New Research

Karim, A, Md., Bhagat, S. R. and Bhowmick, A. R.*, Empirical Detection of Parameter Variation in Growth Curve Models Using Interval Specific Estimators, Chaos, Solitons and Fractals, Volume 157, 111902, 2022. https://doi.org/10.1016/j.chaos.2022.111902  

Quantitative assessment of the growth of biological organisms has produced many mathematical equations, and over time, it has become an independent research area. Many efforts have been given on statistical identification of the correct growth model from a given data set, and have generated many model selection criteria as well. Every growth equation is unique in terms of mathematical structure; however, one model may serve as a close approximation of another equation by some appropriate choice of the parameter(s). It is still an interesting problem to select the best estimating model from a set of models whose shapes are similar in nature. In this manuscript, we utilize an existing model selection criterion which reduces the number of model fitting exercises substantially. By using continuous transformation of parameters, interconnections between many existing equations can be made. We consider four basic models, namely, exponential, logistic, confined exponential, and theta-logistic, as a starting point. Starting with these basic models, we utilize the idea of interval-specific rate parameter (ISRP), proposed by Bhowmick et al. (J. Biol. Phys., Vol 40, pp. 71–95, 2014) to obtain the best model for real data sets. The ISRP profiles of the parameters of simpler models indicate the nature of variation in parameters as a function of time, enabling the experimenter to extrapolate the inference to more complex models. Our proposed methodology significantly reduces the efforts involved in model fitting exercises. Connections have been built amongst many growth equations, which were studied independently to date by researchers. We believe that this work will be helpful for practitioners in the field of growth study. The proposed idea is verified by using simulated, and real data sets. 

Achyut Kumar Banerjee, Jyoti Prajapati, Amiya Ranjan Bhowmick, Yelin Huang, Abhishek Mukherjee, Different factors influence naturalization and invasion processes – A case study of Indian alien flora provides management insights. Journal of Environmental Management, 2021 

Why do some alien plants become naturalized, and some naturalized become invasive? Do different factors determine successful naturalization and invasion? Most, if not all, studies addressing these questions have focused either on the part of the invasion continuum or a specific group of alien species. In this study, we aimed to answer these questions for alien plant invasion in India by considering 13 variables related to biogeography, introduction pathways, uses, functional traits, and distribution for 715 species belonging to three invasion categories. We deciphered the variables’ influence on successful naturalization and invasion through a structural equation modeling framework implemented as path analyses and translated the findings to management implications. 

Our study revealed that the invasive aliens had significantly higher naturalized range size, a greater number of uses, and higher specific leaf area than the naturalized and casual aliens. Path analyses revealed that the native and naturalized range sizes, number of uses, and growth form had a direct influence on naturalization success, whereas longer minimum residence time (MRT) facilitated overcoming of the dispersal barrier for naturalized species. Invasion success was directly influenced by the MRT and number of uses, which were further influenced by the number of native congeners and the naturalized range size, respectively. Plant growth forms indirectly influenced invasion success, whereas the native range sizes had indirect effects on successful naturalization and invasion by strongly influencing the size of the naturalized range.

Mestry, D. V, Bhowmick, A. R.*, On estimating the parameters of generalized logistic model from census data: Drawback of classical approach and reliable inference using Bayesian framework, Ecological Informatics, 2021 (Elsevier)

In ecology, a nonconstant functional relationship between per capita growth rate and population size is referred as density dependence and mathematical models are utilized to detect its presence in natural populations. The theta-logistic model (parameterized by rm: intrinsic growth rate; θ: shape parameter; and K: carrying capacity) has been extensively discussed through various generalizations in the literature due to its flexibility and sound ecological interpretations which can be generalized for many natural populations. In this article, we show that nonlinear least squares approach is not an appropriate choice for estimating the model using real data. Using simulation, we show that the unknown parameters are better estimated under a Bayesian framework. We utilize the Gibbs algorithm for simulating samples from the posterior density, which is approximated by grid approximation and Bayesian credible intervals are obtained. Reliability of the estimation process is shown by using simulated data sets. Rules for choosing the prior distributions are discussed. We also apply the proposed strategy to estimate parameters using real data sets from the global population dynamics database. The robustness of the method with respect to prior distribution of the parameters is investigated by taking different choice of priors. We also establish its effectiveness in estimating parameters in a predator-prey system. The discussed method is computationally intensive, but its simple implementation will be useful for fitting complex models to study growth patterns of natural populations.

Collaborators

• Dr. Abhishek Mukherjee, Agricultural and Ecological Research Unit, Indian Statistical Institute, Kolkata (Giridih)

• Dr. Achyut Kumar Banerjee, Sun Yat-sen University, Guanzhou, China.

• Sabyasachi Bhattacharya, Agricultural and Ecological Research Unit, Indian Statistical Institute, Kolkata

• Dr. Joyita Mukherjee, Department of zoology, Krishna Chandra College, Hetampur, Birbhum.

• Santanu Ray, Department of Zoology, Visva-Bharati University, Santiniketan

• Yun Kang, Department of Mathematics, Arizona State University, USA

• Sourav Kumar Sasmal, Assistant Professor, Department of Mathematics, BITS Pilani

• Sourav Rana, Department of Statistics, Visva-Bharati University, Santiniketan

• Tridip Sardar, Department of Mathematics, Dinabandhu Andrews Collge. 

• Bapi Saha, Assistant Professor, Govt. College of Textile Engineering, Berhampore

• Anindita Chatterjee, Senior Research Fellow, Indian Statistical Institute, Kolkata

• Soumalya Mukhopadhyay, Assistant Professor,  Department of Statistics, Visva-Bharati University, Santiniketan

• Biman Chakraborty, Assistant Professor, Aliah University, Kolkata

Some Research Related Links

• Global Population Dynamics Database, NERC Center for Population Biology, Imperial College London.

Mathematical Biology Activities

• Biomathematical Society of India.

• National Network for Mathematical and Computational Biology

Summary of PhD Research

Growth curve models serve as the mathematical framework for the quantitative studies of growth in many areas of applied science. In Ecology, the density dependent models refer to the population dynamics, where, growth rate depends on some functions of population size. In this thesis we have extended the density dependent model based on the ecological foundation of cooperation. To validate this idea, we used Global Population Dynamics Database, a vast repository of population time series data containing more than 5000 time series, and Sibly’s (Science, 2005) framework of studying growth regulation of animal populations. The proposed methodology helps better explain the extinction risk of species and equip us with a good conservation management tool. In natural populations, the stochastic set up is more appropriate to take into account randomness.  We use a more general form of birth and death process to extend the cooperation model under stochastic environment. We propose a new method for approximating the moments of equilibrium distribution that is valid for both integer and non-integer values of model parameters. Apart from cooperation, memory is also an essential behavioural issue and is an integrated part of living organisms. We extend the density dependent and independent models to include the effect of memory by using fractional calculus. In the next phase of thesis, we introduce the concept of time covariate model, where growth rate is a function of time and discuss its utility to explain a real life biological experiment and study related statistical inference problems. The pillar of the growth models is the relative growth rate (RGR). This RGR is empirically estimated by Fisher (1921)’s average RGR. This average RGR is growth law invariant metric based on the assumption of exponential growth between two consecutive time points. We propose the concept of interval specific rate parameter that captures the proximity of a given data with respect to underlying model even for short time interval. The proposed measure can be used as a more appropriate model selection criterion than the existing summary measures of goodness of fit. Potential future directions are given at the end of the thesis.

Reviewer for Journals

• Bulletin of Mathematical Biology, Springer publications.

• Ecological Modelling, Elsevier publications.

• Mathematical Biosciences, Elsevier publications.

• Stochastic Environmental Research and Risk Assessment, Springer.

• Ecography, Wiley

• Ecological Informatics, Elsevier

• Journal of Avian Biology, Wiley

• Plos One

• Ecological Indicators, Elsevier

• Mathematical Biosciences and Engineering, AIMS

• Expert Systems with Applications, Elsevier

• Environmental Science and Pollution Research, Springer