AICME II abstracts

Mathematical modeling of dynamics population DDE

Partial Functional Mathematical Model of Nitrogen Transformation Cycle Tibor Kme´t1 . We consider a system of parabolic equations with discret time delays describing nitrogen transformation cycle in water environment , which takes the form: ∂xi (p, t) ∂ 2 xi (p, t) = dxi + xi (p, t)Ui (x(p, t))− ∂t ∂p2 xi (p, t − rei )Ei (x(p, t − rei )) − xi (p, t − rmi )Mi (x(p, t − rmi ))) 4 ∂x5 ∂ 2 x5 X = dx5 2 + xi (p, t − rmi )Mi (x(p, t − rmi ) ∂t ∂p i=1

−K5 x5 (p, t − r5 )

∂x6 = ∂t

∂ 2 x6 dx6 2 +K5 x5 (p, t−r5 )−U1 (x)x1 −P6 (x)x4 +E4 (x(p, t−re4 ))x4 (p, t−re4 ) ∂p

∂x7 ∂ 2 y1 = dx7 2 + E1 (x(p, t − re1 ))x1 (p, t − re1 ) + U2 (x)x2 − P7 (x)x4 ∂t ∂p

Mathematical modeling of dynamics population DDE

∂x9 ∂ 2 yi = dx9 2 + x3 (p, t − re3 )E3 (x(p, t − re3 )) − x4 P9 (x) ∂t ∂p where xi , i = 1, ..., 9, are the concentration of the recycling nitrogen in microorganisms,phytoplankton, detritus, DON, ammonium, nitrites, nitrate. From the biological viewpoint the functions Fi , Ui , Ei and Mi describe the growth, uptake, excretion and mortality rate of the living organisms, respectively. The purpose of the paper is to study the dynamic property of the parabolic system in relation to its corresponding elliptic system. We also investigate the dynamics of the coupled parabolic-ordinary system. Stability analysis of equilibria is given.

References [1] Kme´t, T. Material recycling in a closed aquatic ecosystem. I. Nitrogen transformation cycle and preferential utilization of ammonium to nitrate by phytoplankton as an optimal control problem. Bull. Math. Biol. , 58:957 – 982, 1996. [2] Pao, C.V. Dynamic of Nonlinear Parabolic Systems with Time Delays. J. Math. Anal. Applic. , 198:751 – 779, 1996. [3] Pao, C.V. Numerical Methods for Systems of Nonlinear Parabolic Equations with Time Delays. J. Math. Anal. Applic. , 240:249 – 279, 1996. [4] Wu, J. Theory and Applications of Partial Functional Diferential Equations. Spinger, Applied Math. Sciences , 119, 1996.

∂x8 ∂ 2 yi = dx8 2 + x2 (p, t − re2 )E2 (x(p, t − re2 )) − x3 U3 (x) ∂t ∂p 1

Department of Computer Science, Faculty of Natural Scienses, Constantine the Philosopher University in Nitra, Tr. A. Hlinku 1, 949 74 Nitra, Slovak Republic (email: [email protected]).

13-Kme-a

AICME II abstracts

13-Kme-b