P.K. Tapaswi

A delay differential equations model of plankton allelopathy.

In this paper we have studied the dynamics of planktonic growth with special consideration on time dependent fluctuations in density of the species. We propose a modified delay differential equation model of the growth of two species of plankton having competitive and allelopathic effects on each other. The model system shows a stable limit cycle oscillation when the allelopathic effect is of a stimulatory nature.

Effect of Cross-Diffusion on Pattern Formation – a Nonlinear Analysis

In this paper, we consider a model for the switching behaviour (determination) of a cell proposed by Meinhardt (1982) and observe that this two component system can create a pattern only in the presence of cross-diffusion. We also analyse the global behaviour of this model system by the Bendixson–Dulac criteria and Liapunov functional method.

A space-time state-space model of phytoplankton allelopathy

An integro-differential equation system with nonlocal effects of interspecific allelopathic interaction has been studied to investigate the formation of spatio-temporal structures in toxin producing phytoplankton population. The model is inherently more realistic than the usual kind of reaction-diffusion model. Bifurcation from uniform steady-state solution has been examined. Evolution of steady-state spatially periodic structure and periodic standing waves have been studied. The model helps to investigate the blooms, pulses and succession in different patches of phytoplankton population. Numerical simulations for a hypothetical set of parameter values and experimental observations have been presented to substantiate the analytical findings.

Transmission of Japanese Encephalitis in a 3-population model

Abstract The model considered for the spread of Japanese Encephalitis (JE) in a human population of varying size from a reservoir population (pigs, cattle, equines, birds, etc.) through a vector population (particular species of mosquitos) is of SIRS (susceptible-infective-recovered-susceptible) type for the human and reservoir populations and SIS (susceptible-infective-susceptible) type for the vector population. We have considered the logistic differential equation with density-dependent birth rate for the vector population whereas the reservoir population is of constant size. We assume that the human population is regulated by the disease. We also assume that there is a constant recruitment rate of susceptibles into the human population. We perform an equilibrium and stability analysis to find a threshold condition. If the threshold is exceeded, then there is a unique equilibrium with disease present which is locally stable to small perturbations and global stability depends on death rates and the ratio of the equilibrium population sizes of the infected vector and total human populations.

Selective harvesting in a two species fishery model

Abstract A two species combined harvesting fishery model with selective harvesting by incorporating a discrete time delay (τ) in harvesting age and size of both the species have been considered. It has been observed that the otherwise asymptotically stable system undergoes Hopf bifurcation for some value of τ > τ0 giving rise to a small amplitude oscillation around the non-zero equilibrium. Numerical analysis and computer simulation have been performed to investigate the global properties of the system.

EFFECT OF CROSS-DIFFUSION ON A DIFFUSIVE PREY-PREDATOR SYSTEM—A NONLINEAR ANALYSIS

In this paper, we show that a simple prey-predator diffusive system may give rise to spontaneous emergence of dissipative structure (patchiness) only in the presence of cross-diffusion. The critical wave-length just sufficient to drive the system to local instability has been worked out. Moreover, by constructing a suitable Liapunov function, we have also investigated the global behaviour of the dissipative structure.

Immunity boosted by low level of exposure to infection in an SIRS model

Abstract The development of immunity in the susceptible class by a continuous low level of infection is a commonly observed phenomenon in many infectious diseases. This important feature has been incorporated in an SIRS epidemiological model with both the rates of incidence and increase of immunity being nonlinear in nature, instead of being bilinear, of the form β 1 I p S and β 2 I p ′ S (0 p ′ p , p ≠ 1 and p ′ ≠ 1) respectively. The local and global behaviour of the dynamics of the model have been investigated.

Global stability results of a “susceptible-infective-immune-susceptible” (SIRS) epidemic model

Abstract In this paper, we have investigated the global behaviour of the SIRS epidemic model under the assumption that some portion of immunes are infective as proposed by J.L. Aron and the immunity is lost at a constant rate. We have shown that existence of local stability properties guarantees their global stability.

A mathematical model for the occurrences of Japanese Encephalitis

The frequency of occurrences of Japanese Encephalitis (JE) during different months of the years 1986-1990 in Burdwan of West Bengal, India, has been studied with the help of a mathematical model using a third order harmonic Fourier Series having a linear trend. The validity of this model has been tested by available data of five years in Burdwan and of one year in Thailand and by appropriate goodness of fit tests. This model seems to have a wide universal application where predominant seasonal effects prevail and can be satisfactorily used for prediction of JE for the next coming year.

Order and disorder in biological systems through negative cross-diffusion of mitotic inhibitor-a mathematical model

A negative feedback epigenetic control system involving synthesis of the mitotic inducing and inhibiting proteins, simultaneously with the intercellular self-diffusion and cross-diffusion of the latter only have been proposed in this paper. By this model we can explain clearly the mechanisms of three types of cellular development, namely, ordered structure during embryonic development, normal tissue growth and tumor growth, respectively, for different ranges of values of the ratio of self-diffusion and cross-diffusion.