# American Institute of Mathematical Sciences

July  2015, 35(7): 3217-3238. doi: 10.3934/dcds.2015.35.3217

## A nonlocal dispersal logistic equation with spatial degeneracy

 1 School of Mathematics and Statistics, Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, Gansu 730000, China, China, China

Received  April 2014 Revised  November 2014 Published  January 2015

In this paper, we study the nonlocal dispersal Logistic equation \begin{equation*} \begin{cases} u_t=Du+\lambda m(x)u-c(x)u^p &\text{ in }{\Omega}\times(0,+\infty),\\ u(x,0)=u_0(x)\geq0&\text{ in }{\Omega}, \end{cases} \end{equation*} where $\Omega\subset\mathbb{R}^N$ is a bounded domain, $\lambda>0$ and $p>1$ are constants. $Du(x,t)=\int_{\Omega}J(x-y)(u(y,t)-u(x,t))dy$ represents the nonlocal dispersal operator with continuous and nonnegative dispersal kernel $J$, $m\in C(\bar{\Omega})$ and may change sign in $\Omega$. The function $c$ is nonnegative and has a degeneracy in some subdomain of $\Omega$. We establish the existence and uniqueness of positive stationary solution and also consider the effect of degeneracy of $c$ on the long-time behavior of positive solutions. Our results reveal that the necessary condition to guarantee a positive stationary solution and the asymptotic behaviour of solutions are quite different from those of the corresponding reaction-diffusion equation.
Citation: Jian-Wen Sun, Wan-Tong Li, Zhi-Cheng Wang. A nonlocal dispersal logistic equation with spatial degeneracy. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3217-3238. doi: 10.3934/dcds.2015.35.3217
##### References:
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Cosner, Diffusive logistic equations with indefinite weights: Population models in a disrupted environments,, Proc. Roy. Soc. Edinburgh, 112 (1989), 293. doi: 10.1017/S030821050001876X. Google Scholar [12] E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equation,, J. Math. Pures Appl., 86 (2006), 271. doi: 10.1016/j.matpur.2006.04.005. Google Scholar [13] A. Chmaj and X. Ren, Homoclinic solutions of an integral equation: Existence and stability,, J. Differential Equations, 155 (1999), 17. doi: 10.1006/jdeq.1998.3571. Google Scholar [14] S. N. Chow and J. K. Hale, Methods of Bifurcation Theory,, Springer-Verlag, (1982). Google Scholar [15] C. Cortazar, M. Elgueta, J. D. Rossi and N. Wolanski, Boundary fluxes for nonlocal diffusion,, J. Differential Equations, 234 (2007), 360. doi: 10.1016/j.jde.2006.12.002. Google Scholar [16] C. Cortazar, M. Elgueta, J. D. Rossi and N. 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Merino, Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation,, J. Differential Equations, 127 (1996), 295. doi: 10.1006/jdeq.1996.0071. Google Scholar [27] J. García-Melián, R. Gómez-Reñasco, J. López-Gómez and J. C. Sabina de Lis, Pointwise growth and uniqueness of positive solutions for a class of sublinear elliptic problems where bifurcation from infinity occurs,, Arch. Ration. Mech. Anal., 145 (1998), 261. doi: 10.1007/s002050050130. Google Scholar [28] J. García-Melián and J. D. Rossi, A logistic equation with refuge and nonlocal diffusion,, Commun. Pure Appl. Anal., 8 (2009), 2037. doi: 10.3934/cpaa.2009.8.2037. Google Scholar [29] J. García-Melián and J. D. Rossi, Maximum and antimaximum principles for some nonlocal diffusion operators,, Nonlinear Anal., 71 (2009), 6116. doi: 10.1016/j.na.2009.06.004. Google Scholar [30] M. Grinfeld, G. Hines, V. Hutson and K. Mischaikow, Non-local dispersal,, Differential Integral Equations, 18 (2005), 1299. Google Scholar [31] V. Hutson, S. Martinez, K. Mischaikow and G. T. Vickers, The evolution of dispersal,, J. Math. Biol., 47 (2003), 483. doi: 10.1007/s00285-003-0210-1. Google Scholar [32] V. Hutson, W. Shen and G. T. Vickers, Spectral theory for nonlocal dispersal with periodic or almost-periodic time dependence,, Rocky Mountain J. Math., 38 (2008), 1147. doi: 10.1216/RMJ-2008-38-4-1147. Google Scholar [33] L. Ignat, J. D. Rossi and A. San Antolin, Lower and upper bounds for the first eigenvalue of nonlocal diffusion problems in the whole space,, J. Differential Equations, 252 (2012), 6429. doi: 10.1016/j.jde.2012.03.011. Google Scholar [34] C. Y. Kao, Y. Lou and W. Shen, Random dispersal vs non-local dispersal,, Discrete Contin. Dyn. Syst., 26 (2010), 551. doi: 10.3934/dcds.2010.26.551. Google Scholar [35] C. Y. Kao, Y. Lou and W. Shen, Evolution of mixed dispersal in periodic environments,, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2047. doi: 10.3934/dcdsb.2012.17.2047. Google Scholar [36] W. T. Li, Y. J. Sun and Z. C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal,, Nonlinear Anal. Real World Appl., 11 (2010), 2302. doi: 10.1016/j.nonrwa.2009.07.005. Google Scholar [37] T. Ouyang, On the positive solutions of semilinear equations $\Delta u+\lambda u-hu^p=0$,, Trans. Amer. Math. Soc., 331 (1992), 503. doi: 10.2307/2154124. Google Scholar [38] S. Pan, W. T. Li and G. Lin, Travelling wave fronts in nonlocal reaction-diffusion systems and applications,, Z. Angew. Math. Phys., 60 (2009), 377. doi: 10.1007/s00033-007-7005-y. Google Scholar [39] N. Rawal and W. Shen, Criteria for the existence and lower bounds of principal eigenvalues of time periodic nonlocal dispersal operators and applications,, J. Dynam. Differential Equations, 24 (2012), 927. doi: 10.1007/s10884-012-9276-z. Google Scholar [40] W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats,, J. Differential Equations, 249 (2010), 747. doi: 10.1016/j.jde.2010.04.012. Google Scholar [41] W. Shen and A. Zhang, Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats,, Proc. Amer. Math. Soc., 140 (2012), 1681. doi: 10.1090/S0002-9939-2011-11011-6. Google Scholar [42] W. Shen and A. Zhang, Effects of spatial variations and dispersal strategies on the spreading speeds of monostable models in periodic habitats,, Rocky Mountain J. Math., (). Google Scholar [43] J. W. Sun, W. T. Li and F. Y. Yang, Approximate the Fokker-Planck equation by a class of nonlocal dispersal problems,, Nonlinear Anal., 74 (2011), 3501. doi: 10.1016/j.na.2011.02.034. Google Scholar [44] Y. J. Sun, W. T. Li and Z. C. Wang, Entire solutions in nonlocal dispersal equations with bistable nonlinearity,, J. Differential Equations, 251 (2011), 551. doi: 10.1016/j.jde.2011.04.020. Google Scholar [45] K. Taira, Diffusive logistic equations in population dynamics,, Adv. Differential Equations, 7 (2002), 237. Google Scholar [46] X. Wang, Metastability and stability of patterns in a convolution model for phase transitions,, J. Differential Equations, 183 (2002), 434. doi: 10.1006/jdeq.2001.4129. Google Scholar

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##### References:
 [1] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces,, SIAM Rev., 18 (1976), 620. doi: 10.1137/1018114. Google Scholar [2] F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs,, AMS, (2010). doi: 10.1090/surv/165. Google Scholar [3] P. Bates, On some nonlocal evolution equations arising in materials science, Nonlinear Dynamics and Evolution Equations,, in Fields Inst. Commun., 48 (2006), 13. Google Scholar [4] P. Bates, X. Chen and A. Chmaj, Traveling waves of bistable dynamics on a lattice,, SIAM J. Math. Anal., 35 (2003), 520. doi: 10.1137/S0036141000374002. Google Scholar [5] P. Bates, X. Chen and A. Chmaj, Heteroclinic solutions of a van der Waals model with indefinite nonlocal interactions,, Calc. Var. Partial Differential Equations, 24 (2005), 261. doi: 10.1007/s00526-005-0308-y. Google Scholar [6] P. Bates and A. Chmaj, A discrete convolution model for phase transitions,, Arch. Ration. Mech. Anal., 150 (1999), 281. doi: 10.1007/s002050050189. Google Scholar [7] P. Bates and A. Chmaj, An integrodifferential model for phase transitions: Stationary solutions in higher space dimensions,, J. Statist. Phys., 95 (1999), 1119. doi: 10.1023/A:1004514803625. Google Scholar [8] P. Bates, P. Fife, X. Ren and X. Wang, Travelling waves in a convolution model for phase transitions,, Arch. Ration. Mech. Anal., 138 (1997), 105. doi: 10.1007/s002050050037. Google Scholar [9] P. Bates and G. Zhao, Existence, uniquenss, and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal,, J. Math. Anal. Appl., 332 (2007), 428. doi: 10.1016/j.jmaa.2006.09.007. Google Scholar [10] H. Berestycki and N. Rodríguez, Non-local reaction-diffusion equations with a barrier,, preprint, (2013). Google Scholar [11] R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in a disrupted environments,, Proc. Roy. Soc. Edinburgh, 112 (1989), 293. doi: 10.1017/S030821050001876X. Google Scholar [12] E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equation,, J. Math. Pures Appl., 86 (2006), 271. doi: 10.1016/j.matpur.2006.04.005. Google Scholar [13] A. Chmaj and X. Ren, Homoclinic solutions of an integral equation: Existence and stability,, J. Differential Equations, 155 (1999), 17. doi: 10.1006/jdeq.1998.3571. Google Scholar [14] S. N. Chow and J. K. Hale, Methods of Bifurcation Theory,, Springer-Verlag, (1982). Google Scholar [15] C. Cortazar, M. Elgueta, J. D. Rossi and N. Wolanski, Boundary fluxes for nonlocal diffusion,, J. Differential Equations, 234 (2007), 360. doi: 10.1016/j.jde.2006.12.002. Google Scholar [16] C. Cortazar, M. Elgueta, J. D. Rossi and N. Wolanski, How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems,, Arch. Ration. Mech. Anal., 187 (2008), 137. doi: 10.1007/s00205-007-0062-8. Google Scholar [17] J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators,, J. Differential Equations, 249 (2010), 2921. doi: 10.1016/j.jde.2010.07.003. Google Scholar [18] J. Coville, Harnack type inequality for positive solution of some integral equation,, Ann. Mat. Pura Appl., 191 (2012), 503. doi: 10.1007/s10231-011-0193-2. Google Scholar [19] J. Coville, Nonlocal refuge model with a partial control,, Discrete Contin. Dyn. Syst., 35 (2015), 1421. doi: 10.3934/dcds.2015.35.1421. Google Scholar [20] J. Coville, J. Dávila and S. Martínez, Existence and uniqueness of solutions to nonlocal equation with monostable nonlinearity,, SIAM J. Math. Anal., 39 (2008), 1693. doi: 10.1137/060676854. Google Scholar [21] J. Coville, J. Dávila and S. Martínez, Pulsating fronts for nonlocal dispersion and KPP nonlinearity,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 179. doi: 10.1016/j.anihpc.2012.07.005. Google Scholar [22] J. Coville and L. Dupaigne, On a non-local equation arising in population dynamics,, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 727. doi: 10.1017/S0308210504000721. Google Scholar [23] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues,, J. Funct. Anal., 8 (1971), 321. doi: 10.1016/0022-1236(71)90015-2. Google Scholar [24] Y. Du and Y. Yamada, On the long-time limit of positive solutions to the degenerate logistic equation,, Discrete Contin. Dyn. Syst., 25 (2009), 123. doi: 10.3934/dcds.2009.25.123. Google Scholar [25] P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions,, in Trends in Nonlinear Analysis, (2003), 153. Google Scholar [26] J. M. Fraile, P. Koch Medina, J. López-Gómez and S. Merino, Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation,, J. Differential Equations, 127 (1996), 295. doi: 10.1006/jdeq.1996.0071. Google Scholar [27] J. García-Melián, R. Gómez-Reñasco, J. López-Gómez and J. C. Sabina de Lis, Pointwise growth and uniqueness of positive solutions for a class of sublinear elliptic problems where bifurcation from infinity occurs,, Arch. Ration. Mech. Anal., 145 (1998), 261. doi: 10.1007/s002050050130. Google Scholar [28] J. García-Melián and J. D. Rossi, A logistic equation with refuge and nonlocal diffusion,, Commun. Pure Appl. Anal., 8 (2009), 2037. doi: 10.3934/cpaa.2009.8.2037. Google Scholar [29] J. García-Melián and J. D. Rossi, Maximum and antimaximum principles for some nonlocal diffusion operators,, Nonlinear Anal., 71 (2009), 6116. doi: 10.1016/j.na.2009.06.004. Google Scholar [30] M. Grinfeld, G. Hines, V. Hutson and K. Mischaikow, Non-local dispersal,, Differential Integral Equations, 18 (2005), 1299. Google Scholar [31] V. Hutson, S. Martinez, K. Mischaikow and G. T. Vickers, The evolution of dispersal,, J. Math. Biol., 47 (2003), 483. doi: 10.1007/s00285-003-0210-1. Google Scholar [32] V. Hutson, W. Shen and G. T. Vickers, Spectral theory for nonlocal dispersal with periodic or almost-periodic time dependence,, Rocky Mountain J. Math., 38 (2008), 1147. doi: 10.1216/RMJ-2008-38-4-1147. Google Scholar [33] L. Ignat, J. D. Rossi and A. San Antolin, Lower and upper bounds for the first eigenvalue of nonlocal diffusion problems in the whole space,, J. Differential Equations, 252 (2012), 6429. doi: 10.1016/j.jde.2012.03.011. Google Scholar [34] C. Y. Kao, Y. Lou and W. Shen, Random dispersal vs non-local dispersal,, Discrete Contin. Dyn. Syst., 26 (2010), 551. doi: 10.3934/dcds.2010.26.551. Google Scholar [35] C. Y. Kao, Y. Lou and W. Shen, Evolution of mixed dispersal in periodic environments,, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2047. doi: 10.3934/dcdsb.2012.17.2047. Google Scholar [36] W. T. Li, Y. J. Sun and Z. C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal,, Nonlinear Anal. Real World Appl., 11 (2010), 2302. doi: 10.1016/j.nonrwa.2009.07.005. Google Scholar [37] T. Ouyang, On the positive solutions of semilinear equations $\Delta u+\lambda u-hu^p=0$,, Trans. Amer. Math. Soc., 331 (1992), 503. doi: 10.2307/2154124. Google Scholar [38] S. Pan, W. T. Li and G. Lin, Travelling wave fronts in nonlocal reaction-diffusion systems and applications,, Z. Angew. Math. Phys., 60 (2009), 377. doi: 10.1007/s00033-007-7005-y. Google Scholar [39] N. Rawal and W. Shen, Criteria for the existence and lower bounds of principal eigenvalues of time periodic nonlocal dispersal operators and applications,, J. Dynam. Differential Equations, 24 (2012), 927. doi: 10.1007/s10884-012-9276-z. Google Scholar [40] W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats,, J. Differential Equations, 249 (2010), 747. doi: 10.1016/j.jde.2010.04.012. Google Scholar [41] W. Shen and A. Zhang, Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats,, Proc. Amer. Math. Soc., 140 (2012), 1681. doi: 10.1090/S0002-9939-2011-11011-6. Google Scholar [42] W. Shen and A. Zhang, Effects of spatial variations and dispersal strategies on the spreading speeds of monostable models in periodic habitats,, Rocky Mountain J. Math., (). Google Scholar [43] J. W. Sun, W. T. Li and F. Y. Yang, Approximate the Fokker-Planck equation by a class of nonlocal dispersal problems,, Nonlinear Anal., 74 (2011), 3501. doi: 10.1016/j.na.2011.02.034. Google Scholar [44] Y. J. Sun, W. T. Li and Z. C. Wang, Entire solutions in nonlocal dispersal equations with bistable nonlinearity,, J. Differential Equations, 251 (2011), 551. doi: 10.1016/j.jde.2011.04.020. Google Scholar [45] K. Taira, Diffusive logistic equations in population dynamics,, Adv. Differential Equations, 7 (2002), 237. Google Scholar [46] X. Wang, Metastability and stability of patterns in a convolution model for phase transitions,, J. Differential Equations, 183 (2002), 434. doi: 10.1006/jdeq.2001.4129. Google Scholar
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