differentialgleichung mit nacheilendem Argument |
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| volume_link | (DE-604)BV002476038 |
| work_keys_str_mv | AT smithhall anintroductiontodelaydifferentialequationswithapplicationstothelifesciences |
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