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Finite mass solutions for a nonlocal inhomogeneous dispersal equation
Nonlocal refuge model with a partial control
1. | UR 546 Biostatistique et Processus Spatiaux, INRA, Domaine St Paul Site Agroparc, F-84000 Avignon, France |
References:
[1] |
S. Arpaia, Genetically modified plants and non-target organisms: Analysing the functioning of the agro-ecosystem, Collection of Biosafety Reviews, 5 (2010), 12-80. |
[2] |
P. W. 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. |
[3] |
H. Berestycki, Le nombre de solutions de certains problèmes semi-linéaires elliptiques, J. of Funct. Anal., 40 (1981), 1-29, URL http://www.sciencedirect.com/science/article/pii/0022123681900690.
doi: 10.1016/0022-1236(81)90069-0. |
[4] |
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. |
[5] |
H. Berestycki, F. Hamel and L. Rossi, Liouville-type results for semilinear elliptic equations in unbounded domains, Ann. Mat. Pura Appl. (4), 186 (2007), 469-507.
doi: 10.1007/s10231-006-0015-0. |
[6] |
H. Berestycki and L. Rossi, On the principal eigenvalue of elliptic operators in $\mathbb{R}^N2$ and applications, J. Eur. Math. Soc., 8 (2006), 195-215.
doi: 10.4171/JEMS/47. |
[7] |
A. Birch and R. Wheatley, Gm pest-resistant crops: Assessing environmental impacts on non-target organisms, in Sustainability in Agriculture (eds. R. E. Hester and R. M. Harrison), vol. 21, The Royal Society of Chemistry, (2005), 31-57,
doi: 10.1039/9781847552433-00031. |
[8] |
R. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in disrupted environments II, SIAM Journal on Mathematical Analysis, 22 (1991), 1043-1064.
doi: 10.1137/0522068. |
[9] |
R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in disrupted environments, Proc. Roy. Soc. Edinburgh, Section: A Mathematics, 112 (1989), 293-318. URL http://journals.cambridge.org/article_S030821050001876X.
doi: 10.1017/S030821050001876X. |
[10] |
R. S. Cantrell and C. Cosner, The effects of spatial heterogeneity in population dynamics, J. Math. Biol., 29 (1991), 315-338.
doi: 10.1007/BF00167155. |
[11] |
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. |
[12] |
J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Differential Equations, 249 (2010), 2921-2953, URL http://www.sciencedirect.com/science/article/pii/S0022039610002317.
doi: 10.1016/j.jde.2010.07.003. |
[13] |
J. Coville, Harnack type inequality for positive solution of some integral equation, Annali di Matematica Pura ed Applicata, 191 (2012), 503-528.
doi: 10.1007/s10231-011-0193-2. |
[14] |
J. Coville, Singular measure as principal eigenfunction of some nonlocal operators, Applied Mathematics Letters, 26 (2013), 831-835, URL http://www.sciencedirect.com/science/article/pii/S0893965913000864.
doi: 10.1016/j.aml.2013.03.005. |
[15] |
J. Coville, J. Dávila and S. Martínez, Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity, SIAM Journal on Mathematical Analysis, 39 (2008), 1693-1709, URL http://link.aip.org/link/?SJM/39/1693/1.
doi: 10.1137/060676854. |
[16] |
J. Coville, J. Davila and S. Martinez, Pulsating fronts for nonlocal dispersion and kpp nonlinearity, Ann. I. H. Poincare - AN, 30 (2013), 179-223, URL http://www.sciencedirect.com/science/article/pii/S0294144912000674.
doi: 10.1016/j.anihpc.2012.07.005. |
[17] |
D. E. Edmunds, A. J. B. Potter and C. A. Stuart, Non-compact positive operators, Proc. Roy. Soc. London Ser. A, 328 (1972), 67-81.
doi: 10.1098/rspa.1972.0069. |
[18] |
J. M. Fraile, P. Koch Medina, J. López-Gómez and S. Merino, Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation, J. Differential Equations, 127 (1996), 295-319.
doi: 10.1006/jdeq.1996.0071. |
[19] |
J. Garcia-Melian, R. Gomez-Reñasco, J. Lopez-Gomez and J. C. Sabina de Lis, Pointwise growth and uniqueness of positive solutions for a class of sublinear elliptic problems where bifurcation from infinity occurs, Arch. Ration. Mech. Anal., 145 (1998), 261-289.
doi: 10.1007/s002050050130. |
[20] |
J. Garcia-Melian, R. Letelier-Albornoz and J. Sabina de Lis, Uniqueness and asymptotic behaviour for solutions of semilinear problems with boundary blow-up, Proc. Amer. Math. Soc., 129 (2001), 3593-3602.
doi: 10.1090/S0002-9939-01-06229-3. |
[21] |
J. Garcia-Melian 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. |
[22] |
J. García-Melián and J. D. Rossi, On the principal eigenvalue of some nonlocal diffusion problems, Journal of Differential Equations, 246 (2009), 21-38, URL http://www.sciencedirect.com/science/article/B6WJ2-4SM20CB-2/ 2/d5df03a19b1049bde09d05a0c773688b.
doi: 10.1016/j.jde.2008.04.015. |
[23] |
H. P. Hendriksma, S. Haertel, D. Babendreier, W. von der Ohe and I. Steffan-Dewenter, Effects of multiple bt proteins and gna lectin on in vitro-reared honey bee larvae, Apidologie, 43 (2012), 549-560.
doi: 10.1007/s13592-012-0123-3. |
[24] |
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. |
[25] |
V. Hutson, W. Shen and G. Vickers, Spectral theory for nonlocal dispersal with periodic or almost-periodic time dependence, Rocky Mt. J. Math., 38 (2008), 1147-1175.
doi: 10.1216/RMJ-2008-38-4-1147. |
[26] |
C.-Y. Kao, Y. Lou and W. Shen, Random dispersal vs. nonlocal dispersal, Discrete and Continuous Dynamical Systems, 26 (2010), 551-596.
doi: 10.3934/dcds.2010.26.551. |
[27] |
W.-T. Li, J.-W. Sun and F.-Y. Yang, Principal eigenvalues for some nonlocal dispersal dirichlet problems with weight function and applications, Preprint. |
[28] |
T. Ouyang, On the positive solutions of semilinear equations $\delta u + \lambda u - hu^p = 0$ on the compact manifolds, Trans. Amer. Math. Soc., 331 (1992), 503-527, URL http://www.jstor.org/stable/2154124.
doi: 10.2307/2154124. |
[29] |
J. N. Perry, Y. Devos, S. Arpaia, D. Bartsch, C. Ehlert, A. Gathmann, R. S. Hails, N. B. Hendriksen, J. Kiss, A. Messean, S. Mestdagh, G. Neemann, M. Nuti, J. B. Sweet and C. C. Tebbe, Estimating the effects of cry1f bt-maize pollen on non-target lepidoptera using a mathematical model of exposure, J. Applied Ecology, 49 (2012), 29-37.
doi: 10.1111/j.1365-2664.2011.02083.x. |
[30] |
J. N. Perry, Y. Devos, S. Arpaia, D. Bartsch, A. Gathmann, R. S. Hails, J. Kiss, K. Lheureux, B. Manachini, S. Mestdagh, G. Neemann, F. Ortego, J. Schiemann and J. B. Sweet, A mathematical model of exposure of nontarget lepidoptera to bt-maize pollen expressing cry1ab within europe, Proc. Roy. Soc. B-Biological Sciences, 277 (2010), 1417-1425. |
[31] |
J. M. Pleasants and K. S. Oberhauser, Milkweed loss in agricultural fields because of herbicide use: Effect on the monarch butterfly population, Insect Conservation and Diversity, 6 (2013), 135-144.
doi: 10.1111/j.1752-4598.2012.00196.x. |
[32] |
W. Shen and A. Zhang, Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats, Proc. Am. Math. Soc., 140 (2012), 1681-1696.
doi: 10.1090/S0002-9939-2011-11011-6. |
[33] |
W. Shen and A. Zhang, Traveling wave solutions of spatially periodic nonlocal monostable equations, Comm. Appl. Nonlinear Anal., 19 (2012), 73-101. |
show all references
References:
[1] |
S. Arpaia, Genetically modified plants and non-target organisms: Analysing the functioning of the agro-ecosystem, Collection of Biosafety Reviews, 5 (2010), 12-80. |
[2] |
P. W. 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. |
[3] |
H. Berestycki, Le nombre de solutions de certains problèmes semi-linéaires elliptiques, J. of Funct. Anal., 40 (1981), 1-29, URL http://www.sciencedirect.com/science/article/pii/0022123681900690.
doi: 10.1016/0022-1236(81)90069-0. |
[4] |
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. |
[5] |
H. Berestycki, F. Hamel and L. Rossi, Liouville-type results for semilinear elliptic equations in unbounded domains, Ann. Mat. Pura Appl. (4), 186 (2007), 469-507.
doi: 10.1007/s10231-006-0015-0. |
[6] |
H. Berestycki and L. Rossi, On the principal eigenvalue of elliptic operators in $\mathbb{R}^N2$ and applications, J. Eur. Math. Soc., 8 (2006), 195-215.
doi: 10.4171/JEMS/47. |
[7] |
A. Birch and R. Wheatley, Gm pest-resistant crops: Assessing environmental impacts on non-target organisms, in Sustainability in Agriculture (eds. R. E. Hester and R. M. Harrison), vol. 21, The Royal Society of Chemistry, (2005), 31-57,
doi: 10.1039/9781847552433-00031. |
[8] |
R. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in disrupted environments II, SIAM Journal on Mathematical Analysis, 22 (1991), 1043-1064.
doi: 10.1137/0522068. |
[9] |
R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in disrupted environments, Proc. Roy. Soc. Edinburgh, Section: A Mathematics, 112 (1989), 293-318. URL http://journals.cambridge.org/article_S030821050001876X.
doi: 10.1017/S030821050001876X. |
[10] |
R. S. Cantrell and C. Cosner, The effects of spatial heterogeneity in population dynamics, J. Math. Biol., 29 (1991), 315-338.
doi: 10.1007/BF00167155. |
[11] |
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. |
[12] |
J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Differential Equations, 249 (2010), 2921-2953, URL http://www.sciencedirect.com/science/article/pii/S0022039610002317.
doi: 10.1016/j.jde.2010.07.003. |
[13] |
J. Coville, Harnack type inequality for positive solution of some integral equation, Annali di Matematica Pura ed Applicata, 191 (2012), 503-528.
doi: 10.1007/s10231-011-0193-2. |
[14] |
J. Coville, Singular measure as principal eigenfunction of some nonlocal operators, Applied Mathematics Letters, 26 (2013), 831-835, URL http://www.sciencedirect.com/science/article/pii/S0893965913000864.
doi: 10.1016/j.aml.2013.03.005. |
[15] |
J. Coville, J. Dávila and S. Martínez, Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity, SIAM Journal on Mathematical Analysis, 39 (2008), 1693-1709, URL http://link.aip.org/link/?SJM/39/1693/1.
doi: 10.1137/060676854. |
[16] |
J. Coville, J. Davila and S. Martinez, Pulsating fronts for nonlocal dispersion and kpp nonlinearity, Ann. I. H. Poincare - AN, 30 (2013), 179-223, URL http://www.sciencedirect.com/science/article/pii/S0294144912000674.
doi: 10.1016/j.anihpc.2012.07.005. |
[17] |
D. E. Edmunds, A. J. B. Potter and C. A. Stuart, Non-compact positive operators, Proc. Roy. Soc. London Ser. A, 328 (1972), 67-81.
doi: 10.1098/rspa.1972.0069. |
[18] |
J. M. Fraile, P. Koch Medina, J. López-Gómez and S. Merino, Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation, J. Differential Equations, 127 (1996), 295-319.
doi: 10.1006/jdeq.1996.0071. |
[19] |
J. Garcia-Melian, R. Gomez-Reñasco, J. Lopez-Gomez and J. C. Sabina de Lis, Pointwise growth and uniqueness of positive solutions for a class of sublinear elliptic problems where bifurcation from infinity occurs, Arch. Ration. Mech. Anal., 145 (1998), 261-289.
doi: 10.1007/s002050050130. |
[20] |
J. Garcia-Melian, R. Letelier-Albornoz and J. Sabina de Lis, Uniqueness and asymptotic behaviour for solutions of semilinear problems with boundary blow-up, Proc. Amer. Math. Soc., 129 (2001), 3593-3602.
doi: 10.1090/S0002-9939-01-06229-3. |
[21] |
J. Garcia-Melian 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. |
[22] |
J. García-Melián and J. D. Rossi, On the principal eigenvalue of some nonlocal diffusion problems, Journal of Differential Equations, 246 (2009), 21-38, URL http://www.sciencedirect.com/science/article/B6WJ2-4SM20CB-2/ 2/d5df03a19b1049bde09d05a0c773688b.
doi: 10.1016/j.jde.2008.04.015. |
[23] |
H. P. Hendriksma, S. Haertel, D. Babendreier, W. von der Ohe and I. Steffan-Dewenter, Effects of multiple bt proteins and gna lectin on in vitro-reared honey bee larvae, Apidologie, 43 (2012), 549-560.
doi: 10.1007/s13592-012-0123-3. |
[24] |
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. |
[25] |
V. Hutson, W. Shen and G. Vickers, Spectral theory for nonlocal dispersal with periodic or almost-periodic time dependence, Rocky Mt. J. Math., 38 (2008), 1147-1175.
doi: 10.1216/RMJ-2008-38-4-1147. |
[26] |
C.-Y. Kao, Y. Lou and W. Shen, Random dispersal vs. nonlocal dispersal, Discrete and Continuous Dynamical Systems, 26 (2010), 551-596.
doi: 10.3934/dcds.2010.26.551. |
[27] |
W.-T. Li, J.-W. Sun and F.-Y. Yang, Principal eigenvalues for some nonlocal dispersal dirichlet problems with weight function and applications, Preprint. |
[28] |
T. Ouyang, On the positive solutions of semilinear equations $\delta u + \lambda u - hu^p = 0$ on the compact manifolds, Trans. Amer. Math. Soc., 331 (1992), 503-527, URL http://www.jstor.org/stable/2154124.
doi: 10.2307/2154124. |
[29] |
J. N. Perry, Y. Devos, S. Arpaia, D. Bartsch, C. Ehlert, A. Gathmann, R. S. Hails, N. B. Hendriksen, J. Kiss, A. Messean, S. Mestdagh, G. Neemann, M. Nuti, J. B. Sweet and C. C. Tebbe, Estimating the effects of cry1f bt-maize pollen on non-target lepidoptera using a mathematical model of exposure, J. Applied Ecology, 49 (2012), 29-37.
doi: 10.1111/j.1365-2664.2011.02083.x. |
[30] |
J. N. Perry, Y. Devos, S. Arpaia, D. Bartsch, A. Gathmann, R. S. Hails, J. Kiss, K. Lheureux, B. Manachini, S. Mestdagh, G. Neemann, F. Ortego, J. Schiemann and J. B. Sweet, A mathematical model of exposure of nontarget lepidoptera to bt-maize pollen expressing cry1ab within europe, Proc. Roy. Soc. B-Biological Sciences, 277 (2010), 1417-1425. |
[31] |
J. M. Pleasants and K. S. Oberhauser, Milkweed loss in agricultural fields because of herbicide use: Effect on the monarch butterfly population, Insect Conservation and Diversity, 6 (2013), 135-144.
doi: 10.1111/j.1752-4598.2012.00196.x. |
[32] |
W. Shen and A. Zhang, Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats, Proc. Am. Math. Soc., 140 (2012), 1681-1696.
doi: 10.1090/S0002-9939-2011-11011-6. |
[33] |
W. Shen and A. Zhang, Traveling wave solutions of spatially periodic nonlocal monostable equations, Comm. Appl. Nonlinear Anal., 19 (2012), 73-101. |
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