Nonlocal refuge model with a partial control

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

Nonlocal refuge model with a partial control

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

Abstract
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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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