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April  2015, 35(4): 1421-1446. doi: 10.3934/dcds.2015.35.1421

Nonlocal refuge model with a partial control

Jérôme Coville 1,

1. 

UR 546 Biostatistique et Processus Spatiaux, INRA, Domaine St Paul Site Agroparc, F-84000 Avignon, France

Received  May 2013 Revised  January 2014 Published  November 2014

In this paper, we analyse the structure of the set of positive solutions of an heterogeneous nonlocal equation of the form: $$ \int_{\Omega} K(x, y)u(y)\,dy -\int_ {\Omega}K(y, x)u(x)\, dy + a_0u+\lambda a_1(x)u -\beta(x)u^p=0 \quad \text{in}\quad \Omega$$ where $\Omega\subset \mathbb{R}^n$ is a bounded domain, $K\in C(\mathbb{R}^n\times \mathbb{R}^n) $ is non-negative, $a_i,\beta \in C(\Omega)$ and $\lambda\in\mathbb{R}$. Such type of equation appears in some studies of population dynamics where the population evolves in a partially controlled heterogeneous landscape and disperses on long ranges. Under some fairly general assumptions on $K,a_i$ and $\beta$, we first establish a necessary and sufficient condition for the existence of a unique positive solution. Then, we analyse the structure of the set of positive solutions $(\lambda,u_\lambda)$ depending on the presence or absence of a refuge zone (i.e $\omega$ so that $\beta_{|\omega}\equiv 0$).
Keywords: asymptotic behaviour, non trivial solution, principal eigenvalue, Nonlocal diffusion operators, partially controlled refuge model..
Mathematics Subject Classification: Primary: 45G99, 45C05; Secondary: 45K05, 47B6.
Citation: Jérôme Coville. Nonlocal refuge model with a partial control. Discrete and Continuous Dynamical Systems, 2015, 35 (4) : 1421-1446. doi: 10.3934/dcds.2015.35.1421
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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