AICME II abstracts

Poster

Controllability of population systems Z. Varga1 , R. Carre˜ no2 , M. G´amez

3

and I. L´opez 4 .

Poster

AICME II abstracts

except that with indices ij, which is 1, if γij is affected and zero otherP wise. In terms of the matrix Ψ(w) := ni,j=1 wij Rij , the control effect on the community matrix can be described in the form Γ + Ψ(w). Now, considering u = (v, w) as control variable, we have a control-affine system x˙ = Diagx(ε − Γx) + Diagx[Pv − Ψ(w)x].

We start out from the classical n−species Lotka-Volterra system with Malthus parameter vector ε = [ε1 , . . . , εn ]T and community matrix Γ = [γij ]n×n : x˙ = Diagx[ε − Γx]. (1) Mathematical Systems Theory applied to the classical Lotka-Volterra equation provides a model for controlling population systems. Abiotic effects such as human intervention or environmental influence (industrial pollution, climatic changes etc.) are described in terms of time-dependent Malthus parameters or / and interaction coefficients as control functions. Systems-theoretic study of such models (completed with observation) can be found in [2], and [3]. In the present work controllability properties of Lotka-Volterra type control models are studied. In our treatment the time-dependent model parameters will be of the form εi + νi (t) and γij + wij (t). For a structured model we assume first that certain Malthus parameters are affected. Define a matrix P := [pij ]n×n such that pij := 1, if i = j and εi is affected, and 0 otherwise. Then the changed Malthus parameter vector is ε + Pv, where v is an n−dimensional control vector. Similarly, for the structured description of the time-dependent interaction coefficients, for each i, j ∈ 1, n, define an n × n matrix Rij with all items equal to zero 1 Institute of Mathematics and Informatics, Szent Istv´ an University H-2103 G¨ od¨ oll˝ o, P´ ater K. u. 1., Hungary (e-mail: [email protected]). 2 Dept. of Statistics and Applied Mathematics, University of Almer´ıa, Spain (e-mail: [email protected]). 3 Dept. of Statistics and Applied Mathematics, University of Almer´ıa, Spain (e-mail: [email protected] ). 4 Dept. of Statistics and Applied Mathematics, University of Almer´ıa, Spain (e-mail: [email protected] ).

Poster-Var-a

(2)

Controllability properties of system (2) near an equilibrium coexistence state x∗ := Γ−1 ε> 0 of (1) are studied. In particular, applying a sufficient condition for local controllability of nonlinear systems of [1], we can guarantee in different situations that, after a slight perturbation, the population system can be steered back to the equilibrium in given time, using only a “soft intervention”.

References [1] Lee, E. B. and L. Markus, 1971, Foundations of Optimal Control Theory. Wiley, New York - London - Sydney [2] Szigeti, F., C. Vera and Z. Varga, 2002, Nonlinear system inversion applied to ecological monitoring. 15 − th IFAC World Congress on Automatic Control, 2002, Barcelona, Spain (Abstracts p. 210, referred full paper in electronic form) [3] Varga, Z., A. Scarelli and A. Shamandy, 2003, State monitoring of a population system in changing environment. Community Ecology (in press) Acknowledgements: This research was supported by the (Hungarian) National Scientific Research Fund (OTKA No. T037271)

Poster-Var-b