# 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 and Continuous Dynamical Systems, 2015, 35 (7) : 3217-3238. doi: 10.3934/dcds.2015.35.3217
##### References:
 [1] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-709. doi: 10.1137/1018114. [2] F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, AMS, Providence, Rhode Island, 2010. doi: 10.1090/surv/165. [3] P. Bates, On some nonlocal evolution equations arising in materials science, Nonlinear Dynamics and Evolution Equations, in Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 48 (2006), 13-52. [4] P. Bates, X. Chen and A. Chmaj, Traveling waves of bistable dynamics on a lattice, SIAM J. Math. Anal., 35 (2003), 520-546. doi: 10.1137/S0036141000374002. [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), 261-281. doi: 10.1007/s00526-005-0308-y. [6] P. Bates and A. Chmaj, A discrete convolution model for phase transitions, Arch. Ration. Mech. Anal., 150 (1999), 281-305. doi: 10.1007/s002050050189. [7] P. Bates and A. Chmaj, An integrodifferential model for phase transitions: Stationary solutions in higher space dimensions, J. Statist. Phys., 95 (1999), 1119-1139. doi: 10.1023/A:1004514803625. [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), 105-136. doi: 10.1007/s002050050037. [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), 428-440. doi: 10.1016/j.jmaa.2006.09.007. [10] H. Berestycki and N. Rodríguez, Non-local reaction-diffusion equations with a barrier, preprint, 2013. [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), 293-318. doi: 10.1017/S030821050001876X. [12] E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equation, J. Math. Pures Appl., 86 (2006), 271-291. doi: 10.1016/j.matpur.2006.04.005. [13] A. Chmaj and X. Ren, Homoclinic solutions of an integral equation: Existence and stability, J. Differential Equations, 155 (1999), 17-43. doi: 10.1006/jdeq.1998.3571. [14] S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer-Verlag, New York, 1982. [15] C. Cortazar, M. Elgueta, J. D. Rossi and N. Wolanski, Boundary fluxes for nonlocal diffusion, J. Differential Equations, 234 (2007), 360-390. doi: 10.1016/j.jde.2006.12.002. [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), 137-156. doi: 10.1007/s00205-007-0062-8. [17] J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Differential Equations, 249 (2010), 2921-2953. doi: 10.1016/j.jde.2010.07.003. [18] J. Coville, Harnack type inequality for positive solution of some integral equation, Ann. Mat. Pura Appl., 191 (2012), 503-528. doi: 10.1007/s10231-011-0193-2. [19] J. Coville, Nonlocal refuge model with a partial control, Discrete Contin. Dyn. Syst., 35 (2015), 1421-1446. doi: 10.3934/dcds.2015.35.1421. [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), 1693-1709. doi: 10.1137/060676854. [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), 179-223. doi: 10.1016/j.anihpc.2012.07.005. [22] J. Coville and L. Dupaigne, On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 727-755. doi: 10.1017/S0308210504000721. [23] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2. [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), 123-132. doi: 10.3934/dcds.2009.25.123. [25] P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, in Trends in Nonlinear Analysis, Springer, Berlin, 2003, 153-191. [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), 295-319. doi: 10.1006/jdeq.1996.0071. [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), 261-289. doi: 10.1007/s002050050130. [28] J. García-Melián and J. D. Rossi, A logistic equation with refuge and nonlocal diffusion, Commun. Pure Appl. Anal., 8 (2009), 2037-2053. doi: 10.3934/cpaa.2009.8.2037. [29] J. García-Melián and J. D. Rossi, Maximum and antimaximum principles for some nonlocal diffusion operators, Nonlinear Anal., 71 (2009), 6116-6121. doi: 10.1016/j.na.2009.06.004. [30] M. Grinfeld, G. Hines, V. Hutson and K. Mischaikow, Non-local dispersal, Differential Integral Equations, 18 (2005), 1299-1320. [31] V. Hutson, S. Martinez, K. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517. doi: 10.1007/s00285-003-0210-1. [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), 1147-1175. doi: 10.1216/RMJ-2008-38-4-1147. [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), 6429-6447. doi: 10.1016/j.jde.2012.03.011. [34] C. Y. Kao, Y. Lou and W. Shen, Random dispersal vs non-local dispersal, Discrete Contin. Dyn. Syst., 26 (2010), 551-596. doi: 10.3934/dcds.2010.26.551. [35] C. Y. Kao, Y. Lou and W. Shen, Evolution of mixed dispersal in periodic environments, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2047-2072. doi: 10.3934/dcdsb.2012.17.2047. [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), 2302-2313. doi: 10.1016/j.nonrwa.2009.07.005. [37] T. Ouyang, On the positive solutions of semilinear equations $\Delta u+\lambda u-hu^p=0$, Trans. Amer. Math. Soc., 331 (1992), 503-527. doi: 10.2307/2154124. [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), 377-392. doi: 10.1007/s00033-007-7005-y. [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), 927-954. doi: 10.1007/s10884-012-9276-z. [40] W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differential Equations, 249 (2010), 747-795. doi: 10.1016/j.jde.2010.04.012. [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), 1681-1696. doi: 10.1090/S0002-9939-2011-11011-6. [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., (). [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), 3501-3509. doi: 10.1016/j.na.2011.02.034. [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), 551-581. doi: 10.1016/j.jde.2011.04.020. [45] K. Taira, Diffusive logistic equations in population dynamics, Adv. Differential Equations, 7 (2002), 237-256. [46] X. Wang, Metastability and stability of patterns in a convolution model for phase transitions, J. Differential Equations, 183 (2002), 434-461. doi: 10.1006/jdeq.2001.4129.

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##### References:
 [1] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-709. doi: 10.1137/1018114. [2] F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, AMS, Providence, Rhode Island, 2010. doi: 10.1090/surv/165. [3] P. Bates, On some nonlocal evolution equations arising in materials science, Nonlinear Dynamics and Evolution Equations, in Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 48 (2006), 13-52. [4] P. Bates, X. Chen and A. Chmaj, Traveling waves of bistable dynamics on a lattice, SIAM J. Math. Anal., 35 (2003), 520-546. doi: 10.1137/S0036141000374002. [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), 261-281. doi: 10.1007/s00526-005-0308-y. [6] P. Bates and A. Chmaj, A discrete convolution model for phase transitions, Arch. Ration. Mech. Anal., 150 (1999), 281-305. doi: 10.1007/s002050050189. [7] P. Bates and A. Chmaj, An integrodifferential model for phase transitions: Stationary solutions in higher space dimensions, J. Statist. Phys., 95 (1999), 1119-1139. doi: 10.1023/A:1004514803625. [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), 105-136. doi: 10.1007/s002050050037. [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), 428-440. doi: 10.1016/j.jmaa.2006.09.007. [10] H. Berestycki and N. Rodríguez, Non-local reaction-diffusion equations with a barrier, preprint, 2013. [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), 293-318. doi: 10.1017/S030821050001876X. [12] E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equation, J. Math. Pures Appl., 86 (2006), 271-291. doi: 10.1016/j.matpur.2006.04.005. [13] A. Chmaj and X. Ren, Homoclinic solutions of an integral equation: Existence and stability, J. Differential Equations, 155 (1999), 17-43. doi: 10.1006/jdeq.1998.3571. [14] S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer-Verlag, New York, 1982. [15] C. Cortazar, M. Elgueta, J. D. Rossi and N. Wolanski, Boundary fluxes for nonlocal diffusion, J. Differential Equations, 234 (2007), 360-390. doi: 10.1016/j.jde.2006.12.002. [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), 137-156. doi: 10.1007/s00205-007-0062-8. [17] J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Differential Equations, 249 (2010), 2921-2953. doi: 10.1016/j.jde.2010.07.003. [18] J. Coville, Harnack type inequality for positive solution of some integral equation, Ann. Mat. Pura Appl., 191 (2012), 503-528. doi: 10.1007/s10231-011-0193-2. [19] J. Coville, Nonlocal refuge model with a partial control, Discrete Contin. Dyn. Syst., 35 (2015), 1421-1446. doi: 10.3934/dcds.2015.35.1421. [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), 1693-1709. doi: 10.1137/060676854. [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), 179-223. doi: 10.1016/j.anihpc.2012.07.005. [22] J. Coville and L. Dupaigne, On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 727-755. doi: 10.1017/S0308210504000721. [23] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2. [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), 123-132. doi: 10.3934/dcds.2009.25.123. [25] P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, in Trends in Nonlinear Analysis, Springer, Berlin, 2003, 153-191. [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), 295-319. doi: 10.1006/jdeq.1996.0071. [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), 261-289. doi: 10.1007/s002050050130. [28] J. García-Melián and J. D. Rossi, A logistic equation with refuge and nonlocal diffusion, Commun. Pure Appl. Anal., 8 (2009), 2037-2053. doi: 10.3934/cpaa.2009.8.2037. [29] J. García-Melián and J. D. Rossi, Maximum and antimaximum principles for some nonlocal diffusion operators, Nonlinear Anal., 71 (2009), 6116-6121. doi: 10.1016/j.na.2009.06.004. [30] M. Grinfeld, G. Hines, V. Hutson and K. Mischaikow, Non-local dispersal, Differential Integral Equations, 18 (2005), 1299-1320. [31] V. Hutson, S. Martinez, K. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517. doi: 10.1007/s00285-003-0210-1. [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), 1147-1175. doi: 10.1216/RMJ-2008-38-4-1147. [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), 6429-6447. doi: 10.1016/j.jde.2012.03.011. [34] C. Y. Kao, Y. Lou and W. Shen, Random dispersal vs non-local dispersal, Discrete Contin. Dyn. Syst., 26 (2010), 551-596. doi: 10.3934/dcds.2010.26.551. [35] C. Y. Kao, Y. Lou and W. Shen, Evolution of mixed dispersal in periodic environments, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2047-2072. doi: 10.3934/dcdsb.2012.17.2047. [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), 2302-2313. doi: 10.1016/j.nonrwa.2009.07.005. [37] T. Ouyang, On the positive solutions of semilinear equations $\Delta u+\lambda u-hu^p=0$, Trans. Amer. Math. Soc., 331 (1992), 503-527. doi: 10.2307/2154124. [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), 377-392. doi: 10.1007/s00033-007-7005-y. [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), 927-954. doi: 10.1007/s10884-012-9276-z. [40] W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differential Equations, 249 (2010), 747-795. doi: 10.1016/j.jde.2010.04.012. [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), 1681-1696. doi: 10.1090/S0002-9939-2011-11011-6. [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., (). [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), 3501-3509. doi: 10.1016/j.na.2011.02.034. [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), 551-581. doi: 10.1016/j.jde.2011.04.020. [45] K. Taira, Diffusive logistic equations in population dynamics, Adv. Differential Equations, 7 (2002), 237-256. [46] X. Wang, Metastability and stability of patterns in a convolution model for phase transitions, J. Differential Equations, 183 (2002), 434-461. doi: 10.1006/jdeq.2001.4129.
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