1 Introduction  

This work is focused on the application of sustainability ideas in natural ecosystem management. The task of ecosystem management can be facilitated by using sustainability, which accounts for long term effects, as an objective, provided a suitable measure of sustainability is available. Fisher information has been recently proposed as a mathematical measure of sustainability and can be used to track ecosystem changes. At the same time a management strategy based on natural regulation paths of the ecosystem is advantageous for which this work uses and compares the top-down and bottom-up control philosophies. Given the complex nature of natural systems and the objective (sustainability), optimal control theory is used to formulate the control profiles. The aquatic ecosystems are modeled using predator-prey dynamics. Uncertainty, inherent in any natural ecosystem, is incorporated through uncertain mortality rates of species. The uncertainty in mortality rate is modeled as a stochastic Ito process and stochastic maximum principle is used to solve the optimal control problems and derive the control law. To summarize, this work compares the top-down and bottom-up control philosophies, derived using optimal control theory, for an aquatic ecosystem using sustainability, quantified by Fisher Information, as the objective.  

2 Methodology

2.1 Sustainability and Fisher Information  

Cabezas and Fath [1] have proposed to use information theory based Fisher information to quantify sustainability and propose the sustainability hypotheses with particular emphasis on ecological systems. The formal sustainability hypothesis states that: the time-averaged Fisher information of a system in a persistent regime does not change with time. Two additional corollaries to the sustainability hypothesis are: (1) if the Fisher information of a system is increasing with time, then the system is maintaining a state of self-organization and (2) if the Fisher information of a system is decreasing with time, then the system is losing its state of self-organization. From an ecosystem management perspective, two different objectives can be formulated based on these hypotheses:

• Maximization of the time averaged Fisher information which aims to push the system to a more sustainable state but may demand rapid changes
• Minimization of the Fisher information variance over time which aims to maintain the system close to its current state

2.2 Ecosystem management philosophies  

Two different philosophies about the controlling effects of a natural ecosystem exist. The trophic cascade hypothesis [2, 3] for an aquatic ecosystem proposes that the predator-prey interactions are transmitted through food webs to cause variance in phytoplankton biomass and production at constant nutrient load. Another possible regulatory effect in a food web is the effect of available resources on higher level species, e.g. nutrients supporting the phytoplankton in an aquatic food web, which in turn affect the top level species that feed on them. From ecosystem management point of view this has led to the formulation of two control philosophies, the top-down control (consumer control) which refers to controlling the ecosystem through top level predators and the bottom-up control (resource control) which refers to controlling the ecosystem via available resources.

The existence of these two natural regulation paths in ecosystems provides two different avenues to exercise external control of these systems, control using the top predator and using the lowest level resources, which are explored and compared in this work for the given objectives.  

2.3 Predator-prey model  

The model used to represent the aquatic ecosystems is predator-prey model, derived from the more general class of Lotka-Volterra-type models. This work uses the Rosenzweig-MacArthur model, which is frequently used in theoretical ecology [4]. It gives a simplistic representation of the food chain dynamics in an ecosystem through a set of ordinary differential equations. This work considers a three species model. The species are referred to as prey, predator and super-predator in the ascending order of their position in the food chain. The various parameters included in the model are the species populations, prey growth rate (r) and prey carrying capacity (K) of the system, and maximum predation rate, half saturation constant, efficiency, and death rate of the predator and super-predator. The starting populations of the species are known.

Uncertainty is incorporated in the model by considering the predator mortality rate to be uncertain. The mortality rate uncertainty is represented by Ito mean reverting process. The mean reverting process has been used to model many stochastic variables including human mortality rate [5] and is extended to represent predator mortality rate in this work. Although no data is presented to support the validity of such an assumption, its success in modeling human mortality rate motivated its use in the current work.  

2.3 Optimal Control  

Optimal control theory is used in this work to derive the top-down and bottom-up control philosophies. Pontryagin's maximum principle is used to derive the optimal control law equations which are a set of ordinary differential algebraic equations to be solved as a two point boundary value problem. For stochastic models, the stochastic maximum principle, which has been recently proposed [6] is used in this work. In this method the stochastic dynamic programming formulation is converted into a stochastic maximum principle formulation and results in a set of ordinary differential equations to be solved as a boundary value problem. Due to the highly complex nature of these equations for this particular application, numerical method of steepest ascent of Hamiltonian is used for solution.  

3 Results and Conclusions  

In the analysis, both control options and both objectives are tested on deterministic as well as stochastic tri-trophic food chain model. To investigate the approach further, the uncontrolled model is considered to have unstable dynamics. The two cases considered are:

• Super-predator extinction: The super-predator population is going extinct
• Super-predator explosion: The super-predator population is exploding

For the stochastic models, impact of uncertainty is analyzed by studying the Ito process dependence and importance of stochastic optimal control for various objectives and control philosophies.

The results are analyzed in terms of the numerical values of the time averaged FI and FI standard deviation. The dynamic response of various species is also equally important. Hence different cases are also compared in terms of the time dependent variations in species populations. In all the cases it was observed that the desired objectives, that of time averaged FI maximization and FI variance minimization, are achieved. The other results for the deterministic systems force following important conclusions:

• The objective of FI maximization (with top-down or bottom-up control) results in an elevated super-predator population in all cases and the objective of FI variance minimization achieves the desired objective of super-predator population control in all cases.
• Bottom-up control has a greater impact on predator-prey dynamics than the top-down control and this impact is more significant for FI maximization objective.
• Bottom-up controlled systems obtain better objective values (i.e. higher time averaged FI and lower FI standard deviations) than the top-down controlled systems.
• Control variable profiles for the objective of FI variance minimization are much less variable than for FI maximization and hence easier to implement on physical systems.

The results for the stochastic and deterministic models are qualitatively similar. A quantitative comparison indicates that uncertainty impacts the relative extent of success or failure of a management philosophy. It is also shown that the degree of uncertainty is important to rank various management options.

Qualitative similarity between the results confirms the hypothesis that FI variance minimization objective is guaranteed to give a stable response and the FI maximization objective increases super-predator abundance. In this light, the FI maximization objective involves risk of instability while FI variance minimization might be viewed as the ‘safety first' objective. It was also observed that the FI variance minimization objective is not able to recover an uncontrolled system from significant disturbances (such as fast specie extinction). It is likely that some of these trends are system dependent. These findings, obtained for simplistic mathematical representation of complex ecosystems, should guide the experimental biologist managing real ecosystems.  

Reference

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[6] V. Rico-Ramirez and U. Diwekar. Stochastic maximum principle for optimal control under uncertainty. Computers and Chemical Engineering, 28:2845–2845, 2004.