(541f) Application of Optimal Control Theory for Sustainable Ecosystem Management

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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