Reduction to a single closed equation for 2-by-2 reaction-diffusion systems of Lotka-Volterra type. (English) Zbl 1355.35108


35K57 Reaction-diffusion equations
92D25 Population dynamics (general)
35K40 Second-order parabolic systems
Full Text: DOI arXiv


[1] N. Barton, The dynamics of hybrid zone, Heredity, 43 (1979), pp. 341–359.
[2] N. H. Barton and M. Turelli, Spatial waves of advance with bistable dynamics: Cytoplasmic and genetic analogues of Allee effects, Amer. Natural., 178 (2011), pp. E48–E75.
[3] M. H. T. Chan and P. S. Kim, Modeling a Wolbachia invasion using a slow-fast dispersal reaction-diffusion approach, Bull Math Biol, 75 (2013), pp. 1501–1523. · Zbl 1311.92173
[4] H. L. C. Dutra, L. M. Barbosa dos Santos, E. P. Carsagata, J. B. L. Silva, D. A. M. Villela, R. Maciel-de Freitas, and L. A. Moreira, From lab to field: The influence of urban landscapes on the invasive potential of Wolbachia in Brazilian Aedes aegypti mosquitoes, PLoS Neglected Trop. Dis., 9 (2015), 3689.
[5] A. Fenton, K. N. Johnson, J. C. Brownlie, and G. D. D. Hurst, Solving the Wolbachia paradox: Modeling the tripartite interaction between host, Wolbachia, and a natural enemy, Amer. Natural., 178 (2011), pp. 333–342.
[6] P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture notes Biomath. 28, Springer, Berlin, 1979. · Zbl 0403.92004
[7] R. A. Fisher, The advance of advantageous genes, Ann. Eugen., 7 (1937), pp. 355–369. · JFM 63.1111.04
[8] D. A. Focks, D. G. Haile, E. Daniels, and G. A. Mount, Dynamic life table model of a container-inhabiting mosquito, Aedes aegypti (L.) (Diptera: Culicidae). Part 1. Analysis of the literature and model development, J. Med. Entomol., 30 (1993), pp. 1003–1017.
[9] R. A. Gardner, Existence and stability of travelling wave solutions to competition models: A degree theoretic approach, J. Differential Equations, 44 (1982), pp. 343–364. · Zbl 0446.35012
[10] D. Hilhorst, M. Iida, M. Mimura, and H. Ninomiya, Relative compactness in \(L^p\) of solutions of some 2m components competition-diffusion systems, Discrete Contin. Dyn. Syst., 21 (2008), pp. 233–244. · Zbl 1149.35361
[11] D. Hilhorst, S. Martin, and M. Mimura, Singular limit of a competition-diffusion system with large interspecific interaction, J. Math. Anal. Appl., 390 (2012), pp. 488–513. · Zbl 1236.35208
[12] A. Hoffmann, B. Montgomery, J. Popovici, I. Iturbe-Ormaetxe, P. Johnson, F. Muzzi, M. Greenfield, M. Durkan, Y. Leong, and Y. Dong, Successful establishment of Wolbachia in Aedes populations to suppress dengue transmission, Nature, 476 (2011), pp. 454–457.
[13] H. Hughes and N. F. Britton, Modeling the use of Wolbachia to control dengue fever transmission, Bull. Math. Biol., 75 (2013), pp. 796–818. · Zbl 1273.92034
[14] S. Joanne, I. Vythilingam, N. Yugavathy, C. S. Leong, M. Wong, and S. AbuBakar, Distribution and dynamics of Wolbachia infection in Malaysian Aedes albopictus, Acta Trop., 148 (2015), pp. 38–45.
[15] A. Kolmogorov, I. Petrovskii, and N. Piskunov, Etude de l’équation de la chaleur de matière et son application à un problème biologique, Bull. Moskov. Gos. Univ. Mat. Mekh., 1 (1937), pp. 1–25.
[16] L. A. Moreira, I. Iturbe-Ormaetxe, J. A. Jeffery, G. Lu, A. T. Pyke, L. M. Hedges, B. C. Rocha, S. Hall-Mendelin, A. Day, M. Riegler, L. E. Hugo, K. N. Johnson, B. H. Kay, E. A. McGraw, A. F. van den Hurk, P. A. Ryan, and S. L. O’Neill, A Wolbachia symbiont in Aedes aegypti limits infection with dengue, chikungunya, and Plasmodium, Cell, 139 (2009), pp. 1268–1278.
[17] T. Nagylaki, Conditions for existence of clines, Genetics, 80 (1975), pp. 595–615.
[18] M. Otero, N. Schweigmann, and H. G. Solari, A stochastic spatial dynamical model for Aedes aegypti, Bull. Math. Biol., 70 (2008), pp. 1297–325. · Zbl 1142.92028
[19] B. Perthame, Parabolic Equations in Biology, Lect. Notes Math. Model. Life Sci., Springer, Cham, Switzerland, 2015.
[20] J. Schraiber, A. Kaczmarczyk, R. Kwok, M. Park, R. Silverstein, F. Rutaganira, T. Aggarwal, M. Schwemmer, C. Hom, R. Grosberg, and S. Schreiber, Constraints on the use of lifespan-shortening Wolbachia to control dengue fever, J. Theoret. Biol., 297 (2012), pp. 26–32. · Zbl 1336.92085
[21] J. Simon, Compact sets in the space \(L^p (0,T ; B)\)., Ann. Mat. Pura Appl. (4), 146 (1986), pp. 65–96.
[22] A. Volpert, V. Volpert, and V. Volpert, Traveling Wave Solutions of Parabolic Systems, Transl. Math. Monogr. 140, AMS, Providence, RI, 1994. · Zbl 0805.35143
[23] T. Walker, P. H. Johnson, L. A. Moreira, I. Iturbe-Ormaetxe, F. D. Frentiu, C. J. McMeniman, Y. S. Leong, Y. Dong, J. Axford, P. Kriesner, A. L. Lloyd, S. A. Ritchie, S. L. O\'Neill, and A. A. Hoffmann, The wMel Wolbachia strain blocks dengue and invades caged Aedes aegypti populations, Nature, 476 (2011), pp. 450–453.
[24] J. H. Werren, L. Baldo, and M. E. Clark, Wolbachia: master manipulators of invertebrate biology, Nature Rev. Microbiol., 6 (2008), pp. 741–751.
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