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Free boundary problems with nonlocal and local diffusions Ⅱ: Spreading-vanishing and long-time behavior

## Dynamics of positive steady-state solutions of a nonlocal dispersal logistic model with nonlocal terms

 Key Laboratory of Jiangxi Province for Numerical Simulation and Emulation Techniques, College of Mathematics and Computer Science, Gannan Normal University, Ganzhou, Jiangxi 341000, China

* Corresponding author: Li Ma

Received  December 2018 Revised  September 2019 Published  February 2020

Fund Project: The first author is supported by Jiangxi Science and Technology Project (Grant No. GJJ170566) and NSF of China (Grants No. 11801089)

In this paper, we investigate a class of nonlocal dispersal logistic equations with nonlocal terms
 $\left\{ {\begin{array}{*{20}{l}}{{u_t} = Du + {u^q}\left( {\lambda + a(x)\int_\Omega b (x){u^p}} \right),}&{{\rm{\quad\quad in\quad\quad}}\Omega \times (0, + \infty ),}\\{u(x,0) = {u_0}(x) \ge 0}&{{\rm{\quad\quad\quad\quad\quad\quad\quad in\quad\quad}}\Omega ,}\\{u = 0,}&{{\rm{ on\quad\quad}}{{\mathbb{R}}^N}\backslash \Omega \times (0, + \infty ),}\end{array}} \right.$
where
 $\Omega\subset \mathbb{R}^N(N\geq1)$
is a bounded domain,
 $\lambda\in \mathbb{R}$
,
 $0 , $ p>0 $, $ a,b\in C(\overline{\Omega}) $, $ b\geq0 $, $ b\neq0 $and $ a $verifies either $ a>0 $or $ a<0 $. $ Du = \int_\Omega J(x-y)u(y,t){\rm{d}}y-u(x,t) $represents the nonlocal dispersal operators, which is continuous and nonpositive. Under some suitable assumptions we establish the existence, uniqueness or multiplicity and stability of positive stationary solution with nonlocal reaction term by using sub-supersolution methods, Lerray-Schauder degree theory and Lyapunov-Schmidt reduction and so on. Citation: Li Ma, Youquan Luo. Dynamics of positive steady-state solutions of a nonlocal dispersal logistic model with nonlocal terms. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020022 ##### References:  [1] W. Allegretto and A. Barabanova, Existence of positive solutions of semilinear elliptic equations with nonlocal terms, Funkcialaj Ekvacioj, 40 (1997), 395-409. Google Scholar [2] W. Allegretto and P. Nistri, On a class of nonlocal problems with applications to mathematical biology, Differential Equations with Applications to Biology (Halifax, NS, 1997), 1–14, Fields Inst. Commun., 21, Amer. Math. Soc., Providence, RI, 1999. Google Scholar [3] D. Arcoya, J. Carmona and B. Pellacci, Bifurcation for some quasi-linear operators, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 733-765. doi: 10.1017/S0308210500001086. Google Scholar [4] 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. Google Scholar [5] P. Bates and G. Zhao, Existence, uniqueness 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. Google Scholar [6] L. C. Birch, Experimental background to the study of the distribution and aboundance of insects: I. The influence of temperature, moisture and food on the innate capacity for increase of three grain beetles, Ecology, 34 (1953), 698-711. Google Scholar [7] 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. Google Scholar [8] S. S. Chen and J. P. Shi, Stability and hopf bifurcation in a diffusive logistic population model with nonlocal delay effect, J. Differential Equations, 253 (2012), 3440-3470. doi: 10.1016/j.jde.2012.08.031. Google Scholar [9] M. Chipot, Remarks on some class of nonlocal elliptic problems, Recent advances of elliptic and parabolic issues, World Scientific, (2006), 79–102. Google Scholar [10] J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operator, J. Differential Equations, 249 (2010), 2921-2953. doi: 10.1016/j.jde.2010.07.003. Google Scholar [11] F. J. Corrêa, M. Delgado and A. Su$\mathrm{\acute{a}}$rez, Some nonlinear heterogeneous problems with nonlocal reaction term, Adv. in Differential Equations, 16 (2011), 623-641. Google Scholar [12] F. A. Davidson and N. Dodds, Existence of positive solutions due to nonlocal interactions in a class of nonlinear boundary value problems, Methods and Application of Analysis, 14 (2007), 15-27. doi: 10.4310/MAA.2007.v14.n1.a2. Google Scholar [13] B. Fiedler and P. Pol$\mathrm{\acute{a}\check{c}}$ik, Complicated dynamics of scalar reaction diffusion equations with a nonlocal term, Proceedings of The Royal Society of Edinburgh: Section A Mathematics, 115 (1990), 167-192. doi: 10.1017/S0308210500024641. Google Scholar [14] P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, in: Trends in Nonlinear Analysis, Springer, Berlin, 2003,153–191. Google Scholar [15] J. Garc$\mathrm{\acute{i}}$a-Meli$\mathrm{\acute{a}}$n and J. D. Rossi, Maximum and antimaximum principles for some nonlocal diffusion operators, Nonlinear Anal., 71 (2009), 6116-6121. doi: 10.1016/j.na.2009.06.004. Google Scholar [16] J. Garc$\mathrm{\acute{i}}$a-Meli$\mathrm{\acute{a}}$n and J. D. Rossi, A logistic equation with refuge and nonlocal diffusion, Commun. Pure Appl. Anal., 8 (2009), 2037-2053. doi: 10.3934/cpaa.2009.8.2037. Google Scholar [17] J. Garc$\mathrm{\acute{i}}$a-Meli$\mathrm{\acute{a}}$n and J. D. Rossi, On the principal eigenvalue of some nonlocal diffusion problems, J. Differential Equations, 246 (2009), 21-38. doi: 10.1016/j.jde.2008.04.015. Google Scholar [18] S. J. Guo, Stability and bifurcation in a reaction-diffusion model with nonlocal delay effect, J. Differential Equations, 259 (2015), 1409-1448. doi: 10.1016/j.jde.2015.03.006. Google Scholar [19] S. J. Guo and S. J. Yan, Hopf bifurcation in a diffusive Lotka-Volterra type system with nonlocal delay effect, J. Differential Equations, 260 (2016), 781-817. doi: 10.1016/j.jde.2015.09.031. Google Scholar [20] S. J. Guo and L. Ma, Stability and Bifurcation in a Delayed Reaction-Diffusion Equation with Dirichlet Boundary Condition, J. Nonlinear Science, 26 (2016), 545-580. doi: 10.1007/s00332-016-9285-x. Google Scholar [21] G. Hetzer, T. Nguyen and W. Shen, Coexistence and extinction in the Volterra-Lotka competition model with nonlocal dispersal, Commun. Pure Appl. Anal., 11 (2012), 1699-1722. doi: 10.3934/cpaa.2012.11.1699. Google Scholar [22] 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. Google Scholar [23] C. Y. Kao, Y. Lou and W. Shen, Random dispersal vs nonlocal dispersal, Discrete Contin. Dyn. Syst., 26 (2010), 551-596. doi: 10.3934/dcds.2010.26.551. Google Scholar [24] C. Y. Kao, Y. Lou and W. Shen, Evolution of mixed dispersal in periodic environment, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2047-2072. doi: 10.3934/dcdsb.2012.17.2047. Google Scholar [25] 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. Google Scholar [26] Y. Lou and W. M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131. doi: 10.1006/jdeq.1996.0157. Google Scholar [27] Y. Lou, W. M. Ni and S. Yotsutani, On a limiting system in the Lotka-Volterra competition with cross-diffusion, Discrete Contin. Dyn. Syst., 10 (2004), 435-458. doi: 10.3934/dcds.2004.10.435. Google Scholar [28] L. Ma and S. J. Guo, tability and bifurcation in a diffusive Lotka-Volterra system with delay, Comput. Math. Appl., 72 (2016), 147-177. doi: 10.1016/j.camwa.2016.04.049. Google Scholar [29] 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. Google Scholar [30] 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. Google Scholar [31] P. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513. doi: 10.1016/0022-1236(71)90030-9. Google Scholar [32] W. Shen and A. Zhang, Stationary solutions and spreading speeds of nonlocal dispersal in space periodic habitats, Proc. Amer. Math. Soc., 140 (2012), 1681-1696. doi: 10.1090/S0002-9939-2011-11011-6. Google Scholar [33] R. M. Sibly and J. Hone, Population growth rate and its determinants: An overview, Cambridge University Press, 6 (2003), 11-40. doi: 10.1017/CBO9780511615740.002. Google Scholar [34] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, New York, 1983. Google Scholar [35] J. W. Sun, F. Y. Yang and W. T. Li, A nonlocal dispersal equation arising from a selection-migration model in genetics, J. Differential Equations, 257 (2014), 1372-1402. doi: 10.1016/j.jde.2014.05.005. Google Scholar [36] J. W. Sun, W. T. Li and Z. C. Wang, A nonlocal dispersal logistic equation with spatial degeneracy, Discrete Contin. Dyn. Syst., 35 (2015), 3217-3238. doi: 10.3934/dcds.2015.35.3217. Google Scholar [37] L. Sun, J. Shi and Y. Wang, Existence and uniqueness of steady state solutions of a nonlocal diffusive logistic equation, Z. Angew. Math. Phys., 64 (2013), 1267-1278. doi: 10.1007/s00033-012-0286-9. Google Scholar [38] 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. Google Scholar [39] 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. Google Scholar [40] E. Zeidler, Applied Functional Analysis, Main Principle and Their Applications, Appl. Math. Sci. 108, Springer-Verlag, New York, 1995. Google Scholar show all references ##### References:  [1] W. Allegretto and A. Barabanova, Existence of positive solutions of semilinear elliptic equations with nonlocal terms, Funkcialaj Ekvacioj, 40 (1997), 395-409. Google Scholar [2] W. Allegretto and P. Nistri, On a class of nonlocal problems with applications to mathematical biology, Differential Equations with Applications to Biology (Halifax, NS, 1997), 1–14, Fields Inst. Commun., 21, Amer. Math. Soc., Providence, RI, 1999. Google Scholar [3] D. Arcoya, J. Carmona and B. Pellacci, Bifurcation for some quasi-linear operators, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 733-765. doi: 10.1017/S0308210500001086. Google Scholar [4] 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. Google Scholar [5] P. Bates and G. Zhao, Existence, uniqueness 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. Google Scholar [6] L. C. Birch, Experimental background to the study of the distribution and aboundance of insects: I. The influence of temperature, moisture and food on the innate capacity for increase of three grain beetles, Ecology, 34 (1953), 698-711. Google Scholar [7] 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. Google Scholar [8] S. S. Chen and J. P. Shi, Stability and hopf bifurcation in a diffusive logistic population model with nonlocal delay effect, J. Differential Equations, 253 (2012), 3440-3470. doi: 10.1016/j.jde.2012.08.031. Google Scholar [9] M. Chipot, Remarks on some class of nonlocal elliptic problems, Recent advances of elliptic and parabolic issues, World Scientific, (2006), 79–102. Google Scholar [10] J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operator, J. Differential Equations, 249 (2010), 2921-2953. doi: 10.1016/j.jde.2010.07.003. Google Scholar [11] F. J. Corrêa, M. Delgado and A. Su$\mathrm{\acute{a}}$rez, Some nonlinear heterogeneous problems with nonlocal reaction term, Adv. in Differential Equations, 16 (2011), 623-641. Google Scholar [12] F. A. Davidson and N. Dodds, Existence of positive solutions due to nonlocal interactions in a class of nonlinear boundary value problems, Methods and Application of Analysis, 14 (2007), 15-27. doi: 10.4310/MAA.2007.v14.n1.a2. Google Scholar [13] B. Fiedler and P. Pol$\mathrm{\acute{a}\check{c}}$ik, Complicated dynamics of scalar reaction diffusion equations with a nonlocal term, Proceedings of The Royal Society of Edinburgh: Section A Mathematics, 115 (1990), 167-192. doi: 10.1017/S0308210500024641. Google Scholar [14] P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, in: Trends in Nonlinear Analysis, Springer, Berlin, 2003,153–191. Google Scholar [15] J. Garc$\mathrm{\acute{i}}$a-Meli$\mathrm{\acute{a}}$n and J. D. Rossi, Maximum and antimaximum principles for some nonlocal diffusion operators, Nonlinear Anal., 71 (2009), 6116-6121. doi: 10.1016/j.na.2009.06.004. Google Scholar [16] J. Garc$\mathrm{\acute{i}}$a-Meli$\mathrm{\acute{a}}$n and J. D. Rossi, A logistic equation with refuge and nonlocal diffusion, Commun. Pure Appl. Anal., 8 (2009), 2037-2053. doi: 10.3934/cpaa.2009.8.2037. Google Scholar [17] J. Garc$\mathrm{\acute{i}}$a-Meli$\mathrm{\acute{a}}\$n and J. D. Rossi, On the principal eigenvalue of some nonlocal diffusion problems, J. Differential Equations, 246 (2009), 21-38.  doi: 10.1016/j.jde.2008.04.015.  Google Scholar [18] S. J. Guo, Stability and bifurcation in a reaction-diffusion model with nonlocal delay effect, J. Differential Equations, 259 (2015), 1409-1448.  doi: 10.1016/j.jde.2015.03.006.  Google Scholar [19] S. J. Guo and S. J. Yan, Hopf bifurcation in a diffusive Lotka-Volterra type system with nonlocal delay effect, J. Differential Equations, 260 (2016), 781-817.  doi: 10.1016/j.jde.2015.09.031.  Google Scholar [20] S. J. Guo and L. Ma, Stability and Bifurcation in a Delayed Reaction-Diffusion Equation with Dirichlet Boundary Condition, J. Nonlinear Science, 26 (2016), 545-580.  doi: 10.1007/s00332-016-9285-x.  Google Scholar [21] G. Hetzer, T. Nguyen and W. Shen, Coexistence and extinction in the Volterra-Lotka competition model with nonlocal dispersal, Commun. Pure Appl. Anal., 11 (2012), 1699-1722.  doi: 10.3934/cpaa.2012.11.1699.  Google Scholar [22] 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.  Google Scholar [23] C. Y. Kao, Y. Lou and W. Shen, Random dispersal vs nonlocal dispersal, Discrete Contin. Dyn. Syst., 26 (2010), 551-596.  doi: 10.3934/dcds.2010.26.551.  Google Scholar [24] C. Y. Kao, Y. Lou and W. Shen, Evolution of mixed dispersal in periodic environment, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2047-2072.  doi: 10.3934/dcdsb.2012.17.2047.  Google Scholar [25] 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.  Google Scholar [26] Y. Lou and W. M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.  doi: 10.1006/jdeq.1996.0157.  Google Scholar [27] Y. Lou, W. M. Ni and S. Yotsutani, On a limiting system in the Lotka-Volterra competition with cross-diffusion, Discrete Contin. Dyn. Syst., 10 (2004), 435-458.  doi: 10.3934/dcds.2004.10.435.  Google Scholar [28] L. Ma and S. J. Guo, tability and bifurcation in a diffusive Lotka-Volterra system with delay, Comput. Math. Appl., 72 (2016), 147-177.  doi: 10.1016/j.camwa.2016.04.049.  Google Scholar [29] 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.  Google Scholar [30] 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.  Google Scholar [31] P. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9.  Google Scholar [32] W. Shen and A. Zhang, Stationary solutions and spreading speeds of nonlocal dispersal in space periodic habitats, Proc. Amer. Math. Soc., 140 (2012), 1681-1696.  doi: 10.1090/S0002-9939-2011-11011-6.  Google Scholar [33] R. M. Sibly and J. Hone, Population growth rate and its determinants: An overview, Cambridge University Press, 6 (2003), 11-40.  doi: 10.1017/CBO9780511615740.002.  Google Scholar [34] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, New York, 1983.  Google Scholar [35] J. W. Sun, F. Y. Yang and W. T. Li, A nonlocal dispersal equation arising from a selection-migration model in genetics, J. Differential Equations, 257 (2014), 1372-1402.  doi: 10.1016/j.jde.2014.05.005.  Google Scholar [36] J. W. Sun, W. T. Li and Z. C. Wang, A nonlocal dispersal logistic equation with spatial degeneracy, Discrete Contin. Dyn. Syst., 35 (2015), 3217-3238.  doi: 10.3934/dcds.2015.35.3217.  Google Scholar [37] L. Sun, J. Shi and Y. Wang, Existence and uniqueness of steady state solutions of a nonlocal diffusive logistic equation, Z. Angew. Math. Phys., 64 (2013), 1267-1278.  doi: 10.1007/s00033-012-0286-9.  Google Scholar [38] 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.  Google Scholar [39] 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.  Google Scholar [40] E. Zeidler, Applied Functional Analysis, Main Principle and Their Applications, Appl. Math. Sci. 108, Springer-Verlag, New York, 1995.  Google Scholar
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