• Previous Article
    Inside dynamics of solutions of integro-differential equations
  • DCDS-B Home
  • This Issue
  • Next Article
    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

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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[19]

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

[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.

[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.

[22]

J. López-Gómez, Elliptic Operators, World Scientific Publishing, Singapore, 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), 383-398.

[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.

[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.

[26]

D. Sattinger, Topics in Stability and Bifurcation Theory, Lecture Notes in Mathematics 309, Springer, 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), 1199-1228. 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), 459-470. doi: 10.1007/BF01453979.

[29]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 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), 125-146. 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), 336-374. 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), 379-397.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[19]

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

[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.

[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.

[22]

J. López-Gómez, Elliptic Operators, World Scientific Publishing, Singapore, 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), 383-398.

[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.

[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.

[26]

D. Sattinger, Topics in Stability and Bifurcation Theory, Lecture Notes in Mathematics 309, Springer, 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), 1199-1228. 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), 459-470. doi: 10.1007/BF01453979.

[29]

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

[1]

Sébastien Guisset. Angular moments models for rarefied gas dynamics. Numerical comparisons with kinetic and Navier-Stokes equations. Kinetic and Related Models, 2020, 13 (4) : 739-758. doi: 10.3934/krm.2020025

[2]

Weiyi Zhang, Zuhan Liu, Ling Zhou. Dynamics of a nonlocal diffusive logistic model with free boundaries in time periodic environment. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3767-3784. doi: 10.3934/dcdsb.2020256

[3]

Kazuaki Taira. A mathematical study of diffusive logistic equations with mixed type boundary conditions. Discrete and Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021166

[4]

Jesús Ildefonso Díaz, Jesús Hernández. Bounded positive solutions for diffusive logistic equations with unbounded distributed limitations. Discrete and Continuous Dynamical Systems - S, 2022, 15 (10) : 2871-2887. doi: 10.3934/dcdss.2022018

[5]

King-Yeung Lam, Wei-Ming Ni. Limiting profiles of semilinear elliptic equations with large advection in population dynamics. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 1051-1067. doi: 10.3934/dcds.2010.28.1051

[6]

Wenzhen Gan, Peng Zhou. A revisit to the diffusive logistic model with free boundary condition. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 837-847. doi: 10.3934/dcdsb.2016.21.837

[7]

Jumpei Inoue, Kousuke Kuto. On the unboundedness of the ratio of species and resources for the diffusive logistic equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2441-2450. doi: 10.3934/dcdsb.2020186

[8]

Rui Peng, Xiao-Qiang Zhao. The diffusive logistic model with a free boundary and seasonal succession. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 2007-2031. doi: 10.3934/dcds.2013.33.2007

[9]

Juan J. Nieto, M. Victoria Otero-Espinar, Rosana Rodríguez-López. Dynamics of the fuzzy logistic family. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 699-717. doi: 10.3934/dcdsb.2010.14.699

[10]

Xuejun Pan, Hongying Shu, Yuming Chen. Dirichlet problem for a diffusive logistic population model with two delays. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3139-3155. doi: 10.3934/dcdss.2020134

[11]

Qingyan Shi, Junping Shi, Yongli Song. Hopf bifurcation and pattern formation in a delayed diffusive logistic model with spatial heterogeneity. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 467-486. doi: 10.3934/dcdsb.2018182

[12]

Tomomi Yokota, Noriaki Yoshino. Existence of solutions to chemotaxis dynamics with logistic source. Conference Publications, 2015, 2015 (special) : 1125-1133. doi: 10.3934/proc.2015.1125

[13]

Xinfu Chen, King-Yeung Lam, Yuan Lou. Corrigendum: Dynamics of a reaction-diffusion-advection model for two competing species. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4989-4995. doi: 10.3934/dcds.2014.34.4989

[14]

Xinfu Chen, King-Yeung Lam, Yuan Lou. Dynamics of a reaction-diffusion-advection model for two competing species. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 3841-3859. doi: 10.3934/dcds.2012.32.3841

[15]

Bedr'Eddine Ainseba. Age-dependent population dynamics diffusive systems. Discrete and Continuous Dynamical Systems - B, 2004, 4 (4) : 1233-1247. doi: 10.3934/dcdsb.2004.4.1233

[16]

Shuo Zhang, Guo Lin. Propagation dynamics in a diffusive SIQR model for childhood diseases. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3241-3259. doi: 10.3934/dcdsb.2021183

[17]

Nalin Fonseka, Ratnasingham Shivaji, Jerome Goddard, Ⅱ, Quinn A. Morris, Byungjae Son. On the effects of the exterior matrix hostility and a U-shaped density dependent dispersal on a diffusive logistic growth model. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3401-3415. doi: 10.3934/dcdss.2020245

[18]

Siyao Zhu, Jinliang Wang. Asymptotic profiles of steady states for a diffusive SIS epidemic model with spontaneous infection and a logistic source. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3323-3340. doi: 10.3934/cpaa.2020147

[19]

Alan E. Lindsay, Michael J. Ward. An asymptotic analysis of the persistence threshold for the diffusive logistic model in spatial environments with localized patches. Discrete and Continuous Dynamical Systems - B, 2010, 14 (3) : 1139-1179. doi: 10.3934/dcdsb.2010.14.1139

[20]

Qiaoling Chen, Fengquan Li, Feng Wang. A diffusive logistic problem with a free boundary in time-periodic environment: Favorable habitat or unfavorable habitat. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 13-35. doi: 10.3934/dcdsb.2016.21.13

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (134)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]