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. 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.

[2]

G. Aluffi, Per andare a caccia la medusa si muove come un computer, Il Venerdì di Repubblica, August 2014.

[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.

[4]

E. Di NezzaG. 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.

[5]

S. DipierroO. Savin and E. Valdinoci, All functions are locally $ s $-harmonic up to a small error, J. Eur. Math. Soc. (JEMS)., ().

[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.

[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.

[8]

N. E. HumphriesN. QueirozJ. R. M. DyerN. G. PadeM. K. MusylK. M. SchaeferD. W. FullerJ. M. BrunnschweilerT. K. DoyleJ. D. R. HoughtonG. C. HaysC. S. JonesL. R. NobleV. J. WearmouthE. 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.

[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.

[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.

[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.

[12]

E. MontefuscoB. 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.

[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.

[14]

G. NadinL. RossiL. 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.

[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.

[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.

[17]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.

[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.

[19]

G. M. ViswanathanV. AfanasyevS. V. BuldyrevE. J. MurphyP. A. Prince and H. E. Stanley, Lévy flight search patterns of wandering albatrosses, Nature, 381 (1996), 413-415. doi: 10.1038/381413a0.

[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.

show all references

References:
[1]

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.

[2]

G. Aluffi, Per andare a caccia la medusa si muove come un computer, Il Venerdì di Repubblica, August 2014.

[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.

[4]

E. Di NezzaG. 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.

[5]

S. DipierroO. Savin and E. Valdinoci, All functions are locally $ s $-harmonic up to a small error, J. Eur. Math. Soc. (JEMS)., ().

[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.

[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.

[8]

N. E. HumphriesN. QueirozJ. R. M. DyerN. G. PadeM. K. MusylK. M. SchaeferD. W. FullerJ. M. BrunnschweilerT. K. DoyleJ. D. R. HoughtonG. C. HaysC. S. JonesL. R. NobleV. J. WearmouthE. 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.

[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.

[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.

[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.

[12]

E. MontefuscoB. 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.

[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.

[14]

G. NadinL. RossiL. 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.

[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.

[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.

[17]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.

[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.

[19]

G. M. ViswanathanV. AfanasyevS. V. BuldyrevE. J. MurphyP. A. Prince and H. E. Stanley, Lévy flight search patterns of wandering albatrosses, Nature, 381 (1996), 413-415. doi: 10.1038/381413a0.

[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.

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