# American Institute of Mathematical Sciences

September  2019, 24(9): 4929-4936. doi: 10.3934/dcdsb.2019038

## Effects of nonlocal dispersal and spatial heterogeneity on total biomass

 School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

Received  May 2018 Revised  August 2018 Published  September 2019 Early access  February 2019

In this paper, we investigate the effects of nonlocal dispersal and spatial heterogeneity on the total biomass of species via nonlocal dispersal logistic equations. In order to make the model more relevant for real biological systems, we consider a logistic reaction term, with two parameters, $r(x)$ for intrinsic growth rate and $K(x)$ for carrying capacity. We first establish the existence, uniqueness and asymptotic stability of the positive steady state solution for this equation. And then we study the continuous property and asymptotic limit of the positive steady state solution with respect to the dispersal rate. Finally, the function about the total biomass of species is defined by the positive steady state solution. Our results show in a heterogeneous environment, the total biomass is always strictly greater than the total carrying capacity in the special case when the nonlocal dispersal is allowed.

Citation: Yuan-Hang Su, Wan-Tong Li, Fei-Ying Yang. Effects of nonlocal dispersal and spatial heterogeneity on total biomass. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4929-4936. doi: 10.3934/dcdsb.2019038
##### References:
 [1] F. Andreu-Vaillo, J. M. Maz$\acute{o}$n, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, AMS, Providence, Rhode Island, 2010. doi: 10.1090/surv/165. [2] X. Bai and F. Li, Global dynamics of a competition model with nonlocal dispersal Ⅱ: The full system, J. Differential Equations, 258 (2015), 2655-2685.  doi: 10.1016/j.jde.2014.12.014. [3] 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. [4] H. Berestycki, J. Coville and H.-H. Vo, On the definition and the properties of the principal eigenvalue of some nonlocal operators, J. Funct. Anal., 271 (2016), 2701-2751.  doi: 10.1016/j.jfa.2016.05.017. [5] 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. [6] R. S. Cantrell, C. Cosner, Y. Lou and D. Ryan, Evolutionary stability of ideal dispersal strategies: A nonlocal dispersal model, Can. Appl. Math. Q., 20 (2012), 15-38. [7] C. Cortázar, J. Coville, M. Elgueta and S. Martínez, A nonlocal inhomogeneous dispersal process, J.Differential Equations, 241 (2007), 332-358.  doi: 10.1016/j.jde.2007.06.002. [8] C. Cosner, J. Dávila and S. Martínez, Evolutionary stability of ideal free nonlocal dispersal, J. Biol. Dyn., 6 (2012), 395-405.  doi: 10.1080/17513758.2011.588341. [9] 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. [10] J. Coville, Nonlocal refuge model with a partial control, Discrete Contin. Dyn. Syst., 35 (2015), 1421-1446.  doi: 10.3934/dcds.2015.35.1421. [11] 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. [12] 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. [13] D. L. DeAngelis, W. M. Ni and B. Zhang, Dispersal and spatial heterogeneity: Single species, J. Math. Biol., 72 (2016), 239-254.  doi: 10.1007/s00285-015-0879-y. [14] D. L. DeAngelis, W. M. Ni and B. Zhang, Effects of diffusion on total biomass in heterogeneous continuous and discrete-patch systems, Theor. Ecol., 9 (2016), 443-453. [15] P. Fife, Some nonclassical trends in parabolic and parabolic–like evolutions, in: Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153–191. [16] 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. [17] M. Grinfeld, G. Hines, V. Hutson, K. Mischaikow and G. T. Vickers, Non-local dispersal, Diff. Integral Equations, 18 (2005), 1299-1320. [18] V. Hutson and M. Grinfeld, Non-local dispersal and bistability, Euro. J. Appl. Math., 17 (2006), 221-232.  doi: 10.1017/S0956792506006462. [19] 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. [20] 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. [21] C. T. Lee, M. F. Hoopes, J. Diehl, W. Gilliland, G. Huxel, E. V. Leaver, K. Mccann, J. Umbanhowar and A. Mogilner, Non-local concepts and models in biology, J. theor. Biol., 210 (2001), 201-219.  doi: 10.1006/jtbi.2000.2287. [22] S. A. Levin, H. C. Muller-Landau, R. Nathan and J. Chave, The ecology and evolution of seed dispersal: A theoretical perspective, Annu. Rev. Eco. Evol. Syst., 34 (2003), 575-604.  doi: 10.1146/annurev.ecolsys.34.011802.132428. [23] F. Li, Y. Lou and Y. Wang, Global dynamics of a competition model with non-local dispersal Ⅰ: The shadow system, J. Math. Anal. Appl., 412 (2014), 485-497.  doi: 10.1016/j.jmaa.2013.10.071. [24] Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Differential Equations, 223 (2006), 400-426.  doi: 10.1016/j.jde.2005.05.010. [25] Y. Lou, Some challenging mathematical problems in evolution of dispersal and population dynamics, in: Tutorials in Mathematical Biosciences IV, in: Lecture Notes in Mathematics, Springer, Berlin, 1922 (2008), 171–205. doi: 10.1007/978-3-540-74331-6_5. [26] 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. [27] 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. [28] J. W. Sun, Asymptotic behavior for non-homogeneous nonlocal dispersal equations, Appl. Math. Lett., 50 (2015), 64-68.  doi: 10.1016/j.aml.2015.06.007. [29] 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. [30] F. Y. Yang, W. T. Li and J. W. Sun, Principal eigenvalues for some nonlocal eigenvalue problems and applications, Discrete Contin. Dyn. Syst., 36 (2016), 4027-4049.  doi: 10.3934/dcds.2016.36.4027.

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##### References:
 [1] F. Andreu-Vaillo, J. M. Maz$\acute{o}$n, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, AMS, Providence, Rhode Island, 2010. doi: 10.1090/surv/165. [2] X. Bai and F. Li, Global dynamics of a competition model with nonlocal dispersal Ⅱ: The full system, J. Differential Equations, 258 (2015), 2655-2685.  doi: 10.1016/j.jde.2014.12.014. [3] 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. [4] H. Berestycki, J. Coville and H.-H. Vo, On the definition and the properties of the principal eigenvalue of some nonlocal operators, J. Funct. Anal., 271 (2016), 2701-2751.  doi: 10.1016/j.jfa.2016.05.017. [5] 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. [6] R. S. Cantrell, C. Cosner, Y. Lou and D. Ryan, Evolutionary stability of ideal dispersal strategies: A nonlocal dispersal model, Can. Appl. Math. Q., 20 (2012), 15-38. [7] C. Cortázar, J. Coville, M. Elgueta and S. Martínez, A nonlocal inhomogeneous dispersal process, J.Differential Equations, 241 (2007), 332-358.  doi: 10.1016/j.jde.2007.06.002. [8] C. Cosner, J. Dávila and S. Martínez, Evolutionary stability of ideal free nonlocal dispersal, J. Biol. Dyn., 6 (2012), 395-405.  doi: 10.1080/17513758.2011.588341. [9] 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. [10] J. Coville, Nonlocal refuge model with a partial control, Discrete Contin. Dyn. Syst., 35 (2015), 1421-1446.  doi: 10.3934/dcds.2015.35.1421. [11] 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. [12] 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. [13] D. L. DeAngelis, W. M. Ni and B. Zhang, Dispersal and spatial heterogeneity: Single species, J. Math. Biol., 72 (2016), 239-254.  doi: 10.1007/s00285-015-0879-y. [14] D. L. DeAngelis, W. M. Ni and B. Zhang, Effects of diffusion on total biomass in heterogeneous continuous and discrete-patch systems, Theor. Ecol., 9 (2016), 443-453. [15] P. Fife, Some nonclassical trends in parabolic and parabolic–like evolutions, in: Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153–191. [16] 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. [17] M. Grinfeld, G. Hines, V. Hutson, K. Mischaikow and G. T. Vickers, Non-local dispersal, Diff. Integral Equations, 18 (2005), 1299-1320. [18] V. Hutson and M. Grinfeld, Non-local dispersal and bistability, Euro. J. Appl. Math., 17 (2006), 221-232.  doi: 10.1017/S0956792506006462. [19] 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. [20] 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. [21] C. T. Lee, M. F. Hoopes, J. Diehl, W. Gilliland, G. Huxel, E. V. Leaver, K. Mccann, J. Umbanhowar and A. Mogilner, Non-local concepts and models in biology, J. theor. Biol., 210 (2001), 201-219.  doi: 10.1006/jtbi.2000.2287. [22] S. A. Levin, H. C. Muller-Landau, R. Nathan and J. Chave, The ecology and evolution of seed dispersal: A theoretical perspective, Annu. Rev. Eco. Evol. Syst., 34 (2003), 575-604.  doi: 10.1146/annurev.ecolsys.34.011802.132428. [23] F. Li, Y. Lou and Y. Wang, Global dynamics of a competition model with non-local dispersal Ⅰ: The shadow system, J. Math. Anal. Appl., 412 (2014), 485-497.  doi: 10.1016/j.jmaa.2013.10.071. [24] Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Differential Equations, 223 (2006), 400-426.  doi: 10.1016/j.jde.2005.05.010. [25] Y. Lou, Some challenging mathematical problems in evolution of dispersal and population dynamics, in: Tutorials in Mathematical Biosciences IV, in: Lecture Notes in Mathematics, Springer, Berlin, 1922 (2008), 171–205. doi: 10.1007/978-3-540-74331-6_5. [26] 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. [27] 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. [28] J. W. Sun, Asymptotic behavior for non-homogeneous nonlocal dispersal equations, Appl. Math. Lett., 50 (2015), 64-68.  doi: 10.1016/j.aml.2015.06.007. [29] 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. [30] F. Y. Yang, W. T. Li and J. W. Sun, Principal eigenvalues for some nonlocal eigenvalue problems and applications, Discrete Contin. Dyn. Syst., 36 (2016), 4027-4049.  doi: 10.3934/dcds.2016.36.4027.

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