# 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  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 & 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [15] P. Fife, Some nonclassical trends in parabolic and parabolic–like evolutions, in: Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153–191. Google Scholar [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. Google Scholar [17] M. Grinfeld, G. Hines, V. Hutson, K. Mischaikow and G. T. Vickers, Non-local dispersal, Diff. Integral Equations, 18 (2005), 1299-1320. Google Scholar [18] V. Hutson and M. Grinfeld, Non-local dispersal and bistability, Euro. J. Appl. Math., 17 (2006), 221-232. doi: 10.1017/S0956792506006462. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar

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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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [15] P. Fife, Some nonclassical trends in parabolic and parabolic–like evolutions, in: Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153–191. Google Scholar [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. Google Scholar [17] M. Grinfeld, G. Hines, V. Hutson, K. Mischaikow and G. T. Vickers, Non-local dispersal, Diff. Integral Equations, 18 (2005), 1299-1320. Google Scholar [18] V. Hutson and M. Grinfeld, Non-local dispersal and bistability, Euro. J. Appl. Math., 17 (2006), 221-232. doi: 10.1017/S0956792506006462. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar

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