Spatial Dynamics and Pattern Formation in Biological Populations

IntroductionDiffusion, Convection, Advection and Dispersion ProcessesSome Basic Laws of DiffusionFick’s Laws of Diffusion Darcy’s LawDiffusion EquationLinear Diffusion Equation in One DimensionTime-Dependent/Concentration-Dependent Diffusion Coefficient ProblemsLinear Diffusion Equation in Two and Three DimensionsTwo-Dimensional Diffusion on a DiskLinear Diffusion Equation in Three DimensionsReaction–Diffusion Equations in Diffusion ProcessesDiffusion in a Heterogeneous EnvironmentStochastic Reaction–Diffusion (SRD) SystemsHopf Bifurcation AnalysisMultiple-Scale Analysis/Weakly Nonlinear AnalysisLinear Stability Analysis of the Amplitude Equations (1.79)Overview of the BookReferencesReaction–Diffusion ModelingIntroductionReaction–Diffusion EquationsDerivation of Reaction-Diffusion EquationHyperbolic Reaction–Diffusion EquationsSingle-Species Reaction–Diffusion ModelsModel 1: Linear Model of Kierstead and SlobodkinKISS Model in Two DimensionsModel 2: Nonlinear Fisher EquationSpatial Steady-State SolutionSome Analytical SolutionsModel 3: Nagumo EquationNumerical SolutionsTwo-Species Reaction–Diffusion ModelsTuring Instabilities of Two-Species Reaction–Diffusion SystemsPredator–Prey Reaction–Diffusion SystemsApplications in Biochemistry: Belousov–Zhabotinsky Reaction–Diffusion SystemsModel 1: Oregonator ModelModel 2: Brusselator ModelModel 3: Schnakenberg ModelModel 4: Lengyel–Epstein ModelModel 5: Sel’kov ModelModel 6: Gray–Scott ModelMultispecies Reaction–Diffusion ModelsModel 1: Hastings–Powell ModelModel 2: Modified Upadhyay–Rai ModelModel 3: Modified Leslie–Gower-Type Three-Species ModelModeling Virus Dynamics in Time and SpaceIntroductionNext-Generation Operator MethodSusceptible-Infected (SI) ModelsModels with Nonlinear Incidence RateModels with Self and Cross-DiffusionInfluenza Epidemic ModelsA Simple Spatial SI Epidemic ModelTuring InstabilityTwo-Time Scale Influenza ModelsSusceptible-Infected-Susceptible (SIS) ModelsSusceptible-Infected-Removed (SIR) ModelsSusceptible-Infected-Removed-Susceptible (SIRS) ModelsSusceptible-Exposed-Infected-Recovered (SEIR) ModelsInfluenza Model RevisitedExercise 3ReferencesModeling the Epidemic Spread and Outbreak of Ebola VirusIntroductionSource and SymptomsTransmission and Control of EpidemicsFormulation of Ebola Epidemic ModelsModel 1: Ebola Epidemic SEIR ModelSpatial SEIR Ebola Epidemic ModelModel 2: Ebola Epidemic SEIRHD ModelSensitivity Indices of R[sup(0)]Model 3: Ebola Epidemic SEIORD Model and Its ExtensionModel 4: Ebola Epidemic SEIRD Model with Time DelayExistence of Endemic Equilibrium and Stability AnalysisModel 5: General Ebola Transmission Model for Population in a CommunityExercise 4ReferencesModeling the Transmission Dynamics of Zika VirusIntroductionSymptoms and Clinical FeaturesFormulation of Zika Epidemic ModelModel 1: Zika Virus SIR Transmission ModelOptimal Control AnalysisModel 2: Zika Virus SEIR Transmission ModelBifurcation AnalysisOptimal Control AnalysisModel 3: Zika Virus SEIR Horizontal and Vertical Transmission ModelModel 4: Zika Virus with Vertical TransmissionModel 5: Zika Virus SIR Transmission Model with Human and Vector MobilityExistence of Travelling Wave SolutionsModel 6: Zika Virus Transmission with Criss-Cross Interactions ModelModel 7: Zika Virus SEIR Transmission ModelModel with DiffusionExercise 5ReferencesBrain Dynamics: Neural Systems in Space and TimeIntroductionProperties of NeuronsElectrophysiological Properties of NeuronsIonic ConductanceGeneration of Action Potential, Its Activity, and Signal PropagationSynapse and Its Functional MechanismIonic Currents, Neuronal Activity and Neuronal ResponsesHodgkin–Huxley (HH) ModelSimulation ResultsFitzHugh–Nagumo ModelLinear Stability Analysis and Hopf bifurcationAmplitude EquationSecondary Bifurcation of the Turing PatternMorris–Lecar (M–L) ModelStability and Bifurcation AnalysisSpatial Morris–Lecar ModelMultiple-Scale Analysis (Amplitude Equations)Amplitude StabilitySpiking and Bursting in Single M-L Neuron ModelHindmarsh–Rose (H-R) ModelFormulation of a Modified H-R SystemBifurcation AnalysisModified Reaction–Diffusion H-R SystemConstruction of Traveling Front SolutionNumerical ResultsReferencesSolutions to Odd-Numbered Problems
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