BioSystems 100(2010)65-69
BioSystems journal homepage: www.elsevier.com/locate/biosystems
Monitoring in a predator-prey systems via a class of high order observer design Juan Luis M a t a - M a c h u c a a , Rafael M a r t i n e z - G u e r r a a , Ricardo A g u i l a r - ~ 6 p e z ~ ? * a Deportomento de Control Automdtico. Centro de lnvestigocidn y de Estudios Avonzodos del I.P.N.. CINVESTAV-IPN. Av. Instituto Politemico NocionolNo. 2508. Son Pedro Zocotenco, Mexico D.F. 07360.Mexico Deportomentode Biotemologhy Bioingenierio. Centro de Investigacidn y de Estudios Avonzodosdel I.P.N.. CINVESTAV-IPN, Av. lnstituto Politemico NocionolNo. 2508, Son Pedro Zocotenco, Mexico D.F. 07360.Mexico
A R T I C L E
INFO
Article history: Received 22 June2009 Received in revised form 1 1 January 2010 Accepted 12 January 2010 Keywords: Predator-prey systems Monitoring Nonlinear 0b~eNers Polynomial form
A B S T R A C T
The goal of this work is the monitoring of the corresponding species in a class of predator-prey systems. this issue is important from the ecology point of view to analyze the population dynamics.The above is done via a nonlinear observer design which contains on its structure a high order polynomial form of the estimation error. A theoretical frame is provided in order to show the convergence characteristics of the proposed observer, where it can be concluded that the performance of the observer is improved as the order of the polynomial is high.The proposed methodology is applied to a class of Lotka-Volterra systems with two and three species. Finally, numerical simulations present the performance of the proposed observer. 0 2010 Elsevier Ireland Ltd. All rights reserved.
1. Introduction
Ecological systems and their component biological populations exhibit a broad spectrum of non-equilibrium dynamics ranging from characteristic natural cycles to more complex chaotic oscillations (May, 1973; Ranta and Kaitala, 1997; Royama, 1992), a diversity of abiotic variables, spatial and temporal heterogeneity, and most importantly, the presence of other species (as food, as competitors, and as predators) all affect the population dynamics of every species. The monitoring in an ecological system with several populations is generally a difficult task, because only the density of certain populations can be observed or measured. System analysis. either static or dynamic, frequently involves uncertain parameters and inputs. Propagating these uncertainties through a complex model todetermine their effect on system states and outputs can be a challenging problem, especially for dynamic models. From the above. the uncertainties presents on the ecological modeling, together with the corresponding variable measured for these kinds of systems can be very important; in consequence the modeling tasks can be difficult (Gdmez et al.. 2008a). On other hand, control theory provides a useful tool to design mathematical algorithms to infer unmeasured variables from the corresponding measured ones, these algorithms are called state observers, a state observer is a system that models a real system
Corresponding author. Fax: +52 55 57473982. E-moiladdresses:
[email protected] (J.L.Mata-Machuca),
[email protected](R. Martinez-Cuerra),
[email protected] (R. Aguilar-L6pez).
0303-264715- see front matter O 2010 Elsevier Ireland Ltd. All rights resewed. doi:10.1016/j.biosystems.2010.01.003
in order to provide an estimate of its internal state, given measurements of the input and output of the real system. It is typically a computer-implemented mathematical model. Knowing the state system is necessary to solve many control theory problems: for example, stabilizinga system usingstate feedback. Inmost practical cases, the physical stateofthe system cannot be determined by direct observation. Instead, indirect effects of the internal state areobserved by way of the systemoutputs. If a system is observable, it is possible to reconstruct the system state from its output measurements using the state observer. In particular the local observability conditions for several Lotka-Volterra models were analyzed in Ldpez et al. (2007a) and Shamandy (2005). The application of the concept of observability to the monitoring of population systems goes back to Varga (1992)where, concerning frequency-dependent population models, a general sufficient condition for local observability of nonlinear dynamic systems with invariant manifold was developed and applied. Later this method was applied to different models of population genetics and evolutionary dynamics in Gamez et al. (2003) and L6pez et al. (2004, 2005). Observer design in frequency-dependent population models was studied in Ldpez et al. (2008). Different Lotka-Volterra models were considered for observability and observer design in Gdmez et al. (2008a,b). L6pez et al. (2007a.b) and Varga et al. (2003). Monitoring problems of non-Lotka-Volterra type ecological and cell population models were studied in Gdmez et al. (2009). Shamandy (2005)and Varga et al. (2009). An approach based on a new system inversion method was applied in Szigeti et al. (2002) to monitoring of a five-species predator-prey system. Following these ideas, the estimation theory deserves an interesting research field, because the estimation methodolo-
66
j.L. Mota-Machuca et al. /
gies developed are widely employed in online monitoring. fault detection, and control for a wide class of systems. Some of the most important estimation methodologies are related with the observers design, where nonlinear techniques Oarre-Teichmann, 1998; Kalman. 1960: Keller. 1987; Levant. 2001; Rapaport and Harmand. 2002) have been presented in the open literature. these nonlinear approaches provides robustness against modeling errors and noisy measurements. On the other hand some techniques as neural-networks have been successfully used too (Spitz and Lek, 1999). 2. hedator-hey Mathematical Models
As is it well known, the Lotka-Volterra model describes interactions between several species in an ecosystem, predators and preys (Freedman. 1980: Hinrichsen and Pritchard. 2005; May. 1973).This represents our first multi-species model under study. Firstly, let us consider a two-species model, this model involves two equations, one which describes how the prey population changes and the second which describes how the predator population changes. Ifwe let xl and x2 represent the number of preys and predators. respectively, that are alive at time t, then the Lotka-Volterra model is:
parameter model (Eqs. (3a) and (3b)):
Here. C = ( 1 0 . . . 0 ).x E 9tn is the vector of states; f : 9tn -, 9tn is a nonlinear vector field and y E 9im is the system measured output. with m < n. It is necessary the design of an auxiliary system socalled observer system to reconstruct the unknown statesor unmeasurables.Firstly, we give necessary and sufficient conditions to establish whether the system (3a)and (3b) is observable. Now, consider the following assumptions: A l . The system given in Eqs. (3a) and (3b) is locally uniformly observable (Gauthier et al.. 1992). hence for all xc9in,satisfies the observability rank condition
Here I9 is the observability vector function defined as f i = T (dL;h, dL)h,. . . , dL;-'h) , being Ljh the r-order Lie derivatives. which are the directional derivatives of the corresponding state variables along the measured output trajectory. And dLih are the differentials of the rth-order Lie derivatives defined recursively as follows:
where the parameters are defined by: a is the natural growth rate of prey in the absence of predation, c is the natural death rate of predator in the absence of food, b is the death rate per encounter of prey due to predation, e is the efficiency of turning predated prey into predator. Let us consider y =xl as the measured output, that is, we suppose that we observe the total quantity of population preys. Many population cycles have the unusual property that their period length remains remarkably constant while their abundance levels are highly erratic. In L6pez et al. (2007a) and May and Leonard (1975) is shown that such more complexoscillationscan be achieved in simple predator-prey models by including more tropic levels. As a second case, we consider a three-species predator-prey model of two preys and one predator of the form dxldt-Ax), determined by the following system: All the trajectories x(t, xo), xo E!JI" of the system ((3a) and (3b))are bounded.
A2.
Herex1, x3 are prey populations, x2 is the predator population, and a;, bi, ci > 0 for all i = 1,2,3. Moreover, y =xl is the system measured output. Since populations are nonnegative, we will restrict our attention to the nonnegative orthants {(xl.x2. x3) I X I ? 0. x2 1 0 . x3 2 0) c R3 and the positive orthants R3 ={(x1.x2,x3) I X I > O . X ~> O . X ~>O) c R3. It is worthwhile to mention that the both the nonnegative and the positive quadrants are positively invariant for the general Lotka-Volterra model.
Considering the set SZ c 9tnas thecorresponding physically realizable domain, such that: SZ = ((xi)Ll~91:/0 i xi i xmaxI. In most practical cases. SZ will be an open connected relatively compact subset of 9rn, and in the ideal cases, SZ will be positively invariant under the dynamics (3a) and (3b). In order to analyze the estimation error 6 = x - 2 we consider the next assumption. The nonlinear difference vector function A@ =f(x) - f(2) is Lipschitz bounded, i.e. ) A @ )s A
A3.
(1
Condition A3 can be fulfilled satisfied ifthe followingsupremum is finite:
3. Proposed Observer
Now, consider a general representation of the Lotka-Volterra models, which can be generally described by the following lumped
wheref(x) is the Jacobian and 111.1 is the matrix norm associated with the Euclidian vector norm. This is the case in our models (1) and (2),when considered in a bounded part of the positive orthant.
J.L.
Mota-Mochuco et 01./BioSystems 100 (2010) 65-69
Proposition 1. Consider that system (3a) and (3b) satisfies Al, A2 and A3 (with these assumptions is possible to construct an observer) then, there exist kl, k2 such that for any positive odd integer w z 1, system (5) is an asymptotic observerfor system (3a) and (3b)
67
160
-Real
1
i W
250
See the appendix for the corresponding proof. Before designing the asymptotic observer for systems ( I ) and (2). assumptions A1-A3 are checked. for two-species predator-prey system (1 )the observabilitycondition given by assumption A1 is proven. The observability matrix is aI9
X
0
0
50
( -E)
150
time
Its determinant is: det
100
Fis 1. Estimated
states for Lotka-Volterra model.
= -bxl
The observability rank condition is satisfied if and only ifxl # 0. Then by assumption A1 the two-species system ( 1 ) is locally uniformly observable, that is, the whole system can be monitored observing only the prey population (y a x l ) , without any further condition. As will be shown in Section 4, for systems (1) and (2). is not difficult to provide a simple algebraic condition for the existence of an equilibrium in mathematical sense, however its positivity depends on the model parameters, that means, for any set of initial conditions the trajectories for r 1 0 tend or cycle around this equilibrium point in the positive quadrant, in this form, condition A2 is satisfied. In the same manner the three-dimensional predator-prey system (2). the observability matrix is given by:
Corollary 2. The proposed observerfor the three order predator-prey modelgiven by system (2) is:
4. Numerical Examples and Results
In order to measure the performance of the proposed observer under different polynomial degrees is employing the measure the impact of the error, suggested in Ogunnaike and Ray (1994). is the "Integral Time-Weighted Squared Error" (ITSE) defined in (8). ITSE exhibits the advantage of heavy penalization of large errors at long time: therefore is a good measure of resilience of the observer.
where
ITSE =
i
tc2 dt
o
(g)
= c;c2x:x2 Then, det If populations xl, x2 are nonzero. then the observability rank condition is fulfilled. By using assumption A1 the three-species model (2) is locally uniformly observable, in this sense, all state variables can be estimated if we only observe the prey population XI.
Notice also that, ifwe take the initial condition such thatxl (0) = 0, then for all t r 0, the output is zero. In these systems, preys populations cannot grow if they are not present at the beginning of the story.
Corollary 1. The proposed observer for Lotka-Voltera model given by system (1) is as follows:
@I
Firstly, we present some simulations for Lotka-Volterra model given in (1) and its corresponding observer (6). We have taken the parameters values as: a =0.2, b =0.0025, e=0.002, c = 0.1. kl = 1. k2 = 1, and the initial conditions xl = 100. x2 = 90. R1 = 90. 22 = 85. In Fig. 1 are shown the state variables and their corresponding estimated states, using w = 3 in system (6). The model mirrors a qualitative feature which has been observed in many real predator-prey systems: the periodicfluctuations. This is illustrated in Fig. 2, where f = (cle, alb) = (50,80) is an equilibrium point of (1) and any initial state x(0) # i ,xl (0)> 0, ~ ~ (> 00 leads ) to a periodic trajectory cycling around this equilibrium point in the positive quadrant. Moreover, in Fig. 3, is presented the performance index (ITSE) given in (8). We have taken w = 3, 5, 7. for the observer (6). It should be noted that the value of the corresponding performance index decrease as w increase. Now, we present some simulations for three order predator-prey model given in (2) and its corresponding observer (7). We have chosen the parameters values as: a1 = 1, a2 =0.5, a3=3.2, bl =0.01. b2=0.01, b3-0.03, cl -0.01. c2=0.02. c3=0.04. kl -5, k2 = 10, and the initial conditions xl =90, x2 = 100, x3 =80, 2, = 105. 22 = 85. 23 = 89.
j.L Mata-Machuca et al. / BioSysterns 100 (2010) 65-69
68
_,-
120.
-
--
loo 80.
40
60
20.
2 . '
- Real
0.
40 .
Estimated
+
Eqlifibriumpoint
-Real 4
..
L
20
40
Estlmated Equ~libnurnpolnt
60
80
..
0 0
100
xl
Fig. 5. Three order predator-prey model trajectories.
Fig.2. Lotka-Volterra trajectories.
time Fig. 6. Performanceindex for predator-prey model.
time Fi 3. Performance index (Lotka-Voiterra model).
-
a l b 2 ~ 3- a 2 b l ~ 3+ a3b1~2= 84 bl b 3 ~ 2+ b 2 ~ 1 ~ 3
-
a2blb3 + a3b2c1- alb2b3 = 17 b i b 3 ~ 2+ b2cic3
X2 =
In Fig. 4 are shown the state variables and their corresponding estimated states, using w = 3 in system (7). The model has no periodic fluctuations. This is illustrated in Fig. 5. By some algebraic manipulations an asymptotically stable equilibrium point of (2) is obtained as:
-
x1 =
a2clc3 +alb3c2 - a3clc2 = 16 b i b 3 ~ 2+ b 2 ~ 1 ~ 3
X3 =
0 < ~ ~ (#0i)3 Any initial state 0 < xl(0) # t l , 0 < x2(0) # i2. leads to this equilibrium point in the positive quadrant. Furthermore, in Fig. 6, is presented the performance index (ITSE), for three order predator-prey model (2). We have taken w = 3, 5, 7. for the observer (7). It should be noted that the value of the corresponding performance index decrease as w increase.
5. Concluding Remarks
-Real --=--+ Estimated
In this work we have presented a high order polynomial observer to attack the problem of monitoring in predator-prey systems. By some algebraic manipulations the convergence of the corresponding estimation error was proven. The estimation error depends on the observer gains and a Lipschitz constant. The proposed methodology was applied to a class of Lotka-Volterra model with two and three species with success.
Appendix A.
25
time Fig.4. Estimated states for three order predator-prey model.
Proof of Proposition 1. Defining the observation error as 6 = x - R the corresponding dynamic of the estimation error is: = A@(C)
+ klCC + k2CWCW
(9)
1.1. Mata-Machuca et 01. j Bic
Remark 1. Note that the estimation error can be diminished arbitrarily, considering k2 large enough or considering k l small enough. ~ ~ ( ~ A I ~ I + ~ I C= I( ~A I+ +~ I~ ~~ ) C C ~ I I+ ~~ ~~ I~ besides ~ ~ C as~w ~increases the estimation error is diminished as can be noticed from Eq. ( 1 8 ) . (10) Considering assumption
A3:
Considering the i-th coordinate in the above vector inequality and employing the equality:
References Freedman. H.I.. 1980. Deterministic Mathematical Models in Population Ecology. M. Dekker. New York. Gdmez. M.. CarreRo. R.. K6sa. A,. Varga. Z.. 2003. Observability in strategic models of selection. BioSystems 71.249-255. Gdmez. M.. L6pez. I., Molndr. S.. 2008a. Monitoring environmental change in an ecosystem. BioSystems 93.21 1-217. Gdmez. M.. L6pez. I.. Varga. Z.. 2008b. Iterative scheme for the observation ofa competitive Lotka-Volterra system. Applied Mathematics and Computation 201.
It1 = sign(tlt9 the following equation is obtained:
811-818.
Gdmez. M.. Upez. I.. Garay. J.. Varga. Z.. 2009. Observation and control in a model of a cell population affected by radiation. BioSystems 96.172-177. Gauthier.J.P.. Hammouri. H.. Othman. S.. 1992. A simple observer for nonlinear systems. Applications to bioreactors. IEEE Transaction on Automatic Control 37. 875-880.
Hinrichsen. D.. Pritchard. A,. 2005. Mathematical Systems Theory I: Modeling. State Space Analysis. Stability and Robustness. Springer-Verlag. Jarre-Teichmann. A.. 1998. The potential role of mass balance models for the management of upwelling ecosystems. Ecological Applications 8.93-103. Kalman. R.. 1960. A new approach to linear filteringand prediction problems.Transaction of the ASME-Journalof Basic Engineering. Series D 82.35-45. Keller, H.. 1987. Non-linear observer design by transformation into a generalized observer canonical form. International Journal of Control 46. 1915-
defining:
n1i = ( A + k l C ) i s i @ ( t i ) .
n2i =
(k2Cw)isign(Siw
sign(&)
.6i
sign(ii)
The following inequality is obtained:
1930.
-nliti I
(13)
n2i6iw
To solve the above inequality, consider the change of variable:
Yi = t i l - w ,
w>1
(14)
Thus
t i = o = +y i = 0 and
ti # 0 =+ yi # 0 ,
{
i.e.
wodd
Levant. A.. 2001. Universal single-input-single-output (SISO) sliding-mode controllers with finite-time convergence. IEEE Transaction on Automatic Control 46.1447-1451.
Lbpez. I.. G6mez. M.. CarreRo. R.. 2004. Observability in dynamic evolutionary models. BioSystems 73.99-109. Lbpez. l..G6mez. M..Garay.J..Varga.Z..2007a. Monitoring ina Lotka-Volterra model. BioSystems 83.68-74. Lbpez. I.. Gdmez. M.. Molndr. S.. 2007b. Observability and observers in a food web. Applied Mathematics Letters 20.951-957. L6pez. I., GOmez. M., Varga. Z.. 2005. Equilibrium, observability and controllability in selection-mutation models. BioSystems 81.65-75. Upez. I.. G6mez. M.. Varga. Z.. 2008. Observer design for phenotypic observation of genetic processes. Nonlinear Analysis: Real World Applications 9.270. ~
302.
=+yi>O
Hereafter we consider 6i # 0 with w e Z + , w odd, w > 1 . Therefore the following first order ordinary differential inequality is generated:
May. R.. 1973. Stability and Complexity in Model Ecosystems. Princeton University Press. May. R.. Leonard. W.. 1975. Nonlinear aspectsofcompetition between three species. SlAM Journal on Applied Mathematics 29.243-253. Ogunnaike. B.. Ray. W.. 1994. Process Dynamics. Modeling and Control. Oxford University Press. USA. Rapaport. A,. Harmand. 1.. 2002. Robust regulation of a class of partially observer nonlinear continuous bioreactors. Journal of Process Control 12.291302.
Ranta. E.. Kaitala. V.. 1997. Travelling waves in vole population dynamics. Nature ;i-(1
-w)nliyi 2
(1
-w)n2i
By solving the inequality above:
390.456.
Royama. T.. 1992. Analytical Population Dynamics. Chapman & Hall. London. Shamandy. A.. 2005. Monitoring of trophic chains. BioSystems 81.43-48. Spitz. F., Lek. S.. 1999. Environmental impact prediction using neural network modeling: an example in wildlife damage. Journal of Applied Ecology 36. 317326.
Fort -+ oc we get:
In terms of the original variable (observation error):
Szigeti. F.. Vera. C.. Varga. Z..2002. Nonlinear system inversion applied ecological monitoring. In: 15-th IFAC World Congress of Automatic Control. Barcelona. Varga. 2..1992. On observability of Fisher's model of selection. Pure Mathematics and Applications. Series B 3. 15-25. Varga. Z.. Gdmez. M., Lbpez. 1.. 2009. Observer design for open and closed trophic chains. Nonlinear Analysis: Real World Applications. doi:10.1016/j.nonrwa.2009.04.015. Varga, Z..Scarelli, A.. Shamandy. A.. 2003. State monitoring of a population system in changing environment. Community Ecology 4-73-78.
