BioSystems 100 (2010) 65–69 Contents lists available at ScienceDirect BioSystems journal homepage: www.elsevier.com/locate/biosystems Monitoring in a predator–prey systems via a class of high order observer design Juan Luis Mata-Machuca a , Rafael Martínez-Guerra a , Ricardo Aguilar-López b,∗ a Departamento de Control Automático, Centro de Investigación y de Estudios Avanzados del I.P.N., CINVESTAV-IPN, Av. Instituto Politécnico Nacional No. 2508, San Pedro Zacatenco, México D.F. 07360, Mexico b Departamento de Biotecnología y Bioingeniería, Centro de Investigación y de Estudios Avanzados del I.P.N., CINVESTAV-IPN, Av. Instituto Politécnico Nacional No. 2508, San Pedro Zacatenco, México D.F. 07360, Mexico article info Article history: Received 22 June 2009 Received in revised form 11 January 2010 Accepted 12 January 2010 Keywords: Predator–prey systems Monitoring Nonlinear observers Polynomial form abstract 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. © 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 oscil- lations (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 to determine their effect on system states and outputs can be a challenging problem, especially for dynamic models. From the above, the uncertainties presents on the ecolog- ical modeling, together with the corresponding variable measured for these kinds of systems can be very important; in consequence the modeling tasks can be difficult (Gámez 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-mail addresses: jmata@ctrl.cinvestav.mx (J.L. Mata-Machuca), rguerra@ctrl.cinvestav.mx (R. Martínez-Guerra), raguilar@cinvesta.mx (R. Aguilar-López). in order to provide an estimate of its internal state, given measure- ments 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, stabilizing a system using state feed- back. In most practical cases, the physical state of the system cannot be determined by direct observation. Instead, indirect effects of the internal state are observed by way of the system outputs. 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 López 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 con- dition 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 evo- lutionary dynamics in Gámez et al. (2003) and López et al. (2004, 2005). Observer design in frequency-dependent population models was studied in López et al. (2008). Different Lotka–Volterra models were considered for observability and observer design in Gámez et al. (2008a,b), López et al. (2007a,b) and Varga et al. (2003). Mon- itoring problems of non-Lotka–Volterra type ecological and cell population models were studied in Gámez 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- 0303-2647/$ – see front matter © 2010 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.biosystems.2010.01.003