Abstract
This paper deals with entire solutions of a nonlocal dispersal epidemic model. Unlike local (random) dispersal problems, a nonlocal dispersal operator is not compact and the solutions of nonlocal dispersal system studied here lack regularity in suitable spaces, which affects the uniform convergence of the solution sequences and the technique details in constructing the entire solutions. In the monostable case, some new types of entire solutions are constructed by combining leftward and rightward traveling fronts with different speeds and a spatially independent solution. In the bistable case, the existence of many different entire solutions with merging fronts are proved by constructing different sub- and super-solutions. Various qualitative features of the entire solutions are also investigated. A key idea is to characterize the asymptotic behaviors of the traveling wave solutions at infinite in terms of appropriate sub- and super-solutions. Finally, we also obtain the smoothness of the entire solutions in space, i.e., the solutions established in our paper are global Lipschitz continuous in space.
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References
Andreu-Vaillo, F., Mazón, J.M., Rossi, J.D., Toledo-Melero, J.: Nonlocal Diffusion Problems, Mathematical Surveys and Monographs. AMS, Providence (2010)
Bates, P.W.: On some nonlocal evolution equations arising in materials science. In: Brunner, H., Zhao, X.Q., Zou, X. (eds.) Nonlinear Dynamics and Evolution Equations. Fields Institute Communications, vol. 48, pp. 13–52. AMS, Providence (2006)
Bates, P., Fife, P., Ren, X., Wang, X.: Traveling waves in a convolution model for phase transitions. Arch. Ration. Mech. Anal. 138, 105–136 (1997)
Carr, J., Chmaj, A.: Uniqueness of travelling waves for nonlocal monostable equations. Proc. Am. Math. Soc. 132, 2433–2439 (2004)
Capasso, V.: Mathematical Structures of Epidemic Systems, Lecture Notes in Biomath, vol. 97. Springer-Verlag, Heidelberg (1993)
Capasso, V., Kunisch, K.: A reaction-diffusion system arising in modelling man-environment diseases. Q. Appl. Math. 46, 431–450 (1988)
Capasso, V., Maddalena, L.: Convergence to equilibrium states for a reaction-diffusion system modeling the spatial spread of a class of bacterial and viral diseases. J. Math. Biol. 13, 173–184 (1981)
Capasso, V., Maddalena, L.: Saddle point behavior for a reaction-diffusion system: application to a class of epidemic models. Math. Comput. Simul. 24, 540–547 (1982)
Capasso, V., Paveri-Fontana, S.: A mathematical model for the 1973 cholera epidemic in the European Mediterranean region. Revue d’Epidemical. et de Santé Publique. 27, 121–132 (1979)
Coville, J., Dupaigne, L.: On a nonlocal reaction-diffusion eqution arising in population dynamics. Proc. R. Soc. Edinburgh 137A, 727–755 (2007)
Crooks, E.C.M., Tsai, J.C.: Front-like entire solutions for equations with convection. J. Differ. Equ. 253, 1206–1249 (2012)
Chen, X.: Existence, uniqueness and asymptotical stability of travelling fronts in non-local evolution equations. Adv. Differ. Equ. 2, 125–160 (1997)
Chen, X., Guo, J.S.: Existence and uniqueness of entire solutions for a reaction-diffusion equation. J. Differ. Equ. 212, 62–84 (2005)
Chen, X., Guo, J.S., Ninomiya, H.: Entire solutions of reaction-diffusion equations with balanced bistable nonlinearity. Proc. R. Soc. Edinburgh 136A, 1207–1237 (2006)
Ermentrout, B., Mcleod, J.: Existence and uniqueness of traveling waves for a neural network. Proc. R. Soc. Edinburgh 123A, 461–478 (1994)
Fang, J., Zhao, X.Q.: Bistable traveling waves for monotone semiflows with applications. J. Eur. Math. Soc. 7, 173–213 (2005)
Fife, P.C., McLeod, J.B.: The approach of solutions of nonlinear diffusion equations to travelling front solutions. Arch. Ration. Mech. Anal. 65, 335–361 (1977)
Fukao, Y., Morita, Y., Ninomiya, H.: Some entire solutions of the Allen–Cahn equation. Taiwan. J. Math. 8, 15–32 (2004)
Gourley, S.A., Wu, J.: Delayed nonlocal diffusive systems in biological invasion and disease spread. Fields Inst. Commun. 48, 137–200 (2006)
Guo, J.S., Morita, Y.: Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations. Discret. Contin. Dyn. Syst. 12, 193–212 (2005)
Guo, J.S., Wu, C.H.: Entire solutions for a two-component competition system in a lattice. Tohoku Math. J. 62, 17–28 (2010)
Hamel, F., Nadirashvili, N.: Entire solution of the KPP eqution. Commun. Pure Appl. Math. 52, 1255–1276 (1999)
Hamel, F., Nadirashvili, N.: Travelling fronts and entire solutions of the Fisher-KPP equation in \(R^{N}\). Arch. Ration. Mech. Anal. 157, 91–163 (2001)
Kao, C.Y., Lou, Y., Shen, W.: Random dispersal vs non-local dispersal. Discret. Contin. Dyn. Syst. 26, 551–596 (2010)
Lee, C.T., et al.: Non-local concepts in models in biology. J. Theor. Biol. 210, 201–219 (2001)
Li, W.T., Sun, Y.J., Wang, Z.C.: Entire solutions in the Fisher-KPP equation with nonlocal dispersal. Nonlinear Anal. Real World Appl. 11, 2302–2313 (2010)
Li, W.T., Wang, Z.C., Wu, J.: Entire solutions in monostable reaction-diffusion eqautions with delayed nonlinearity. J. Differ. Equ. 245, 102–129 (2008)
Li, W.T., Liu, N.W., Wang, Z.C.: Entire solutions in reaction-advection-diffusion equations in cylinders. J. Math. Pures Appl. 90, 492–504 (2008)
Li, W.T., Zhang, L., Zhang, G.B. : Invasion entire solutions in a competition system with nonlocal dispersal. Discrete Contin. Dyn. Syst. 35, 1531–1560 (2015)
Liu, N.W., Li, W.T., Wang, Z.C.: Entire solutions of reaction-advection-diffusion equations with bistable nonlinearity in cylinders. J. Differ. Equ. 246, 4249–4267 (2009)
Martin, R.H., Smith, H.L.: Abstract functional differential equations and reaction-diffusion equations. Trans. Am. Math. Soc. 321, 1–44 (1990)
Morita, Y., Ninomiya, H.: Entire solutions with merging fronts to reaction-diffusion equations. J. Dyn. Diff. Eqns. 18, 841–861 (2006)
Morita, Y., Tachibana, K.: An entire solution to the Lotka–Volterra competition-diffusion equations. SIAM J. Math. Anal. 40, 2217–2240 (2009)
Murray, J.: Mathematical Biology, 3rd edn. Springer, Berlin, Heidelberg, New York (1993)
Pan, S., Li, W.T., Lin, G.: Travelling wave fronts in nonlocal reaction-diffusion systems and applications. Z. Angew. Math. Phys. 60, 377–392 (2009)
Roquejoffre, J.M.: Eventual monotonicity and convergence to travelling fronts for the solutions of parabolic equations in cylinders. Ann. Inst. H. Poincaré Anal. Non Linéaire 14, 499–552 (1997)
Schumacher, K.: Traveling-front solutions for integro-differential equations. I. J. Reine Angew. Math. 316, 54–70 (1980)
Sun, Y.J., Li, W.T., Wang, Z.C.: Entire solutions in nonlocal dispersal equations with bistable nonlinearity. J. Differ. Equ. 251, 551–581 (2011)
Wang, M., Lv, G.: Entire solutions of a diffusion and competitive Lotka–Volterra type system with nonlocal delayed. Nonlinearity 23, 1609–1630 (2010)
Wang, Z.C., Li, W.T., Ruan, S.: Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity. Trans. Am. Math. Soc. 361, 2047–2084 (2009)
Wang, Z.C., Li, W.T., Wu, J.: Entire solutions in delayed lattice differential equations with monostable nonlinearity. SIAM J. Math. Anal. 40, 2392–2420 (2009)
Weinberger, H.F.: Long-time behavior of a class of biological models. SIAM J. Math. Anal. 13, 353–396 (1982)
Widder, D.V.: The Laplace Transform. Princeton University Press, Princeton (1941)
Wu, S.L.: Entire solutions in a bistable reaction-diffusion system modeling man-environment-man epidemics. Nonlinear Anal. Real World Appl. 13, 1991–2005 (2012)
Wu, S.L., Wang, H.Y.: Front-like entire solutions for monostable reaction-diffusion systems. J. Dyn. Differ. Equ. 25, 505–533 (2013)
Wu, S.L., Sun, Y.J., Liu, S.Y.: Traveling fronts and entire solutions in partially degenerate reaction-diffusion systems with monostable nonlinearity. Discret. Contin. Dyn. Syst. 33, 921–946 (2013)
Xu, D., Zhao, X.Q.: Erratum to Bistable waves in an epidemic model. J. Dyn. Differ. Equ. 17, 219–247 (2005)
Yagisita, H.: Back and global solutions characterizing annihilation dynamics of traveling fronts. Publ. Res. Inst. Math. Sci. 39, 117–164 (2003)
Yu, Z., Yuan, R.: Existence of traveling wave solutions in nonlocal reaction-diffusion systems with delays and applications. ANZIAM. J. 51, 49–66 (2009)
Yu, Z., Yuan, R.: Existence and asymptotics of traveling waves for nonlocal diffusion systems. Chaos Solitons Fractals 45, 1361–1367 (2012)
Zhao, X.Q., Wang, W.: Fisher waves in an epidemic model. Discret. Contin. Dyn. Syst. 4B, 1117–1128 (2004)
Acknowledgments
Wan-Tong Li: Supported by the NSF of China (11271172). Shi-Liang Wu: Supported by the NSF of China (11301407).
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Zhang, L., Li, WT. & Wu, SL. Multi-type Entire Solutions in a Nonlocal Dispersal Epidemic Model. J Dyn Diff Equat 28, 189–224 (2016). https://doi.org/10.1007/s10884-014-9416-8
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DOI: https://doi.org/10.1007/s10884-014-9416-8