We continue studying systems whose state depends on time and whose resources are renewably based on functional operators with shift. In previous articles, we considered the term which described results of reproductive processes as a linear expression or as a shift summand. In this work, the reproductive term is represented using an integral with a degenerate kernel. A cyclic model of evolution of the system with a renewable resource is developed. We propose a method for solving the balance equation and we determine an equilibrium state of the system. Having applied this model, we can investigate problems of natural systems and their resource production.
Cite this paper
Karelin, O. , Tarasenko, A. and Gonzalez-Hernandez, M. (2013) Application of Functional Operators with Shift to the Study of Renewable Systems When the Reproductive Process Is Described by Integrals with Degenerate Kernels. Applied Mathematics
, 1376-1380. doi: 10.4236/am.2013.410186
 W. Clark, “Mathematical Bioeconomics: The Optimal Management of Renewable Resources,” 2nd Edition, Wiley, New York, 1990.
 Y. N. Xiao, D. Z. Cheng and H. S. Qin, “Optimal Impul sive Control in Periodic Ecosystem,” Systems & Control Letters, Vol. 55, No. 7, 2006, pp. 558-565.
 C. Castilho and P. Srinivasu, “Bio-Economics of a Re newable Resource in a Seasonally Varying Environment,” Mathematical Biosciences, Vol. 205, No. 1, 2007, pp. 1-18. http://dx.doi.org/10.1016/
 Yu. Karlovich and V. Kravchenko, “Singular Integral Equations with Non-Carleman Shift on an Open Con tour,” Differential Equations, Vol. 17, No. 12, 1981, pp. 1408-1417.
 V. G. Kravchenko and G. S. Litvinchuk, “Introduction to the Theory of Singular Integral Operators with Shift,” Kluwer Academic Publishers, Dordrecht, Boston, London, 1994. http://dx.doi.org/10.1007/978-94-011-1180-5
 A. B. Antonevich, “Linear Functional Equations. Opera tor Approach,” Birkhauser, Basel, 1996.
 A. Tarasenko, A. Karelin, G. P. Lechuga and M. G. Her nandez, “Modelling Systems with Renewable Resources Based on Functional Operators with Shift,” Applied Mathematics and Computation, Vol. 216, No. 7, 2010, pp. 1938-1944.
 A. J. Jerri, “Introduction to Integral Equations with Ap plication,” 2nd Edition, John Wiley & Sons Ltd., Hobo ken, 1999.
 K. E. Atkinson, “The Numerical Solution of Integral Equations of the Second Kind,” Cambridge University Press, Cambridge, 1997.