Coexistence and optimal control problems for a degenerate predator-prey model. (English) Zbl 1210.49042
Summary: We present a predator-prey mathematical model for two biological populations which dislike crowding. The model consists of a system of two degenerate parabolic equations with nonlocal terms and drifts. We provide conditions on the system ensuring the periodic coexistence, namely the existence of two non-trivial non-negative periodic solutions representing the densities of the two populations. We assume that the predator species is harvested if its density exceeds a given threshold. A minimization problem for a cost functional associated with this process and with some other significant parameters of the model is also considered.
MSC:
| 49N75 | Pursuit and evasion games |
| 91A24 | Positional games (pursuit and evasion, etc.) |
| 35K65 | Degenerate parabolic equations |
References:
| [1] | Martínez, S., The effect of diffusion for the multispecies Lotka-Volterra competition model, Nonlinear Anal. Real World Appl., 4, 409-436 (2003) · Zbl 1015.35039 |
| [2] | Jia, Y.; Wu, J.; Nie, H., The coexistence states of a predator-prey model with nonmonotonic functional response and diffusion, Acta Appl. Math., 108, 413-428 (2009) · Zbl 1180.35087 |
| [3] | Lou, Y.; Martínez, S.; Poláčik, P., Loops and branches of coexistence states in a Lotka-Volterra competition model, J. Differential Equations, 230, 720-742 (2006) · Zbl 1154.35011 |
| [4] | Wang, M.; Pang, P. Y.H., Global asymptotic stability of positive steady states of a diffusive ratio-dependent prey-predator model, Appl. Math. Lett., 21, 1215-1220 (2008) · Zbl 1171.34310 |
| [5] | Hirano, N.; Rybicki, S., Existence of periodic solutions for semilinear reaction diffusion systems, Nonlinear Anal., 59, 931-949 (2004) · Zbl 1082.35020 |
| [6] | Xu, R.; Ma, Z., Global stability of a reaction-diffusion predator-prey model with a nonlocal delay, Math. Comput. Modelling, 50, 194-206 (2009) · Zbl 1185.35130 |
| [7] | Brown, K. J.; Hess, P., Positive periodic solutions of predator-prey reaction-diffusion systems, Nonlinear Anal., 16, 1147-1158 (1991) · Zbl 0743.35030 |
| [8] | Liang, X.; Jiang, J., Discrete infinite-dimensional type-\(K\) monotone dynamical systems and time-periodic reaction-diffusion systems, J. Differential Equations, 189, 318-354 (2003) · Zbl 1036.35091 |
| [9] | Liu, X.; Huang, L., Permanence and periodic solutions for a diffusive ratio-dependent predator-prey system, Appl. Math. Model., 33, 683-691 (2009) · Zbl 1168.35377 |
| [10] | Liu, Y.; Li, Z.; Ye, Q., The existence, uniqueness and stability of positive periodic solution for periodic reaction-diffusion system, Acta Math. Appl. Sin. Engl. Ser., 17, 1-13 (2001) · Zbl 1158.35009 |
| [11] | Pao, C. V., Stability and attractivity of periodic solutions of parabolic systems with time delays, J. Math. Anal. Appl., 304, 423-450 (2005) · Zbl 1063.35020 |
| [12] | Tineo, A.; Rivero, J., Permanence and asymptotic stability for competitive and Lotka-Volterra systems with diffusion, Nonlinear Anal. Real World Appl., 4, 615-624 (2003) · Zbl 1088.35028 |
| [13] | Wang, C., Existence and stability of periodic solutions for parabolic systems with time delays, J. Math. Anal. Appl., 339, 1354-1361 (2008) · Zbl 1130.35075 |
| [14] | Wang, Y., Convergence to periodic solutions in periodic quasimonotone reaction-diffusion systems, J. Math. Anal. Appl., 268, 25-40 (2002) · Zbl 1042.35030 |
| [15] | Gurtin, E.; McCamy, R. C., On the diffusion of biological populations, Math. Biosci., 33, 35-49 (1977) · Zbl 0362.92007 |
| [16] | Gurtin, E.; McCamy, R. C., Diffusion models for age-structured populations, Math. Biosci., 54, 49-59 (1981) · Zbl 0459.92015 |
| [17] | Okubo, A., Diffusion and Ecological Problems: Mathematical Models, Biomathematics, vol. 10 (1980), Springer-Verlag: Springer-Verlag Berlin, New York · Zbl 0422.92025 |
| [18] | Schigesada, M.; Kawasaki, K.; Teramoto, E., Spatial segregation of interacting species, J. Theoret. Biol., 79, 83-99 (1979) |
| [19] | Badii, M., Periodic solutions for a class of degenerate evolution problem, Nonlinear Anal., 44, 499-508 (2001) · Zbl 0984.35085 |
| [20] | Badii, M., Existence and uniqueness of periodic solutions for a model of contaminant flow in porous medium, Rend. Semin. Mat. Univ. Politec. Torino, 61, 1-11 (2003) · Zbl 1098.35088 |
| [21] | Barbu, V.; Favini, A., Periodic problems for degenerate differential equations, Rend. Istit. Mat. Univ. Trieste, 28, 29-57 (1997) · Zbl 0892.35012 |
| [22] | Favini, A.; Marinoschi, G., Periodic behavior for a degenerate fast diffusion equation, J. Math. Anal. Appl., 351, 509-521 (2009) · Zbl 1160.35456 |
| [23] | Giga, Y.; Mizoguchi, N., On time periodic solutions of the Dirichlet problem for degenerate parabolic equations of nondivergence type, J. Math. Anal. Appl., 201, 396-416 (1996) · Zbl 0864.35058 |
| [24] | Hess, P.; Pozio, M. A.; Tesei, A., Time periodic solutions for a class of degenerate parabolic problems, Houston J. Math., 21, 367-394 (1995) · Zbl 0837.35075 |
| [25] | Liu, Z., Periodic solutions for double degenerate quasilinear parabolic equations, Nonlinear Anal., 51, 1245-1257 (2002) · Zbl 1012.35050 |
| [26] | Marinoschi, G., Periodic solutions to fast diffusion equations with nonLipschitz convective terms, Nonlinear Anal. Real World Appl., 10, 1048-1067 (2009) · Zbl 1167.35400 |
| [27] | Mizoguchi, N., Periodic solutions for degenerate diffusion equations, Indiana Univ. Math. J., 44, 413-432 (1995) · Zbl 0842.35049 |
| [28] | Nakao, M., Periodic solutions of some nonlinear degenerate parabolic equations, J. Math. Anal. Appl., 104, 554-567 (1984) · Zbl 0565.35057 |
| [29] | Sun, J.; Wu, B.; Zhang, D., Asymptotic behavior of solutions of a periodic diffusion equation, J. Inequal. Appl., 2010 (2010), article ID 597569 · Zbl 1190.35021 |
| [30] | Wang, Y.; Yin, J.; Wu, Z., Periodic solutions of porous medium equations with weakly nonlinear sources, Northeast. Math. J., 16, 475-483 (2000) · Zbl 1021.35007 |
| [31] | Huang, R.; Wang, Y.; Ke, Y., Existence of non-trivial non-negative periodic solutions for a class of degenerate parabolic equations with nonlocal terms, Discrete Contin. Dyn. Syst. Ser. B, 5, 1005-1014 (2005) · Zbl 1090.35021 |
| [32] | Ke, Y.; Huang, R.; Sun, J., Periodic solutions for a degenerate parabolic equation, Appl. Math. Lett., 22, 910-915 (2009) · Zbl 1171.35416 |
| [33] | Wang, C.; Yin, J.; Wen, M., Periodic optimal control for a degenerate nonlinear diffusion equation, Comput. Math. Model., 17, 364-375 (2006) · Zbl 1132.49028 |
| [34] | Zhou, Q.; Ke, Y.; Wang, Y.; Yin, J., Periodic \(p\)-Laplacian with nonlocal terms, Nonlinear Anal., 66, 442-453 (2007) · Zbl 1110.35039 |
| [35] | Wang, J.; Gao, W., Existence of nontrivial nonnegative periodic solutions for a class of doubly degenerate parabolic equation with nonlocal terms, J. Math. Anal. Appl., 331, 481-498 (2007) · Zbl 1156.35413 |
| [36] | Vazquez, J. L., The Porous Medium Equation. Mathematical Theory, Oxford Math. Monogr. (2007), The Clarendon Press, Oxford University Press: The Clarendon Press, Oxford University Press Oxford · Zbl 1107.35003 |
| [37] | DiBenedetto, E., Degenerate Parabolic Equations (1993), Springer-Verlag: Springer-Verlag New York · Zbl 0794.35090 |
| [38] | Allegretto, W.; Nistri, P., Existence and optimal control for periodic parabolic equations with nonlocal term, IMA J. Math. Control Inform., 16, 43-58 (1999) · Zbl 0926.49002 |
| [39] | He, F.; Leung, A.; Stojanovic, S., Periodic optimal control for competing parabolic Volterra-Lotka-type systems. Oscillations in nonlinear systems: applications and numerical aspects, J. Comput. Appl. Math., 52, 199-217 (1994) · Zbl 0811.49005 |
| [40] | He, F.; Leung, A.; Stojanovic, S., Periodic optimal control for parabolic Volterra-Lotka type equations, Math. Methods Appl. Sci., 18, 127-146 (1995) · Zbl 0818.49002 |
| [41] | G. Fragnelli, P. Nistri, D. Papini, Positive periodic solutions and optimal control for a distributed biological model of two interacting species, Nonlinear Anal. Real World Appl., doi:10.1016/j.nonrwa.2010.10.002; G. Fragnelli, P. Nistri, D. Papini, Positive periodic solutions and optimal control for a distributed biological model of two interacting species, Nonlinear Anal. Real World Appl., doi:10.1016/j.nonrwa.2010.10.002 · Zbl 1215.35020 |
| [42] | Pao, C. V.; Ruan, W. H., Positive solutions of quasilinear parabolic systems with nonlinear boundary conditions, J. Math. Anal. Appl., 333, 472-499 (2007) · Zbl 1120.35049 |
| [43] | Pao, C. V.; Ruan, W. H., Positive solutions of quasilinear parabolic systems with Dirichlet boundary condition, J. Differential Equations, 248, 1175-1211 (2010) · Zbl 1188.35100 |
| [44] | Murray, J. D., Mathematical Biology, Biomathematics, vol. 19 (1993), Springer-Verlag: Springer-Verlag Berlin · Zbl 0779.92001 |
| [45] | Nanbu, T., Some degenerate nonlinear parabolic equations, Math. Rep. Kyushu Univ., 14, 91-110 (1984) · Zbl 0575.35044 |
| [46] | Ladyzenskaja, O.; Solonnikov, V.; Uraltseva, N., Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monogr., vol. 23 (1967), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0164.12302 |
| [47] | Lieberman, G. M., Second Order Parabolic Differential Equations (1996), World Scientific Publishing Co., Inc.: World Scientific Publishing Co., Inc. River Edge, NJ · Zbl 0884.35001 |
| [48] | Porzio, M. M.; Vespri, V., Hölder estimates for local solution of some double degenerate parabolic equation, J. Differential Equations, 103, 146-178 (1993) · Zbl 0796.35089 |
| [49] | DiBenedetto, E.; Gianazza, U.; Vespri, V., Harnack estimates for quasi-linear degenerate parabolic differential equations, Acta Math., 200, 181-209 (2008) · Zbl 1221.35213 |
| [50] | Fornaro, S.; Sosio, M., Intrinsic Harnack estimates for some Doubly nonlinear degenerate parabolic equations, Adv. Differential Equations, 13, 139-168 (2008) · Zbl 1160.35039 |
| [51] | F. Ragnedda, S. Vernier-Piro, V. Vespri, Large time behaviour of solutions to a class of nonautonomous degenerate parabolic equations, preprint, Dip. Matematica e Inf., Università di Cagliari, 2008.; F. Ragnedda, S. Vernier-Piro, V. Vespri, Large time behaviour of solutions to a class of nonautonomous degenerate parabolic equations, preprint, Dip. Matematica e Inf., Università di Cagliari, 2008. · Zbl 1204.35049 |
| [52] | Ragnedda, F.; Vernier-Piro, S.; Vespri, V., Asymptotic time behaviour for non-autonomous degenerate parabolic problems with forcing term, Nonlinear Anal., 71, e2316-e2321 (2009) · Zbl 1239.35080 |
| [53] | Rockafellar, R. T.; Wets, R. J.-B., Variational Analysis, Grundlehren Math. Wiss., vol. 317 (1998), Springer-Verlag: Springer-Verlag Berlin · Zbl 0888.49001 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
