N. Bairagi

Pelicans at risk in Salton Sea-an eco-epidemiological model-II

The Salton Sea which is located in the Southeast desert of California is becoming a dangerous habitat for birds. It is supposed that elevated salinity, accelerated eutrophication, blooms of Avian botulism and dramatic water quality fluctuation are the key factors for massive die-off of Tilapia (prey) and Pelican (predator) in the Salton sea. Chattopadhyay and Bairagi [Ecol. Model. 136 (2001) 103] proposed and analyzed a three-component eco-epidemiological model consisting of susceptible fish population, infected fish population and their predator, the Pelican population. We modify their model from more biologically realistic point of view and then analyze it. The main objective of the work is to find out conditions for which the modified system becomes disease free. Numerical simulations for a hypothetical set of parameter values are presented to illustrate the analytical findings. It is observed that if the initial value of the system is contained in the invariant set which contain the disease-free equilibrium, the solution will approach the disease-free equilibrium under suitable parametric conditions. If the initial value of the system is not in the invariant set, impulsive harvesting strategies can be used to change the initial state to the desired disease-free equilibrium state.

Harvesting as a disease control measure in an eco-epidemiological system - a theoretical study

Epidemiology and ecology are traditionally treated as independent research areas, but there are many commonalities between these two fields. It is frequently observed in nature that the former has an encroachment into the later and changes the system dynamics significantly. In population ecology, in particular, the predator-prey interaction in presence of parasites can produce more complex dynamics including switching of stability, extinction and oscillations. On the other hand, harvesting practices may play a crucial role in a host-parasite system. Reasonable harvesting can remove a parasite, in principle, from their host. In this paper, we study theoretically the role of harvesting in a predator-prey-parasite system. Our study shows that, using impulsive harvesting effort as control parameter, it is not only possible to control the cyclic behavior of the system populations leading to the persistence of all species, but other desired stable equilibrium including disease-free can also be obtained.

Variability in the secretion of corticotropin-releasing hormone, adrenocorticotropic hormone and cortisol and understandability of the hypothalamic-pituitary-adrenal axis dynamics—a mathematical study based on clinical evidence

In this article, we have developed a simple mathematical model that captures the vital mechanisms of the hypothalamic-pituitary-adrenal (HPA) axis self-regulatory activities. For this, a system of three-component non-linear delay differential equations has been proposed and analysed to observe the ultradian and circadian variabilities of the hormone secretion of the HPA axis in normal subjects. Our analysis reveals that a feedback mechanism is sufficient to show the ultradian variability of the hormone secretion pattern but fails to show the circadian variability. A central nervous system-driven pulse generator coupled with the primary feedback mechanism can exhibit the ultradian as well as circadian variability in the hormone secretion of the HPA axis. The model can also predict different dynamics of the normal HPA axis following physiological changes (viz. adrenalectomy and hypophysectomy) and pathological changes (viz. infusion of different hormones).

Chaos in eco-epidemiological problem of the Salton Sea and its possible control

Abstract The Salton Sea, which is located in the southeast desert of California, came into the limelight due to deaths of fish and fish catching birds on a massive scale. Recently, Chattopadhyay and Bairagi [J. Chattopadhyay, N. Bairagi, Pelicans at risk in Salton Sea – an eco-epidemiological model, Ecol. Model. 136 (2001) 103–112] proposed and analysed an eco-epidemiological model on Salton Sea. In the present paper, we modified their model by taking into account the bilinear mass action incidence rate and performed extensive numerical simulations. Our studies show that the system exhibits chaotic dynamics when some key parameters attain their critical values. We have tried to explain the unusual deaths of fish and fish eating birds in the Salton Sea using the simulation results. We have also suggested some possible measures to avoid chaos in such natural systems.

Age-structured predator–prey model with habitat complexity: oscillations and control

In this article, we study a predator–prey interaction in a homogeneously complex habitat where predator takes a fixed time to develop from immature to its mature stage. The age-structure of the predator and its interaction with the prey is framed in a system of delay differential equations. The objective is to study the role of habitat complexity and the maturation delay of the predator on the overall dynamics of the model system. Different interesting dynamical behaviours can be obtained by regulating two key parameters, namely the degree of habitat complexity and the maturation delay. It is observed that the system becomes unstable from its stable condition when the maturation delay crosses some critical value. The periodic solutions bifurcated from the interior equilibrium is found to be supercritical and stable. Synchronization of population fluctuations is, however, possible by increasing the strength of habitat complexity. The predator population goes to extinction and the prey population reaches to its maximum, irrespective of the length of maturation delay, when the habitat complexity crosses some upper critical value. The qualitative dynamical behaviours of the model system are verified with the data of Paramecium aurelia (prey) and Didinium nasutum (predator) interaction.

Nutrient, Non-toxic Phytoplankton, Toxic Phytoplankton and Zooplankton Interaction in an Open Marine System

In this paper we propose a mathematical model for the interaction of nutrient, non-toxic phytoplankton, toxic phytoplankton and their predator zooplankton population in an open marine system. For a realistic representation of the open marine plankton ecosystem, we have incorporated various natural phenomena such as spatial flow, nutrient recycling, toxin effects, inter-species competition and grazing at a higher level. Nutrient-phytoplankton-zooplankton interactions are observed to be very complex and situation specific. Different exciting results, ranging from stable situation to cyclic blooms or monospecies bloom, may occur under different favourable conditions, which may give some insights for predictive management.

The evolution on eco-epidemiological systems theory and evidence

In this article, we have briefly reviewed the development of eco-epidemiology since its initiation about two decades before. Different studies of eco-epidemiological models reveal that a predator-prey system in presence of parasites may exhibit complex but biologically relevant dynamics including switching of stability, extinction and oscillations or even unpredictable dynamics under some parametric conditions. This article also points out some important aspects that should be addressed with great importance.

A predator-prey model with Beddington-DeAngelis functional response: a non-standard finite-difference method

In this paper, we transform a continuous-time predator-prey model with Beddington–DeAngelis functional response into a discrete-time model by nonstandard finite difference scheme (NSFD). The NSFD model shows complete dynamic consistency with its continuous counterpart for any step size. However, the discrete model of same continuous system obtained by Euler forward method shows dynamic inconsistency for larger step size. Extensive numerical simulations have been done to compare the dynamics of NSFD system and Euler system. Our analysis reveals that dynamics of NSFD model is independent of the step-size, whereas the dynamics of the standard discrete model completely depends on the step-size and produces spurious dynamics like chaos.

Switching from simple to complex dynamics in a predator–prey–parasite model: An interplay between infection rate and incubation delay

Parasites play a significant role in trophic interactions and can regulate both predator and prey populations. Mathematical models might be of great use in predicting different system dynamics because models have the potential to predict the system response due to different changes in system parameters. In this paper, we study a predator-prey-parasite (PPP) system where prey population is infected by some micro parasites and predator-prey interaction occurs following Leslie-Gower model with type II response function. Infection spreads following SI type epidemic model with standard incidence rate. The infection process is not instantaneous but mediated by a fixed incubation delay. We study the stability and instability of the endemic equilibrium point of the delay-induced PPP system with respect to two parameters, viz., the force of infection and the length of incubation delay under two cases: (i) the corresponding non-delayed system is stable and (ii) the corresponding non-delayed system is unstable. In the first case, the system populations coexist in stable state for all values of delay if the force of infection is low; or show oscillatory behavior when the force of infection is intermediate and the length of delay crosses some critical value. The system, however, exhibits very complicated dynamics if the force of infection is high, where the system is unstable in absence of delay. In this last case, the system shows oscillatory, stable or chaotic behavior depending on the length of delay.

Pelican at Risk in Salton Sea – a Delay-Induced Eco-Epidemiological Model

Elevated salinity, accelerated eutrophication, blooms of Avian botulism and dramatic water quality fluctuation are supposed to be the key factors for massive die-off of Tilapia (prey) and Pelican (predator) in the Salton sea. We modify the model of Chattopadhyay and Bairagi [Ecological Modelling 136 (2001), pp. 103-112] with an assumption that the growth rate of susceptible fish population is very high and study the dynamics of the system by introducing a delay factor in the predator response function. It is observed that the otherwise stable system exhibit a stable limit cycle solution when the lag factor attains its critical value. It is also observed that there is a high possibility of an epidemic out break in the fish as well as in the Pelican population if the predation process is delayed by a considerable amount of time. Numerical simulations for a hypothetical set of parameter values are presented to illustrate the analytical findings.