1. Introduction

Mercury has been recognized as a global threat to our ecosystem, and it is fast becoming a major concern to environmentalists and policy makers. Mercury can cycle in the environment in all media as part of both natural and anthropogenic activities. Once present in water, mercury is highly dangerous not only to the aquatic communities, but also to humans through direct and indirect effects [1]. Mercury, mostly in the form of methyl mercury, accumulates up the aquatic food chains, so that organisms in higher trophic levels (e.g. fishes) have higher mercury concentrations [2]. As a result, contaminated fish consumption is the most predominant path of human exposure to mercury.

            It has been illustrated that a major portion of mercury found in the tissues of various aquatic organisms enters through food (ingestion). As a consequence, the eating habits of these organisms are expected to have a significant impact on the mercury intake by these organisms. The eating habits depend to quite an extend on the various species populations and their pattern of fluctuations at a given time in the water body. In ecological literature, these different patterns are referred to as regimes. A regime, therefore, if maintained for sufficient duration, is expected to affect the steady state mercury bioaccumulation levels in different species. As a result, manipulation of the regimes of these species populations presents a tool to control mercury bioaccumulation levels.

            This work performs an optimal control analysis to achieve regime shifts in a predator-prey model and analyzes these regime shifts from mercury bioaccumulation point of view. The predator-prey model, known as the Canale's model, models three species and nutrients in a water body. Mercury bioaccumulation along the food chain is modeled using a bioenergetics model that accounts for mercury intake through water as well as ingestion (food). The predator-prey model and the bioenergetics model are connected by correlating the food intake (and hence the mercury intake) of a particular species with the current population of the prey for that species. Thus, any fluctuations in the species populations reflect in the varying mercury bioaccumulation in those species.          

2. Models The predator-prey model analyzed in this work in an extension of the three-species predator prey model, know as the Rosenzweig–MacArthur model [3]. The Rosenzweig–MacArthur tritrophic food chain model comprises of three species, called as the prey, predator and super-predator, which are modeled using a set of ordinary differential equations. This model is complemented by an extra differential equation for the nutrient. Such an equation is simply the balance of the various flows regulating the nutrient concentration, namely inflow and outflow rates, nutrient recycling due to decomposition of dead individuals of the three populations and nutrient uptake rate of the prey population. This resulting model is referred to as the Canale's model is this work. The model has been extensively studied in bifurcation literature, and a detailed exposition of this model can be found in those reference [3, 4].

            Estimating metal bioaccumulation in organisms has been a topic of intense research. Since the actual bioaccumulation of a metal (such as mercury) along a food chain depends to quite an extent on the site specific conditions, a generalized model to predict metal bioaccumulation has been difficult to formulate. Out of the various types of models that have been proposed, the bioenergetics based toxicokinetic models, which are a type of general kinetic models, have been quite promising [5]. The bioenergetics based models are appropriate to model mercury bioaccumulation since they model the metal intake through water as well as food, and the model parameters are relatively easy to determine. The basic equation that is applicable to each species is the simple unsteady state mass balance equation where accumulation (of mercury) in a species is the difference between the sum of the total input and total generation and sum of the total output and total consumption. It assumes that mercury uptake is proportional to the flux, and uses uptake parameters such as food rate and assimilation efficiency for computation. Certain assumptions of the model are: elimination is not related to organism's metabolism, equation parameters are assumed to be constant, and physiological parameters are known. A detailed explanation of the bioenergetics model can be found in [5]. Various studies have been conducted to estimate different model parameters specific to mercury [6, 7]. In this work, the literature has been reviewed and published data has been used to identify appropriate model parameters.

            The predator-prey model and the bioaccumulation model are inter-related by correlating the food intake of any particular species with the mercury intake for the bioaccumulation model. Changes in the dynamics of the Canale's model change the instantaneous food intake for the predators and super-predators (due to changing predation rates). This affects the total mercury that is taken by these species through food. Hence, any regime shift in the predator-prey model, which affects the predation rates, affects the mercury intake by the species. If the particular regime is maintained for a sufficient duration, the steady state mercury concentration in these species can alter. This is the basic foundation for the proposed work.          

3. Regime change and optimal control Optimal control theory presents an option to derive time dependent management strategies that can effectively achieve regime shifts in food chain models. Past work by the authors has illustrated the success of this approach [8, 9].  That work uses Fisher information based sustainability hypothesis, proposed by Cabezas and Fath [10], to formulate time dependent objective functions for the control problem. A similar approach has been used in this work. The regime shift is to be achieved by minimizing the variation of the time averaged Fisher information around the constant Fisher information of the targeted regime. More information about this approach can be found in [8, 9].

Canale's model exhibits various regimes such as cyclic low frequency, cyclic high frequency, stationary, chaotic etc. [3]. The idea proposed in this work is to achieve regime shift from a regime leading to high mercury bioaccumulation to a regime resulting in low mercury bioaccumulation. The control variables to achieve the regime shift are: nutrient inflow rate and nutrient input concentration.  

4. Results and discussion Simulations for the integrated model (Canale's model and the bioaccumulation model) illustrate that there is a strong correlation between the regime and steady state mercury bioaccumulation in predator and super-predators. Hence, the objective of causing a regime change in justified.

The solution of various control problems indicate that the some regime shifts are easily achievable, while others are quite difficult to achieve. Preliminary studies show that shifting the model from low bioaccumulation to high bioaccumulation regime is possible. This, however, is not the desired change. Attempts to cause shift from higher bioaccumulation to lower bioaccumulation regime led to mixed results. This emphasizes the severe nonlinearities in these population models, and thereby highlighting the fact that a systematic study of these issues is essential.

Further investigations in this field are expected to put forth various possibilities that might exist to effectively achieve stable population dynamics, while at the same time achieve reduced mercury bioaccumulation levels in the aquatic species.  

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