Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-06T01:08:10.455Z Has data issue: false hasContentIssue false
  • English
  • Français

An Optimal Control Problem for a Predator-PreyReaction-Diffusion System

Published online by Cambridge University Press:  13 September 2010

N. C. Apreutesei
N. C. Apreutesei*
Affiliation:
Department of Mathematics, Technical University "Gh. Asachi" Iasi, 11, Bd. Carol I 700506 Iasi, Romania
*
*Corresponding author: E-mail:[email protected]
Get access

Abstract

An optimal control problem is studied for a predator-prey system of PDE, with a logisticgrowth rate of the prey and a general functional response of the predator. The controlfunction has two components. The purpose is to maximize a mean density of the two speciesin their habitat. The existence of the optimal solution is analyzed and some necessaryoptimality conditions are established. The form of the optimal control is found in someparticular cases.

Keywords

adjoint systemfunctional response of the predatorlogistic growth ratemaximum principleoptimality conditions
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 5 , Issue 6: Ecology (Part 2) , 2010 , pp. 180 - 195
Copyright
© EDP Sciences, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Apreutesei, N.. Necessary optimality conditions for a Lotka-Volterra three species system . Math. Model. Nat. Phen., 1 (2006), 123-135.Google Scholar
Apreutesei, N.. An optimal control problem for prey-predator system with a general functional response . Appl. Math. Letters, 22 (2009), no. 7, 1062-1065. CrossRefGoogle Scholar
V. Barbu. Mathematical methods in optimization of differential systems. Kluwer Academic Publishers, Dordrecht, 1994.
Feichtinger, G. ,Tragler, G., Veliov, V.. Optimality conditions for age-structured control systems . J. Math. Anal. Appl., 288 (2003), no. 1, 47-68. CrossRefGoogle Scholar
Garvie, M., Trenchea, C.. Optimal control of a nutrient-phytoplankton-zooplankton-fish system . SIAM J. Control Optim., 46 (2007), no. 3, 775-791. CrossRefGoogle Scholar
He, Z.,Hong, S., Zhang, C.. Double control problems of age-distributed population dynamics . Nonlinear Anal., Real World Appl., 10 (2009), no. 5, 3112-3121. CrossRefGoogle Scholar
Hrinca, I., An optimal control problem for the Lotka-Volterra system with diffusion . Panam. Math. J., 12 (2002), no. 3, 23-46. Google Scholar
Kato, N.. Maximum principle for optimal harvesting in linear size-structured population . Math. Popul. Stud., 15 (2008), no. 2, 123-136. CrossRefGoogle Scholar
Kuang, Y.. Some mechanistically derived population models . Math. Biosci. Eng., 4 (2007), no. 4, 1-11. Google Scholar
J. D. Murray. Mathematical Biology. Springer Verlag, Berlin-Heidelberg-New York, third edition, 2002.
A. Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, 44. New York etc., Springer- Verlag, 1983.
Xu, S.. Existence of global solutions for a predator-prey model with cross-diffusion . Electron. J. Diff. Eqns., (2008), 1-14. Google Scholar
Yosida, S.. Optimal control of prey-predator systems with Lagrange type and Bolza type cost functionals . Proc. Faculty Science Tokai Univ., 18 (1983), 103-118.Google Scholar