Observer design for phenotypic observation of genetic processes

In the paper a single-locus sexual population is considered, where the phenotypic selection process is described by an evolutionary game. First, in order to recover the genetic process from different observations, observability is guaranteed by the linearization method developed in earlier papers of the authors for systems with invariant manifold. Then, based on a known result of nonlinear systems theory, an observer system is constructed that makes it possible to asymptotically recover the solution of the original system from the observation. In the numerical illustrations the selection is described by a "rock-scissors-paper" type game widely studied in evolutionary game theory. For the corresponding evolutionary dynamics a Hopf bifurcation result is also obtained

OBSERVABILITY AND OBSERVERS IN A FOOD WEB

The problem of the possibility to recover the time-dependent state of a whole population system out of the observation of certain components has been studied in earlier publications, in terms of the observability concept of mathematical systems theory. In the present note a method is proposed to effectively calculate the state process. For an illustration an observer system for a simple food web is numerically constructed

Recent Developments in Monitoring of Complex Population Systems

The paper is an update of two earlier review papers concerning the application of the methodology of mathematical systems theory to population ecology, a research line initiated two decades ago. At the beginning the research was con- centrated on basic qualitative properties of ecological models, such as observability and controllability. Observability is closely related to the monitoring problem of ecosystems, while controllability concerns both sustainable harvesting of population systems and equilibrium control of such systems, which is a major concern of conservation biology. For population system, observability means that, e.g. from partial observation of the system (observing only certain indica- tor species), in principle the whole state process can be recovered. Recently, for different ecosystems, the so-called ob- server systems (or state estimators) have been constructed that enable us to effectively estimate the whole state process from the observation. This technique offers an efficient methodology for monitoring of complex ecosystems (including spatially and stage-structured population systems). In this way, from the observation of a few indicator species the state of the whole complex system can be monitored, in particular certain abiotic effects such as environmental contamina- tion can be identified. In this review, with simple and transparent examples, three topics illustrate the recent develop- ments in monitoring methodology of ecological systems: stock estimation of a fish population with reserve area; and observer construction for two vertically structured population systems (verticum-type systems): a four-level ecological chain and a stage-structured fishery model with reserve area

Observer design for open and closed trophic chains

Monitoring of ecological systems is one of the major issues in ecosystem research. The concepts and methodology of mathematical systems theory provide useful tools to face this problem. In many cases, state monitoring of a complex ecological system consists in observation (measurement) of certain state variables, and the whole state process has to be determined from the observed data. The solution proposed in the paper is the design of an observer system, which makes it possible to approximately recover the state process from its partial observation. Such systems-theoretical approach has been applied before by the authors to Lotka–Volterra type population systems. In the present paper this methodology is extended to a non-Lotka–Volterra type trophic chain of resource–producer–primary consumer type and numerical examples for different observation situations are also presented

Iterative scheme for the observation of a competitive Lotka-Volterra system

In this work, in terms of the model parameters, suffcient conditions are established to construct a sequence of approximate observers for a two-species competitive Lotka-Volterra system. This iterative approach makes it possible to localize the solution of the system, and reveal its long-term behaviour. The main results are also illustrated by numerical simulations

Equilibrium, observability and controllability in selection-mutation models

In this paper we shortly discuss the problem of the equilibrium in the well-known Fisher type selection-model, also providing a formula for particular three-allele models. The considered continuous-time dynamics is a known extension of the classical model of natural selection given by Fisher. We also extend the existing investigation of the observability of Fisher’s model to the case when another evolutionary factor, mutation is also present. Moreover, we prove a result of technical character, which makes it possible to apply the methodology of nonlinear systems with invariant manifold, to models of artificial selection. For an illustration, a class of three-allele systems is presented in which the controllability into equilibrium is guaranteed without any condition on the biological parameters

MONITORING ENVIRONMENTAL CHANGE IN AN ECOSYSTEM

The monitoring and analysis of the processes taking place in an ecosystem is a key issue for a sustainable human activity. A system of populations, as the biotic component of a complex ecosystem is usually affected by the variation of its abiotic environment. Even in nearly natural ecosystems an abiotic effect like climatic implications of global warming may cause important changes in the dynamics of the population system. In ecosystems involving field cultivation or any industrial activity; the abiotic parameter in question may be the concentration of a substance, changing e.g. as a result of pollution, application of a pesticide, or a fertilizer, etc. In many cases the observation of the densities of each population may be technically complicated or expensive, therefore the question arises whether from the observation of the densities of certain (indicator) populations, the whole state process of the population system can be uniquely recovered. The paper is aimed at a methodological development of the state monitoring, under the conditions of a changing environment. It is shown, how the technique of mathematical systems theory can be applied not only for the approximate calculation of the state process on the basis of the observed data, even under the effect of an exogene abiotic change with known dynamics; but in certain cases, also for the estimation of the unknown biological effect of the change of an abiotic parameter. The proposed methodology is applied to simple illustrative examples concerning a three-species predator-prey system

Observability in dynamic evolutionary models

In the paper observability problems are considered in basic dynamic evolutionary models for sexual and asexual populations. Observability means that from the (partial) knowledge of certain phenotypic characteristics the whole evolutionary process can be uniquely recovered. Sufficient conditions are given to guarantee observability for both sexual and asexual populations near an evolutionarily stable state

OPEN- AND CLOSED-LOOP EQUILIBRIUM CONTROL OF TROPHIC CHAINS

If a nearly natural population system is deviated from its equilibrium, an important task of conservation ecology may be to control it back into equilibrium. In the paper a trophic chain is considered, and control systems are obtained by changing certain model parameters into control variables. For the equilibrium control two approaches are proposed. First, for a fixed time interval, local controllability into equilibrium is proved, and applying tools of optimal control, it is also shown how an appropriate open-loop control can be determined that actually controls the system into the equilibrium in given time. Another considered problem is to control the system to a new desired equilibrium. The problem is solved by the construction of a closed-loop control which asymptotically steers the trophic chain into this new equilibrium. In this way, actually, a controlled regime shift is realized

Monitoring in a Lotka-Volterra model

The problem of monitoring arises when in an ecosystem, in particular in a system of several populations, observing some components, we want to recover the state of the whole system in function of time. Due to the difficulty to construct exactly this state process, we look for an auxiliary system called observer, the solution of which reproduces this process with certain approximation. This means, that the solution of the observer tends to that of the original system. For this work an important concept is observability which means, that from the observation it is possible to recover the state process in a unique way, however without determining a constructive method to obtain it. If observability holds for the original system, it guarantees the existence of an auxiliary matrix which makes it possible to construct an observer of the system. The considered system of populations is described by the classical Lotka-Volterra model with one predator and two preys and the construction of its observer is illustrated with a numerical example. Finally, it is shown how the observer can be used for the estimation of the level of an abiotic effect on the population system