OBSERVER DESIGN FOR OPEN AND CLOSED TROPHIC CHAINS

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1 OBSERVER DESIGN FOR OPEN AND CLOSED TROPHIC CHAINS Z. Vrg M. Gáez b, I. López b Institute of Mthetics nd Infortics, Szent István University, Páter K. u.., H-3 Godollo, Hungry. (Vrg.Zoltn@gek.szie.hu) b Deprtent of Sttistics nd Applied Mthetics, University of Alerí, L Cñd de Sn Urbno, 4 Alerí, Espñ. (gez@ul.es, ilopez@ul.es) Abstrct: Monitoring of ecologicl systes is one of the jor issues in ecosyste reserch. The concepts nd ethodology of theticl systes theory provide useful tools to fce this proble. In ny cses, stte onitoring of cople ecologicl syste consists in observtion (esureent) of certin stte vribles, nd the whole stte process hs to be deterined fro the observed dt. The solution proposed in the pper is the design of n observer syste, which kes it possible to pproitely recover the stte process fro its prtil observtion. Such systes-theoreticl pproch hs been pplied before by the uthors to Lotk-Volterr type popultion systes. In the present pper this ethodology is etended to non Lotk-Volterr type trophic chin of resource producer priry consuer type nd nuericl eples for different observtion situtions re lso presented. Keywords: ecologicl onitoring, observer design, trophic chin. Introduction The proble of sustinbility of econoic nd socil developent in broder sense lso involves conservtion spects of ecology. The proble of stte onitoring of popultion systes, even under nturl conditions, is n iportnt issue in conservtion ecology. Nerly nturl popultions re often eposed to strong hun intervention, e.g. by wildlife ngeent, fisheries or environentl pollution. This ens tht hun ctivity y iprove or brek the equilibriu of the popultion syste in question, it y lso increse or decrese the genetic vribility of the given popultions. One of the in tsks of conservtion biology is to preserve the diversity of popultion systes nd genetic vribility of certin popultions. These probles ke it necessry to etend the trditionl pproch of theoreticl biology focusing only on biologicl object, to the study of the syste biologicl object n. This, in dynic sitution, i.e. in cse of long-ter hun intervention, typiclly requires the

2 pproch of theticl systes theory (in frequently used ters, stte-spce odelling). On the stte-spce pproch to odelling in popultion biology, Metz (977) is n erly reference, see lso Metz nd Dicknn (986). Mtheticl systes theory offers ethodology to nswer this question. This discipline hd been developed by the 96s to solve vriety of probles fced in engineering nd industry. A bsic reference is Kln et l. (969), see lso Zdeh nd Desoer (963). A recent reference is Chen et l. (4). While by now, theticl systes theory bece quite filir to syste engineers, observbility nd controllbility nlysis of dynic odels in popultion biology is reltively new. In ny cses, stte onitoring of cople ecologicl syste consists in observtion (esureent) of certin stte vribles, nd the whole stte process hs to be deterined fro the observed dt. In ore generl setting, the stte process is syste of differentil equtions, nd insted of its concrete solution only trnsfor (in prticulr subset of the coponents) of it is known (esured). The considered syste is clled (loclly) observble, if fro the observtion, the underlying stte process cn be uniquely recovered (ner n equilibriu stte). Bsed on the sufficient condition for nonliner observtion systes published in Lee nd Mrkus (97), for different coeisting Lotk-Volterr type popultion systes, locl observbility results hve been obtined in prt by soe of the couthors of the of the present pper in López et l. (7,b). Lter on, in ddition to these theoreticl results, for Lotk-Volterr systes even so-clled observer systes hs been constructed tht de it possible to nuericlly recover the stte process fro the observtion dt, see López et l. (7, b), Gáez et l. (8,b) nd Vrg (8). We lso ention tht, bsed on n observbility result of Vrg (99) for nonliner observtion systes with invrint nifold, in López et l. (8) n observer syste ws designed for the frequencydependent odel of phenotypic observtion of genetic processes. In the present pper ecologicl systes of non Lotk-Volterr type will be considered. Until now in Shndy (5), only observbility results hve been obtined for systes of type resource producer priry consuer. In Section, fro Shndy (5), the odel setup nd bsic conditions for the eistence of n equilibriu of the syste re shortly reclled. Section 3 is the in body of the pper. First the theoreticl bckground of the observer design is set up. Then the construction of the observer nd the syptotic recovery of the stte process is illustrted with

3 nuericl eples for different observtion situtions. Section 4 is devoted to the discussion of the results.. Description of the dynic odel In order to illustrte the ppliction of the ethodology of theticl systes theory, reltively siple food web, trophic chin hs been chosen, tht in ddition to popultions lso involves resource (energy or nutrient). In the following, the odel setup is shortly reclled fro Shndy (5), see lso Svirezhev nd Logofet (983), Jorgensen nd Svirezhev (4). For further detils on trophic chins (nd genrl food webs) see e.g. Odu (97) nd Yodzis (989) The considered odel describes how resource oves through trophic chin. A typicl terrestril trophic chins consists in the following coponents: resource, the th trophic level (solr energy or inorgnic nutrient), which is incorported by plnt popultion, the st trophic level (producer), which trnsfers it to herbivorous nil popultion, the nd trophic level (priry consuer). Let it be noted tht, in longer trophic chin, the herbivores cn be consued by predtor popultion the 3 rd trophic level (secondry consuer), which cn be followed by top predtor popultion (tertiry consuers). In the present pper, for technicl siplicity only trophic chins of the type resource producer priry consuer will be studied. According to the possible types of th level (energy or nutrient), two types of trophic chins will be considered: open chins (without recycling) nd closed chins (with recycling). At the th trophic level, resource is the coon ter for energy nd nutrient. Let denote of the tie-vrying quntity of resource present in the syste, nd, in function of tie, the bioss (or density) of the producer (species ) nd the priry consuer (species ), respectively. Let Q be the resource supply considered constnt in the odel. Let be the velocity t which unit of bioss of species consues the resource, nd it is ssued tht this consuption increses the bioss of this species t rte k. A unit of bioss of species consues the bioss of species 3

4 t velocity, converting it into bioss t rte k. Both the plnt nd the nil popultions re supposed to decrese eponentilly in the bsence of the resource nd the other species, with respective rtes of decrese (Mlthus preters) nd. Finlly, in closed syste the ded individuls of species nd re recycled into nutrient t respective rtes nd, while for n open syste (where there is no nturl recycling), holds. Then with odel preters Q,,, ; k k ],[;, [,[,,, for the trophic chin the following dynic odel cn be set up: (.) Q ( k ) ( k ) (.). (.3) Let function f be defined in ters of the right-hnd side of this syste: Q 3 3 f : R R, f ( ) f (,, ) : ( k ). ( k ) In Shndy (5), necessry nd sufficient condition were found for the eistence of non-trivil ecologicl equilibriu of dynic syste (.)-(.3), where ll coponents re present: syste (.)-(.3) hs unique equilibriu (,, ) if nd only if the resource supply is high enough, i.e. Q Q :. (.4) kk k Throughout the pper condition (.4) will be supposed. Rerk.. For the threshold Q is lower thn for. Clerly, in the ltter cse the lck of recycling fro species, higher vlue of resource supply is necessry to produce the required positive equilibriu. Rerk.. Moreover, this equilibriu is syptoticlly stble. In order to gurntee n ecologicl positive equilibriu we shll suppose in this pper tht condition (.3) holds. 4

5 5 3. Construction of n observer syste for trophic chin For syste (.)-(.3) the observbility (see Appendi) of it, when we observe seprtely ech one of its vribles, ws proved in Shndy (5). Now, following the Theore of Sundrpndin () (see Appendi), we shll construct, in n eplicit wy, the locl eponentil observer for the three cses considered by Shndy. With this i we clculte the corresponding tri of the linerized syste of (.)-(.3) t equilibriu ) ( k k f A. Cse. We consider the observtion of the resources of syste (.)-(.3), where the observtion function is (,,). ) ( : ) : ( h C h (3.) In order to construct the locl observer for the considered observtion syste, we need to deterine tri ),, ( 3 h h h col H such tht tri A-HC ws Hurwitz, i.e. ll its eigenvlues hve negtive rel prts. According to the Hurwitz criterion (see e.g. Chen et l. (4)), in ters of the nored chrcteristic polynoil of A-HC, the following necessry nd sufficient condition holds:.,, ) ( 3 nd Hurwitz is p (3.) This tri H cn be deterined fro the following theore: Theore 3.. Let us supposed tht the resource supply is high enough, k k Q nd tri h H,

6 is such tht h {, }. Then dynic syste defined by k z f ( z) H[ y h( z)] is locl eponentil observer for syste (.)-(.3) with the observtion of the resource (3.). Proof. It is sufficient to show tht under the conditions of the theore, is Lypunov stble equilibriu of syste (.)-(.3), nd tri A-HC is Hurwitz. Then the proof cn be concluded pplying Theore of Sundrpndin () (see Appendi). First, fro Q k k inequlity Q Q lso follows, which on the one hnd, s quoted t the end of Section, iplies the eistence of unique positive equilibriu. On the other hnd, in Shndy (5), Svirezhev nd Logofet it ws proved, tht both in open systes (with, ) nd in prtilly or totlly closed systes (t lest one of inequlities nd holds) condition Q Q lso iplies (syptotic) stbility of the equilibriu. Fro (.)-(.3) the coordintes of the positive equilibriu Q k, k k, k k Q k k k. k k. re 6

7 Now it will be proved tht for the coefficients of the nored chrcteristic polynoil of A-HC conditions (3.) hold. To cut short the rther tedious clcultions, the following stteents cn be checked: Hypothesis Q nd k k k k ],[;, [,[ iply, Q nd lso, furtherore, the ltter is sufficient for nd lso used in the proof of. On the other hnd, h h k hk h, to be used in the proof of. Fro k inequlity k k k cn nd, k ],[; [,[ k be derived, which iplies. Finlly, inequlities h,, directly iply. Suing up, ll inequlities conditions (3.) hold for p ( ). Therefore tri A-HC is Hurwitz, nd concludes the proof. Eple 3.. As nuericl eple, we consider the following Q ; :.3; :.; : :.; :.3; :.; :.4; k :.5; k :.5. In this cse the considered syste (.)-(.3) hs positive equilibriu (4.5, 8, 5.78), which is syptoticlly stble (see Figure ) Figure. Soe solutions of systes (.)-(.3) 7

8 nd with, H. Conditions of Theore 3. re verified, therefore we cn construct the following observer syste z z z.3z z..z z (..5.3z z (.4.5.z.z.3.4z ) ) [ y ( z [ y ( z )] )] (3.3) If we suppose the initil condition : (3, 7, ), ner the equilibriu of syste (.)- (.3), nd siilrly, we consider nother nerby initil condition, z : (.9, 7.,.8) for the observer syste (3.), Figure shows tht the corresponding solution z tends to the solution of the originl syste. Figure. Solutions of systes (.)-(.3) nd (3.3) Cse. Now we consider the cse when the plnt of syste (.)-(.3) is observed. The observtion function then is h h ( ) : C : ( ) (,,). (3.4) 8

9 Siilrly to Cse, we cn prove the following theore providing n observer for the cse (3.4). Theore 3.3. Given tri h H h, with h nd h, dynic syste defined by z f ( z) H[ y h( z)] is locl eponentil observer for syste (.)-(.3) with the observtion of the plnt, s given in (3.4). Proof. The schee of the proof is siilr to tht of the previous ones. We only hve to prove tht tri A-HC is Hurwitz nd the ppliction Theore of Sundrpndin () will conclude the proof. Since fro Shndy (5),, nd k k, k ],[; [,[ we obtin tht k. Moreover, s [, [ nd h we hve tht h. Applying these inequlities nd tking into ccount tht the cse of positive equilibriu is considered nd h, it is esy to check tht conditions (3.) hold, therefore tri A-HC is Hurwitz, nd the proof is coplete. Eple 3.4. With the se odel preters s Eple 3., we consider.5 H.. Then, conditions of Theore 3.3 re verified nd therefore we cn construct the following observer syste z z z.3z z..z z (..5.3z z (.4.5.z ).z.3.4z ).[ y ( z.5[ y ( z )] )] (3.5) If we suppose gin s initil condition : (3, 7, ), ner the equilibriu of syste (.)-(.3), nd siilrly, we consider nother nerby initil condition, z : (.9, 7.,.8) for observer syste (3.5), Figure 3 shows tht the corresponding solution z tends to the solution of the originl syste. 9

10 Figure 3. Solutions of systes (.)-(.3) nd (3.5) Cse 3. Let us finlly consider the observtion of the herbivorous species of syste (.)-(.3), where the observtion function is h h ( ) : C : ( ) (,,). (3.6) Siilrly to Theores 3. nd 3.3, it is not hrd to prove the following theore providing n observer for the cse (3.6). Theore 3.5. Given Q k k nd tri h H h, where h nd h, then dynic syste defined by z f ( z) H[ y h( z)] is locl eponentil observer for syste (.)-(.3) with the observtion h of the plnt. Eple 3.6. For the odel preters of the previous eples, with.5 H.5,

11 conditions of Theore 3.5 hold, nd hence we obtin the following observer syste z z z.3z z..z z (..5.3z z (.4.5.z ).z.3.4z ).5[ y ( z.5[ y ( z )] )] (3.7) Set gin initil condition : (3, 7, ), close to the equilibriu of syste (.)-(.3), nd s nerby initil condition for the observer syste (3.7) lso choose z : (.9, 7.,.8). Now Figure 4 shows tht the corresponding solution z tends gin to the solution of the originl syste. Figure 4. Solutions of systes (.)-(.3) nd (3.7) 4. Discussion In the pper the construction of n observer syste ws pplied for the stte onitoring of siple trophic chin of the type resource producer priry consuer, recovering the whole stte process fro the only observtion of different coponents of the systes, such s the resource, the plnt (producer) nd herbivorous nil. The pplied ethodology cn lso be etended to ore cople odels of food webs, involving the observtion of certin biotic environentl coponents nd/or certin indictor species. A siilr pproch y be lso useful for the onitoring of popultion systes in chnging environent, where the chnge of certin biotic preters of the ecosyste is governed by n eternl dynic syste (describing n industril pollution or clitic chnges).

12 5. Acknowledgeents The uthors wish to thnk the Ministry of Eduction nd Science of Spin for the finncil support of the project TIN C3-, which hs prtilly supported this work. The reserch ws lso supported by the Hungrin Ntionl Scientific Reserch Fund (OTKA 6 nd 6887), nd bilterl project funded by the Scientific nd Technologicl Innovtion Fund (of Hungry) nd the Ministry of Eduction nd Sciences (of Spin HH8-3). Appendi Given positive integers, n, let f : R n n R, h : R n R be continuously differentible functions nd for soe nd h ( ). R n we hve tht f ( ) We consider the following observtion syste where y is clled the observed function. f () (A.) y h(), (A.) Definition A.. Observtion syste (A.)-(A.) is clled loclly observble ner equilibriu, over given tie intervl, ] [ T, if there eists, such tht for ny two different solutions nd of syste () with ( t) nd ( t) ( t [, T ]), the observed functions h nd h re different. ( denotes the coposition of functions. For brevity, the reference to [, T ] is suppressed). For the forultion of sufficient condition for locl observbility consider the lineriztion of the observtion syste (A.)-(A.), consisting in the clcultion of the Jcobins A : f ( ) nd C : h( ). Theore A.. (Lee nd Mrkus, 97). Suppose tht rnk[ C CA CA... CA ] n T n. (A.3)

13 Then the observtion syste (A.)-(A.) is loclly observble ner equilibriu. Now, the construction of n observer syste will be bsed on Sundrpndin (). Let us consider observtion syste (A.)-(A.). Definition A.3. Given continuously differentible function syste n G : R R R, z G( z, y) (A.4) n is clled locl syptotic (respectively, eponentil) observer for observtion syste (A.)-(A.) if the coposite syste (A.)-(A.), (A.4) stisfies the following two requireents: i) If ( ) z(), then ( t) z( t), for ll t. n ii) There eists neighbourhood V of the equilibriu of R such tht for ll ( ), z() V, the estition error z( t) ( t) decys syptoticlly (respectively, eponentilly) to zero. Theore A.4. (Sundrpndin, ). Suppose tht equilibriu (A.) is Lypunov stble, nd tht there eists tri K such tht tri of syste (A.)- A KC is Hurwitz (i.e. its eigenvlues hve negtive rel prts), where A f ( ) nd C h( ). Then dynic syste defined by z f ( z) K[ y h( z)] (A.5) is locl eponentil observer for observtion syste (A.)-(A.). References Chen, Ben M.; Lin, Zongli; Shesh, Ycov A., 4. Liner Systes Theory. A Structurl Decoposition Approch. Birkhuser, Boston. Gáez, M., López, I. nd Vrg, Z., 8?. Itertive schee for the observtion of copetitive Lotk Volterr syste. Applied Mthetics nd Coputtion Gáez, M.; López, I. nd Molnár, S., 8b?. Monitoring environentl chnge in n ecosyste. Biosystes, 93, -7. 3

14 Jorgensen, S., Svirezhev, Y. (Eds.), 4. Towrds Therodynic Theory for Ecologicl Systes Pergon. Kln, R. E., Flb, P. L., Arbib, M. A., 969. Topics in Mtheticl Syste Theory. McGrw-Hill, New York. Lee, E.B. nd Mrkus, L., 97. Foundtions of Optil Control Theory. New York- London-Sydney : Wiley. López I, Gáez M, Molnár, S., 7. Observbility nd observers in food web. Applied Mthetics Letters (8): López, I., Gáez, M., Gry, J. nd Vrg, Z., 7b. Monitoring in Lotk-Volterr odel. Biosystes, 83, López, I., Gáez, M. nd Vrg, Z. 8, Observer design for phenotypic observtion of genetic processes. Nonliner Anlysis: Rel World Applictions 9, 9 3. Metz, J. A. J Stte spce odel for nil behviour. Ann. Syst. Res. 6: Metz, J. A. J. nd Dicknn O. (Eds), 986. The Dinics of Physiologiclly structured Popultions, Springer Lecture Notes in Bioth. 68. Pgins?? Odu, E. P. 97. Fundentls of Ecology. 3rd ed. Sunders, Phildelphi. 574 pp. Shndy, A., 5. Monitoring of trophic chins. Biosystes, Vol. 8, Issue, Sundrpndin, V.,. Locl observer design for nonliner systes. Mtheticl nd coputer odelling 35, Svirezhev, Yu.M. nd D.O. Logofet (983). Stbility of biologicl counities. Mir Publishers, Moscow. Vrg, Z., 99. On Observbility of Fisher's odel of selection, Pure Mthetics nd Applictions, Ser. B. Vol. 3, No, 5-5. Vrg, Z., 8. Applictions of theticl systes theory in popultion biology. Periodic Mthetic Hungric. 5 (), Yodzis, P. (989). Introduction to Theoreticl Ecology. Hrper & Row. New York. Zdeh, L. A. nd Desoer, C. A., 963. Liner Syste Theory-The Stte Spce Approch, New York: McGrw-Hill Book Co. 4