Joydev Chattopadhyay

A predator—prey model with disease in the prey

After the seminal models of Vito Volterra and Alfred James Lotka in the mid 1920s for predator-prey interactions, mutualist and competitive mechanisms have been studied extensively in the recent years by researchers. There are so many references in this context, we have just cited here some books (e.g. see, [14, 16{18, ?] and the refer- ences therein). Similarly, after the pioneering work of Kermack{McKendrick on SIRS (susceptible-infective-removal-susceptible) epidemiological models have also received much attention from scientists. Relevant references in this context are also vast and we shall again mention here some books (see [1, 2, 4], to mention a few). But little attention has been paid so far to merge these two important areas of research (see [7, 21]). In this paper, we shall put emphasis in such an eco-epidemiological system. We consider a three species eco-epidemiological system, namely, sound prey (suscep- tible), infected prey (infective) and predator. We consider the case when the predator mainly eats the infected prey. This is in accordance with a previous model by Hadeler and Freedman [7] which describes a predator-prey model where the prey is infected by a parasite, and the prey in turn infects the predator with that parasite. The infec- tion weakens the prey and increases its susceptibility to predation, while no predator impairing elect is accounted for. While the paper is mainly theoretical and does not address any specific situation, the reader may and several examples in [7]. We derive persistence and extinction conditions of the populations and we also determine conditions for which the system enters a Hopf-type bifurcation. Moreover, we observe that the bifurcated branches are supercritical in some parametric region space in a special case when the predator response function is a Holling-type II function.

Effect of toxic substances on a two-species competitive system

In this paper we have considered a two-species competitive system which is also affected by toxic substances. It has been observed that the ratio of the toxic substances of the two species plays a crucial role in shaping the dynamics of the system. Lastly, by using a suitable Liapunov function we have observed that the toxic substances have some stabilizing effect on the system.

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.

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.

Infection in prey population may act as a biological control in ratio-dependent predator–prey models

A ratio-dependent predator-prey model with infection in prey population is proposed and analysed. The behaviour of the system near the biological feasible equilibria is observed. The conditions for which no trajectory can reach the origin following any fixed direction or spirally are worked out. We investigate the criteria for which the system will persist. It is observed that the introduction of an infected population in the classical ratio-dependent predator-prey model may act as a biological control to save the population from extinction.

Viral infection on phytoplankton-zooplankton system: a mathematical model

The present paper deals with the problem of a phytoplankton-zooplankton system in which some of the phytoplankton cells are infected by some viral infection and forming an infected group. The proposed model is similar to the model of Beltrami and Carroll (J. Math. Biol. 32 (1994) 857) and the model of Venturino (Math. Popul. Dyn. Anal. Heterogeneity 1 (1995) 381). The conditions for the coexistence of the populations are presented. The essential mathematical features are analyzed with the help of local stability, global stability and numerical analysis to compare the findings of Beltrami-Carroll and Venturino. Our conclusion is that the role of viral infection in plankton community is very much unpredictable and model dependent.

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.

The stability of ecosystems: A brief overview of the paradox of enrichment

In theory, enrichment of resource in a predator-prey model leads to destabilization of the system, thereby collapsing the trophic interaction, a phenomenon referred to as “the paradox of enrichment”. After it was first proposed by Rosenzweig (1971), a number of subsequent studies were carried out on this dilemma over many decades. In this article, we review these theoretical and experimental works and give a brief overview of the proposed solutions to the paradox. The mechanisms that have been discussed are modifications of simple predator-prey models in the presence of prey that is inedible, invulnerable, unpalatable and toxic. Another class of mechanisms includes an incorporation of a ratio-dependent functional form, inducible defence of prey and density-dependent mortality of the predator. Moreover, we find a third set of explanations based on complex population dynamics including chaos in space and time. We conclude that, although any one of the various mechanisms proposed so far might potentially prevent destabilization of the predator-prey dynamics following enrichment, in nature different mechanisms may combine to cause stability, even when a system is enriched. The exact mechanisms, which may differ among systems, need to be disentangled through extensive field studies and laboratory experiments coupled with realistic theoretical models.

Effect of awareness programs by media on the epidemic outbreaks: A mathematical model

We propose and analyze a mathematical model to assess the effect of awareness programs by media on the prevalence of infectious diseases. Such programs may induce behavioral changes in the population, and divide the susceptible class into two subclasses with different infectivity rates. The biologically feasible equilibria and their stability properties are analyzed and discussed. The model analysis reveals that the rate of executing awareness programs has a substantial effect over the system and sustained oscillation may arise with increasing its value above a threshold. This threshold poses a challenge to control the epidemic. Numerical simulation also supports the analytical findings.

Chaos to order: preliminary experiments with a population dynamics models of three trophic levels

Hastings and Powell [Ecology 72(3) (1991) 896] produced a new example of a chaotic population system in a simple tri-trophic food chain with Holling type II functional response. Toxin producing phytoplankton (TPP) reduces the grazing pressure of zooplankton. The present note modifies Hastings and Powell model by introducing an extra mortality term in zooplankton population. Our result suggests that chaotic behaviour less likely occurs in a real food chain dynamics.