Spatial Dynamics and Pattern Formation in Biological Populations

✍ Scribed by Ranjit Kumar Upadhyay, Satteluri R. K. Iyengar

• Publisher: Chapman and Hall/CRC
• Year: 2021
• Tongue: English
• Leaves: 449
• Edition: 1
• Category: Library

✦ Synopsis

The book provides an introduction to deterministic (and some stochastic) modeling of spatiotemporal phenomena in ecology, epidemiology, and neural systems. A survey of the classical models in the fields with up to date applications is given.

The book begins with detailed description of how spatial dynamics/diffusive processes influence the dynamics of biological populations. These processes play a key role in understanding the outbreak and spread of pandemics which help us in designing the control strategies from the public health perspective. A brief discussion on the functional mechanism of the brain (single neuron models and network level) with classical models of neuronal dynamics in space and time is given. Relevant phenomena and existing modeling approaches in ecology, epidemiology and neuroscience are introduced, which provide examples of pattern formation in these models. The analysis of patterns enables us to study the dynamics of macroscopic and microscopic behaviour of underlying systems and travelling wave type patterns observed in dispersive systems. Moving on to virus dynamics, authors present a detailed analysis of different types models of infectious diseases including two models for influenza, five models for Ebola virus and seven models for Zika virus with diffusion and time delay. A Chapter is devoted for the study of Brain Dynamics (Neural systems in space and time).

Significant advances made in modeling the reaction-diffusion systems are presented and spatiotemporal patterning in the systems is reviewed. Development of appropriate mathematical models and detailed analysis (such as linear stability, weakly nonlinear analysis, bifurcation analysis, control theory, numerical simulation) are presented.

Key Features

• Covers the fundamental concepts and mathematical skills required to analyse reaction-diffusion models for biological populations.
• Concepts are introduced in such a way that readers with a basic knowledge of differential equations and numerical methods can understand the analysis. The results are also illustrated with figures.
• Focuses on mathematical modeling and numerical simulations using basic conceptual and classic models of population dynamics, Virus and Brain dynamics.
• Covers wide range of models using spatial and non-spatial approaches.
• Covers single, two and multispecies reaction-diffusion models from ecology and models from bio-chemistry. Models are analysed for stability of equilibrium points, Turing instability, Hopf bifurcation and pattern formations.
• Uses Mathematica for problem solving and MATLAB for pattern formations.
• Contains solved Examples and Problems in Exercises.

The Book is suitable for advanced undergraduate, graduate and research students. For those who are working in the above areas, it provides information from most of the recent works. The text presents all the fundamental concepts and mathematical skills needed to build models and perform analyses.

✦ Table of Contents

Cover
Half Title
Title Page
Copyright Page
Table of Contents
Foreword
Preface
About Authors
1 Introduction to Diffusive Processes
1.1 Introduction
1.2 Diffusion, Convection, Advection and Dispersion Processes
1.3 Some Basic Laws of Diffusion
1.3.1 Fick’s Laws of Diffusion
1.3.2 Darcy’s Law
1.4 Diffusion Equation
1.4.1 Linear Diffusion Equation in One Dimension
1.4.1.1 Time-Dependent/Concentration-Dependent Diffusion Coefficient Problems
1.4.2 Linear Diffusion Equation in Two and Three Dimensions
1.4.2.1 Two-Dimensional Diffusion on a Disk
1.4.2.2 Linear Diffusion Equation in Three Dimensions
1.4.2.3 Reaction–Diffusion Equations in Diffusion Processes
1.4.3 Diffusion in a Heterogeneous Environment
1.5 Stochastic Reaction–Diffusion (SRD) Systems
1.6 Hopf Bifurcation Analysis
1.7 Multiple-Scale Analysis/Weakly Nonlinear Analysis
1.7.1 Linear Stability Analysis of the Amplitude Equation
1.8 Overview of the Book
References
2 Reaction–Diffusion Modeling
2.1 Introduction
2.2 Reaction–Diffusion Equations
2.2.1 Derivation of Reaction-Diffusion Equation
2.3 Hyperbolic Reaction–Diffusion Equations
2.4 Single-Species Reaction–Diffusion Models
2.4.1 Model 1: Linear Model of Kierstead and Slobodkin
2.4.1.1 KISS Model in Two Dimensions
2.4.2 Model 2: Nonlinear Fisher Equation
2.4.2.1 Spatial Steady-State Solution
2.4.2.2 Some Analytical Solutions
2.4.3 Model 3: Nagumo Equation
2.4.3.1 Numerical Solutions
2.5 Two-Species Reaction–Diffusion Models
2.5.1 Turing Instabilities of Two-Species Reaction–Diffusion Systems
2.5.1.1 Predator–Prey Reaction–Diffusion Systems
2.6 Applications in Biochemistry: Belousov–Zhabotinsky Reaction–Diffusion Systems
2.6.1 Model 1: Oregonator Model
2.6.2 Model 2: Brusselator Model
2.6.3 Model 3: Schnakenberg Model
2.6.4 Model 4: Lengyel–Epstein Model
2.6.5 Model 5: Sel’kov Model
2.6.6 Model 6: Gray–Scott Model
2.7 Multispecies Reaction–Diffusion Models
2.7.1 Model 1: Hastings–Powell Model
2.7.2 Model 2: Modified Upadhyay–Rai Model
2.7.3 Model 3: Modified Leslie–Gower-Type Three-Species Model
References
3 Modeling Virus Dynamics in Time and Space
3.1 Introduction
3.1.1 Next-Generation Operator Method
3.2 Susceptible-Infected (SI) Models
3.2.1 Models with Nonlinear Incidence Rate
3.2.2 Models with Self and Cross-Diffusion
3.2.3 Influenza Epidemic Models
3.2.3.1 A Simple Spatial SI Epidemic Model
3.2.3.2 Turing Instability
3.2.3.3 Two-time Scale Influenza Models
3.3 Susceptible-Infected-Susceptible (SIS) Models
3.4 Susceptible-Infected-Removed (SIR) Models
3.4.1 SIR Models with Vital Dynamics
3.4.2 SIR Models with Treatment Rate
3.5 Susceptible-Infected-Removed-Susceptible (SIRS) Models
3.6 Susceptible-Exposed-Infected-Recovered (SEIR) Models
3.6.1 Influenza Model Revisited
Exercise 3
References
4 Modeling the Epidemic Spread and Outbreak of Ebola Virus
4.1 Introduction
4.1.1 Source and Symptoms
4.1.2 Transmission and Control of Epidemics
4.2 Formulation of Ebola Epidemic Models
4.3 Model 1: Ebola Epidemic SEIR Model
4.3.1 Spatial SEIR Ebola Epidemic Model
4.4 Model 2: Ebola Epidemic SEIRHD Model
4.4.1 Sensitivity Indices of R[sub(0)]
4.5 Model 3: Ebola Epidemic SEIORD Model and Its Extension
4.6 Model 4: Ebola Epidemic SEIRD Model with Time Delay
4.6.1 Existence of Endemic Equilibrium and Stability Analysis
4.7 Model 5: General Ebola Transmission Model for Population in a Community
Exercise 4
References
5 Modeling the Transmission Dynamics of Zika Virus
5.1 Introduction
5.1.1 Symptoms and Clinical Features
5.2 Formulation of Zika Epidemic Model
5.3 Model 1: Zika Virus SIR Transmission Model
5.3.1 Optimal Control Analysis
5.4 Model 2: Zika Virus SEIR Transmission Model
5.4.1 Bifurcation Analysis
5.4.2 Optimal Control Analysis
5.5 Model 3: Zika Virus SEIR Horizontal and Vertical Transmission Model
5.6 Model 4: Zika Virus with Vertical Transmission
5.7 Model 5: Zika Virus SIR Transmission Model with Human and Vector Mobility
5.7.1 Existence of Travelling Wave Solutions
5.8 Model 6: Zika Virus Transmission with Criss-Cross Interactions Model
5.9 Model 7: Zika Virus SEIR Transmission Model
5.9.1 Model with Diffusion
Exercise 5
References
6 Brain Dynamics: Neural Systems in Space and Time
6.1 Introduction
6.2 Properties of Neurons
6.2.1 Electrophysiological Properties of Neurons
6.2.2 Ionic Conductance
6.2.3 Generation of Action Potential, Its Activity, and Signal Propagation
6.2.3.1 Synapse and Its Functional Mechanism
6.2.4 Ionic Currents, Neuronal Activity and Neuronal Responses
6.3 Hodgkin–Huxley (HH) Model
6.3.1 Simulation Results
6.4 FitzHugh-Nagumo (FHN) Model
6.4.1 Linear Stability Analysis and Hopf Bifurcation
6.4.2 Amplitude Equation
6.4.2.1 Linear Stability Analysis of the Amplitude Equation
6.4.3 Secondary Bifurcation of the Turing Pattern
6.4.3.1 Dynamics of 1D Diffusion in FHN Model
6.5 Morris–Lecar (M–L) Model
6.5.1 Stability and Bifurcation Analysis
6.5.1.1 Bifurcation Analysis
6.5.2 Spatial Morris–Lecar Model
6.5.3 Multiple-Scale Analysis (Amplitude Equations)
6.5.3.1 Amplitude Stability
6.5.4 Spiking and Bursting in Single M-L Neuron Model
6.6 Hindmarsh–Rose (H-R) Model
6.6.1 Formulation of a Modified H-R System
6.6.2 Bifurcation Analysis
6.6.3 Modified Reaction–Diffusion H-R System
6.6.4 Construction of Traveling Front Solution
6.6.4.1 Numerical Results
References
Solutions to Odd-Numbered Problems
Index