SOME ECOLOGICAL MODELS: EFFECTS OF TOXICANT AND DELAY

Abstract

The biological populations in both terrestrial and aquatic ecosystems are affected when their habitats are stressed by presence of toxicant or pollutant. Numerous experiments have been carried out worldwide to study these effects; however few attempts have been made to analyze these and related phenomena mathematically. In this thesis, attempts have been made to study some of the ecological, eco-toxicological and virological problems through mathematical models. Most of these mathematical models incorporate time delay such as delayed effect of toxicant and delay due to latency period of bacteriophage. The underlying system of delay differential equations are much more challenging to analyze compared to corresponding system of ODE's. The non-linear delay models with seasonal variation of parameters have lead to complexities in the system. The emphasis is to explore the complex dynamical behavior in delay models. Some models for bacteria and bacteriophage interactions have also been developed and analyzed. The Chapter 1 gives an overview of the literature related to this work. It also gives the basic concepts related to the dynamics of ecological, eco-toxicological and virological systems. Brief discussions on relevant tools/techniques are also included. A delayed prey-predator with Holling type II functional response is discussed in Chapter 2. The model incorporates delay both in prey and predator equations. The local stability analysis and Hopf bifurcation has been carried out. The analytical studies of the model show that the existence of positive equilibrium point rules out the stability of predator free equilibrium point. The complex dynamical behavior arising due to seasonality in the growth function of prey has been explored. The numerical studies in case of seasonal growth of prey have revealed that both delays can give rise to a chaotic solution. Further, the increase in the death rate of predator can control the chaotic solution to stable periodic solutions. In Chapter 3, a model is proposed to study the delay effect of toxicant on a biological species. The Hopf bifurcation analysis suggests that the otherwise stable system can be destabilized by the introduction of delay. The periodic toxicant input into the system lead to a chaotic system. The numerical simulations reveal that the system can be controlled by introduction of some constant negative feedback on the environmental toxicant. Further, the increase in the negative feedback increases the equilibrium level of the species and decreases that of toxicant level. The Chapter 4 deals with the study of delayed effect of toxicant on species-biomass interaction. It has been also assumed that the increased level of toxicant has adverse effect on its intrinsic growth. The introduction of delay in the model can lead to instability in the system. A 1-lopf bifurcation analysis has been performed with respect to key parameters for non-trivial equilibrium point. Numerical simulations suggest that the system may become chaotic with periodic input of toxicant into the system. Further, the negative feedback may control the chaos in the system In Chapter 5, mathematical models for the effects of bacterial debris, Monod type interaction and mutation on bacteria-bacteriophage interaction have been discussed. A model for impulsive input of phage to cure the bacterial infection has iv also been proposed and analyzed. The analytical results in the case of mutation have shown that the equilibrium state consisting of only mutant bacteria can be achieved under certain conditions on the system parameters. The effect of b i.e. Monod interaction can bring about the Hopf bifurcation in the system. The effect of debris is to stabilize the system. Effect of impulsive dose of phage in bacterial infection may have a complex behavior including chaos. Further, a suitable amount of phage dose can eliminate the pathogen bacteria. Chapter 6 proposes a model incorporating the lysogenic life cycle of the temperate phage. Due to lysogenic growth of infected bacteria, the existence of otherwise non-existing susceptible bacteria free equilibrium point is possible. The coexistence of bacteria and bacteriophage is possible through the stable positive equilibrium point and through limit cycle. A delay differential model for dynamics of bacteria and bacteriophage interaction has been developed and analyzed in Chapter 7. The time lag is assumed because of latency period of infected bacteria. Due to lysogenic growth of infected bacteria the existence of susceptible bacteria free equilibrium point is possible which can be stabilized by suitable choice of parameters. The numerical simulations have revealed that the otherwise unstable system can be stabilized by the lysogenic growth of infected bacteria. The increase in the fraction of infected bacteria going into lytic phage can destabilized the system.