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    Preface on the special issue of Discrete and Continuous Dynamical Systems- Series B in honor of Chris Cosner on the occasion of his 60th birthday
December  2014, 19(10): 3031-3056. doi: 10.3934/dcdsb.2014.19.3031

Some paradoxical effects of the advection on a class of diffusive equations in Ecology

David Aleja 1, and  Julián López-Gómez 2,

1. 

Department of Mathematics, University Carlos III of Madrid, Leganés (Madrid), 28911, Spain
2. 

Department of Applied Mathematics, Complutense University of Madrid, Madrid, 28040, Spain

Received  July 2013 Revised  September 2013 Published  October 2014

In this paper we refine in a substantial way part of the materials of the celebrated paper of Belgacem and Cosner [3] by considering a rather general class of degenerate diffusive logistic equations in the presence of advection. Rather paradoxically, a large advection can provoke the stabilization to an steady state of a former explosive solution. Similarly, even with a severe taxis down the environmental gradient the species can be permanent.
Keywords: diffusive logistic equations, dynamics., Advection.
Mathematics Subject Classification: Primary: 35K20, 35K58; Secondary: 35P1.
Citation: David Aleja, Julián López-Gómez. Some paradoxical effects of the advection on a class of diffusive equations in Ecology. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3031-3056. doi: 10.3934/dcdsb.2014.19.3031
References:
[1]

H. Amann, On the existence of positive solutions of nonlinear elliptic boundary value problems, Indiana J. Mathematics, 21 (1972), 125-146. doi: 10.1512/iumj.1972.21.21012.  Google Scholar

[2]

H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Diff. Equations, 146 (1998), 336-374. doi: 10.1006/jdeq.1998.3440.  Google Scholar

[3]

F. Belgacem and C. Cosner, The effects of dispersal along environmental gradients on the dynamics of populations in heterogeneous environments, Can. Appl. Math. Quart., 3 (1995), 379-397.  Google Scholar

[4]

H. Berestycki, F. Hamel and N. Nadirashvili, Elliptic eigenvalue problems with latge drift and applications to nonlinear propagation phenomena, Commun. Math. Phys., 253 (2005), 451-480. doi: 10.1007/s00220-004-1201-9.  Google Scholar

[5]

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

[6]

S. Cano-Casanova and J. López-Gómez, Properties of the principal eigenvalues of a general class of non-classical mixed boundary value problems, J. Diff. Equations, 178 (2002), 123-211. doi: 10.1006/jdeq.2000.4003.  Google Scholar

[7]

X. Chen and Y. Lou, Principal eigenvaue and eigenfunctions of an elliptic operator with large advection and its application to a competition model, Indiana Univ. Math. J., 57 (2008), 627-658. doi: 10.1512/iumj.2008.57.3204.  Google Scholar

[8]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[9]

E. N. Dancer and J. López-Gómez, Semiclassical analysis of general second order elliptic operators on bounded domains, Trans. Amer. Math. Soc., 352 (2000), 3723-3742. doi: 10.1090/S0002-9947-00-02534-4.  Google Scholar

[10]

J. E. Furter and J. López-Gómez, Diffusion mediated permanence problem for a heterogeneous Lotka-Volterra competition model, Proc. Royal Soc. Edinburgh, A, 127 (1997), 281-336. doi: 10.1017/S0308210500023659.  Google Scholar

[11]

R. Gómez-Reñasco, The Effects of Varying Coefficents in Semilinear Elliptic Boundary Value Problems. From Classical Solutons to Metasolutions, Ph.D Thesis, University of La Laguna (Tenerife, Canary Islands), February 1999. Google Scholar

[12]

R. Gómez-Reñasco and J. López-Gómez, On the existence and numerical computation of classical and non-classical solutions for a family of elliptic boundary value problems, Nonl. Anal. T.M.A., 48 (2002), 567-605. doi: 10.1016/S0362-546X(00)00208-X.  Google Scholar

[13]

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

[14]

V. Hutson, J. López-Gómez, K. Mischaikow and G. Vickers, Limit behaviour for a competing species problem with diffusion, in World Scientific Series in Applied Analysis, (ed. R. Agarwal), World Scientific Publishing, 4 (1995), 343-358. doi: 10.1142/9789812796417_0022.  Google Scholar

[15]

T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer, Berlin, 1995.  Google Scholar

[16]

J. López-Gómez, On linear weighted boundary value problems, in Partial Differential Equations, Models in Physics and Biology, Mathematical Research, (eds. G. Lumer, S. Nicaise and B. W. Schulze), Akademie Verlag Berlin, 82 (1994), 188-203.  Google Scholar

[17]

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

[18]

J. López-Gómez, Large solutions, metasolutions, and asymptotic behaviour of the regular positive solutions of sublinear parabolic problems, El. J. Diff. Equations, Conf., 5 (2000), 135-171.  Google Scholar

[19]

J. López-Gómez, Approaching metasolutions by solutions, Diff. Int. Eqns., 14 (2001), 739-750.  Google Scholar

[20]

J. López-Gómez, Metasolutions: Malthus versus Verhulst in population dynamics. A dream of Volterra, in Handbook of Differential Equations, Stationary Partial Differential Equations, (eds. M. Chipot and P. Quittner), Elsevier, 2 (2005), 211-309. doi: 10.1016/S1874-5733(05)80012-9.  Google Scholar

[21]

J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, Research Notes in Mathematics 426, Chapman & Hall/CRC Press, Boca Raton (FL), 2001. doi: 10.1201/9781420035506.  Google Scholar

[22]

J. López-Gómez, Elliptic Operators, World Scientific Publishing, Singapore, 2013. Google Scholar

[23]

J. López-Gómez and M. Molina-Meyer, The maximum principle for cooperative weakly coupled elliptic systems and some applications, Diff. Int. Eqns., 7 (1994), 383-398.  Google Scholar

[24]

J. López-Gómez and M. Montenegro, The effects of transport on the maximum principle, J. Math. Anal. Appl., 403 (2013), 547-557. doi: 10.1016/j.jmaa.2013.02.049.  Google Scholar

[25]

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

[26]

D. Sattinger, Topics in Stability and Bifurcation Theory, Lecture Notes in Mathematics 309, Springer, 1973.  Google Scholar

[27]

S. Senn, On a semilinear elliptic eigenvalue problem with Neumann boundary conditions, with an application to population genetics, Comm. Part. Diff. Eqns., 8 (1983), 1199-1228. doi: 10.1080/03605308308820300.  Google Scholar

[28]

S. Senn and P. Hess, On positive solutions of a linear elliptic boundary value problem with Neumann boundary conditions, Math. Annalen, 258 (1982), 459-470. doi: 10.1007/BF01453979.  Google Scholar

[29]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970.  Google Scholar

show all references

References:
[1]

H. Amann, On the existence of positive solutions of nonlinear elliptic boundary value problems, Indiana J. Mathematics, 21 (1972), 125-146. doi: 10.1512/iumj.1972.21.21012.  Google Scholar

[2]

H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Diff. Equations, 146 (1998), 336-374. doi: 10.1006/jdeq.1998.3440.  Google Scholar

[3]

F. Belgacem and C. Cosner, The effects of dispersal along environmental gradients on the dynamics of populations in heterogeneous environments, Can. Appl. Math. Quart., 3 (1995), 379-397.  Google Scholar

[4]

H. Berestycki, F. Hamel and N. Nadirashvili, Elliptic eigenvalue problems with latge drift and applications to nonlinear propagation phenomena, Commun. Math. Phys., 253 (2005), 451-480. doi: 10.1007/s00220-004-1201-9.  Google Scholar

[5]

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

[6]

S. Cano-Casanova and J. López-Gómez, Properties of the principal eigenvalues of a general class of non-classical mixed boundary value problems, J. Diff. Equations, 178 (2002), 123-211. doi: 10.1006/jdeq.2000.4003.  Google Scholar

[7]

X. Chen and Y. Lou, Principal eigenvaue and eigenfunctions of an elliptic operator with large advection and its application to a competition model, Indiana Univ. Math. J., 57 (2008), 627-658. doi: 10.1512/iumj.2008.57.3204.  Google Scholar

[8]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[9]

E. N. Dancer and J. López-Gómez, Semiclassical analysis of general second order elliptic operators on bounded domains, Trans. Amer. Math. Soc., 352 (2000), 3723-3742. doi: 10.1090/S0002-9947-00-02534-4.  Google Scholar

[10]

J. E. Furter and J. López-Gómez, Diffusion mediated permanence problem for a heterogeneous Lotka-Volterra competition model, Proc. Royal Soc. Edinburgh, A, 127 (1997), 281-336. doi: 10.1017/S0308210500023659.  Google Scholar

[11]

R. Gómez-Reñasco, The Effects of Varying Coefficents in Semilinear Elliptic Boundary Value Problems. From Classical Solutons to Metasolutions, Ph.D Thesis, University of La Laguna (Tenerife, Canary Islands), February 1999. Google Scholar

[12]

R. Gómez-Reñasco and J. López-Gómez, On the existence and numerical computation of classical and non-classical solutions for a family of elliptic boundary value problems, Nonl. Anal. T.M.A., 48 (2002), 567-605. doi: 10.1016/S0362-546X(00)00208-X.  Google Scholar

[13]

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

[14]

V. Hutson, J. López-Gómez, K. Mischaikow and G. Vickers, Limit behaviour for a competing species problem with diffusion, in World Scientific Series in Applied Analysis, (ed. R. Agarwal), World Scientific Publishing, 4 (1995), 343-358. doi: 10.1142/9789812796417_0022.  Google Scholar

[15]

T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer, Berlin, 1995.  Google Scholar

[16]

J. López-Gómez, On linear weighted boundary value problems, in Partial Differential Equations, Models in Physics and Biology, Mathematical Research, (eds. G. Lumer, S. Nicaise and B. W. Schulze), Akademie Verlag Berlin, 82 (1994), 188-203.  Google Scholar

[17]

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

[18]

J. López-Gómez, Large solutions, metasolutions, and asymptotic behaviour of the regular positive solutions of sublinear parabolic problems, El. J. Diff. Equations, Conf., 5 (2000), 135-171.  Google Scholar

[19]

J. López-Gómez, Approaching metasolutions by solutions, Diff. Int. Eqns., 14 (2001), 739-750.  Google Scholar

[20]

J. López-Gómez, Metasolutions: Malthus versus Verhulst in population dynamics. A dream of Volterra, in Handbook of Differential Equations, Stationary Partial Differential Equations, (eds. M. Chipot and P. Quittner), Elsevier, 2 (2005), 211-309. doi: 10.1016/S1874-5733(05)80012-9.  Google Scholar

[21]

J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, Research Notes in Mathematics 426, Chapman & Hall/CRC Press, Boca Raton (FL), 2001. doi: 10.1201/9781420035506.  Google Scholar

[22]

J. López-Gómez, Elliptic Operators, World Scientific Publishing, Singapore, 2013. Google Scholar

[23]

J. López-Gómez and M. Molina-Meyer, The maximum principle for cooperative weakly coupled elliptic systems and some applications, Diff. Int. Eqns., 7 (1994), 383-398.  Google Scholar

[24]

J. López-Gómez and M. Montenegro, The effects of transport on the maximum principle, J. Math. Anal. Appl., 403 (2013), 547-557. doi: 10.1016/j.jmaa.2013.02.049.  Google Scholar

[25]

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

[26]

D. Sattinger, Topics in Stability and Bifurcation Theory, Lecture Notes in Mathematics 309, Springer, 1973.  Google Scholar

[27]

S. Senn, On a semilinear elliptic eigenvalue problem with Neumann boundary conditions, with an application to population genetics, Comm. Part. Diff. Eqns., 8 (1983), 1199-1228. doi: 10.1080/03605308308820300.  Google Scholar

[28]

S. Senn and P. Hess, On positive solutions of a linear elliptic boundary value problem with Neumann boundary conditions, Math. Annalen, 258 (1982), 459-470. doi: 10.1007/BF01453979.  Google Scholar

[29]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970.  Google Scholar

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