基本介紹
內容簡介,目錄,
內容簡介
以隨機擾動項分別為Browan運動、分數Brown運動、Markovian過程和Poisson過程為主線，對種群模型進行數值計算理論的研究；主要針對年齡相關的種群模型、年齡相關擴散的種群模型和神經網路模型開展數值方法研究。採用Euler和半隱式Euler等數值方法，研究年齡相關隨機種群模型數值計算方法，給出數值解收斂和指數穩定的充分條件，並通過大量的數值算例驗證算法的有效性。為隨機種群發展系統求解構造出穩定的求解算法。主要包括四部分內容，一、預備知識；二、年齡相關隨機種群模型解的存在性；三、年齡相關隨機種群模型的數值計算；四、隨機神經網路模型的數值計算。《生物數學叢書：隨機年齡結構種群系統（英文版）》的內容全部是新研究成果。
目錄
Contents
《生物數學叢書》序
Preface
Chapter 1 Introduction 1
1.1 Introduction 1
1.2 Basic notations of probability theory 2
1.3 Stochastic processes 9
1.4 Brownian motions 15
1.5 Stochastic integrals 18
1.6 It？o’s formula 31
1.7 Moment inequalities 40
1.8 Gronwall-type inequalities 45
Chapter 2 Existence， uniqueness and exponential stability for stochastic age-dependent population 48
2.1 Introduction 48
2.2 Assumptions and preliminaries 49
2.3 Existence and uniqueness of solutions 52
2.3.1 Uniqueness of solutions 52
2.3.2 Existence of strong solutions 53
2.4 Stability of strong solutions 59
Chapter 3 Existence and uniqueness for stochastic age-structured population system with diffusion 64
3.1 Introduction 64
3.2 Euler approximation and main result 66
3.3 Existence and uniqueness of solutions 68
3.3.1 Uniqueness of solutions 68
3.3.2 Existence of strong solutions 70
3.4 Numerical simulation example 76
Chapter 4 Existence and uniqueness for stochastic age-dependent population with fractional Brownian motion 79
4.1 Introduction 79
4.2 Preliminaries 81
4.3 Existence and uniqueness of solutions 84
Chapter 5 Convergence of the Euler scheme for stochastic functional partial differential equations 90
5.1 Introduction 90
5.2 Preliminaries and the Euler approximation 91
5.3 The main results 93
5.4 Numerical simulation example 99
Chapter 6 Numerical analysis for stochastic age-dependent population equations 101
6.1 Introduction 101
6.2 Preliminaries and the Euler approximation 102
6.3 The main results 105
Chapter 7 Convergence of numerical solutions to stochastic age-structured population system with diffusion 116
7.1 Introduction 116
7.2 Preliminaries and approximation 118
7.3 The main results 121
7.4 Numerical simulation example 126
Chapter 8 Exponential stability of numerical solutions to a stochastic age-structured population system with diffusion 128
8.1 Introduction 128
8.2 Preliminaries and Euler approximation 130
8.3 The main results 132
8.4 Numerical simulation example 137
Chapter 9 Numerical analysis for stochastic age-dependent population equations with fractional Brownian motion 140
9.1 Introduction 140
9.2 Preliminaries and the Euler approximation 141
9.3 The main results 144
9.4 Numerical simulation example 154
Chapter 10 Convergence of the semi-implicit Euler method for stochastic age-dependent population equations with Markovian switching 156
10.1 Introduction 156
10.2 Preliminaries and semi-implicit approximation 157
10.3 Several lemmas 159
10.4 Main results 165
Chapter 11 Convergence of numerical solutions to stochastic age-dependent population equations with Poisson jump and Markovian switching 170
11.1 Introduction 170
11.2 Preliminaries and semi-implicit approximation 171
11.3 Several lemmas 173
11.4 Main results 179
Chapter 12 Numerical analysis for stochastic delay neural networks with Poisson jump 184
12.1 Introduction 184
12.2 Preliminaries and the Euler approximation 185
12.3 The main results 187
12.4 Numerical simulation example 195
Chapter 13 Convergence of numerical solutions to stochastic delay neural networks with Poisson jump and Markov switching 197
13.1 Introduction 197
13.2 Preliminaries and the Euler approximation 198
13.3 Lemmas and corollaries 201
13.4 Convergence with the local Lipschitz condition 205
Chapter 14 Exponential stability of numerical solutions to a stochastic delay neural networks 211
14.1 Introduction 211
14.2 Preliminaries and approximation 212
14.3 Lemmas 214
14.4 Numerical simulation example 220
Bibliography 222
Index 228
