# American Institute of Mathematical Sciences

May  2021, 26(5): 2645-2676. 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  May 2021 Early access  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 and Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199
##### 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. [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. [3] N. Apreutesei, N. Bessonov, V. 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. [4] N. Apreutesei, A. 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. [5] H. Berestycki, G. Nadin, B. 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. [6] H. Berestycki, J. 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. [7] H. Berestycki, F. 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. [8] H. Berestycki, F. 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. [9] S. Bian, L. 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. [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. [11] L. Caffarelli, S. Dipierro and E. Valdinoci, A logistic equation with nonlocal interactions, Kinet. Relat. Models, 10 (2017), 141-170.  doi: 10.3934/krm.2017006. [12] L. Caffarelli, S. Patrizi and V. Quitalo, On a long range segregation model, J. Eur. Math. Soc. (JEMS), 19 (2017), 3575-3628.  doi: 10.4171/JEMS/747. [13] F. Corrêa, M. Delgado and A. Suárez, Some nonlinear heterogeneous problems with nonlocal reaction term, Adv. Differ. Equat., 16 (2011), 623-641. [14] J. Coville, J. 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. [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. [16] M. Delgado, G. M. Figueiredo, M. 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. [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. [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. [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. [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. [21] D. Finkelshtein, Y. Kondratiev and P. Tkachov, Doubly nonlocal Fisher-KPP equation: Front propagation, Appl. Anal., (2019), 1–24. [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. [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. [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. [25] S. Genieys, V. 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. [26] S. A. Gourley, Travelling front solutions of a nonlocal Fisher equation, J. Math. Biol., 41 (2000), 272-284.  doi: 10.1007/s002850000047. [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. [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. [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. [30] V. Hutson and M. Grinfeld, Non-local dispersal and bistability, Eur. J. Appl. Math., 17 (2006), 221-232.  doi: 10.1017/S0956792506006462. [31] A. Kolmogorov, I. 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. [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. [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. [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. [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. [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. [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. [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. [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. [40] N. Rawal, W. 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. [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. [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. [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. [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. [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. [46] W. Shen and A. Zhang, Traveling wave solutions of spatially periodic nonlocal monostable equations, Commun. Appl. Nonlinear Anal., 19 (2012), 73-101. [47] L. N. Sun, J. 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. [48] J.-W. Sun, W.-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.

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##### 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. [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. [3] N. Apreutesei, N. Bessonov, V. 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. [4] N. Apreutesei, A. 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. [5] H. Berestycki, G. Nadin, B. 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. [6] H. Berestycki, J. 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. [7] H. Berestycki, F. 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. [8] H. Berestycki, F. 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. [9] S. Bian, L. 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. [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. [11] L. Caffarelli, S. Dipierro and E. Valdinoci, A logistic equation with nonlocal interactions, Kinet. Relat. Models, 10 (2017), 141-170.  doi: 10.3934/krm.2017006. [12] L. Caffarelli, S. Patrizi and V. Quitalo, On a long range segregation model, J. Eur. Math. Soc. (JEMS), 19 (2017), 3575-3628.  doi: 10.4171/JEMS/747. [13] F. Corrêa, M. Delgado and A. Suárez, Some nonlinear heterogeneous problems with nonlocal reaction term, Adv. Differ. Equat., 16 (2011), 623-641. [14] J. Coville, J. 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. [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. [16] M. Delgado, G. M. Figueiredo, M. 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. [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. [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. [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. [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. [21] D. Finkelshtein, Y. Kondratiev and P. Tkachov, Doubly nonlocal Fisher-KPP equation: Front propagation, Appl. Anal., (2019), 1–24. [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. [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. [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. [25] S. Genieys, V. 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. [26] S. A. Gourley, Travelling front solutions of a nonlocal Fisher equation, J. Math. Biol., 41 (2000), 272-284.  doi: 10.1007/s002850000047. [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. [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. [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. [30] V. Hutson and M. Grinfeld, Non-local dispersal and bistability, Eur. J. Appl. Math., 17 (2006), 221-232.  doi: 10.1017/S0956792506006462. [31] A. Kolmogorov, I. 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. [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. [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. [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. [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. [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. [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. [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. [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. [40] N. Rawal, W. 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. [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. [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. [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. [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. [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. [46] W. Shen and A. Zhang, Traveling wave solutions of spatially periodic nonlocal monostable equations, Commun. Appl. Nonlinear Anal., 19 (2012), 73-101. [47] L. N. Sun, J. 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. [48] J.-W. Sun, W.-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.
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