May  2017, 22(3): 1023-1047. doi: 10.3934/dcdsb.2017051

Spectraltheory for nonlocal dispersal operators with time periodic indefinite weight functions and applications

1. 

Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA

2. 

Department of Mathematics and Statistisc, Idaho State University, Pocatello, ID 83209, USA

* Corresponding author: Wenxian Shen

Dedicate to Professor Stephen Cantrell on the occasion of his 60th birthday

Received  November 2015 Revised  July 2016 Published  January 2017

In this paper, we study the spectral theory for nonlocal dispersal operators with time periodic indefinite weight functions subject to Dirichlet type, Neumann type and spatial periodic type boundary conditions. We first obtain necessary and sufficient conditions for the existence of a unique positive principal spectrum point for such operators. We then investigate upper bounds of principal spectrum points and sufficient conditions for the principal spectrum points to be principal eigenvalues. Finally we discuss the applications to nonlinear mathematical models from biology.

Citation: Wenxian Shen, Xiaoxia Xie. Spectraltheory for nonlocal dispersal operators with time periodic indefinite weight functions and applications. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 1023-1047. doi: 10.3934/dcdsb.2017051
References:
[1]

A. AlvinoG. TrombettiP.-L. Lions and S. Matarasso, Comparison results for solutions of elliptic problems via symmetrization, Ann. Inst. H. Poincaré Anal. Non Linéaire, 16 (1999), 167-188. doi: 10.1016/S0294-1449(99)80011-0. Google Scholar

[2]

F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, AMS, Providence, Rhode Island, 2010.Google Scholar

[3]

P. 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. Google Scholar

[4]

A. Beltramo and P. Hess, On the principal eigenvalue of a periodic-parabolic operator, Comm. Partial Differential Equations, 9 (1984), 919-941. doi: 10.1080/03605308408820351. Google Scholar

[5]

J. Bochenek, On some linear eigenvalue problems with an indefinite weight function, Univ. Iagel. Acta Math., 27 (1988), 315-323. Google Scholar

[6]

M. Bȏcher, The smallest characteristic numbers in a certain exceptional case, Proc. Amer. Math. Soc., 21 (1914), 6-9. doi: 10.1090/S0002-9904-1914-02560-1. Google Scholar

[7]

K. J. BrownC. Cosner and J. Fleckinger, Principal eigenvalues for problems with indefinite weight function on $\mathbb{R}^n$, Proc. Amer. Math. Soc., 109 (1990), 147-155. doi: 10.2307/2048374. Google Scholar

[8]

K. J. Brown and S. S. Lin, On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight function, J. Math. Anal. Appl., 75 (1980), 112-120. doi: 10.1016/0022-247X(80)90309-1. Google Scholar

[9]

R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in disrupted environments. Ⅱ, SIAM J. Math. Anal., 22 (1991), 1043-1064. doi: 10.1137/0522068. Google Scholar

[10]

R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations, John Wiley & Sons, Ltd. , Chichester, 2003.Google Scholar

[11]

R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in disrupted environments, Proc. Roy. Soc. Edinburgh Sect. A, 112 (1989), 293-318. doi: 10.1017/S030821050001876X. Google Scholar

[12]

A. Chmaj and X. Ren, Homoclinic solutions of an integral equation: Existence and stability, J. Differential Equations, 155 (1999), 17-43. doi: 10.1006/jdeq.1998.3571. Google Scholar

[13]

C. Cosner, Eigenvalue problems with indefinite weights and reaction-diffusion models in population dynamics, Oxford Sci. Publ. , Oxford Univ. Press, New York, 1990,117–137.Google Scholar

[14]

C. CosnerF. Cuccu and G. Porru, Optimization of the first eigenvalue of equations with indefinite weights, Adv. Nonlinear Stud., 13 (2013), 79-95. doi: 10.1515/ans-2013-0105. Google Scholar

[15]

J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Differential Equations, 249 (2010), 2921-2953. doi: 10.1016/j.jde.2010.07.003. Google Scholar

[16]

J. CovilleJ. Dávila and S. Martínez, Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity, SIAM J. Math. Anal., 39 (2008), 1693-1709. doi: 10.1137/060676854. Google Scholar

[17]

D. Daners, Periodic-parabolic eigenvalue problems with indefinite weight functions, Arch. Math., 68 (1997), 388-397. doi: 10.1007/s000130050071. Google Scholar

[18] D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, The Clarendon Press Oxford University Press, New York, 1987. Google Scholar
[19]

L. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998.Google Scholar

[20]

E. FeireislF. Issard-Roch and H. Petzeltová, A non-smooth version of the Lojasiewicz-Simon theorem with applications to non-local phase-field systems, J. Differential Equations, 199 (2004), 1-21. doi: 10.1016/j.jde.2003.10.026. Google Scholar

[21]

P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, in Trends in nonlinear analysis, Springer, Berlin, (2003), 153–191.Google Scholar

[22]

R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugen., 7 (1937), 335-369. doi: 10.1111/j.1469-1809.1937.tb02153.x. Google Scholar

[23]

J. García-Melán and J. D. Rossi, On the principal eigenvalue of some nonlocal diffusion problems, J. Differential Equations, 246 (2009), 21-38. doi: 10.1016/j.jde.2008.04.015. Google Scholar

[24]

J.-P. Gossez and E. Lami Dozo, On the principal eigenvalue of a second order linear elliptic problem, Arch. Rational Mech. Anal., 89 (1985), 169-175. doi: 10.1007/BF00282330. Google Scholar

[25]

M. GrinfeldG. HinesV. HutsonK. Mischaikow and G. T. Vickers, Non-local dispersal, Differential Integral Equations, 18 (2005), 1299-1320. Google Scholar

[26]

K. P. Hadeler, Reaction transport systems in biological modelling, mathematics inspired by biology, Lecture Notes in Math. , Springer, Berlin, 1714 (1999), 95-150.Google Scholar

[27]

P. Hess, On the spectrum of elliptic operators with respect to indefinite weights, Proceedings of the symposium on operator theory (Athens, 1985), Linear Algebra Appl., 84 (1986), 99-109. doi: 10.1016/0024-3795(86)90309-5. Google Scholar

[28]

P. Hess, Periodic-parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematics Series, 247, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc. , New York, 1991.Google Scholar

[29]

P. Hess and T. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Differential Equations, 5 (1980), 999-1030. doi: 10.1080/03605308008820162. Google Scholar

[30]

P. Hess and H. Weinberger, Convergence to spatial-temporal clines in the Fisher equation with time-periodic fitnesses, J. Math. Biol., 28 (1990), 83-98. doi: 10.1007/BF00171520. Google Scholar

[31]

G. HetzerW. Shen and A. Zhang, Effects of spatial variations and dispersal strategies on principal eigenvalues of dispersal operators and spreading speeds of monostable equations, ocky Mountain Journal of Mathematics, 43 (2013), 489-513. doi: 10.1216/RMJ-2013-43-2-489. Google Scholar

[32]

M. HintermüllerC.-Y. Kao and A. Laurain, Principal eigenvalue minimization for an elliptic problem with indefinite weight and Robin boundary conditions, Appl. Math. Optim., 65 (2012), 111-146. doi: 10.1007/s00245-011-9153-x. Google Scholar

[33]

V. Hutson and M. Grinfeld, Non-local dispersal and bistability, European J. Appl. Math., 17 (2006), 221-232. doi: 10.1017/S0956792506006462. Google Scholar

[34]

V. HutsonS. MartinezK. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517. doi: 10.1007/s00285-003-0210-1. Google Scholar

[35]

V. HutsonW. Shen and G. T. Vickers, Spectraltheory for nonlocal dispersal with periodic or almost-periodic time dependence, Rocky Mountain Journal of Mathematics, 38 (2008), 1147-1175. doi: 10.1216/RMJ-2008-38-4-1147. Google Scholar

[36]

C.-Y. KaoY. Lou and W. Shen, Random dispersal vs non-Local dispersal, Discrete and Continuous Dynamical Systems, 26 (2010), 551-596. doi: 10.3934/dcds.2010.26.551. Google Scholar

[37]

C.-Y. KaoY. Lou and E. Yanagida, Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains, Math. Biosci. Eng., 5 (2008), 315-335. doi: 10.3934/mbe.2008.5.315. Google Scholar

[38]

A. KolmogorovI. Petrowsky and N. Piscunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Byul. Moskovskogo Gos, Univ. 1, 6 (1937), 1-25. Google Scholar

[39]

J. López-Gómez, On linear weighted boundary value problems, Partial differential equations, (Han-sur-Lesse, 1993), 82 (1994), 188-203. Google Scholar

[40]

J. López-Gómez, The maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems, J. Differential Equations, 127 (1996), 263-294. doi: 10.1006/jdeq.1996.0070. Google Scholar

[41]

Y. Lou and E. Yanagida, Minimization of the principal eigenvalue for an elliptic boundary value problem with indefinite weight, and applications to population dynamics, Japan J. Indust. Appl. Math., 23 (2006), 275-292. doi: 10.1007/BF03167595. Google Scholar

[42]

A. Manes and A. M. Micheletti, Un'estensione della teoria variazionale classica degli autovalori per operatori ellittici del secondo ordine, Boll. Un. Mat. Ital.(4), 7 (1973), 285-301. Google Scholar

[43]

G. Nadin, Existence and uniqueness of the solutions of a space-time periodic reaction-diffusion equation, J. Differential Equation, 249 (2010), 1288-1304. doi: 10.1016/j.jde.2010.05.007. Google Scholar

[44]

G. Nadin, The principal eigenvalue of a space-time periodic parabolic operator, Ann. Mat. Pura Appl., 188 (2009), 269-295. doi: 10.1007/s10231-008-0075-4. Google Scholar

[45]

H. G. OthmerS. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-295. doi: 10.1007/BF00277392. Google Scholar

[46]

N. Rawal and W. Shen, Criteria for the existence and lower bounds of principal eigenvalues of time periodic nonlocal dispersal operators and applications, J. Dynam. Differential Equations, 24 (2012), 927-954. doi: 10.1007/s10884-012-9276-z. Google Scholar

[47]

S. Senn and P. Hess, On positive solutions of a linear elliptic eigenvalue problem with Neumann boundary conditions, Math. Ann., 258 (1981), 459-470. doi: 10.1007/BF01453979. Google Scholar

[48]

W. Shen and X. Xie, On principal spectrum points/principal eigenvalues of nonlocal dispersal operators and applications, Discrete Contin. Dyn. Syst., 35 (2015), 1665-1696. doi: 10.3934/dcds.2015.35.1665. Google Scholar

[49]

W. Shen and X. Xie, Approximations of random dispersal operators/equations by nonlocal dispersal operators/equations, Journal of Differential Equations, 259 (2015), 7375-7405. doi: 10.1016/j.jde.2015.08.026. Google Scholar

[50]

W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, Journal of Differential Equations, 249 (2010), 747-795. doi: 10.1016/j.jde.2010.04.012. Google Scholar

[51]

I. Stoleriu, Integro-differential Equations in Materials Science, Ph. D thesis, University of Strathclyde in Glasgow, 2001.Google Scholar

[52]

X.-Q. Zhao, Global attractivity and stability in some monotone discrete dynamical systems, Bull. Austral. Math. Soc., 53 (1996), 305-324. doi: 10.1017/S0004972700017032. Google Scholar

show all references

References:
[1]

A. AlvinoG. TrombettiP.-L. Lions and S. Matarasso, Comparison results for solutions of elliptic problems via symmetrization, Ann. Inst. H. Poincaré Anal. Non Linéaire, 16 (1999), 167-188. doi: 10.1016/S0294-1449(99)80011-0. Google Scholar

[2]

F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, AMS, Providence, Rhode Island, 2010.Google Scholar

[3]

P. 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. Google Scholar

[4]

A. Beltramo and P. Hess, On the principal eigenvalue of a periodic-parabolic operator, Comm. Partial Differential Equations, 9 (1984), 919-941. doi: 10.1080/03605308408820351. Google Scholar

[5]

J. Bochenek, On some linear eigenvalue problems with an indefinite weight function, Univ. Iagel. Acta Math., 27 (1988), 315-323. Google Scholar

[6]

M. Bȏcher, The smallest characteristic numbers in a certain exceptional case, Proc. Amer. Math. Soc., 21 (1914), 6-9. doi: 10.1090/S0002-9904-1914-02560-1. Google Scholar

[7]

K. J. BrownC. Cosner and J. Fleckinger, Principal eigenvalues for problems with indefinite weight function on $\mathbb{R}^n$, Proc. Amer. Math. Soc., 109 (1990), 147-155. doi: 10.2307/2048374. Google Scholar

[8]

K. J. Brown and S. S. Lin, On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight function, J. Math. Anal. Appl., 75 (1980), 112-120. doi: 10.1016/0022-247X(80)90309-1. Google Scholar

[9]

R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in disrupted environments. Ⅱ, SIAM J. Math. Anal., 22 (1991), 1043-1064. doi: 10.1137/0522068. Google Scholar

[10]

R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations, John Wiley & Sons, Ltd. , Chichester, 2003.Google Scholar

[11]

R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in disrupted environments, Proc. Roy. Soc. Edinburgh Sect. A, 112 (1989), 293-318. doi: 10.1017/S030821050001876X. Google Scholar

[12]

A. Chmaj and X. Ren, Homoclinic solutions of an integral equation: Existence and stability, J. Differential Equations, 155 (1999), 17-43. doi: 10.1006/jdeq.1998.3571. Google Scholar

[13]

C. Cosner, Eigenvalue problems with indefinite weights and reaction-diffusion models in population dynamics, Oxford Sci. Publ. , Oxford Univ. Press, New York, 1990,117–137.Google Scholar

[14]

C. CosnerF. Cuccu and G. Porru, Optimization of the first eigenvalue of equations with indefinite weights, Adv. Nonlinear Stud., 13 (2013), 79-95. doi: 10.1515/ans-2013-0105. Google Scholar

[15]

J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Differential Equations, 249 (2010), 2921-2953. doi: 10.1016/j.jde.2010.07.003. Google Scholar

[16]

J. CovilleJ. Dávila and S. Martínez, Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity, SIAM J. Math. Anal., 39 (2008), 1693-1709. doi: 10.1137/060676854. Google Scholar

[17]

D. Daners, Periodic-parabolic eigenvalue problems with indefinite weight functions, Arch. Math., 68 (1997), 388-397. doi: 10.1007/s000130050071. Google Scholar

[18] D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, The Clarendon Press Oxford University Press, New York, 1987. Google Scholar
[19]

L. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998.Google Scholar

[20]

E. FeireislF. Issard-Roch and H. Petzeltová, A non-smooth version of the Lojasiewicz-Simon theorem with applications to non-local phase-field systems, J. Differential Equations, 199 (2004), 1-21. doi: 10.1016/j.jde.2003.10.026. Google Scholar

[21]

P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, in Trends in nonlinear analysis, Springer, Berlin, (2003), 153–191.Google Scholar

[22]

R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugen., 7 (1937), 335-369. doi: 10.1111/j.1469-1809.1937.tb02153.x. Google Scholar

[23]

J. García-Melán and J. D. Rossi, On the principal eigenvalue of some nonlocal diffusion problems, J. Differential Equations, 246 (2009), 21-38. doi: 10.1016/j.jde.2008.04.015. Google Scholar

[24]

J.-P. Gossez and E. Lami Dozo, On the principal eigenvalue of a second order linear elliptic problem, Arch. Rational Mech. Anal., 89 (1985), 169-175. doi: 10.1007/BF00282330. Google Scholar

[25]

M. GrinfeldG. HinesV. HutsonK. Mischaikow and G. T. Vickers, Non-local dispersal, Differential Integral Equations, 18 (2005), 1299-1320. Google Scholar

[26]

K. P. Hadeler, Reaction transport systems in biological modelling, mathematics inspired by biology, Lecture Notes in Math. , Springer, Berlin, 1714 (1999), 95-150.Google Scholar

[27]

P. Hess, On the spectrum of elliptic operators with respect to indefinite weights, Proceedings of the symposium on operator theory (Athens, 1985), Linear Algebra Appl., 84 (1986), 99-109. doi: 10.1016/0024-3795(86)90309-5. Google Scholar

[28]

P. Hess, Periodic-parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematics Series, 247, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc. , New York, 1991.Google Scholar

[29]

P. Hess and T. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Differential Equations, 5 (1980), 999-1030. doi: 10.1080/03605308008820162. Google Scholar

[30]

P. Hess and H. Weinberger, Convergence to spatial-temporal clines in the Fisher equation with time-periodic fitnesses, J. Math. Biol., 28 (1990), 83-98. doi: 10.1007/BF00171520. Google Scholar

[31]

G. HetzerW. Shen and A. Zhang, Effects of spatial variations and dispersal strategies on principal eigenvalues of dispersal operators and spreading speeds of monostable equations, ocky Mountain Journal of Mathematics, 43 (2013), 489-513. doi: 10.1216/RMJ-2013-43-2-489. Google Scholar

[32]

M. HintermüllerC.-Y. Kao and A. Laurain, Principal eigenvalue minimization for an elliptic problem with indefinite weight and Robin boundary conditions, Appl. Math. Optim., 65 (2012), 111-146. doi: 10.1007/s00245-011-9153-x. Google Scholar

[33]

V. Hutson and M. Grinfeld, Non-local dispersal and bistability, European J. Appl. Math., 17 (2006), 221-232. doi: 10.1017/S0956792506006462. Google Scholar

[34]

V. HutsonS. MartinezK. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517. doi: 10.1007/s00285-003-0210-1. Google Scholar

[35]

V. HutsonW. Shen and G. T. Vickers, Spectraltheory for nonlocal dispersal with periodic or almost-periodic time dependence, Rocky Mountain Journal of Mathematics, 38 (2008), 1147-1175. doi: 10.1216/RMJ-2008-38-4-1147. Google Scholar

[36]

C.-Y. KaoY. Lou and W. Shen, Random dispersal vs non-Local dispersal, Discrete and Continuous Dynamical Systems, 26 (2010), 551-596. doi: 10.3934/dcds.2010.26.551. Google Scholar

[37]

C.-Y. KaoY. Lou and E. Yanagida, Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains, Math. Biosci. Eng., 5 (2008), 315-335. doi: 10.3934/mbe.2008.5.315. Google Scholar

[38]

A. KolmogorovI. Petrowsky and N. Piscunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Byul. Moskovskogo Gos, Univ. 1, 6 (1937), 1-25. Google Scholar

[39]

J. López-Gómez, On linear weighted boundary value problems, Partial differential equations, (Han-sur-Lesse, 1993), 82 (1994), 188-203. Google Scholar

[40]

J. López-Gómez, The maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems, J. Differential Equations, 127 (1996), 263-294. doi: 10.1006/jdeq.1996.0070. Google Scholar

[41]

Y. Lou and E. Yanagida, Minimization of the principal eigenvalue for an elliptic boundary value problem with indefinite weight, and applications to population dynamics, Japan J. Indust. Appl. Math., 23 (2006), 275-292. doi: 10.1007/BF03167595. Google Scholar

[42]

A. Manes and A. M. Micheletti, Un'estensione della teoria variazionale classica degli autovalori per operatori ellittici del secondo ordine, Boll. Un. Mat. Ital.(4), 7 (1973), 285-301. Google Scholar

[43]

G. Nadin, Existence and uniqueness of the solutions of a space-time periodic reaction-diffusion equation, J. Differential Equation, 249 (2010), 1288-1304. doi: 10.1016/j.jde.2010.05.007. Google Scholar

[44]

G. Nadin, The principal eigenvalue of a space-time periodic parabolic operator, Ann. Mat. Pura Appl., 188 (2009), 269-295. doi: 10.1007/s10231-008-0075-4. Google Scholar

[45]

H. G. OthmerS. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-295. doi: 10.1007/BF00277392. Google Scholar

[46]

N. Rawal and W. Shen, Criteria for the existence and lower bounds of principal eigenvalues of time periodic nonlocal dispersal operators and applications, J. Dynam. Differential Equations, 24 (2012), 927-954. doi: 10.1007/s10884-012-9276-z. Google Scholar

[47]

S. Senn and P. Hess, On positive solutions of a linear elliptic eigenvalue problem with Neumann boundary conditions, Math. Ann., 258 (1981), 459-470. doi: 10.1007/BF01453979. Google Scholar

[48]

W. Shen and X. Xie, On principal spectrum points/principal eigenvalues of nonlocal dispersal operators and applications, Discrete Contin. Dyn. Syst., 35 (2015), 1665-1696. doi: 10.3934/dcds.2015.35.1665. Google Scholar

[49]

W. Shen and X. Xie, Approximations of random dispersal operators/equations by nonlocal dispersal operators/equations, Journal of Differential Equations, 259 (2015), 7375-7405. doi: 10.1016/j.jde.2015.08.026. Google Scholar

[50]

W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, Journal of Differential Equations, 249 (2010), 747-795. doi: 10.1016/j.jde.2010.04.012. Google Scholar

[51]

I. Stoleriu, Integro-differential Equations in Materials Science, Ph. D thesis, University of Strathclyde in Glasgow, 2001.Google Scholar

[52]

X.-Q. Zhao, Global attractivity and stability in some monotone discrete dynamical systems, Bull. Austral. Math. Soc., 53 (1996), 305-324. doi: 10.1017/S0004972700017032. Google Scholar

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