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APPM MODELLING COURSE NOTES
- Institution
- University Of The Witwatersrand (wits)
MODELLING COURSE NOTES WITH EXAMPLES AND QUESTION WITH SOLUTIONS
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A PPMZOO?NTeRMEDIATE ffiATHEmATICAL T[CbEIl1
NG
UR SE. NorEs zo zo
, Conterets
0.1 Course Objectives i
02 Activities i
A .2.1 Lectures i
A.2.2 Tutoriais and Consulting
0.3 Assessment II
4.4 An overview of the course "i
0.5 Resources iv
1 Introduction -t
J-
1 .1 Assumed background for the course 1
1.2 Review of the Modelling Process 1
1.3 Some popular real life systems 2
.3. 1 Physical systems
1 2
1.3.2 Biological Systems e
tJ
1.4 Systems of Differential Equations 4
1.4.1 Categories of ODE Systems 4
1 .4.2 Solutions of ODE Systems L.
d
1.5 Basic terminology . 5
1 6 'Qualitative Analysis r
U
2 First Order Systems 6
2.1 Introduction . 6
2.2 Phase diagrams of 1D Systems 6
2,3 Stability of critical points aI
2.4 Stability condition for ID systerns B
2.5 Solutions of 1D systerns 8
2.6 Exercises I
3 Plane (zI.J-) Autonomous Systems 10
3.1 Equilibrium solutions lcritical points) 10
3.2 .tnalytical Concepts 11
:) 2.1 Direction fielC 11
,2.2 Simple Isoclines
:') L2
.2.3 Separatrices
'',
12
1
i
, 3.3 Phasc Diagrams of 2D Systerus 13
3.4 Characterising Critical Points 14
3.5 Finding tire shape of trajectories analytically 15
3,6 Obtaining solution curves from the phase diagram 1r
1d
,)a
d.I Exercises 19
4 Solutions of Linear Systerns 20
4,1 Method 1: Elimination 20
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4.2 fulethod 2: Systems Meihod
4.3 Exercises 24
5 Linear Stability 26
5.1 Introduction 25
5.2 T5,pes of equilibria (Linear systems) 25
5,3 Stability . , 28
5.4 Eigenvalues and linear stability 29
5.5 Exercises 31
6 Nonlinear stability 32
6,1 Linearisation . , 32
6.1.1 Taylor's Theorern .
,.)
04
6.L.2'Linearisation Theorem 34
6.2 Local stability . . . 34
6.3 Limit Cycles . . 35
6.4 Exercises . . 37
7 Modelling'Apptications d 1Il Systems 38
7.1 The Logistic Population Model 38
7.2 Solutions of 1D systems 39
7.2.1 Solution of the Logistic Mode1 39
7 ,3 Exercises 40
8 Modelling with ZD Systems 42
8.1 Tire linear Oscillator 42
8.1.1 fulodel Formulation 42
8.1.2 Qualitative analysis 42
,3.2 Nonlinear L{odels 43
l.t
, 8. 2. i hile rar:t iirg PoJ:ulatir:ns 43
8.2.2 The Noirlinear Oscillator (Sirr:ple pendulum) 44
8.2.3 Eiectrical circuits 4tr.
.lL,
8.2.4 ;\ gerrcral epidenric nrodel 46
83 Flxercises +o
I lliscrete Systenls 48
3. i \iili]' I)iscrete l'Jodcis . 48
9.2 Frorn Difierence Equations tr: Discrete Systerns 48
ll.3 Discrete First Order Eqiiati()ns 49
9.3.1 Equilibrir-rnr Solutions 49
9,3.2 General Solutions 49
9.3.3 Stability Analysis 50
9,4 Discrete Second Order Systerns 52
9.4.1 Linear Stability Anaiysis . 52
9.4.2 Nonlinear Stability Analysis 55
10 Iliscrete Models 56
10.1 Density-independent gt'ov,,tlt', ,.. . 56
i0.2 The Logistic Difrerence Equation 56
10.3 Interest and Loan reparylnents 58
10.4 An Economic Mode} of Supply and Demand 5g
10.1,1 Cell division 60
10.5 Plant Populations 60
10.6 Structured populations in discrete time 61
10.7 The notion of Chaos 62
10.8 Exercises 62
A Conversion of higher order OBEs 65
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