zbMATH — the first resource for mathematics

Reduction to a single closed equation for 2-by-2 reaction-diffusion systems of Lotka-Volterra type. (English) Zbl 1355.35108
Summary: We consider general models of coupled reaction-diffusion systems for interacting variants of a species. When the total population becomes large with intensive competition, we prove that the frequency (i.e., proportion) of a variant can be approached by the solution of a single reaction-diffusion equation, through a singular limit method and a relative compactness argument. Applying this result, we retrieve the classical bistable equation for Wolbachia’s spread into an arthropod population from a system modeling interaction between infected and uninfected individuals.
Reviewer: Reviewer (Berlin)

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