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The book provides an introduction to deterministic (and some stochastic) modelling 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.
Spatial Dynamics and Pattern Formation in Biological Populations begins with a 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 modelling 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, the authors present a detailed analysis of different types of 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 to the study of brain dynamics (neural systems in space and time).
Significant advances made in modelling 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 modelling and numerical simulations using basic conceptual and classic models of population dynamics, virus and brain dynamics.
- Covers a wide range of models using spatial and non-spatial approaches.
- Covers single, two and multispecies reaction-diffusion models from ecology and models from biochemistry. Models are analysed for the 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.
- Introduction to Diffusive Processes
- Reaction-Diffusion Modeling
- Modeling Virus Dynamics in Time and Space
- Modeling the Epidemic Spread and Outbreak of Ebola Virus
- Modeling the Transmission Dynamics of Zika Virus
- Brain Dynamics: Neural Systems in Space and Time
Dr Ranjit Kumar Upadhyay is a Professor in the Department of Mathematics & Computing, Indian Institute of Technology (Indian School of Mines), Dhanbad, India. He is an acknowledged researcher and has contributed to various areas of applied mathematics, mathematical modelling and nonlinear dynamics. His research areas are interdisciplinary in nature like dynamical systems theory; chaotic dynamics of real-world situations; population dynamics for marine and terrestrial ecosystems; spatial dynamics in epidemics (influenza, Ebola and Zika viruses), e-epidemic and neural models; and reaction-diffusion modelling. He has published 150 research papers in different international journals of repute and a number of these publications are with international collaborators. He has supervised 9 students for their PhD. Currently, 6 students are working under his guidance for their PhD degree. He has handled a number of sponsored R&D projects. He is the co-author of Introduction to Mathematical Modeling and Chaotic Dynamics published by CRC Press. He is on the editorial board for many reputed journals. He is an Associate Editor of the Elsevier journal Food Webs and Springer journal Advances in Difference Equations and Differential Equations and Dynamical Systems. He is a Guest Editor of a Special Issue entitled Nonlinear Models in Biosignaling, Biosensor and Neural Systems – Modeling, Simulations, and Applications being brought out by the Springer journal Differential Equations and Dynamical Systems. He was a visiting research fellow under the Indo-Hungarian Educational exchange program in Eötvös University, Budapest, Hungary. He was a visiting scientist and delivered invited lectures in many renowned institutions like the University of Cambridge; the University of Leicester, UK; the University of Le Havre Normandie, France. He is a member of the International Society of Computational Ecology, Hong Kong.
Dr Satteluri R. K. Iyengar was a Professor and former Head of the Department of Mathematics, Indian Institute of Technology (IIT), New Delhi, India. He worked as a Professor for more than 22 years. His areas of research work are Numerical Analysis and Mathematical Modeling. He is the co-author of Introduction to Mathematical Modeling and Chaotic Dynamics published by CRC Press and Advanced Engineering Mathematics, published by Narosa Publishing House. He is a co-author of a number of books on numerical analysis like Numerical Methods for Scientific and Engineering Computation etc. He has many research publications in international journals of repute. He was a post-doctoral fellow at Oxford University Computing Laboratory, Oxford, United Kingdom, and the University of Saskatchewan, Canada. He was awarded the Distinguished Service Award by the Indian Institute of Technology, New Delhi, India during its Golden Jubilee Year in 2011. He was also awarded the Distinguished Indian Award in 2007 by the Pentagram Research Center (P) Limited, Hyderabad, India for his contributions.
