Optimal control for an age-dependent predator-prey system in a polluted environment. (English) Zbl 1297.35251
Summary: In this paper, we investigate an optimal control problem for an age-dependent predator-prey hybrid system in a polluted environment. Namely, we establish a new toxicant-population model with age-dependence in an environment with small toxicant capacity. By using a fixed point theorem, we obtain the existence and uniqueness of nonnegative solution of the toxicant-population model with age-dependence. We employ tangent-normal cone techniques in nonlinear functional analysis to deduct the optimality conditions for the control problem. The existence of optimal control policy is carefully verified by means of Ekeland’s variational principle.
MSC:
| 35Q93 | PDEs in connection with control and optimization |
| 35F25 | Initial value problems for nonlinear first-order PDEs |
| 35M10 | PDEs of mixed type |
| 92B05 | General biology and biomathematics |
| 92D25 | Population dynamics (general) |
| 35A15 | Variational methods applied to PDEs |
| 93C20 | Control/observation systems governed by partial differential equations |
| 92D40 | Ecology |
Keywords:
optimal control; age-dependence; predator-prey; hybrid system; pollution; Ekeland’s variational principleReferences:
| [1] | Hallam, T.G., Clark, C.E., Lassider, R.R.: Effects of toxicants on population: a qualitative approach. I. Equilibrium environmental exposure. Ecol. Model. 8, 291–304 (1983) · doi:10.1016/0304-3800(83)90019-4 |
| [2] | Hallam, T.G., Clark, C.E., Jordan, G.S.: Effects of toxicants on populations: a qualitative approach. II. First order kinetics. J. Math. Biol. 18, 25–37 (1983) · Zbl 0548.92019 · doi:10.1007/BF00275908 |
| [3] | Hallam, T.G., de Luna, J.L.: Effects of toxicants on populations: a qualitative approach. III. Environmental and food chain pathways. J. Theor. Biol. 109, 411–429 (1984) · doi:10.1016/S0022-5193(84)80090-9 |
| [4] | Hallam, T.G., Ma, Z.: Persistence in population models with demographic fluctuations. J. Math. Biol. 24, 327–339 (1986) · Zbl 0606.92022 · doi:10.1007/BF00275641 |
| [5] | Ma, Z., Cui, G., Wang, W.: Persistence and extinction of a population in a polluted environment. Math. Biosci. 101, 75–97 (1990) · Zbl 0714.92027 · doi:10.1016/0025-5564(90)90103-6 |
| [6] | Wang, W., Ma, Z.: Permanence of populations in a polluted environment. Math. Biosci. 122(2), 235–248 (1994) · Zbl 0817.92018 · doi:10.1016/0025-5564(94)90060-4 |
| [7] | Ma, Z., Zong, W., Luo, Z.: The threshold of survival for an n-dimensional food chain in a polluted environment. J. Math. Anal. Appl. 210, 440–458 (1997) · Zbl 0905.92030 · doi:10.1006/jmaa.1997.5387 |
| [8] | Pan, J., Jin, Z., Ma, Z.: The threshold of survival for an n-dimensional Volterra mutualistic system in a polluted environment. J. Math. Anal. Appl. 252(2), 519–531 (2000) · Zbl 0965.92030 · doi:10.1006/jmaa.2000.6853 |
| [9] | Dubey, B., Hussain, J.: Modelling the survival of species dependent on a resource in a polluted environment. Nonlinear Anal., Real World Appl. 7(2), 187–210 (2006) · Zbl 1088.92054 · doi:10.1016/j.nonrwa.2005.02.003 |
| [10] | Chen, L., Song, X.: Extinction and permanence of Chemostat model with pulsed input in a polluted environment. Commun. Nonlinear Sci. Numer. Simul. 14(4), 1737–1745 (2009) · Zbl 1221.37209 · doi:10.1016/j.cnsns.2008.01.009 |
| [11] | Liu, M., Wang, K.: Survival analysis of stochastic single-species population models in polluted environment. Ecol. Model. 220(9–10), 1347–1357 (2009) · doi:10.1016/j.ecolmodel.2009.03.001 |
| [12] | Liu, B., Zhang, L.: Dynamics of a two-species Lotka-Volterra competition system in a polluted environment with pulse toxicant input. Appl. Math. Comput. 214(1), 155–162 (2009) · Zbl 1172.92036 · doi:10.1016/j.amc.2009.03.065 |
| [13] | Sinha, S., Misra, O.P., Dhar, J.: Modelling a predator-prey system with infected prey in polluted environment. Appl. Math. Model. 34(7), 1861–1872 (2010) · Zbl 1193.34071 · doi:10.1016/j.apm.2009.10.003 |
| [14] | Chan, W.L., Guo, B.Z.: Optimal birth control of population dynamics. J. Math. Anal. Appl. 144, 532–552 (1989) · Zbl 0708.92017 · doi:10.1016/0022-247X(89)90350-8 |
| [15] | Brokate, M.: Pontryagin’s principle for control problems in age-dependent population dynamics. J. Math. Biol. 23, 75–101 (1985) · Zbl 0599.92017 · doi:10.1007/BF00276559 |
| [16] | Murphy, L.F., Smith, S.J.: Optimal harvesting of an age-structured population. J. Math. Biol. 29, 77–90 (1990) · Zbl 0737.92025 · doi:10.1007/BF00173910 |
| [17] | Clark, C.W.: Mathematical Bioeconomics: the Optimal Management of Renewable Resources, 2 edn. Wiley, New York (1990) · Zbl 0712.90018 |
| [18] | Busoni, G., Matucci, S.: A problem of optimal harvesting policy in two-stage age-dependent population. Math. Biosci. 143, 1–33 (1997) · Zbl 0924.92019 · doi:10.1016/S0025-5564(97)00011-4 |
| [19] | Anita, S.: Optimal harvesting for a nonlinear age-dependent population dynamics. J. Math. Anal. Appl. 226, 6–22 (1998) · Zbl 0980.92027 · doi:10.1006/jmaa.1998.6064 |
| [20] | Anita, S., Iannelli, M., Kim, M.Y., Park, E.J.: Optimal harvesting for periodic age-dependent population dynamics. SIAM J. Appl. Math. 58(5), 1648–1666 (1998) · Zbl 0935.92030 · doi:10.1137/S0036139996301180 |
| [21] | Anita, S.: Analysis and Control of Age-Dependent Population Dynamics. Kluwer Academic, Dordrecht (2000) |
| [22] | Luo, Z., Li, W.-T., Wang, M.: Optimal harvesting control problem for linear periodic age-dependent population dynamics. Appl. Math. Comput. 151, 789–800 (2004) · Zbl 1043.92039 · doi:10.1016/S0096-3003(03)00536-8 |
| [23] | He, Z.-R., Xu, J.-J.: Overtaking optimal harvesting of age-structured predator-prey systems. Int. J. Biomath. 3(3), 277–298 (2010) · Zbl 1403.91268 · doi:10.1142/S1793524510001008 |
| [24] | He, Z.-R., Liu, L.-L., Luo, Z.: Global positive periodic solutions of age-dependent competing systems. Appl. Math. J. Chin. Univ. Ser. A 26(1), 38–46 (2011) · Zbl 1234.92064 · doi:10.1007/s11766-011-2503-2 |
| [25] | Luo, Z., Xing, T., Li, X.: Optimal birth control of free horizon problems for predator-prey system with age-structure. J. Appl. Math. Comput. 34, 19–84 (2010) · Zbl 1203.35286 · doi:10.1007/s12190-009-0302-1 |
| [26] | Barbu, V., Iannelli, M.: Optimal control of population dynamics. J. Optim. Theory Appl. 102, 1–14 (1999) · Zbl 0984.92022 · doi:10.1023/A:1021865709529 |
| [27] | Barbu, V.: Mathematical Methods in Optimization of Differential Systems. Kluwer Academic, Dordrecht (1994) · Zbl 0819.49002 |
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