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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 concentrated 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 indicator species), in principle the whole state process can be recovered. Recently, for different ecosystems, the so-called observer 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 contamination can be identified. In this review, with simple and transparent examples, three topics illustrate the recent developments 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.

Mathematical Systems Theory (MST) looks back on several decades of history. In engineering practice, it is a typical situation that an object (e.g. a machine or electronic circuit) is controlled by a human intervention to influence the state of the object, or observing a transform of the state the task is to recover the state process of the object. The corresponding concepts of controllability, observability and the related state space model played an important role in the development of MST. The first comprehensive monograph of this discipline, dealing only with linear systems, was [

For population system, observability means that, from partial observation of the system, in principle, the whole state process can be recovered. Recently, for different ecosystems, the so-called observer systems (or state estimators) have been constructed that enables 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 contamination may be identified.

In fact, the systems-theoretical study of the considered nonlinear frequency-dependent population models required the generalization of general sufficient conditions for controllability and observability to the case of nonlinear systems with invariant manifold (see [4,5]). These results have been applied to a control-theoretical model of artificial selection, phenotypic observation of genetic processes and evolutionary game dynamics, as well as to systems-theoretical models of reaction kinetics, see [6- 15].

Later on, the methodology of MST was used for monitoring of different population systems: observability and system inversion were investigated in density-dependent models of population ecology, ranging from Lotka-Volterra-type ([16-18]) and non Lotka-Volterratype population systems to monitoring of environmental change in a complex ecosystem ([

In the present survey we report on recent developments in the methodology and application areas of monitoring in complex ecological systems. Although the presented methodology can be applied for the monitoring of large, but appropriately structured complex population systems, for the sake of simplicity and transparency we illustrate the procedures on observation systems of low dimensions. A particular attention is paid to recent results concerning the so-called verticum-type observation systems. The linear version of such systems have been introduced for modelling certain industrial systems and studied for controllability and observability in [30-38]. Verticum-type systems are composed from several “subsystems” connected sequentially in a particular way: a part of the state variables of each “subsystem” also appears in the next “subsystem” as an “exogenous variable” which can be also interpreted as a control generated by an “exosystem”. Therefore, these “subsystems” are not observation systems, but formally can be considered as controlobservation systems. The problem of observability of such systems can be reduced to rank conditions on the “subsystems”, which is a kind of decoupling of a complex system into simpler parts. Since most dynamic models of population biology are nonlinear, for the application in this field, it was necessary to extend the basic concepts and theorems of the theory of linear verticumtype systems to the nonlinear case, which has been done recently in [19,39-41].

The paper is organized as follows. In Section 2, based on [

For the basic model of this section, from [_{1}(t) and x_{2}(t) be the respective biomass densities of the same fish population inside the unreserved and reserved areas, respectively. Assume that the fish subpopulation of the unreserved area migrates into the reserved area at a rate m_{12}, and there is also an inverse migration at rate m_{21}. Let E be a constant fishing effort applied for harvesting in the unreserved area and let us assume that in each area the growth of the fish population follows a logistic model. The dynamics of the fish subpopulations in unreserved and reserved areas are then assumed to be governed by the following system of differential Equations (1) and (2):

where r_{1} and r_{2} are the intrinsic growth rates of the corresponding subpopulations, K_{1} and K_{2} are the carrying capacities for the fish species in the unreserved and reserved areas, respectively; q is the catchability coefficient in the unreserved area. All parameters r_{1}, r_{2}, q, m_{12}, m_{21}, E, K_{1} and K_{2} are positive constants.

In [

Furthermore, the Lyapunov function

also implies asymptotic stability of equilibrium for system (1)-(2), globally with respect to the positive orthant of. Throughout the section we shall suppose conditions (3a)-(3c) to guarantee the stable coexistence of the system applying a constant reference fishing effort.

From [

Let

and consider observation system

where function (5) defines the transform of the state, observed instead of the state itself.

Definition 2.1. System (4)-(5) is called locally observable at on, if for any solutions of (4) defined on, initially close enough to,

.

Linearizing (4)-(5) around, we get

.

Theorem 2.1. ([

system (4)-(5) is locally observable at on.

Now, let us consider the problem of stock estimation in the reserve area on the basis of the biomass harvested in the free area. (For technical reason its difference from the equilibrium value is supposed to be observed.) To this end, in addition to dynamics (1)-(2) we introduce an observation equation

representing the observation of the biomass harvested in the free fishing area. Then linearizing observation system (1)-(2), (6), we get the Jacobian of the right-hand side of (1)-(2)

and the observation matrix

Now, for the linearized system we obviously have Hence, Theorem 2.1 implies local observability of the system near the equilibrium. In other words, in principle the whole system state (in particular the stock of the species in the reserve area) as function of time can be uniquely recovered, observing the biomass harvested per unit time. In the following illustrative example we will see how the state of the system (and hence the total stock) can be effectively calculated from the catch realized in the fishing area, applying the methodology of [

Recently, for different ecosystems, the so-called observer system (or state estimators) have been constructed that enables us to effectively estimate the whole state process from the observation. Here we remind the methodology that will be used in this subsection and in the following sections. Consider again observation system (4)-(5)

.

Definition 2.2. Given, system

is called a local (exponential) observer for system (4)-(5) at, if for the composite system (4)-(5), (7) we have 1)2) there exists a neighbourhood of such that (exponentially).

Theorem 2.2. ([

is a local exponential observer for observation system (4)-(5).

Example 2.1. For a possible comparison, in this numerical example we use the same parameters as [

and.

Now the positive equilibrium is, and with

matrix is Hurwitz; therefore by Theorem 2.2 we have the following observer system

If we take an initial condition for system (9), and similarly, we consider another nearby initial condition, for the observer system (10), then the corresponding solution of the observer tends to the solution of the original system, as shown in

As a modification of the well-known three-level trophic chain consisting of resource-producer-primary consumer studied in [

level 0: a resource;

level 1: the producer is a plant, supposed to die out without the resource, and the positive effect of the latter is proportional to the quantity of the resource present in the system;

level 2: the primary user (instead of consumer), i.e. a commensalist animal, making use of the plant as part of its habitat without harming it (e.g. an insect species hosted by the plant), displaying a logistic dynamics in absence of the plant and the secondary consumer;

level 3: the secondary consumer is a monophagous predator of the primary user (e.g. an insectivorous singing bird species), with intraspecific competition.

(For more details on the role of commensalism in ecological communities, we supposed between the producer and the primary user, see e.g. [

For a dynamical model let be the time-dependent quantity, with a constant supply of the resource pre-

sent in the system, and the time-varying population size (biomass or density) of the producer, the primary user and the secondary consumer, respectively. Assume that a unit of biomass of the plant consumes the resource at velocity; however, it increases the biomass of the plant at rate. The relative rate of increase in biomass of the primary user, due to the presence of the plant is. While the plant population is supposed to die out exponentially in the absence of the resource, with Malthus parameter, the primary user displays a logistic growth with Malthus parameter and is limited by a carrying capacity. Furthermore, the secondary consumer would die out at rate, without the presence of the primary user, and there is an intraspecific competition among predators with rate. We will consider a partially closed system, where the dead plants may be recycled into nutrient resource with rate. Then with parameters

we have the following dynamic model for the considered interaction chain:

Theorem 3.1. ([

Then, both the open and the partially closed ecological chains stably coexist in the sense there exists a positive equilibrium of system calculated in [

Remark 3.1. The conditions of Theorem 3.1 can also be formulated conversely: Given a resource supply Q, biological parameters satisfying condition (15) imply the stable coexistence of the considered ecological chain.

Let us consider now the following two auxiliary 2-dimension systems

and

In ecological terms (16) is a subsystem of the original chain (11)-(14), while in (17) the positive effect of the plant on the animal species 2 appears with the equilibrium value of the plant. We note that by setting (i.e. considering the original system without commensalisms), the original ecological chain is split up into two components without interaction.

Remark 3.2. The biological interpretation of system (17) is the following: Suppose that system (11)-(14) is in equilibrium, and the two animal species, by an external disturbance, deviate from their equilibrium densities. Then the resource-primary consumer subsystem can maintain its equilibrium, and the predator-prey subsystem will be governed by system (17).

Continuing the study of systems (16) and (17), we can easily check that they have respective equilibria and. For system (16) with notation, let us consider observation function

This means that the deviation of the resource from its equilibrium value is observed. In order to check local observability, we calculate the linearization of system (16) at equilibrium:

Hence we easily calculate

provided. From the classical sufficient condition for the local observability of nonlinear systems, [

Similarly, suppose that in system (17) the deviation of the density of the prey from its equilibrium value is observed, i.e., with notation we consider the observation function

The linearization of system (17) at equilibrium is

Checking again the rank condition, by we get

implying local observability of system (17)-(20) near. Now, let us observe that with definition

system matrix

together with observation matrix

define a verticum-type linear observation system in the sense defined in the Appendix. Applying Theorem A.2 of the Appendix, we obtain that the linear observation system

is observable. Since is just the Jacobian of the righthand side of system (11)-(14), therefore (22) is just the linearization of system (11)-(14). Furthermore, (23) is the linearization of observation function

which can be associated with system (11)-(14). Finally, applying again the classical rank condition of [

Theorem 3.2. Let us suppose that ecological chain (11)-(14) is partially closed.Then with observation function (24), system (11)-(14) is locally observable near equilibrium calculated in [

Following the procedure of [

For matrices and, figuring in (19), we have to find a matrix such that

is a Hurwitz matrix, i.e. all roots of the characteristic polynomial of matrix have real negative parts. It is easy to check that the latter condition is satisfied if and only if the following inequalities hold:

Simple sufficient conditions for (25) and (26) are and, respectively. By the Theorem of [

Similarly, for matrices and, figuring in (21), we need to find a matrix such that all roots of the characteristic polynomial of matrix

have real negative parts. Now a straightforward checking shows that the latter condition is satisfied if and only if satisfy the following inequalities:

Similarly to the previous case, in order to satisfy conditions (27) and (28), it is sufficient to set and, and again by the Theorem of [

Finally, based on the above reasoning, it will be easy to prove the following result:

Theorem 3.3. ([

with and, and function defined as the right-hand side of system (11)-(14), system

is a local exponential observer for system (11)-(14) with observation equation, where is defined in (24).

Example 3.1. We consider the following system

System (29) has a positive equilibrium , which is asymptotically stable, because conditions of Theorem 3.1 are satisfied. In

Consider now system (29) with observation

.

Since matrix

satisfies the conditions of Theorem 3.3, we can construct the following observer

Solving (30) with initial condition near the equilibrium, we can check how this solution tends to recover the corresponding solution of system (29), see

Let us consider a modification of the stage-structured fishery model of [

where

natural mortality rate of class ,

linear aging coefficient in areas

juvenile competition parameter in areas

fecundity rate of adult fish in areas

predation rate of class 1 on class 0 in areas

catchability coefficient of class 1 in the unreserved area,

migration rate of the second class from reserved area to unreserved area,

constant fishing effort.

From [

system (31) has a unique positive equilibrium, which is asymptotically stable under conditions

and (33)

Remark 4.1. Since asymptotic stability implies Lyapunov stability, in the next section we can apply Theorem A.3 of the Appendix to the corresponding nonlinear verticum-type observation system.

Let with, and we consider the observation function defined by

Now the observability of observation system (31)-(34) will be analyzed using the results of the Appendix. Consider systems

and

Given observation

we calculate its linearization

It is easy to check that, where is the linearization of (35), therefore by Theorem A.1 of the Appendix we can guarantee local observability of system (35)-(37).

Analogously, for observation

of system (36) calculate

Again we have , where is the linearization of (36), therefore from Theorem A.1 of the Appendix we have local observability of system (36)- (38). Since under the appropriate conditions equilibrium is asymptotically stable and hence also Lyapunov stable, applying Theorem A.3 we obtain Theorem 4.1. Suppose that conditions (32) and (33) hold. Then observation system (31)-(34) is locally observable near the asymptotically stable equilibrium.

Given the observation system (35)-(37), using the corresponding observer design of [

is appropriate.

Analogously, for observation systems (36)-(38), with,

,

is Hurwitz, guaranteeing the construction of the observer system.

From these results, for

we can check that is Hurwitz, which allows the construction of an observer for system (31)-(34) moreover, this observer is composed of the observers constructed for the two subsystems.

Example 4.1. Consider the following model parameters of [

To construct the observer system for (35)-(37) we can take

.

Then the observer system is

Considering as initial value for the system (35), and for the observer (39), in

To construct the observer of system (36)-(38) take

Then the observer system is

If we consider as initial value for system (36), and for the observer (40)we obtain the result plotted in

Now the observer for system (31)-(34) can be simply composed from the single observers (39) and (40). In

Observation problems arise in many fields of human activity, when state of an object can be characterized by several numbers (i.e. by a state vector), and it is impossible or too expensive to measure all state variables. Then

we may want to recover the whole state vector. In a static situation this is clearly impossible, since projection is not invertible. However, in dynamic situation the concepts of observability and observer design of Mathematical Systems Theory turned out to be efficient tools for monitoring of ecological systems, as well. We presented some recent developments in concrete applications to population systems. These systems are not only simple sets of populations, but each of them has a particular structure. In the first case (Section 2) a single species has a spatially structured habitat (with a reserve area, where observation of density by harvesting is not allowed). In the other two cases, verticum-type, i.e. vertically organized dynamic population systems (ecological chains in Section 3, and stage structure of a single species in Section 4), for monitoring purpose “decoupled” observer design may be efficient even in large systems.

These examples anticipate the application of the presented methodology in similar situations. Furthermore, in multispecies models of evolutionary ecology it also opens the way to the monitoring in behaviour-structured population systems. In case of density dependent models, for the monitoring of propagation or extinction of a species we may want to recover the time-dependent density of scarce species, observing a more abundant species of the system. This idea may be applied to the dynamic models of [49-53]. In ecological games the dynamics depends on the behavior types present in the populations, see [54-56]. Then the convergence towards a stable coexistence can be monitored from the observation of certain phenotypes.

Finally, we note that recent papers also show how observer design can be efficiently applied for the monitoring of particular engineering systems. For example, in [

The present research has been supported in part by the Hungarian Scientific Research Fund OTKA (K81279) and by the Excellence Project Programme of the Ministry of Economy, Innovation and Science of the Andalusian Regional Government, supported by FEDER Funds (P09-AGR-5000).

First, we recall the extension of local observability to the case of a control-observation system.

Suppose

such that and.

Remark A.1. It is known (see e.g. [

, for all

Definition A.1. With the above notation, consider the control-observation system in

System (A.1)-(A.2) is said to be locally observable near the equilibrium if there exists such that

and

for all

imply that

Theorem A.1. ([

Assume

Then system is locally observable near the equilibrium.

Remark A.2. The theorem similar to the previous one is also valid for function not depending on control, as we have shown in Section 2.

Now, based on [

Let

,

, and consider the nonlinear system

and for all

Denoting

let with

and with

We shall suppose that there exists

such that

and.

Definition A.2. Observation system

is said to be of verticum type.

Remark A.3. Equations do not define a standard observation system in this setting, because of the presence of the “exogenous” variable connecting it to system.

Remark A.4. It is known that near equilibrium all solutions of system (V) can be defined on the same time interval. In what follows will be considered fixed and concerning observability, the reference to T will be suppressed.

For the analysis of observability of system (V), let us linearize systems (V_{i}), at the respective equilibria , obtaining the linearized systems

and for all

where

Define matrices as follows:

,

obtaining linear observation system

of verticum type (see [

Theorem A.2. ([

Then the linear verticum-type system (LV) is observable.

Remark A.5. If is a Lyapunov stable equilibrium of system

then can be considered as a control-observation system with “small” controls in the following sense. By the Lyapunov stability of, for all , there exists such that implies

(for). In particular,

for all.

Considering as a control for system (V_{i}) becomes a controlobservation system in the sense of this Appendix. Suppose that for each

then by Theorem A.2 the verticum-type system (LV) is observable.

Hence, the linearization of the observation system (V) is observable. Therefore, by Kalman’s theorem on observability of linear systems (see [

The above reasoning can be summarized in the following theorem:

Theorem A.3. If equilibrium is Lyapunov stable for system , and

then observation system (V) is observable near its equilibrium.