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Identification of the initial population of a nonlinear predator-prey system backwards in time. (English) Zbl 1447.35391

Summary: We study for the first time the ill-posed backward problem for a contaminated nonlinear predator-prey system whose velocities of migration depend on the total average populations in the considered space domain. We propose a new regularized problem for which we are able to prove its unique solvability in Theorem 1. Moreover, under some mild assumptions on the true solution, we give useful and rigorous error estimates and convergence rates in both the \(L^2\)- and \(H^1\)-norms in Theorems 2 and 3, respectively. Furthermore, numerical simulations are performed to illustrate the accuracy and stability of the regularized solution.

MSC:

35R30 Inverse problems for PDEs
35R25 Ill-posed problems for PDEs
35K57 Reaction-diffusion equations
35K51 Initial-boundary value problems for second-order parabolic systems
35R09 Integro-partial differential equations
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[1] Almeida, R. M.P.; Antonsev, S. N.; Duque, J. C.M.; Ferreira, J., A reaction-diffusion for the nonlocal coupled system: existence, uniqueness, long-time behaviour and localization properties of solutions, IMA J. Appl. Math., 81, 344-364 (2016) · Zbl 1335.35111
[2] Anaya, V.; Bendahmane, M.; Sepúlveda, M., Mathematical and numerical analysis for predator-prey system in a polluted environment, Netw. Heterog. Media, 5, 813-847 (2010) · Zbl 1262.35123
[3] Burger, R.; Baier, R.; Tian, C., Stability analysis and finite volume element discretization for delay-driven spatio-temporal patterns in a predator-prey model, Math. Comput. Simulation, 132, 28-52 (2017) · Zbl 1519.92185
[4] Camliyurt, G.; Kukavica, I.; Vicol, V., Gevrey regularity for the Navier-Stokes equations in a half-space, J. Differential Equations, 265, 4052-4075 (2018) · Zbl 1397.35049
[5] Cao, C.; Rammaha, M. A.; Titi, E. S., Gevrey regularity for nonlinear analytic parabolic equations on the sphere, J. Dynam. Differential Equations, 12, 411-433 (2000) · Zbl 0967.35063
[6] Chipot, M.; Lovat, B., Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal.: Theory Methods Appl., 30, 4619-4627 (1997) · Zbl 0894.35119
[7] Chipot, M.; Lovat, B., On the asymptotic behaviour of some nonlocal problems, Positivity, 3, 6-81 (1999) · Zbl 0921.35071
[8] Chipot, M.; Lovat, B., Existence and uniqueness results for a class of nonlocal elliptic and parabolic problems. Advances in quenching, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 8, 35-51 (2001) · Zbl 0984.35066
[9] Chipot, M.; Molinet, L., Asymptotic behaviour of some nonlocal diffusion problems, Appl. Anal., 80, 279-315 (2001) · Zbl 1023.35016
[10] Coayla-Teran, E. A.; Ferreira, J.; de Magalhaes, P. M.D.; de Oliveira, H. B., On a stochastic coupled system of reaction-diffusion of nonlocal type, (Bernardin, C.; Goncalves, P., From Particle Systems to Partial Differential Equations. From Particle Systems to Partial Differential Equations, Springer Proceedings in Mathematics & Statistics, vol. 75 (2014), Springer-Verlag: Springer-Verlag Berlin), 301-320 · Zbl 1323.35227
[11] Dubey, B.; Hussain, J., Modeling the interaction of two biological species in a polluted environment, J. Math. Anal. Appl., 246, 58-79 (2000) · Zbl 0952.92030
[12] Ferarri, A. B.; Titi, E. S., Gevrey regularity for nonlinear analytic parabolic equations, Comm. Partial Differential Equations, 23, 1-16 (1998) · Zbl 0907.35061
[13] Ferreira, J.; Oliveira, H. B., Parabolic reaction-diffusion systems with nonlocal coupled diffusivity terms, Discrete Contin. Dyn. Syst. Ser. A, 37, 2431-2453 (2017) · Zbl 1357.35186
[14] Freedman, H. I.; Shukla, J. B., Models for the effects of toxicant in single-species and predator-prey systems, J. Math. Biol., 30, 15-30 (1991) · Zbl 0825.92125
[15] Hallam, T. G.; Clark, C. E.; Lassider, R. R., Effects of toxicants on populations: a qualitative approach I. Equilibrium environment exposured, Ecol. Model., 18, 291-304 (1983)
[16] 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
[17] Hallam, T. G.; Luna, J. T., Effects of toxicants on populations: a qualitative approach III. Environment and food chains pathways, J. Theoret. Biol., 109, 11-29 (1984)
[18] Ivanchov, M., A nonlocal inverse problem for the diffusion equation, Visnyk Lviv Univ. Ser. Mech. Math., 77, 103-108 (2012) · Zbl 1289.35333
[19] Lions, J. L., Quelques méthodes de résolution des problèmes aux limites nonlinéaires (1969), Dunod, Gauthier-Villars: Dunod, Gauthier-Villars Paris · Zbl 0189.40603
[20] Shangerganesh, L.; Barani, N. B.; Balachandran, K., Weak-renormalized solutions for predator-prey system, Appl. Anal., 92, 441-459 (2013) · Zbl 1263.35135
[21] Shukla, J. B.; Dubey, B., Simultaneous effect of two toxicants on biological species: a mathematical model, J. Biol. Systems, 4, 109-130 (1996)
[22] Tuan, N. H.; Au, V. V.; Khoa, V. A.; Lesnic, D., Identification of the population density of a species model with nonlocal diffusion and nonlinear reaction, Inverse Probl., 33, Article 055019 pp. (2017), (40 pp) · Zbl 1515.35357
[23] Au, Vo Van; Can, Nguyen Huu; Tuan, Nguyen Huy; Binh, Tran Thanh, Regularization of a backward problem for Lotka-Volterra competition system, Comput. Math. Appl. (2019), (in press) · Zbl 1442.92122
[24] Yang, X.; Jin, Z.; Xue, Y., Weak average persistence and extinction of a predator-prey system in a polluted environment with impulsive toxicant input, Chaos Solitons Fractals, 31, 726-735 (2007) · Zbl 1133.92032
[25] Zheng, S.; Chipot, M., Asymptotic behavior of solutions to nonlinear parabolic equations with nonlocal terms, Asymptot. Anal., 45, 301-312 (2005) · Zbl 1089.35027
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