Harvesting renewable resources of population with size structure and diffusion. (English) Zbl 1406.92702

Summary: The aim of this work is to explore the optimal exploitation way for a biological resources model incorporating individual’s size difference and spatial effects. The existence of a unique nonnegative solution to the state system is shown by means of Banach’s fixed point theorem, and the continuous dependence of the population density with the harvesting effort is given. The optimal harvesting strategy is established via normal cone and adjoint system technique. Some conditions are found to assure that there is only one optimal policy.


92D40 Ecology
92D25 Population dynamics (general)
91B76 Environmental economics (natural resource models, harvesting, pollution, etc.)
49N90 Applications of optimal control and differential games
35Q92 PDEs in connection with biology, chemistry and other natural sciences
Full Text: DOI


[1] Skellam, J. G., Random dispersal in theoretical populations, Biometrika, 38, 1-2, 196-218 (1951) · Zbl 0043.14401
[2] Okubo, A.; Levin, S. A., Diffusion and Ecological Problems: Modern Perspectives (2001), New York, NY, USA: Springer, New York, NY, USA · Zbl 1027.92022
[3] Nathan, R.; Kokko, H.; Lpez-Sepulcre, A.; Alerstam, T.; Holland, R. A.; Wikelski, M.; Wilcove, D. S., Migration and Dispersal: Perspectives, Science, Special Section, 313, 5788, 786-795 (2006)
[4] Ebenman, B.; Persson, L., Size-Structured Populations: Ecology and Evolution (1988), Berlin, Germany: Springer, Berlin, Germany · Zbl 0688.92014
[5] Webb, G. F.; Magal, P.; Ruan, S., Population models structured by age, size, and spatial position, Structured Population Models in Biology and Epidemiology. Structured Population Models in Biology and Epidemiology, Lecture Notes in Math, 1-49 (2008), Berlin, Germany: Springer, Berlin, Germany
[6] Aniţa, S., Analysis and Control of Age-Dependent Population Dynamics (2000), Dordrecht, The Netherlands: Kluwer, Dordrecht, The Netherlands · Zbl 0960.92026
[7] Zhao, C.; Wang, M.; Zhao, P., Optimal harvesting problems for age-dependent interacting species with diffusion, Applied Mathematics and Computation, 163, 1, 117-129 (2005) · Zbl 1066.92062
[8] Luo, Z., Optimal harvesting problem for an age-dependent \(n\)-dimensional food chain diffusion model, Applied Mathematics and Computation, 186, 2, 1742-1752 (2007) · Zbl 1111.92061
[9] Botsford, L. W., Optimal fishery policy for size-specific, density-dependent population models, Journal of Mathematical Biology, 12, 3, 265-293 (1981) · Zbl 0469.92010
[10] Hadeler, K. P.; Müller, J., Optimal harvesting and optimal vaccination, Mathematical Biosciences, 206, 2, 249-272 (2007) · Zbl 1114.92066
[11] Kato, N., Maximum principle for optimal harvesting in linear size-structured population, Mathematical Population Studies, 15, 2, 123-136 (2008) · Zbl 1166.92322
[12] Tarniceriu, O. C.; Veliov, V. M.; Lirkov, I.; Margenov, S.; Wasniewski, J., Optimal control of a class of size-structured systems, Large-Scale Scientific Computing. Large-Scale Scientific Computing, Lecture Notes in Computer Science, 4818, 366-373 (2008), Berlin, Germany: Springer, Berlin, Germany · Zbl 1229.49022
[13] Hritonenko, N.; Yatsenko, Y.; Goetz, R.-U.; Xabadia, A., Maximum principle for a size-structured model of forest and carbon sequestration management, Applied Mathematics Letters, 21, 10, 1090-1094 (2008) · Zbl 1152.49020
[14] Gasca-Leyva, E.; Hernandez, J. M.; Veliov, V. M., Optimal harvesting time in a sizeheterogeneous population, Ecological Modelling, 210, 1-2, 161-168 (2008)
[15] Davydov, A. A.; Platov, A. S., Optimization of stationary solution of a model of size-structured population exploitation, Journal of Mathematical Sciences, 176, 6, 860-869 (2011) · Zbl 1290.92026
[16] Kato, N.; Boucekkine; Hritoenko, N.; Yatsenko, Y., Optimal harvesting in a two-species model of size-structured population, Optimal Control of Age-Structured Populations in Economy, Demography, and the Environment, 229-252 (2011), London, UK: Routledge Explorations in Environmental Economics, London, UK
[17] He, Z.-R.; Liu, Y., An optimal birth control problem for a dynamical population model with size-structure, Nonlinear Analysis: Real World Applications, 13, 3, 1369-1378 (2012) · Zbl 1239.49053
[18] Liu, Y.; Cheng, X.-L.; He, Z.-R., On the optimal harvesting of size-structured population dynamics, Applied Mathematics B: A Journal of Chinese Universities, 28, 2, 173-186 (2013) · Zbl 1299.49027
[19] Faugeras, B.; Maury, O., An advection-diffusion-reaction size-structured fish population dynamics model combined with a statistical parameter estimation procedure: application to the Indian Ocean skipjack tuna fishery, Mathematical Biosciences and Engineering, 2, 4, 719-741 (2005) · Zbl 1097.92058
[20] Hadeler, K. P., Structured populations with diffusion in state space, Mathematical Biosciences and Engineering, 7, 1, 37-49 (2010) · Zbl 1184.92039
[21] Barbu, V., Partial Differential Equations and Boundary Value Problems (1998), Dordrecht, The Netherlands: Kluwer, Dordrecht, The Netherlands
[22] Aubin, J.-P.; Ekeland, I., Applied Nonlinear Analysis (1984), New York, NY, USA: John Wiley & Sons, New York, NY, USA
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.