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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 & Continuous Dynamical Systems - A, 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.   Google Scholar

[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.  doi: 10.1016/j.jmaa.2006.09.007.  Google Scholar

[3]

H. Berestycki, Le nombre de solutions de certains problèmes semi-linéaires elliptiques,, J. of Funct. Anal., 40 (1981), 1.  doi: 10.1016/0022-1236(81)90069-0.  Google Scholar

[4]

H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model. I. Species persistence,, J. Math. Biol., 51 (2005), 75.  doi: 10.1007/s00285-004-0313-3.  Google Scholar

[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.  doi: 10.1007/s10231-006-0015-0.  Google Scholar

[6]

H. Berestycki and L. Rossi, On the principal eigenvalue of elliptic operators in $\mathbbR^N$ and applications,, J. Eur. Math. Soc., 8 (2006), 195.  doi: 10.4171/JEMS/47.  Google Scholar

[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), (2005), 31.  doi: 10.1039/9781847552433-00031.  Google Scholar

[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.  doi: 10.1137/0522068.  Google Scholar

[9]

R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in disrupted environments,, Proc. Roy. Soc. Edinburgh, 112 (1989), 293.  doi: 10.1017/S030821050001876X.  Google Scholar

[10]

R. S. Cantrell and C. Cosner, The effects of spatial heterogeneity in population dynamics,, J. Math. Biol., 29 (1991), 315.  doi: 10.1007/BF00167155.  Google Scholar

[11]

C. Cortázar, J. Coville, M. Elgueta and S. Martínez, A nonlocal inhomogeneous dispersal process,, J. Differential Equations, 241 (2007), 332.  doi: 10.1016/j.jde.2007.06.002.  Google Scholar

[12]

J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators,, J. Differential Equations, 249 (2010), 2921.  doi: 10.1016/j.jde.2010.07.003.  Google Scholar

[13]

J. Coville, Harnack type inequality for positive solution of some integral equation,, Annali di Matematica Pura ed Applicata, 191 (2012), 503.  doi: 10.1007/s10231-011-0193-2.  Google Scholar

[14]

J. Coville, Singular measure as principal eigenfunction of some nonlocal operators,, Applied Mathematics Letters, 26 (2013), 831.  doi: 10.1016/j.aml.2013.03.005.  Google Scholar

[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.  doi: 10.1137/060676854.  Google Scholar

[16]

J. Coville, J. Davila and S. Martinez, Pulsating fronts for nonlocal dispersion and kpp nonlinearity,, Ann. I. H. Poincare - AN, 30 (2013), 179.  doi: 10.1016/j.anihpc.2012.07.005.  Google Scholar

[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.  doi: 10.1098/rspa.1972.0069.  Google Scholar

[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.  doi: 10.1006/jdeq.1996.0071.  Google Scholar

[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.  doi: 10.1007/s002050050130.  Google Scholar

[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.  doi: 10.1090/S0002-9939-01-06229-3.  Google Scholar

[21]

J. Garcia-Melian and J. D. Rossi, A logistic equation with refuge and nonlocal diffusion,, Commun. Pure Appl. Anal., 8 (2009), 2037.  doi: 10.3934/cpaa.2009.8.2037.  Google Scholar

[22]

J. García-Melián and J. D. Rossi, On the principal eigenvalue of some nonlocal diffusion problems,, Journal of Differential Equations, 246 (2009), 21.  doi: 10.1016/j.jde.2008.04.015.  Google Scholar

[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.  doi: 10.1007/s13592-012-0123-3.  Google Scholar

[24]

V. Hutson, S. Martinez, K. Mischaikow and G. T. Vickers, The evolution of dispersal,, J. Math. Biol., 47 (2003), 483.  doi: 10.1007/s00285-003-0210-1.  Google Scholar

[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.  doi: 10.1216/RMJ-2008-38-4-1147.  Google Scholar

[26]

C.-Y. Kao, Y. Lou and W. Shen, Random dispersal vs. nonlocal dispersal,, Discrete and Continuous Dynamical Systems, 26 (2010), 551.  doi: 10.3934/dcds.2010.26.551.  Google Scholar

[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., ().   Google Scholar

[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.  doi: 10.2307/2154124.  Google Scholar

[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.  doi: 10.1111/j.1365-2664.2011.02083.x.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1111/j.1752-4598.2012.00196.x.  Google Scholar

[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.  doi: 10.1090/S0002-9939-2011-11011-6.  Google Scholar

[33]

W. Shen and A. Zhang, Traveling wave solutions of spatially periodic nonlocal monostable equations,, Comm. Appl. Nonlinear Anal., 19 (2012), 73.   Google Scholar

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.   Google Scholar

[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.  doi: 10.1016/j.jmaa.2006.09.007.  Google Scholar

[3]

H. Berestycki, Le nombre de solutions de certains problèmes semi-linéaires elliptiques,, J. of Funct. Anal., 40 (1981), 1.  doi: 10.1016/0022-1236(81)90069-0.  Google Scholar

[4]

H. Berestycki, F. Hamel and L. Roques, Analysis of the periodically fragmented environment model. I. Species persistence,, J. Math. Biol., 51 (2005), 75.  doi: 10.1007/s00285-004-0313-3.  Google Scholar

[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.  doi: 10.1007/s10231-006-0015-0.  Google Scholar

[6]

H. Berestycki and L. Rossi, On the principal eigenvalue of elliptic operators in $\mathbbR^N$ and applications,, J. Eur. Math. Soc., 8 (2006), 195.  doi: 10.4171/JEMS/47.  Google Scholar

[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), (2005), 31.  doi: 10.1039/9781847552433-00031.  Google Scholar

[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.  doi: 10.1137/0522068.  Google Scholar

[9]

R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in disrupted environments,, Proc. Roy. Soc. Edinburgh, 112 (1989), 293.  doi: 10.1017/S030821050001876X.  Google Scholar

[10]

R. S. Cantrell and C. Cosner, The effects of spatial heterogeneity in population dynamics,, J. Math. Biol., 29 (1991), 315.  doi: 10.1007/BF00167155.  Google Scholar

[11]

C. Cortázar, J. Coville, M. Elgueta and S. Martínez, A nonlocal inhomogeneous dispersal process,, J. Differential Equations, 241 (2007), 332.  doi: 10.1016/j.jde.2007.06.002.  Google Scholar

[12]

J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators,, J. Differential Equations, 249 (2010), 2921.  doi: 10.1016/j.jde.2010.07.003.  Google Scholar

[13]

J. Coville, Harnack type inequality for positive solution of some integral equation,, Annali di Matematica Pura ed Applicata, 191 (2012), 503.  doi: 10.1007/s10231-011-0193-2.  Google Scholar

[14]

J. Coville, Singular measure as principal eigenfunction of some nonlocal operators,, Applied Mathematics Letters, 26 (2013), 831.  doi: 10.1016/j.aml.2013.03.005.  Google Scholar

[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.  doi: 10.1137/060676854.  Google Scholar

[16]

J. Coville, J. Davila and S. Martinez, Pulsating fronts for nonlocal dispersion and kpp nonlinearity,, Ann. I. H. Poincare - AN, 30 (2013), 179.  doi: 10.1016/j.anihpc.2012.07.005.  Google Scholar

[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.  doi: 10.1098/rspa.1972.0069.  Google Scholar

[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.  doi: 10.1006/jdeq.1996.0071.  Google Scholar

[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.  doi: 10.1007/s002050050130.  Google Scholar

[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.  doi: 10.1090/S0002-9939-01-06229-3.  Google Scholar

[21]

J. Garcia-Melian and J. D. Rossi, A logistic equation with refuge and nonlocal diffusion,, Commun. Pure Appl. Anal., 8 (2009), 2037.  doi: 10.3934/cpaa.2009.8.2037.  Google Scholar

[22]

J. García-Melián and J. D. Rossi, On the principal eigenvalue of some nonlocal diffusion problems,, Journal of Differential Equations, 246 (2009), 21.  doi: 10.1016/j.jde.2008.04.015.  Google Scholar

[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.  doi: 10.1007/s13592-012-0123-3.  Google Scholar

[24]

V. Hutson, S. Martinez, K. Mischaikow and G. T. Vickers, The evolution of dispersal,, J. Math. Biol., 47 (2003), 483.  doi: 10.1007/s00285-003-0210-1.  Google Scholar

[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.  doi: 10.1216/RMJ-2008-38-4-1147.  Google Scholar

[26]

C.-Y. Kao, Y. Lou and W. Shen, Random dispersal vs. nonlocal dispersal,, Discrete and Continuous Dynamical Systems, 26 (2010), 551.  doi: 10.3934/dcds.2010.26.551.  Google Scholar

[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., ().   Google Scholar

[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.  doi: 10.2307/2154124.  Google Scholar

[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.  doi: 10.1111/j.1365-2664.2011.02083.x.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1111/j.1752-4598.2012.00196.x.  Google Scholar

[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.  doi: 10.1090/S0002-9939-2011-11011-6.  Google Scholar

[33]

W. Shen and A. Zhang, Traveling wave solutions of spatially periodic nonlocal monostable equations,, Comm. Appl. Nonlinear Anal., 19 (2012), 73.   Google Scholar

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