Nitu Kumari

Turing patterns and long-time behavior in a three-species food-chain model

We consider a spatially explicit three-species food chain model, describing generalist top predator-specialist middle predator-prey dynamics. We investigate the long-time dynamics of the model and show the existence of a finite dimensional global attractor in the product space, L(2)(Ω). We perform linear stability analysis and show that the model exhibits the phenomenon of Turing instability, as well as diffusion induced chaos. Various Turing patterns such as stripe patterns, mesh patterns, spot patterns, labyrinth patterns and weaving patterns are obtained, via numerical simulations in 1d as well as in 2d. The Turing and non-Turing space, in terms of model parameters, is also explored. Finally, we use methods from nonlinear time series analysis to reconstruct a low dimensional chaotic attractor of the model, and estimate its fractal dimension. This provides a lower bound, for the fractal dimension of the attractor, of the spatially explicit model.

Finite Time Blowup in a Realistic Food-Chain Model

We investigate a realistic three-species food-chain model, with generalist top predator. The model based on a modified version of the Leslie-Gower scheme incorporates mutual interference in all the three populations and generalizes several other known models in the ecological literature. We show that the model exhibits finite time blowup in certain parameter range and for large enough initial data. This result implies that finite time blowup is possible in a large class of such three-species food-chain models. We propose a modification to the model and prove that the modified model has globally existing classical solutions, as well as a global attractor. We reconstruct the attractor using nonlinear time series analysis and show that it pssesses rich dynamics, including chaos in certain parameter regime, whilst avoiding blowup in any parameter regime. We also provide estimates on its fractal dimension as well as provide numerical simulations to visualise the spatiotemporal chaos.

A remark on “Study of a Leslie–Gower-type tritrophic population model” [Chaos, Solitons and Fractals 14 (2002) 1275–1293]

In Aziz-Alaoui (2002) a three species ODE model, based on a modified Leslie–Gower scheme is investigated. It is shown that under certain restrictions on the parameter space, the model has bounded solutions for all positive initial conditions, which eventually enter an invariant attracting set. We show that this is not true. To the contrary, solutions to the model can blow up in finite time, even under the restrictions derived in Aziz-Alaoui (2002), if the initial data is large enough. We also prove similar results for the spatially extended system. We validate all of our results via numerical simulations.

Pattern Formation in Spatially Extended Tritrophic Food Chain Model Systems: Generalist versus Specialist Top Predator

The complex dynamics of two types of tritrophic food chain model systems when the species undergo spatial movements, modeling two real situations of marine ecosystem, are investigated in this study analytically and using numerical simulations. The study has been carried out with the objective to explore and compare the competitive effects of fish and molluscs species being the top predators, when phytoplankton and zooplankton species are undergoing spatial movements in the subsurface water. Reaction diffusion systems have been used to represent temporal evolution and spatial interaction among the species. The two model systems differ in an essential way that the top predators are generalist and specialist, respectively, in two models. “Wave of Chaos” mechanism is found to be the responsible factor for the pattern (non-Turing) formation in one dimension seen in the food chain ending with top generalist predator. In the present work we have reported WOC phenomenon, for the first time in the literature, in a three-species spatially extended food chain model system. The numerical simulation leads to spontaneous and interesting pattern formation in two dimensions. Constraints on different parameters under which Turing and non-Turing patterns may be observed are obtained analytically. Diffusion-driven analysis is carried out, and the effect of diffusion on the chaotic dynamics of the model systems is studied. The existence of chaotic attractor and long-term chaotic behavior demonstrate the effect of diffusion on the dynamics of the model systems. It is observed from numerical study that food chain model system with top predator as generalist has very rich dynamics and shows very interesting patterns. An ecosystem having top predator as specialist leads to the stability of the system.

Modeling the spread of bird flu and predicting outbreak diversity

Abstract Avian influenza, commonly known as bird flu, is an epidemic caused by H5N1 virus that primarily affects birds like chickens, wild water birds, etc. On rare occasions, these can infect other species including pigs and humans. In the span of less than a year, the lethal strain of bird flu is spreading very fast across the globe mainly in South East Asia, parts of Central Asia, Africa and Europe. In order to study the patterns of spread of epidemic, we made an investigation of outbreaks of the epidemic in one week, that is from February 13–18, 2006, when the deadly virus surfaced in India. We have designed a statistical transmission model of bird flu taking into account the factors that affect the epidemic transmission such as source of infection, social and natural factors and various control measures are suggested. For modeling the general intensity coefficient f ( r ) , we have implemented the recent ideas given in the article Fitting the Bill, Nature [R. Howlett, Fitting the bill, Nature 439 (2006) 402], which describes the geographical spread of epidemics due to transportation of poultry products. Our aim is to study the spread of avian influenza, both in time and space, to gain a better understanding of transmission mechanism. Our model yields satisfactory results as evidenced by the simulations and may be used for the prediction of future situations of epidemic for longer periods. We utilize real data at these various scales and our model allows one to generalize our predictions and make better suggestions for the control of this epidemic.

Modeling spatiotemporal dynamics of vole populations in Europe and America

The mathematical models proposed and studied in the present paper provide a unified framework to understand complex dynamical patterns in vole populations in Europe and North America. We have extended the well-known model provided by Hanski and Turchin by incorporating the diffusion term and spatial heterogeneity and performed several mathematical and numerical analyses to explore the dynamics in space and time of the model. These models successfully predicted the observed rodent dynamics in these regions. An attempt has been made to bridge the gap between the field and theoretical studies carried out by Turchin and Hanski (1997) and Turchin and Ellner (2000). Simulation experiments, mainly two-dimensional parameter scans, show the importance of spatial heterogeneity in order to understand the poorly understood fluctuations in population densities of voles in Fennoscandia and Northern America. This study shed new light upon the dynamics of voles in these regions. The nonlinear analysis of vole data suggests that the dynamical shift is from stability to chaos. Diffusion driven model systems predict a new type of dynamics not yet observed in the field studies of vole populations carried out so far. This has been termed as chaotic in time and regular in space (CTRS). We observed CTRS dynamics in several simulation experiments. This directs us to expect that dynamics of this animal would be de-correlated in time and simultaneously mass extinctions might be possible at many spatial locations.

Long time dynamics of a three-species food chain model with Allee effect in the top predator

The Allee effect is an important phenomenon in population biology characterized by positive density dependence, that is a positive correlation between population density and individual fitness. However, the effect is not well studied in multi-level trophic food chains. We consider a ratio dependent spatially explicit three species food chain model, where the top predator is subjected to a strong Allee effect. We show the existence of a global attractor for the model, that is upper semicontinuous in the Allee threshold parameter m . Next, we numerically investigate the decay rate to a target attractor, that is when m = 0 , in terms of m . We find decay estimates that are O ( m γ ) , where γ is found explicitly. Furthermore, we prove various overexploitation theorems for the food chain model, showing that overexploitation has to be driven by the middle predator. In particular overexploitation is not possible without an Allee effect in place. We also uncover a rich class of Turing patterns in the model which depend significantly on the Allee threshold parameter m . Our results have potential applications to trophic cascade control, conservation efforts in food chains, as well as Allee mediated biological control.

A comment on “Mathematical study of a Leslie-Gower type tritrophic population model in a polluted environment” [Modeling in Earth Systems and Environment 2 (2016) 1–11]

In the current manuscript we comment on (Misra and Babu, Model Earth Syst Environ 2(1):1–11, 2016), where two novel five-species ODE models are proposed and analyzed, in order to investigate the population dynamics of a three-species food chain, in a polluted environment. It is shown in Misra and Babu (Model Earth Syst Environ 2(1):1–11, 2016) that under certain restrictions on the parameters, the models have bounded solutions for all positive initial conditions. Furthermore, a globally attracting set is explicitly constructed for initial conditions in

Backward Bifurcation in a Cholera Model: A Case Study of Outbreak in Zimbabwe and Haiti

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