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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

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.
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. doi: 10.1512/iumj.1972.21.21012.

[2]

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

[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.

[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. doi: 10.1007/s00220-004-1201-9.

[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. doi: 10.1016/0022-247X(80)90309-1.

[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. doi: 10.1006/jdeq.2000.4003.

[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. doi: 10.1512/iumj.2008.57.3204.

[8]

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

[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. doi: 10.1090/S0002-9947-00-02534-4.

[10]

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

[11]

R. Gómez-Reñasco, The Effects of Varying Coefficents in Semilinear Elliptic Boundary Value Problems. From Classical Solutons to Metasolutions,, Ph.D Thesis, (1999).

[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. doi: 10.1016/S0362-546X(00)00208-X.

[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. doi: 10.1080/03605308008820162.

[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, 4 (1995), 343. doi: 10.1142/9789812796417_0022.

[15]

T. Kato, Perturbation Theory for Linear Operators,, Classics in Mathematics, (1995).

[16]

J. López-Gómez, On linear weighted boundary value problems,, in Partial Differential Equations, 82 (1994), 188.

[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. doi: 10.1006/jdeq.1996.0070.

[18]

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

[19]

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

[20]

J. López-Gómez, Metasolutions: Malthus versus Verhulst in population dynamics. A dream of Volterra,, in Handbook of Differential Equations, 2 (2005), 211. doi: 10.1016/S1874-5733(05)80012-9.

[21]

J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis,, Research Notes in Mathematics 426, (2001). doi: 10.1201/9781420035506.

[22]

J. López-Gómez, Elliptic Operators,, World Scientific Publishing, (2013).

[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.

[24]

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

[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.

[26]

D. Sattinger, Topics in Stability and Bifurcation Theory,, Lecture Notes in Mathematics 309, (1973).

[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. doi: 10.1080/03605308308820300.

[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. doi: 10.1007/BF01453979.

[29]

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

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. doi: 10.1512/iumj.1972.21.21012.

[2]

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

[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.

[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. doi: 10.1007/s00220-004-1201-9.

[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. doi: 10.1016/0022-247X(80)90309-1.

[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. doi: 10.1006/jdeq.2000.4003.

[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. doi: 10.1512/iumj.2008.57.3204.

[8]

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

[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. doi: 10.1090/S0002-9947-00-02534-4.

[10]

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

[11]

R. Gómez-Reñasco, The Effects of Varying Coefficents in Semilinear Elliptic Boundary Value Problems. From Classical Solutons to Metasolutions,, Ph.D Thesis, (1999).

[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. doi: 10.1016/S0362-546X(00)00208-X.

[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. doi: 10.1080/03605308008820162.

[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, 4 (1995), 343. doi: 10.1142/9789812796417_0022.

[15]

T. Kato, Perturbation Theory for Linear Operators,, Classics in Mathematics, (1995).

[16]

J. López-Gómez, On linear weighted boundary value problems,, in Partial Differential Equations, 82 (1994), 188.

[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. doi: 10.1006/jdeq.1996.0070.

[18]

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

[19]

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

[20]

J. López-Gómez, Metasolutions: Malthus versus Verhulst in population dynamics. A dream of Volterra,, in Handbook of Differential Equations, 2 (2005), 211. doi: 10.1016/S1874-5733(05)80012-9.

[21]

J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis,, Research Notes in Mathematics 426, (2001). doi: 10.1201/9781420035506.

[22]

J. López-Gómez, Elliptic Operators,, World Scientific Publishing, (2013).

[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.

[24]

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

[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.

[26]

D. Sattinger, Topics in Stability and Bifurcation Theory,, Lecture Notes in Mathematics 309, (1973).

[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. doi: 10.1080/03605308308820300.

[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. doi: 10.1007/BF01453979.

[29]

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

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