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Abstract: In this paper, we have proposed and analysed a mathematical model to figure out possible ways to rescue a damaged eco-epidemiological system. Our strategy of rescue is based on the realization of the fact that chaotic dynamics often associated with excursions of system dynamics to extinction–sized densities. Chaotic dynamics of the model is depicted by 2D scans, bifurcation analysis, largest Lyapunov exponent and basin boundary calculations. 2D scan results show that μ, the total death rate of infected prey should be brought down in order to avoid chaotic dynamics. We have carried out linear and nonlinear stability analysis and obtained Hopf-bifurcation and persistence criteria of the proposed model system. The other outcome of this study is a suggestion which involves removal of infected fishes at regular interval of time. The estimation of timing and periodicity of the removal exercises would be decided by the nature of infection more than anything else. If this suggestion is carefully worked out and implemented, it would be most effective in restoring the health of the ecosystem which has immense ecological, economic and aesthetic potential. We discuss the implications of this result to Salton Sea, California, USA. The restoration of the Salton Sea provides a perspective for conservation and management strategy. [Copyright &y& Elsevier]

Abstract: In this paper, a three species food chain model, consisting of a hybrid type of prey-dependent and predator-dependent functional responses, is investigated analytically as well as numerically. The local and global stability analysis is carried out. The persistence conditions are established. Bifurcation diagrams are obtained for biologically feasible parameters. The results show that the system exhibits rich complexity features such as stable, periodic and chaotic dynamics. [Copyright &y& Elsevier]

Abstract: In this paper, dynamical complexities in two reaction–diffusion (RD) model systems are explored. A spatial heterogeneity in the form of linear spatial gradient in the reproductive growth rate of the phytoplankton is incorporated in both the model systems. Extra mortality of the zooplankton due to toxin production by the phytoplankton is included in the second reaction diffusion model system. Effect of toxin production and spatial heterogeneity in the model systems are studied. Toxin production does not seem to have an appreciable effect on the asymptotic dynamics of the model systems. On the other hand, spatial heterogeneity does influence the dynamics. In particular, it increases the frequency of occurrence of chaos as evident from two dimensional parameter scans. Both these model systems display short term recurrent chaos [Rai V. Chaos in natural populations: edge or wedge? Ecol Complex 2004;1: 127–38] as they reside on ‘edges of chaos’ (EOC) [Rai V, Upadhyay RK. Evolving to the edge of chaos: chance or necessity? Chaos, Solitons and Fractals 2006;30:1074–87]. This suggests that the ecological systems have a tendency to evolve to EOC. The study corroborates the inferences drawn from an earlier study by Rai and Upadhyay [Rai V, Upadhyay RK. Evolving to the edge of chaos: chance or necessity? Chaos, Solitons and Fractals 2006;30:1074–87]. The system’s dynamics is largely unpredictable and admits bursts of short-term predictability. [Copyright &y& Elsevier]

Abstract: Synchronization is a natural phenomenon in non-linear dynamical systems. The relative importance of various mechanisms of population synchrony has been debated by population ecologists. The debate revolves around the issue whether the regionally extrinsic or locally intrinsic agents are more potent. In the present paper, we have attempted to demonstrate that a local intrinsic mechanism, predation, can be more common cause of population synchrony than is believed. Two chaotic food chains having different kinds of top-predators are synchronized using a recently proposed algorithm by Lu and Cao [Lu J, Cao J. Adaptive complete synchronization of two identical or different chaotic (hyperchaotic) dynamical systems with fully unknown parameters. Chaos 2005;15(043901):1–10]. The idiosyncracy of this approach is that it takes care of the uncertainties involved in the parameter estimation. The complete synchronization achieved is robust to noise present in the system. We suggest that local intrinsic causes of population synchrony should be given more attention. [Copyright &y& Elsevier]

Abstract: We show that wave of chaos (WOC) can generate two-dimensional time-independent spatial patterns which can be a potential candidate for understanding planktonic patchiness observed in marine environments. These spatio-temporal patterns were obtained in computer simulations of a minimal model of phytoplankton–zooplankton dynamics driven by forces of diffusion. We also attempt to figure out the average lifetimes of these non-linear non-equilibrium patterns. These spatial patterns serve as a realistic model for patchiness found in aquatic systems (e.g., marine and oceanic). Additionally, spatio-temporal chaos produced by bi-directional WOCs is robust to changes in key parameters of the system; e.g., intra-specific competition among individuals of phytoplankton and the rate of fish predation. The ideas contained in the present paper may find applications in diverse fields of human endeavor. [Copyright &y& Elsevier]

Abstract: We propose a new mathematical model for aquatic populations. This model incorporates mutual interference in all the three populations and an extra mortality term in zooplankton population and also taking into account the toxin liberation process of TPP population. The proposed model generalizes several other known models in the literature. The principal interest in this paper is in a numerical study of the model’s behaviour. It is observed that both types of food chains display same type of chaotic behaviour, short-term recurrent chaos, with different generating mechanisms. Toxin producing phytoplankton (TPP) reduces the grazing pressure of zooplankton. To observe the role of TPP, we consider Holling types I, II and III functional forms for this process. Our study suggests that toxic substances released by TPP population may act as bio-control by changing the state of chaos to order and extinction. [Copyright &y& Elsevier]

An attempt has been made to identify the mechanism, which is responsible for the existence of chaos in narrow parameter range in a realistic ecological model food-chain. Analytical and numerical studies of a three species food-chain model similar to a situation likely to be seen in terrestrial ecosystems has been carried out. The study of the model food chain suggests that the existence of chaos in narrow parameter ranges is caused by the crisis-induced sudden death of chaotic attractors. Varying one of the critical parameters in its range while keeping all the others constant, one can monitor the changes in the dynamical behaviour of the system, thereby fixing the regimes in which the system exhibits chaotic dynamics. The computed bifurcation diagrams and basin boundary calculations indicate that crisis is the underlying factor which generates chaotic dynamics in this model food-chain. We investigate sudden qualitative changes in chaotic dynamical behaviour, which occur at a parameter value a1=1.7804 at which the chaotic attractor destroyed by boundary crisis with an unstable periodic orbit created by the saddle-node bifurcation. Multiple attractors with riddled basins and fractal boundaries are also observed. If ecological systems of interacting species do indeed exhibit multiple attractors etc., the long term dynamics of such systems may undergo vast qualitative changes following epidemics or environmental catastrophes due to the system being pushed into the basin of a new attractor by the perturbation. Coupled with stochasticity, such complex behaviours may render such systems practically unpredictable. [Copyright &y& Elsevier]