doi: 10.3934/dcdsb.2020199

Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media

1. 

School of Mathematics, Hunan University, Changsha, Hunan 410082, China

2. 

Department of Mathematics and Statistics, Auburn University, AL 36849, USA

3. 

School of Mathematics and Physics, China University of Geosciences, Wuhan, Hubei 430074, China

* Corresponding author: Jianping Gao

Received  December 2019 Revised  April 2020 Published  June 2020

In this paper, we investigate the asymptotic dynamics of Fisher-KPP equations with nonlocal dispersal operator and nonlocal reaction term in time periodic and space heterogeneous media. We first show the global existence and boundedness of nonnegative solutions. Next, we obtain some sufficient conditions ensuring the uniform persistence. Finally, we study the existence, uniqueness and global stability of positive time periodic solutions under several different conditions.

Citation: Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020199
References:
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M. Alfaro and J. Coville, Rapid traveling waves in the nonlocal Fisher equation connect two unstable states, Appl. Math. Lett., 25 (2012), 2095-2099.  doi: 10.1016/j.aml.2012.05.006.  Google Scholar

[2]

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N. ApreuteseiA. Ducrot and V. Volpert, Travelling waves for integro-differential equations in population dynamics, Discrete Contin. Dyn. Syst.-Ser. B, 11 (2009), 541-561.  doi: 10.3934/dcdsb.2009.11.541.  Google Scholar

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H. BerestyckiG. NadinB. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: Travelling waves and steady states, Nonlinearity, 22 (2009), 2813-2844.  doi: 10.1088/0951-7715/22/12/002.  Google Scholar

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H. BerestyckiJ. Coville and H.-H. Vo, Persistence criteria for populations with non-local dispersion, J. Math. Biol., 72 (2016), 1693-1745.  doi: 10.1007/s00285-015-0911-2.  Google Scholar

[7]

H. BerestyckiF. Hamel and L. Roques, Analysis of the periodically fragmented environment model. I. Species persistence, J. Math. Biol., 51 (2005), 75-113.  doi: 10.1007/s00285-004-0313-3.  Google Scholar

[8]

H. BerestyckiF. Hamel and L. Roques, Analysis of the periodically fragmented environment model. II. Biological invasions and pulsating travelling fronts, J. Math. Pures Appl., 84 (2005), 1101-1146.  doi: 10.1016/j.matpur.2004.10.006.  Google Scholar

[9]

S. BianL. Chen and E. A. Latos, Global existence and asymptotic behavior of solutions to a nonlocal Fisher-KPP type problem, Nonlinear Anal., 149 (2017), 165-176.  doi: 10.1016/j.na.2016.10.017.  Google Scholar

[10]

J. Billingham, Dynamics of a strongly nonlocal reaction-diffusion population model, Nonlinearity, 17 (2003), 313-346.  doi: 10.1088/0951-7715/17/1/018.  Google Scholar

[11]

L. CaffarelliS. Dipierro and E. Valdinoci, A logistic equation with nonlocal interactions, Kinet. Relat. Models, 10 (2017), 141-170.  doi: 10.3934/krm.2017006.  Google Scholar

[12]

L. CaffarelliS. Patrizi and V. Quitalo, On a long range segregation model, J. Eur. Math. Soc. (JEMS), 19 (2017), 3575-3628.  doi: 10.4171/JEMS/747.  Google Scholar

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F. CorrêaM. Delgado and A. Suárez, Some nonlinear heterogeneous problems with nonlocal reaction term, Adv. Differ. Equat., 16 (2011), 623-641.   Google Scholar

[14]

J. CovilleJ. Dávila and S. Martínez, Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity, SIAM J. Math. Anal., 39 (2008), 1693-1709.  doi: 10.1137/060676854.  Google Scholar

[15]

J. Coville and L. Dupaigne, Propagation speed of travelling fronts in non local reaction-diffusion equations, Nonlinear Anal., 60 (2005), 797-819.  doi: 10.1016/j.na.2003.10.030.  Google Scholar

[16]

M. DelgadoG. M. FigueiredoM. T. O. Pimenta and A. Suárez, Study of a logistic equation with local and non-local reaction terms, Topol. Methods Nonlinear Anal., 47 (2016), 693-713.  doi: 10.12775/TMNA.2016.026.  Google Scholar

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K. Deng, On a nonlocal reaction-diffusion population model, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 65-73.  doi: 10.3934/dcdsb.2008.9.65.  Google Scholar

[18]

K. Deng and Y. X. Wu, Global stability for a nonlocal reaction–diffusion population model, Nonlinear Anal.-Real World Appl., 25 (2015), 127-136.  doi: 10.1016/j.nonrwa.2015.03.006.  Google Scholar

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D. Finkelshtein, Y. Kondratiev and P. Tkachov, Traveling waves and long-time behavior in a doubly nonlocal Fisher-KPP equation, J. Math. Anal. Appl., 475 (2019), 94–122, arXiv: 1508.02215v2. doi: 10.1016/j.jmaa.2019.02.010.  Google Scholar

[20]

D. Finkelshtein, Y. Kondratiev, S. Molchanov and P. Tkachov, Global stability in a nonlocal reaction-diffusion equation, Stoch. Dyn., 18 (2018), 1850037, 15 pp. doi: 10.1142/S0219493718500375.  Google Scholar

[21]

D. Finkelshtein, Y. Kondratiev and P. Tkachov, Doubly nonlocal Fisher-KPP equation: Front propagation, Appl. Anal., (2019), 1–24. Google Scholar

[22]

D. Finkelshtein, Y. Kondratiev and P. Tkachov, Existence and properties of traveling waves for doubly nonlocal Fisher-KPP equations, Electron. J. Differ. Equ., 2019 (2019), 27 pp.  Google Scholar

[23]

R. A. Fisher, The wave of advance of advantageous genes, Ann. Hum. Genet., 7 (1937), 355-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

[24]

J. Furter and M. Grinfeld, Local vs. non-local interactions in population dynamics, J. Math. Biol., 27 (1989), 65-80.  doi: 10.1007/BF00276081.  Google Scholar

[25]

S. GenieysV. Volpert and P. Auger, Pattern and waves for a model in population dynamics with nonlocal consumption of resources, Math. Model. Nat. Phenom., 1 (2006), 65-82.  doi: 10.1051/mmnp:2006004.  Google Scholar

[26]

S. A. Gourley, Travelling front solutions of a nonlocal Fisher equation, J. Math. Biol., 41 (2000), 272-284.  doi: 10.1007/s002850000047.  Google Scholar

[27]

J. K. Hale and H. Koçak, Dynamics and Bifurcations, Texts in Applied Mathematics, 3. Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-4426-4.  Google Scholar

[28]

F. Hamel and L. Ryzhik, On the nonlocal Fisher-KPP equation: Steady states, spreading speed and global bounds, Nonlinearity, 27 (2014), 2735-2753.  doi: 10.1088/0951-7715/27/11/2735.  Google Scholar

[29]

P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematics Series, 247. Longman Scientific & Technical, Harlow, copublished in the United States with John Wiley & Sons, Inc., New York, 1991.  Google Scholar

[30]

V. Hutson and M. Grinfeld, Non-local dispersal and bistability, Eur. J. Appl. Math., 17 (2006), 221-232.  doi: 10.1017/S0956792506006462.  Google Scholar

[31]

A. KolmogorovI. Petrovskii and N. Piscunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Byul. Moskovskogo Gos. Univ., 1 (1937), 1-26.   Google Scholar

[32]

C. Kuehn and P. Tkachov, Pattern formation in the doubly-nonlocal Fisher-KPP equation, Discrete Contin. Dyn. Syst. Ser. B, 39 (2019), 2077-2100.  doi: 10.3934/dcds.2019087.  Google Scholar

[33]

L. Ma, S. J. Guo and T. Chen, Dynamics of a nonlocal dispersal model with a nonlocal reaction term, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1850033, 18 pp. doi: 10.1142/S0218127418500335.  Google Scholar

[34]

L. Ma and Y. Q. Luo, Dynamics of positive steady-state solutions of a nonlocal dispersal logistic model with nonlocal terms, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 2555-2582.  doi: 10.3934/dcdsb.2020022.  Google Scholar

[35]

G. Nadin, Existence and uniqueness of the solution of a space-time periodic reaction-diffusion equation, J. Differ. Equ., 249 (2010), 1288-1304.  doi: 10.1016/j.jde.2010.05.007.  Google Scholar

[36]

G. Nadin, Reaction-diffusion equations in space-time periodic media, C. R. Math. Acad. Sci. Paris, 345 (2007), 489-493.  doi: 10.1016/j.crma.2007.10.004.  Google Scholar

[37]

G. Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Appl., 92 (2009), 232-262.  doi: 10.1016/j.matpur.2009.04.002.  Google Scholar

[38]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  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. Dyn. Differ. Equ., 24 (2012), 927-954.  doi: 10.1007/s10884-012-9276-z.  Google Scholar

[40]

N. RawalW. Shen and A. Zhang, Spreading speeds and traveling waves of nonlocal monostable equations in time and space periodic habitats, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), 1609-1640.  doi: 10.3934/dcds.2015.35.1609.  Google Scholar

[41]

R. B. Salako and W. Shen, Parabolic-elliptic chemotaxis model with space-time dependent logistic sources on $\mathbb{R}^N$. I. Persistence and asymptotic spreading, Math. Models Methods Appl. Sci., 28 (2018), 2237-2273.  doi: 10.1142/S0218202518400146.  Google Scholar

[42]

W. Shen, Stability of transition waves and positive entire solutions of Fisher-KPP equations with time and space dependence, Nonlinearity, 30 (2017), 3466-3491.  doi: 10.1088/1361-6544/aa7f08.  Google Scholar

[43]

W. Shen and X. Xie, Spectral theory for nonlocal dispersal operators with time periodic indefinite weight functions and applications, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1023-1047.  doi: 10.3934/dcdsb.2017051.  Google Scholar

[44]

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.  Google Scholar

[45]

W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differ. Equ., 249 (2010), 747-795.  doi: 10.1016/j.jde.2010.04.012.  Google Scholar

[46]

W. Shen and A. Zhang, Traveling wave solutions of spatially periodic nonlocal monostable equations, Commun. Appl. Nonlinear Anal., 19 (2012), 73-101.   Google Scholar

[47]

L. N. SunJ. P. Shi and Y. W. 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

[48]

J.-W. SunW.-T. Li and Z.-C. Wang, The periodic principal eigenvalues with applications to the nonlocal dispersal logistic equation, J. Differ. Equ., 263 (2017), 934-971.  doi: 10.1016/j.jde.2017.03.001.  Google Scholar

show all references

References:
[1]

M. Alfaro and J. Coville, Rapid traveling waves in the nonlocal Fisher equation connect two unstable states, Appl. Math. Lett., 25 (2012), 2095-2099.  doi: 10.1016/j.aml.2012.05.006.  Google Scholar

[2]

F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, 165. American Mathematical Society, Providence, RI, Real Sociedad Matemática Española, Madrid, 2010. doi: 10.1090/surv/165.  Google Scholar

[3]

N. ApreuteseiN. BessonovV. Volpert and V. Vougalter, Spatial structures and generalized travelling waves for an integro-differential equation, Discrete Contin. Dyn. Syst.-Ser. B, 13 (2010), 537-557.  doi: 10.3934/dcdsb.2010.13.537.  Google Scholar

[4]

N. ApreuteseiA. Ducrot and V. Volpert, Travelling waves for integro-differential equations in population dynamics, Discrete Contin. Dyn. Syst.-Ser. B, 11 (2009), 541-561.  doi: 10.3934/dcdsb.2009.11.541.  Google Scholar

[5]

H. BerestyckiG. NadinB. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: Travelling waves and steady states, Nonlinearity, 22 (2009), 2813-2844.  doi: 10.1088/0951-7715/22/12/002.  Google Scholar

[6]

H. BerestyckiJ. Coville and H.-H. Vo, Persistence criteria for populations with non-local dispersion, J. Math. Biol., 72 (2016), 1693-1745.  doi: 10.1007/s00285-015-0911-2.  Google Scholar

[7]

H. BerestyckiF. Hamel and L. Roques, Analysis of the periodically fragmented environment model. I. Species persistence, J. Math. Biol., 51 (2005), 75-113.  doi: 10.1007/s00285-004-0313-3.  Google Scholar

[8]

H. BerestyckiF. Hamel and L. Roques, Analysis of the periodically fragmented environment model. II. Biological invasions and pulsating travelling fronts, J. Math. Pures Appl., 84 (2005), 1101-1146.  doi: 10.1016/j.matpur.2004.10.006.  Google Scholar

[9]

S. BianL. Chen and E. A. Latos, Global existence and asymptotic behavior of solutions to a nonlocal Fisher-KPP type problem, Nonlinear Anal., 149 (2017), 165-176.  doi: 10.1016/j.na.2016.10.017.  Google Scholar

[10]

J. Billingham, Dynamics of a strongly nonlocal reaction-diffusion population model, Nonlinearity, 17 (2003), 313-346.  doi: 10.1088/0951-7715/17/1/018.  Google Scholar

[11]

L. CaffarelliS. Dipierro and E. Valdinoci, A logistic equation with nonlocal interactions, Kinet. Relat. Models, 10 (2017), 141-170.  doi: 10.3934/krm.2017006.  Google Scholar

[12]

L. CaffarelliS. Patrizi and V. Quitalo, On a long range segregation model, J. Eur. Math. Soc. (JEMS), 19 (2017), 3575-3628.  doi: 10.4171/JEMS/747.  Google Scholar

[13]

F. CorrêaM. Delgado and A. Suárez, Some nonlinear heterogeneous problems with nonlocal reaction term, Adv. Differ. Equat., 16 (2011), 623-641.   Google Scholar

[14]

J. CovilleJ. Dávila and S. Martínez, Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity, SIAM J. Math. Anal., 39 (2008), 1693-1709.  doi: 10.1137/060676854.  Google Scholar

[15]

J. Coville and L. Dupaigne, Propagation speed of travelling fronts in non local reaction-diffusion equations, Nonlinear Anal., 60 (2005), 797-819.  doi: 10.1016/j.na.2003.10.030.  Google Scholar

[16]

M. DelgadoG. M. FigueiredoM. T. O. Pimenta and A. Suárez, Study of a logistic equation with local and non-local reaction terms, Topol. Methods Nonlinear Anal., 47 (2016), 693-713.  doi: 10.12775/TMNA.2016.026.  Google Scholar

[17]

K. Deng, On a nonlocal reaction-diffusion population model, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 65-73.  doi: 10.3934/dcdsb.2008.9.65.  Google Scholar

[18]

K. Deng and Y. X. Wu, Global stability for a nonlocal reaction–diffusion population model, Nonlinear Anal.-Real World Appl., 25 (2015), 127-136.  doi: 10.1016/j.nonrwa.2015.03.006.  Google Scholar

[19]

D. Finkelshtein, Y. Kondratiev and P. Tkachov, Traveling waves and long-time behavior in a doubly nonlocal Fisher-KPP equation, J. Math. Anal. Appl., 475 (2019), 94–122, arXiv: 1508.02215v2. doi: 10.1016/j.jmaa.2019.02.010.  Google Scholar

[20]

D. Finkelshtein, Y. Kondratiev, S. Molchanov and P. Tkachov, Global stability in a nonlocal reaction-diffusion equation, Stoch. Dyn., 18 (2018), 1850037, 15 pp. doi: 10.1142/S0219493718500375.  Google Scholar

[21]

D. Finkelshtein, Y. Kondratiev and P. Tkachov, Doubly nonlocal Fisher-KPP equation: Front propagation, Appl. Anal., (2019), 1–24. Google Scholar

[22]

D. Finkelshtein, Y. Kondratiev and P. Tkachov, Existence and properties of traveling waves for doubly nonlocal Fisher-KPP equations, Electron. J. Differ. Equ., 2019 (2019), 27 pp.  Google Scholar

[23]

R. A. Fisher, The wave of advance of advantageous genes, Ann. Hum. Genet., 7 (1937), 355-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

[24]

J. Furter and M. Grinfeld, Local vs. non-local interactions in population dynamics, J. Math. Biol., 27 (1989), 65-80.  doi: 10.1007/BF00276081.  Google Scholar

[25]

S. GenieysV. Volpert and P. Auger, Pattern and waves for a model in population dynamics with nonlocal consumption of resources, Math. Model. Nat. Phenom., 1 (2006), 65-82.  doi: 10.1051/mmnp:2006004.  Google Scholar

[26]

S. A. Gourley, Travelling front solutions of a nonlocal Fisher equation, J. Math. Biol., 41 (2000), 272-284.  doi: 10.1007/s002850000047.  Google Scholar

[27]

J. K. Hale and H. Koçak, Dynamics and Bifurcations, Texts in Applied Mathematics, 3. Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-4426-4.  Google Scholar

[28]

F. Hamel and L. Ryzhik, On the nonlocal Fisher-KPP equation: Steady states, spreading speed and global bounds, Nonlinearity, 27 (2014), 2735-2753.  doi: 10.1088/0951-7715/27/11/2735.  Google Scholar

[29]

P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematics Series, 247. Longman Scientific & Technical, Harlow, copublished in the United States with John Wiley & Sons, Inc., New York, 1991.  Google Scholar

[30]

V. Hutson and M. Grinfeld, Non-local dispersal and bistability, Eur. J. Appl. Math., 17 (2006), 221-232.  doi: 10.1017/S0956792506006462.  Google Scholar

[31]

A. KolmogorovI. Petrovskii and N. Piscunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Byul. Moskovskogo Gos. Univ., 1 (1937), 1-26.   Google Scholar

[32]

C. Kuehn and P. Tkachov, Pattern formation in the doubly-nonlocal Fisher-KPP equation, Discrete Contin. Dyn. Syst. Ser. B, 39 (2019), 2077-2100.  doi: 10.3934/dcds.2019087.  Google Scholar

[33]

L. Ma, S. J. Guo and T. Chen, Dynamics of a nonlocal dispersal model with a nonlocal reaction term, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1850033, 18 pp. doi: 10.1142/S0218127418500335.  Google Scholar

[34]

L. Ma and Y. Q. Luo, Dynamics of positive steady-state solutions of a nonlocal dispersal logistic model with nonlocal terms, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 2555-2582.  doi: 10.3934/dcdsb.2020022.  Google Scholar

[35]

G. Nadin, Existence and uniqueness of the solution of a space-time periodic reaction-diffusion equation, J. Differ. Equ., 249 (2010), 1288-1304.  doi: 10.1016/j.jde.2010.05.007.  Google Scholar

[36]

G. Nadin, Reaction-diffusion equations in space-time periodic media, C. R. Math. Acad. Sci. Paris, 345 (2007), 489-493.  doi: 10.1016/j.crma.2007.10.004.  Google Scholar

[37]

G. Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Appl., 92 (2009), 232-262.  doi: 10.1016/j.matpur.2009.04.002.  Google Scholar

[38]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  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. Dyn. Differ. Equ., 24 (2012), 927-954.  doi: 10.1007/s10884-012-9276-z.  Google Scholar

[40]

N. RawalW. Shen and A. Zhang, Spreading speeds and traveling waves of nonlocal monostable equations in time and space periodic habitats, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), 1609-1640.  doi: 10.3934/dcds.2015.35.1609.  Google Scholar

[41]

R. B. Salako and W. Shen, Parabolic-elliptic chemotaxis model with space-time dependent logistic sources on $\mathbb{R}^N$. I. Persistence and asymptotic spreading, Math. Models Methods Appl. Sci., 28 (2018), 2237-2273.  doi: 10.1142/S0218202518400146.  Google Scholar

[42]

W. Shen, Stability of transition waves and positive entire solutions of Fisher-KPP equations with time and space dependence, Nonlinearity, 30 (2017), 3466-3491.  doi: 10.1088/1361-6544/aa7f08.  Google Scholar

[43]

W. Shen and X. Xie, Spectral theory for nonlocal dispersal operators with time periodic indefinite weight functions and applications, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1023-1047.  doi: 10.3934/dcdsb.2017051.  Google Scholar

[44]

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.  Google Scholar

[45]

W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differ. Equ., 249 (2010), 747-795.  doi: 10.1016/j.jde.2010.04.012.  Google Scholar

[46]

W. Shen and A. Zhang, Traveling wave solutions of spatially periodic nonlocal monostable equations, Commun. Appl. Nonlinear Anal., 19 (2012), 73-101.   Google Scholar

[47]

L. N. SunJ. P. Shi and Y. W. 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

[48]

J.-W. SunW.-T. Li and Z.-C. Wang, The periodic principal eigenvalues with applications to the nonlocal dispersal logistic equation, J. Differ. Equ., 263 (2017), 934-971.  doi: 10.1016/j.jde.2017.03.001.  Google Scholar

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