ABSTRACT

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.

1. Introduction and Historical Overview

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 [1], a more recent reference is [2]. Generalizations of controllability and observability theory to nonlinear systems can be found in [3]. Following a successful development of MST for engineering purpose, as a new research line, in [4,5] the application MST to the study of population systems was proposed.

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 ([19]). In particular, in [20] the optimal control software developed in [21] and [22], was also used for equilibrium control of a trophic chain. In [23] a new nonlinear system inversion method was applied for the reconstruction of time-dependent abiotic environmental changes, from the observation of certain indicator species. Furthermore, both monitoring and control were studied in a systems-theoretical model of radiotherapy in [24], while in [25-27] tools of MST were applied to biological pest control. Most of these topics and results have been reviewed in the survey papers [28] and [29].

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 [42], following a necessary stability analysis according to [43], the stock estimation of a fish population with reserved area is presented, using an appropriate observer design. In Section 3 results from [40] are recalled concerning the monitoring of ecological interaction chains of the type resource-producer-primary user-secondary consumer. The dynamic behaviour of these four-level chains is modelled by a system of differential equations, the linearization of which is a verticum-type system. Section 4 is devoted to the general concept of a nonlinear verticum-type observation system and the corresponding general sufficient condition of observability obtained in [41]. As an application, observer design is also presented for a stage-structured population, decomposing the state estimation according to the verticum structure. In Section 5 further possible application fields of the presented monitoring methodology are summarized. Finally, in the Appendix the theoretical background necessary for the monitoring of nonlinear verticum-type systems is recalled.

2. Stock Estimation of a Fish Population with Reserved Area

For the basic model of this section, from [43] we recall the dynamics of a fish population moving between two areas, the first, an unreserved one where fishing is allowed, and the second, a reserved one where fishing is prohibited. At time t, let x₁(t) and x₂(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 m₁₂, and there is also an inverse migration at rate m₂₁. 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):

(1)

, (2)

where r₁ and r₂ are the intrinsic growth rates of the corresponding subpopulations, K₁ and K₂ 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 r₁, r₂, q, m₁₂, m₂₁, E, K₁ and K₂ are positive constants.

In [43], it was checked that for a unique positive equilibrium of the dynamic model (1)-(2) the following set of inequalities are sufficient:

(3a)

(3b)

(3c)

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.

2.1. Observability of the Model

From [3], we recall the basic concept of local observability of nonlinear systems and a sufficient condition for a system to have this property, in order to apply it to the considered model and in order to be used in the following sections.  

Let

and consider observation system

(4)

, (5)

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. ([3])

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

(6)

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 [44] that we will recall next.

2.2. Observer System

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

(7)

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. ([44]) Suppose equilibrium of system (4) is Lyapunov stable, and there exists nxn matrix such that is stable. Then system

(8)

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 [45]:

and.

(9)

Now the positive equilibrium is, and with

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

(10)

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 Figure 1. We note that in this particular case the convergence is much faster than that of the observer constructed in [45].

3. Monitoring of a Four-Level Ecological Chain

As a modification of the well-known three-level trophic chain consisting of resource-producer-primary consumer studied in [46], we consider the following four-level ecological interaction chain:

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. [47]).

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:

(11)

(12)

(13)

(14)

Theorem 3.1. ([40]) Let us suppose that for given biological parameters, the resource supply is high enough,

. (15)

Then, both the open and the partially closed ecological chains stably coexist in the sense there exists a positive equilibrium of system calculated in [40], which is asymptotically stable.

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.

3.1. Observability of the Ecological Chain

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

(16)

and

. (17)

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

. (18)

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:

(19)

Hence we easily calculate

provided. From the classical sufficient condition for the local observability of nonlinear systems, [3], we obtain local observability of system (16) near the equilibrium, with observation (18).

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

. (20)

The linearization of system (17) at equilibrium is

. (21)

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

(22)

(23)

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

(24)

which can be associated with system (11)-(14). Finally, applying again the classical rank condition of [3], we can summarize the reasoning of this subsection in the following theorem.

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 [40].  

3.2. Construction of an Observer System

Following the procedure of [44], let us first determine conditions for the construction of observers for systems (16) and (17), with respective observation functions (18) and (20).

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:

(25)

. (26)

Simple sufficient conditions for (25) and (26) are and, respectively. By the Theorem of [44], the observer for system (16) with observation function (18) can be determined.

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:

(27)

(28)

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 [44], the observer for system (17) with observation function (20) can be determined.

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

Theorem 3.3. ([40]) Given

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

(29)

System (29) has a positive equilibrium , which is asymptotically stable, because conditions of Theorem 3.1 are satisfied. In Figure 2 it can be seen how, e.g. from initial condition near the equilibrium, the solution of system (29) tends to this positive equilibrium, see Figure 2.

Consider now system (29) with observation

.

Since matrix

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

(30)

Solving (30) with initial condition near the equilibrium, we can check how this solution tends to recover the corresponding solution of system (29), see Figure 3.

4. A Stage-Structured Fishery Model with Reserve Area

Let us consider a modification of the stage-structured fishery model of [45], supposing that there is reserve area where fishing is not allowed. In what follows, the first index of the biomass density will indicate the area: for the reserve and for the free area; the second index will refer to the development stage: for the pre-recruits, i.e. the eggs, larvae and the juveniles together, and the exploited stage of the population. The dynamics of the system is modeled by the following autonomous system of differential equations

(31a)

(31b)

(31c)

(31d)

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 [41] we know that if

(32)

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.

4.1. Observability of the Model

Let with, and we consider the observation function defined by

(34)

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

(35)

and

. (36)

Given observation

(37)

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

(38)

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.

4.2. Construction of an Observer System

Given the observation system (35)-(37), using the corresponding observer design of [44], it is sufficient to find a matrix such that  is Hurwitz. It is easy to check that with ,

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 [48]:

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

.

Then the observer system is

(39)

Considering as initial value for the system (35), and for the observer (39), in Figure 4 we can see how the solution of the observer system approaches the solution of the original system.

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

Then the observer system is

(40)

If we consider as initial value for system (36), and for the observer (40)we obtain the result plotted in Figure 5.

Now the observer for system (31)-(34) can be simply composed from the single observers (39) and (40). In Figure 6 we can see how the solution of the observer (39)-(40) with initial value, estimates the solution of system (31) with initial value.

5. Discussion and Outlook

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 [57] a real-time local observer was constructed for a linear model of a solar thermal heating system. With different algorithm, a global real-time observer was designed for a more precise nonlinear model for the same solar thermal heating system in [58].

6. Acknowledgements

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).

REFERENCES

Appendix

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. [3]), given a fixed , there exists an  such that for all with there exists a unique continuously differentiable function such that

, for all

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

(A.1)

. (A.2)

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. ([3]) Consider the control-observation system (A.1)-(A.2) in with

Assume

(A.3)

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 [36], we summarize some concepts, notation and a basic sufficient condition for observability of verticum-type systems, in a simplified form used in the present paper.

Let

,

, and consider the nonlinear system

, (V₀)

and for all

. (V_i)

Denoting

let  with

and  with

We shall suppose that there exists

such that

and.

Definition A.2. Observation system

(V)

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

, (LV₀)

and for all

, (LV_i)

where

Define matrices as follows:

,

obtaining linear observation system

(LV)

of verticum type (see [36]). In the latter paper, a Kalman-type necessary and sufficient condition for observability of linear verticum-type systems was obtained. Here we recall only its “sufficient part” to be applied below.

Theorem A.2. ([36]) Suppose that

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 [1]), the rank condition is fulfilled, which by Theorem A.1 implies local observability of system (V) near equilibrium.

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.

NOTES

^*Corresponding author.