Recent Developments in Monitoring of Complex Population Systems

Zoltán Varga, Manuel Gámez, Inmaculada López

Department of Mathematics, University of Almería, Almería, Spain.
Institute of Mathematics and Informatics, Szent István University, Godollo, Hungary.

DOI: 10.4236/ajor.2013.31A016                

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.

Keywords

Ecological Chain; Fishery with Reserve Area; Stable Coexistence; Ecosystem Monitoring; Verticum-Type System; Nonlinear System; Observer Design

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	R. E. Kalman, P. L. Falb and M. A. Arbib, “Topics in Mathematical System Theory,” McGraw-Hill, New York, 1969.
[2]	B. M. Chen, Z. Lin and Y. Shamesh, “Linear Systems Theory. A Structural Decomposition Approach,” Birkhauser, Boston, 2004. doi:10.1007/978-1-4612-2046-6
[3]	E. B. Lee and L. Markus, “Foundations of Optimal Control Theory,” Wiley, New York, London, Sydney, 1971.
[4]	Z. Varga, “On Controllability of Fisher’s Model of Selection,” In: C. M. Dafermos, G. Ladas and G. Papanicolau, Eds, Differential Equations, Marcel Dekker, New York, 1989, pp. 717-723.
[5]	Z. Varga, “On Observability of Fisher’s Model of Selection,” Pure Mathematics and Applications, Series B, Vol. 3, No. 1, 1992, pp. 15-25.
[6]	Gy. Farkas, “Local Controllability of Reactions,” Journal of Mathematical Chemistry, Vol. 24, No. 1, 1998, pp. 1-14. doi:10.1023/A:1019150014783
[7]	Gy. Farkas, “On Local Observability of Chemical Systems,” Journal of Mathematical Chemistry, Vol. 24, No. 1-3, 1998, pp. 15-22. doi:10.1023/A:1019158316600
[8]	A. Scarelli and Z. Varga, “Controllability of Selection-Mutation Systems,” BioSystems, Vol. 65, No. 2-3, 2002, pp. 113-121. doi:10.1016/S0303-2647(02)00012-6
[9]	I. López, “Observabilidad y Controlabilidad en Modelos de Evolución,” Ph.D. Dissertation, Universidad de Almería, Espana, 2003.
[10]	I. López, M. Gámez, R. Carreno and Z. Varga, “Recovering Genetic Processes from Phenotypic Observation,” In: V. Capasso, Ed., Mathematical Modelling & Computing in Biology and Medicine, MIRIAM, Milan, 2003, pp. 356-361.
[11]	I. López, M. Gámez, R. Carreno and Z. Varga, “Optimization of Mean Fitness of a Population via Artificial Selection,” In: R. Bars, Ed., Control Applications of Optimisation, Elsevier, Amsterdam, 2003, pp. 147-150.
[12]	I. López, M. Gámez and R. Carreno, “Observability in Dynamic Evolutionary Models,” BioSystems, Vol. 73, No. 2, 2004, pp. 99-109. doi:10.1016/j.biosystems.2003.10.003
[13]	I. López, M. Gámez and Z. Varga, “Equilibrium, Observability and Controllability in Selection-Mutation Models,” BioSystems, Vol. 81, No. 1, 2005, pp. 65-75. doi:10.1016/j.biosystems.2005.02.006
[14]	I. López, M. Gámez and Z. Varga, “Observer Design for Phenotypic Observation of Genetic Processes,” Nonlinear Analysis: Real World Applications, Vol. 9, No. 2, 2008, pp. 290-302. doi:10.1016/j.nonrwa.2006.10.004
[15]	M. Gámez, R. Carreno, A. Kósa and Z. Varga, “Observability in Strategic Models of Selection,” BioSystems, Vol. 71, No. 3, 2003, pp. 249-255. doi:10.1016/S0303-2647(03)00072-8
[16]	Z. Varga, A. Scarelli and A. Shamandy, “State Monitoring of a Population System in Changing Environment,” Community Ecology, Vol. 4, No. 1, 2003, pp. 73-78. doi:10.1556/ComEc.4.2003.1.11
[17]	I. López, M. Gámez, J. Garay and Z. Varga, “Monitoring in a Lotka-Volterra Model,” BioSystems, Vol. 87, No. 1, 2007, pp. 68-74. doi:10.1016/j.biosystems.2006.03.005
[18]	M. Gámez, I. López and Z. Varga, “Iterative Scheme for the Observation of a Competitive Lotka-Volterra System,” Applied Mathematics and Computation, Vol. 201, No. 1-2, 2008, pp. 811-818. doi:10.1016/j.amc.2007.11.049
[19]	M. Gámez, I. López and S. Molnár, “Monitoring Environmental Change in an Ecosystem,” BioSystems, Vol. 93, No. 3, 2008, pp. 211-217. doi:10.1016/j.biosystems.2008.04.012
[20]	M. Gámez, I. López and A. Shamandy, “Open-and Closed-Loop Equilibrium Control of Trophic Chains,” Ecological Modelling, Vol. 221, No. 16, 2010, pp. 1839-1846. doi:10.1016/j.ecolmodel.2010.04.011
[21]	J. R. Banga, E. Balsa-Canto, C. G. Moles and A. A. Alonso, “Dynamic Optimization of Bioprocesses: Efficient and Robust Numerical Strategies,” Journal of Biotechnology, Vol. 117, No. 4, 2005, pp. 407-419. doi:10.1016/j.jbiotec.2005.02.013
[22]	T. Hirmajer, E. Balsa-Canto and J. R. Banga, “DOTcvpSB, a Software Toolbox for Dynamic Optimization in Systems Biology,” BMC Bioinformatics, Vol. 10, 2009, p. 199. doi:10.1186/1471-2105-10-199
[23]	F. Szigeti, C. Vera and Z. Varga, “Nonlinear System Inversion Applied to Ecological Monitoring,” Proceedings of the 15th IFAC World Congress on Automatic Control, Barcelona, 21-26 July 2002, pp. 1-5. http://www.nt.ntnu.no/users/skoge/prost/proceedings/ifac2002/data/content/01758/1758.pdf
[24]	M. Gámez, I. López, J. Garay and Z. Varga, “Observation and Control in a Model of a Cell Population Affected by Radiation,” BioSystems, Vol. 96, No. 2, 2009, pp. 172-177. doi:10.1016/j.biosystems.2009.01.004
[25]	M. Rafikov, J. M. Balthazar and H. F. von Bremen, “Mathematical Modelling and Control of Population Systems: Applications in Biological Pest Control,” Applied Mathematics and Computation, Vol. 200, No. 2, 2008, pp. 557-573. doi:10.1016/j.amc.2007.11.036
[26]	M. Rafikov and E. H. Limeira, “Mathematical Modelling of the Biological Pest Control of the Sugarcane Borer,” International Journal of Computer Mathematics, Vol. 89, No. 3, 2012, pp. 390-401. doi:10.1080/00207160.2011.587873
[27]	M. Rafikov, A. Del Sole Lordelo and E. Rafikova, “Impulsive Biological Pest Control Strategies of the Sugarcane,” Mathematical Problems in Engineering, Vol. 2012, 2012, pp. 1-14. doi:10.1155/2012/726783
[28]	Z. Varga, “Applications of Mathematical Systems Theory in Population Biology,” Periodica Mathematica Hungarica, Vol. 56, No. 1, 2008, pp. 157-168. doi:10.1007/s10998-008-5157-0
[29]	M. Gámez, “Observation and Control in Density and Frequency-dependent Population Models,” In: W. J. Zhang, Ed., Ecological Modeling, Nova Science Publishers, New York, 2011, pp. 285-306.
[30]	S. Molnár, “Model Runs for the Definition of the Most Advantageous Integrated Energetical Verticum in the National Economy,” Publications of Central Mining Development Institute, Vol. 30, 1887, pp. 121-127.
[31]	S. Molnár, “Realization of Verticum-Type Systems,” Mathematical Analysis and Systems Theory, Vol. 5, 1988, pp. 11-30.
[32]	S. Molnár, “Optimization of Realization-Independent Cost Functions,” Mathematical Analysis and Systems Theory, Vol. 5, 1988, pp. 1-10.
[33]	S. Molnár, “Observability and Controllability of Decomposed Systems I,” Mathematical Analysis and Systems Theory, Vol. 5, 1988, pp. 57-66.
[34]	S. Molnár, “Observability and Controllability of Decomposed Systems II,” Mathematical Analysis and Systems Theory, Vol. 5, 1988, pp. 67-72.
[35]	S. Molnár, “Observability and Controllability of Decomposed Systems III,” Mathematical Analysis and Systems Theory, Vol. 5, 1988, pp. 73-80.
[36]	S. Molnár, “A Special Decomposition of Linear Systems,” Belgian Journal Operations Reseach, Statistics and Computation Science, Vol. 29, No. 4, 1989, pp. 1-19.
[37]	S. Molnár, “Stabilization of Verticum-Type Systems,” Pure Mathematics and Applications, Vol. 4, No. 4, 1993, pp. 493-499.
[38]	S. Molnár and F. Szigeti, “On Verticum-Type Linear Systems with Time-Dependent Linkage,” Applied Mathematics and Computation, Vol. 60, No. 1, 1994, 89-102. doi:10.1016/0096-3003(94)90208-9
[39]	I. López, M. Gámez and S. Molnár, “Observability and Observers in a Food Web,” Applied Mathematics Letters, Vol. 20, No. 8, 2007, pp. 951-957. doi:10.1016/j.aml.2006.09.007
[40]	M. Gámez, I. López, I. Szabó and Z. Varga, “Verticum-Type Systems Applied to Ecological Monitoring,” Applied Mathematics and Computation, Vol. 215, No. 9, 2010, pp. 3230-3238. doi:10.1016/j.amc.2009.10.010
[41]	S. Molnár, M. Gámez and I. López, “Observation of Nonlinear Verticum-Type Systems Applied to Ecological Monitoring,” International Journal of Biomathematics, Vol. 5, No. 6, 2012, pp. 1-15. doi:10.1142/S1793524512500519
[42]	M. Gámez, I. López, Z. Varga and J. Garay, “Stock Estimation, Environmental Monitoring and Equilibrium Control of a Fish Population with Reserve Area,” Reviews in Fish Biology and Fisheries, Vol. 22, No. 3, 2012, pp. 751-766. doi:10.1007/s11160-012-9253-y
[43]	B. Dubey, P. Chandra and P. Sinha, “A Model for Fishery Resource with Reserve Area,” Nonlinear Analysis. Real World Applications, Vol. 4, No. 4, 2003, pp. 625-637. doi:10.1016/S1468-1218(02)00082-2
[44]	V. Sundarapandian, “Local Observer Design for Nonlinear Systems,” Mathematical and Computer Modelling, Vol. 35, No. 1, 2002, pp. 25-36. doi:10.1016/S0895-7177(01)00145-5
[45]	A. Guiro, A. Iggidr, D. Ngom and H. Touré, “On the Stock Estimation for Some Fishery Systems,” Review in Fish Biology and Fisheries, Vol. 19, No. 3, 2009, pp. 313-327. doi:10.1007/s11160-009-9104-7
[46]	A. Shamandy, “Monitoring of Trophic Chains,” Biosystems, Vol. 81, No. 1, 2005, pp. 43-48. doi:10.1016/j.biosystems.2005.02.005
[47]	P. J. Morin and P. Morin, “Community Ecology,” Wiley-Blackwell, Hoboken, 1991.
[48]	A. Ouahbi, A. Iggidr and M. El Bagdouri, “Stabilization of an Exploited Fish Population,” Systems Analysis Modelling Simulation, Vol. 43, No. 4, 2003, pp. 513-524. doi:10.1080/02329290290028543
[49]	R. Cressman, J. Garay and J. Hofbauer, “Evolutionary Stability Concepts for N-species Frequency-Dependent Interactions,” Journal of Theoretical Biology, Vol. 211, No. 1, 2001, pp. 1-10. doi:10.1006/jtbi.2001.2321
[50]	J. Garay, “Many Species Partial Adaptive Dynamics,” BioSystems, Vol. 65, No. 1, 2002, pp. 19-23. doi:10.1016/S0303-2647(01)00196-4
[51]	R. Cressman and J. Garay, “Evolutionary Stability in Lotka-Volterra Systems,” Journal of Theoretical Biology, Vol. 222, No. 2, 2003, pp. 233-245. doi:10.1016/S0022-5193(03)00032-8
[52]	R. Cressman and J. Garay, “Stablility N-Species Coevolutionary Systems,” Theoretical Population Biology, Vol. 64, No. 4, 2003, pp. 519-533. doi:10.1016/S0040-5809(03)00101-1
[53]	R. Cressman and J. Garay, “A Game-Theoretical Model for Punctuated Equilibrium: Species Invasion and Stasis through Coevolution,” BioSystem, Vol. 84, No. 1, 2006, pp. 1-14. doi:10.1016/j.biosystems.2005.09.006
[54]	R. Cressman, V. Krivan and J. Garay, “Ideal Free Distributions, Evolutionary Games, and Population Dynamics in Multiple-Species Environments,” The American Naturalist, Vol. 164, No. 4, 2004, pp. 473-489. doi:10.1086/423827
[55]	R. Cressman and J. Garay, “A Predator-Prey Refuge System: Evolutionary Stability in Ecological Systems,” Theoretical Population Biology, Vol. 76, No. 4, 2009, pp. 248-257. doi:10.1016/j.tpb.2009.08.005
[56]	R. Cressman and J. Garay, “The Effects of Opportunistic and Intentional Predators on the Herding Behavior of Prey,” Ecology, Vol. 92, No. 2, 2011, pp. 432-440. doi:10.1890/10-0199.1
[57]	R. Kicsiny and Z. Varga, “Real-Time State Observer Design for Solar Thermal Heating Systems,” Applied Mathematics and Computation, Vol. 218, No. 23, 2012, pp. 11558-11569. doi:10.1016/j.amc.2012.05.040
[58]	R. Kicsiny and Z. Varga, “Real-Time Nonlinear Global State Observer Design for Solar Heating Systems,” Nonlinear Analysis: Real World Applications, Vol. 14, 2013, pp. 1247-1264. doi:10.1016/j.nonrwa.2012.09.017