Robust set-point regulation for ecological models with multiple management goals (original) (raw)

Integral control for population management

Journal of Mathematical Biology, 2014

We present a novel management methodology for restocking a declining population. The strategy uses integral control, a concept ubiquitous in control theory which has not been applied to population dynamics. Integral control is based on dynamic feedback-using measurements of the population to inform management strategies and is robust to model uncertainty, an important consideration for ecological models. We demonstrate from first principles why such an approach to population management is suitable via theory and examples.

Predictive PI-control of linear plants under positional and incremental input saturations

Automatica, 2000

A predictive control strategy is developed for set-point tracking of LTI plants in the presence of joint positional and incremental (rate) input saturation constraints. The resulting control algorithm is built so as to provide an integral action capable to asymptotically reject arbitrary constant disturbances. This task is achieved by possibly modifying on-line the set-point in the case this becomes incompatible with constraints and disturbances. When applied to asymptotically null controllable with bounded inputs (ANCBI) plants, viz. all stabilizable linear systems with eigenvalues in the closed unit circle, the control law is shown to globally provide set-point tracking in the presence of arbitrary constant disturbances and initial state. For non-ANCBI plants, viz. systems with exponentially unstable modes, the control law features the mentioned properties locally, viz. in bounded regions of initial states and disturbances.

Feedback control systems analysis of density dependent population dynamics

Systems & Control Letters, 2012

We use feedback control methods to prove a trichotomy of stability for nonlinear (density dependent) discrete-time population dynamics defined on a natural state space of non-negative vectors. Specifically, using comparison results and small gain techniques we obtain a computable formula for parameter ranges when one of the following must hold: there is a positive, globally asymptotically stable equilibrium; zero is globally asymptotically stable or all solutions with non-zero initial conditions diverge. We apply our results to a model for Chinook Salmon.

Control theory and the management of ecosystems: A threshold policy with hysteresis is robust

Applied Mathematics and Computation, 2010

Threshold policy with delay Threshold policy with hysteresis and delay a b s t r a c t Control theory is much used in engineering to stabilize a given dynamical system at a desired equilibrium point or to confine its trajectories to a region. Extinction of species is one of the most serious problems facing fishery and, to avoid it, different policies are applied. The threshold policy (TP) is a harvesting strategy commonly used in fisheries all over the world, and also is a special and simple case of the variable structure control (VSC). In this paper, the concept of virtual equilibrium point is used to design three different kinds of threshold policies; the standard one (TP), one with delay (TPD) and finally with hysteresis and delay (TPHD), for logistic models subject to Euler as well as two different nonstandard discretization schemes. Uncertainties in the intrinsic growth rate, in the carrying capacity, in the population density, and in the effort policy as well as an overexploitation situation are considered. A time lag in the control policies is also introduced. The important novel characteristic of the TPD and TPHD is that both ensure that, even though the system is subjected to uncertainties and a period of overexploitation, the system eventually stabilizes in bounded oscillations in a desired safe region of the state space. In addition, the sustainable yield under the proposed policies is compared with that of the proportional policy proposed in [1], in order to discuss economic aspects of the proposed threshold policies.

Optimal population stabilization and control using the Leslie matrix model

Bulletin of Mathematical Biology, 1995

We consider the problem of optimal stabilization and control of populations which follow the Leslie model dynamics, within state space and control systems theory and methodology. Various types of culling strategies are formulated and introduced into the Leslie model as control inputs, and their effect on global asymptotic stability is investigated. Our new approach provides answers to several unexplored problems. We show that in general it is possible to achieve a desired stable equilibrium population level, through the design of a class of shifted-proportional stabilizing culling policies. Further, we formulate general non-linear constrained optimization problems, for obtaining the cost-optimal policy among this generally infinite class of such stabilizing policies. The theoretical findings are illustrated through the solution of the problem over an infinite planning horizon for a numerical example. A comparative study of the costs and dynamic effects of various culling strategies also supports the mathematical results.

Set-Point Regulation of Constrained Strongly Monotone Systems

Proceedings of the 18th IFAC World Congress, 2011

To date, monotone systems have been basically studied from the analysis viewpoint, whereas control design issues for such dynamical systems have not been considerably remarked. This paper focuses on set-point regulation of output of strongly monotone systems using a static output feedback under input constraints. The design procedure, in the cast of a new theorem, is derived from Hirsch theorem for autonomous systems. The proposed theorem ensures that the closed-loop system, even under input saturation, remains strongly monotone. The resulting control law will guarantee the closed-loop stability and offset-free output regulation.

On stabilizing PI controller ranges for multivariable systems

Chaos, Solitons & Fractals, 2008

The paper is concerned with the computation of the maximum ranges of stabilizing proportional-integral (PI) controllers for multiple-input and multiple-output (MIMO) systems. A time-domain scheme is proposed by converting the considered problem to a robust stability problem for a polytopic system. An algorithm based on linear matrix inequality (LMI) is established to find the maximum ranges.

Stabilization of an uncertain competing species system

Computers & Mathematics with Applications, 1996

Many real world problems concerning the management of natural renewable resources are appropriately modeled and formulated as stabilization problems of uncertain dynamical systems subject to control constraints. Based on the recent results of Corless and Leitmann [1] on the design of bounded controllers for a class of nonlinear uncertain systems that assure robust exponential convergence to a neighborhood of the origin, a componentwise bounded harvest strategy is proposed for the management of an ecological system of two competing species at some desired prescribed level in the presence of bounded system and input uncertainties. Keywords-Ecological modelling, Bounded stabilizing harvest strategies.

Monotonic properties for the viable control of discrete time systems

Citeseer

This paper deals with the control of nonlinear systems in the presence of state and control constraints for discrete time dynamics in finite dimensional spaces. The viability kernel is known to play a basic role for the analysis of such problems and the design of viable control feedbacks. Unfortunately, this kernel may display very non regular geometry and its computation is not an easy task in general. In the present paper, we show how monotonic properties of both dynamics and constraints allow for relevant analytical upper and lower approximations of the viability kernel through weakly and strongly invariant sets. An example on fish harvesting management illustrates some of the assertions.

A comparison of six methods for stabilizing population dynamics

Journal of Theoretical Biology, 2014

Over the last two decades, several methods have been proposed for stabilizing the dynamics of biological populations. However, these methods have typically been evaluated using different population dynamics models and in the context of very different concepts of stability, which makes it difficult to compare their relative efficiencies. Moreover, since the dynamics of populations are dependent on the life-history of the species and its environment, it is conceivable that the stabilizing effects of control methods would also be affected by such factors, a complication that has typically not been investigated. In this study we compare six different control methods with respect to their efficiency at inducing a common level of enhancement (defined as 50% increase) for two kinds of stability (constancy and persistence) under four different life history/ environment combinations. Since these methods have been analytically studied elsewhere, we concentrate on an intuitive understanding of realistic simulations incorporating noise, extinction probability and lattice effect. We show that for these six methods, even when the magnitude of stabilization attained is the same, other aspects of the dynamics like population size distribution can be very different. Consequently, correlated aspects of stability, like the amount of persistence for a given degree of constancy stability (and vice versa) or the corresponding effective population size (a measure of resistance to genetic drift) vary widely among the methods. Moreover, the number of organisms needed to be added or removed to attain similar levels of stabilization also varies for these methods, a fact that has economic implications. Finally, we compare the relative efficiency of these methods through a composite index of various stability related measures. Our results suggest that restocking to a constant lower threshold seems to be the optimal method under most conditions, with the recently proposed Adaptive Limiter Control (ALC) being a close second.