# American Institute of Mathematical Sciences

March  2017, 10(1): 141-170. doi: 10.3934/krm.2017006

## A logistic equation with nonlocal interactions

 1 The University of Texas at Austin, Department of Mathematics and Institute for Computational Engineering and Sciences, 2515 Speedway, Austin, TX 78751, USA 2 School of Mathematics and Statistics, University of Melbourne, 813 Swanston St, Parkville VIC 3010, Australia 3 School of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, Crawley, Perth, WA 6009, Australia 4 Weierstraß-Institut für Angewandte Analysis und Stochastik, Hausvogteiplatz 5/7,10117 Berlin, Germany 5 CNR, Istituto di Matematica Applicata e Tecnologie Informatiche, via Ferrata 1,27100 Pavia, Italy 6 Dipartimento di Matematica, Università degli studi di Milano, Via Saldini 50,20133 Milan, Italy

*Corresponding author: Luis Caffarelli

Received  February 2016 Revised  May 2016 Published  November 2016

We consider here a logistic equation, modeling processes of nonlocal character both in the diffusion and proliferation terms.

More precisely, for populations that propagate according to a Lévy process and can reach resources in a neighborhood of their position, we compare (and find explicit threshold for survival) the local and nonlocal case.

As ambient space, we can consider:
$\bullet$bounded domains,
$\bullet$periodic environments,
$\bullet$transition problems, where the environment consists of a block of infinitesimal diffusion and an adjacent nonlocal one.

In each of these cases, we analyze the existence/nonexistence of solutions in terms of the spectral properties of the domain. In particular, we give a detailed description of the fact that nonlocal populations may better adapt to sparse resources and small environments.

Citation: Luis Caffarelli, Serena Dipierro, Enrico Valdinoci. A logistic equation with nonlocal interactions. Kinetic & Related Models, 2017, 10 (1) : 141-170. doi: 10.3934/krm.2017006
##### References:
 [1] 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. Google Scholar [2] G. Aluffi, Per andare a caccia la medusa si muove come un computer, Il Venerdì di Repubblica, August 2014.Google Scholar [3] L. Caffarelli and L. Silvestre, Regularity results for nonlocal equations by approximation, Arch. Ration. Mech. Anal., 200 (2011), 59-88. doi: 10.1007/s00205-010-0336-4. Google Scholar [4] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004. Google Scholar [5] S. Dipierro, O. Savin and E. Valdinoci, All functions are locally $s$-harmonic up to a small error, J. Eur. Math. Soc. (JEMS)., (). Google Scholar [6] S. Dipierro, O. Savin and E. Valdinoci, Local approximation of arbitrary functions by solutions of nonlocal equations, Arxiv Preprint, https://arxiv.org/pdf/1609.04438.pdf, 2016.Google Scholar [7] B. Dyda, Fractional calculus for power functions and eigenvalues of the fractional Laplacian, Fract. Calc. Appl. Anal., 15 (2012), 536-555. doi: 10.2478/s13540-012-0038-8. Google Scholar [8] N. E. Humphries, N. Queiroz, J. R. M. Dyer, N. G. Pade, M. K. Musyl, K. M. Schaefer, D. W. Fuller, J. M. Brunnschweiler, T. K. Doyle, J. D. R. Houghton, G. C. Hays, C. S. Jones, L. R. Noble, V. J. Wearmouth, E. J. Southall and D. W. Sims, Environmental context explains Lévy and Brownian movement patterns of marine predators, Nature, 465 (2010), 1066-1069. doi: 10.1038/nature09116. Google Scholar [9] 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 [10] D. Kriventsov, Regularity for a local-nonlocal transmission problem, Arch. Ration. Mech. Anal., 217 (2015), 1103-1195. doi: 10.1007/s00205-015-0851-4. Google Scholar [11] A. G. McKendrick and M. Kesava Pai, The rate of multiplication of micro-organisms: A mathematical study, Proceedings of the Royal Society of Edinburgh, 31 (1912), 649-653. doi: 10.1017/S0370164600025426. Google Scholar [12] E. Montefusco, B. Pellacci and G. Verzini, Fractional diffusion with Neumann boundary conditions: The logistic equation, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2175-2202. doi: 10.3934/dcdsb.2013.18.2175. Google Scholar [13] A. Massaccesi and E. Valdinoci, Is a nonlocal diffusion strategy convenient for biological populations in competition?, J. Math. Biol., http://link.springer.com/article/10.1007%2Fs00285-016-1019-z, DOI:10.1007/s00285-016-1019-z, 2016. doi: 10.1007/s00285-016-1019-z. Google Scholar [14] G. Nadin, L. Rossi, L. Ryzhik and B. Perthame, Wave-like solutions for nonlocal reaction-diffusion equations: A toy model, Math. Model. Nat. Phenom., 8 (2013), 33-41. doi: 10.1051/mmnp/20138304. Google Scholar [15] R. Pearl and L. J Reed, On the rate of growth of the population of the United States since 1790 and its mathematical representation, Proc. Natl. Acad. Sci. U.S.A., 6 (1977), 341-347. doi: 10.1007/978-3-642-81046-6_38. Google Scholar [16] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl. (9), 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003. Google Scholar [17] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137. Google Scholar [18] R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat., 58 (2014), 133-154. doi: 10.5565/PUBLMAT_58114_06. Google Scholar [19] G. M. Viswanathan, V. Afanasyev, S. V. Buldyrev, E. J. Murphy, P. A. Prince and H. E. Stanley, Lévy flight search patterns of wandering albatrosses, Nature, 381 (1996), 413-415. doi: 10.1038/381413a0. Google Scholar [20] P. F. Verhulst, Recherches mathématiques sur la loi d'accroissement de la population, Nouveaux mémoires de l'Académie Royale des Sciences et Belles-Lettres de Bruxelles, 18 (1845), 14-54. Google Scholar

show all references

##### References:
 [1] 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. Google Scholar [2] G. Aluffi, Per andare a caccia la medusa si muove come un computer, Il Venerdì di Repubblica, August 2014.Google Scholar [3] L. Caffarelli and L. Silvestre, Regularity results for nonlocal equations by approximation, Arch. Ration. Mech. Anal., 200 (2011), 59-88. doi: 10.1007/s00205-010-0336-4. Google Scholar [4] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004. Google Scholar [5] S. Dipierro, O. Savin and E. Valdinoci, All functions are locally $s$-harmonic up to a small error, J. Eur. Math. Soc. (JEMS)., (). Google Scholar [6] S. Dipierro, O. Savin and E. Valdinoci, Local approximation of arbitrary functions by solutions of nonlocal equations, Arxiv Preprint, https://arxiv.org/pdf/1609.04438.pdf, 2016.Google Scholar [7] B. Dyda, Fractional calculus for power functions and eigenvalues of the fractional Laplacian, Fract. Calc. Appl. Anal., 15 (2012), 536-555. doi: 10.2478/s13540-012-0038-8. Google Scholar [8] N. E. Humphries, N. Queiroz, J. R. M. Dyer, N. G. Pade, M. K. Musyl, K. M. Schaefer, D. W. Fuller, J. M. Brunnschweiler, T. K. Doyle, J. D. R. Houghton, G. C. Hays, C. S. Jones, L. R. Noble, V. J. Wearmouth, E. J. Southall and D. W. Sims, Environmental context explains Lévy and Brownian movement patterns of marine predators, Nature, 465 (2010), 1066-1069. doi: 10.1038/nature09116. Google Scholar [9] 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 [10] D. Kriventsov, Regularity for a local-nonlocal transmission problem, Arch. Ration. Mech. Anal., 217 (2015), 1103-1195. doi: 10.1007/s00205-015-0851-4. Google Scholar [11] A. G. McKendrick and M. Kesava Pai, The rate of multiplication of micro-organisms: A mathematical study, Proceedings of the Royal Society of Edinburgh, 31 (1912), 649-653. doi: 10.1017/S0370164600025426. Google Scholar [12] E. Montefusco, B. Pellacci and G. Verzini, Fractional diffusion with Neumann boundary conditions: The logistic equation, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2175-2202. doi: 10.3934/dcdsb.2013.18.2175. Google Scholar [13] A. Massaccesi and E. Valdinoci, Is a nonlocal diffusion strategy convenient for biological populations in competition?, J. Math. Biol., http://link.springer.com/article/10.1007%2Fs00285-016-1019-z, DOI:10.1007/s00285-016-1019-z, 2016. doi: 10.1007/s00285-016-1019-z. Google Scholar [14] G. Nadin, L. Rossi, L. Ryzhik and B. Perthame, Wave-like solutions for nonlocal reaction-diffusion equations: A toy model, Math. Model. Nat. Phenom., 8 (2013), 33-41. doi: 10.1051/mmnp/20138304. Google Scholar [15] R. Pearl and L. J Reed, On the rate of growth of the population of the United States since 1790 and its mathematical representation, Proc. Natl. Acad. Sci. U.S.A., 6 (1977), 341-347. doi: 10.1007/978-3-642-81046-6_38. Google Scholar [16] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl. (9), 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003. Google Scholar [17] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137. Google Scholar [18] R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat., 58 (2014), 133-154. doi: 10.5565/PUBLMAT_58114_06. Google Scholar [19] G. M. Viswanathan, V. Afanasyev, S. V. Buldyrev, E. J. Murphy, P. A. Prince and H. E. Stanley, Lévy flight search patterns of wandering albatrosses, Nature, 381 (1996), 413-415. doi: 10.1038/381413a0. Google Scholar [20] P. F. Verhulst, Recherches mathématiques sur la loi d'accroissement de la population, Nouveaux mémoires de l'Académie Royale des Sciences et Belles-Lettres de Bruxelles, 18 (1845), 14-54. Google Scholar
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