doi: 10.3934/dcdss.2021166
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

A mathematical study of diffusive logistic equations with mixed type boundary conditions

Institute of Mathematics, University of Tsukuba, Tsukuba 305–8571, Japan

Dedicated to the memory of Professor Rosella Mininni (1963–2020)

Received  August 2021 Early access December 2021

The purpose of this paper is to provide a careful and accessible exposition of static bifurcation theory for a class of mixed type boundary value problems for diffusive logistic equations with indefinite weights, which model population dynamics in environments with spatial heterogeneity. We discuss the changes that occur in the structure of the positive solutions as a parameter varies near the first eigenvalue of the linearized problem, and prove that the most favorable situations will occur if there is a relatively large favorable region (with good resources and without crowding effects) located some distance away from the boundary of the environment. A biological interpretation of main theorem is that an initial population will grow exponentially until limited by lack of available resources if the diffusion rate is below some critical value; this idea is generally credited to the English economist T. R. Malthus. On the other hand, if the diffusion rate is above this critical value, then the model obeys the logistic equation introduced by the Belgian mathematical biologist P. F. Verhulst. The approach in this paper is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in partial differential equations.

Citation: Kazuaki Taira. A mathematical study of diffusive logistic equations with mixed type boundary conditions. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021166
References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd edition, Pure and Applied Mathematics, Vol. 140, Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar

[2]

G. A. Afrouzi and K. J. Brown, On principal eigenvalues for boundary value problems with indefinite weight and Robin boundary conditions, Proc. Amer. Math. Soc., 127 (1999), 125-130.  doi: 10.1090/S0002-9939-99-04561-X.  Google Scholar

[3]

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-709.  doi: 10.1137/1018114.  Google Scholar

[4]

H. Amann, Nonlinear elliptic equations with nonlinear boundary conditions, North-Holland Math. Studies, 21 (1976), 43-63.   Google Scholar

[5] A. Ambrosetti and A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge Studies in Advanced Mathematics, No. 104, Cambridge University Press, Cambridge, 2007.  doi: 10.1017/CBO9780511618260.  Google Scholar
[6] A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis, Cambridge Studies in Advanced Mathematics, No. 34, Cambridge University Press, Cambridge, 1995.   Google Scholar
[7]

J. Bergh and J. Löfström, Interpolation Spaces, An Introduction, Springer-Verlag, Berlin Heidelberg New York, 1976.  Google Scholar

[8]

J.-M. Bony, Principe du maximum dans les espaces de Sobolev, C. R. Acad. Sc. Paris, 265 (1967), 333-336.   Google Scholar

[9]

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

[10]

R. F. Brown, A Topological Introduction to Nonlinear Analysis, 3$^{rd}$ edition, Springer, Cham, 2014. doi: 10.1007/978-3-319-11794-2.  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]

K.-C. Chang, Methods in Nonlinear Analysis, Springer Monogr. Math., Springer-Verlag, Berlin, 2005.  Google Scholar

[13] I. Chavel, Eigenvalues in Riemannian Geometry, Pure and Applied Mathematics, 115. Academic Press, Inc., Orlando, FL, 1984.   Google Scholar
[14]

J. Chazarain and A. Piriou, Introduction à La Théorie Des Équations Aux Dérivées Partielles Linéaires, Gauthier-Villars, Paris, 1981.  Google Scholar

[15]

S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[16]

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

[17]

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180.  doi: 10.1007/BF00282325.  Google Scholar

[18]

E. N. Dancer, Global solution branches for positive mappings, Arch. Rational Mech. Anal., 52 (1973), 181-192.  doi: 10.1007/BF00282326.  Google Scholar

[19]

D. G. de Figueiredo, Positive solutions of semilinear elliptic problems, Differential Equations, 957 (1982), 34-87.   Google Scholar

[20]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar

[21]

P. Drábek and J. Milota, Methods of Nonlinear Analysis, Applications to Differential Equations, 2$^{nd}$ edition, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser/Springer Basel AG, Basel, 2013. doi: 10.1007/978-3-0348-0387-8.  Google Scholar

[22]

W. H. Fleming, A selection-migration model in population genetics, J. Math. Biol., 2 (1975), 219-233.  doi: 10.1007/BF00277151.  Google Scholar

[23]

J. M. FraileP. Koch MedinaJ. López-Gómez and S. Merino, Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation, J. Differential Equations, 127 (1996), 295-319.  doi: 10.1006/jdeq.1996.0071.  Google Scholar

[24]

J. García-MeliánR. Gómez-ReñascoJ. López-Gómez 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. Rational Mech. Anal., 145 (1998), 261-289.  doi: 10.1007/s002050050130.  Google Scholar

[25]

J. García-MeliánJ. D. Rossi and J. C. Sabina de Lis, Existence and uniqueness of positive solutions to elliptic problems with sublinear mixed boundary conditions, Commun. Contemp. Math., 11 (2009), 585-613.  doi: 10.1142/S0219199709003508.  Google Scholar

[26]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001.  Google Scholar

[27]

I. C. Gohberg and M. G. Kreĭn, The basic propositions on defect numbers, root numbers and indices of linear operators, Uspehi Mat. Nauk., 12 (1957), 43–118; English translation: Amer. Math. Soc. Transl., 13 (1960), 185–264. doi: 10.1090/trans2/013/08.  Google Scholar

[28]

P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematical Series, 247, Longman Scientific & Technical, Harlow, 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]

L. Hörmander, The Analysis of Linear Partial Differential Operators III, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 274. Springer-Verlag, Berlin, 1994.  Google Scholar

[31]

M. A. Krasnosel'skii, Positive Solutions of Operator Equations, P. Noordhoff, Groningen, 1964. Google Scholar

[32]

M. G. Kreĭn and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Amer. Math. Soc. Transl., 1950 (1950), 128 pp.  Google Scholar

[33]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, Rhode Island 1968.  Google Scholar

[34] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Equations of Elliptic Type, Nauka, Moscow, 1964 (Russian), English translation; Academic Press, New York London, 1968.   Google Scholar
[35]

J. López-Gómez, Metasolutions: Malthus versus Verhulst in population dynamics, A dream of Volterra, Elsevier/North-Holland, Amsterdam, Stationary Partial Differential Equations, 2 (2005), 211-309.  doi: 10.1016/S1874-5733(05)80012-9.  Google Scholar

[36]

J. López-Gómez, Linear Second Order Elliptic Operators, World Scientific Publishing, Co. Pte. Ltd., Hackensack, NJ, 2013. doi: 10.1142/8664.  Google Scholar

[37]

J. López-Gómez and J. C. Sabina de Lis, First variations of principal eigenvalues with respect to the domain and point-wise growth of positive solutions for problems where bifurcation from infinity occurs, J. Differential Equations, 148 (1998), 47-64.  doi: 10.1006/jdeq.1998.3456.  Google Scholar

[38]

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

[39]

J. Moser, A new proof of de Giorgi's theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math., 13 (1960), 457-468.  doi: 10.1002/cpa.3160130308.  Google Scholar

[40]

L. Nirenberg, Topics in Nonlinear Functional Analysis, New York University, Courant Institute of Mathematical Sciences, New York; revised reprint of the 1974 original, Courant Lecture Notes in Mathematics, No, 6, American Mathematical Society, Providence, Rhode Island, 2001. doi: 10.1090/cln/006.  Google Scholar

[41]

T.-C. 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), 503-527.  doi: 10.2307/2154124.  Google Scholar

[42] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York London, 1992.   Google Scholar
[43]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Corrected reprint of the 1967 original. Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5.  Google Scholar

[44]

P. H. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math., 3 (1973), 161-202.  doi: 10.1216/RMJ-1973-3-2-161.  Google Scholar

[45]

R. Redlinger, Über die $C^{2}$-Kompaktheit der Bahn von Lösungen semilinearer parabolischer systeme, Proc. Roy. Soc. Edinburgh, 93 (1982/83), 99-103.  doi: 10.1017/S0308210500031693.  Google Scholar

[46] M. Reed and B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators, Academic Press, New York San Francisco London, 1978.   Google Scholar
[47]

T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, De Gruyter Series in Nonlinear Analysis and Applications, 3. Walter de Gruyter & Co., Berlin, 1996. doi: 10.1515/9783110812411.  Google Scholar

[48]

D. H. Sattinger, Topics in Stability and Bifurcation Theory, Lecture Notes in Mathematics, No. 309, Springer-Verlag, New York Heidelberg Berlin, 1973.  Google Scholar

[49]

J. C. Saut and B. Scheurer, Remarks on a non linear equation arising in population genetics, Comm. Partial Differential Equations, 3 (1978), 907-931.  doi: 10.1080/03605307808820080.  Google Scholar

[50]

M. Schechter, Principles of Functional Analysis, 2$^{nd}$ edition, Graduate Studies in Mathematics, 36. American Mathematical Society, Providence, RI, 2002. doi: 10.1090/gsm/036.  Google Scholar

[51] N. N. Semenov, Chemical Kinetics and Chain Reactions, Clarendon Press, Oxford, 1935.   Google Scholar
[52]

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

[53]

S. Senn and P. Hess, On positive solutions of a linear elliptic eigenvalue problem with Neumann boundary conditions, Math. Ann., 258 (1981/82), 459-470.  doi: 10.1007/BF01453979.  Google Scholar

[54]

K. Taira, The Yamabe problem and nonlinear boundary value problems, J. Differential Equations, 122 (1995), 316-372.  doi: 10.1006/jdeq.1995.1151.  Google Scholar

[55]

K. Taira, Bifurcation for nonlinear elliptic boundary value problems I, Collect. Math., 47 (1996), 207-229.   Google Scholar

[56]

K. Taira, Boundary value problems for elliptic integro-differential operators, Math. Z., 222 (1996), 305-327.  doi: 10.1007/BF02621868.  Google Scholar

[57]

K. Taira, Introduction to semilinear elliptic boundary value problems, Taiwanese J. Math., 2 (1998), 127-172.  doi: 10.11650/twjm/1500406929.  Google Scholar

[58]

K. Taira, Positive solutions of diffusive logistic equations, Taiwanese J. Math., 5 (2001), 117-140.  doi: 10.11650/twjm/1500574891.  Google Scholar

[59]

K. Taira, Diffusive logistic equations in population dynamics}, Adv. Differential Equations, 7 (2002), 237-256.   Google Scholar

[60]

K. Taira, Introduction to diffusive logistic equations in population dynamics, Korean J. Comput. Appl. Math., 9 (2002), 289-347.  doi: 10.1007/BF03021545.  Google Scholar

[61]

K. Taira, Logistic Dirichlet problems with discontinuous coefficients, J. Math. Pures Appl., 82 (2003), 1137-1190.  doi: 10.1016/S0021-7824(03)00058-8.  Google Scholar

[62]

K. Taira, Diffusive logistic equations with degenerate boundary conditions, Mediterr. J. Math., 1 (2004), 315-365.  doi: 10.1007/s00009-004-0018-2.  Google Scholar

[63]

K. Taira, Degenerate elliptic eigenvalue problems with indefinite weights, Mediterr. J. Math., 5 (2008), 133-162.  doi: 10.1007/s00009-008-0140-7.  Google Scholar

[64]

K. Taira, Degenerate elliptic boundary value problems with asymmetric nonlinearity, J. Math. Soc. Japan, 62 (2010), 431-465.   Google Scholar

[65]

K. Taira, Semigroups, Boundary Value Problems and Markov Processes, 2$^{nd}$ edition, Springer Monographs in Mathematics, Springer, Heidelberg, 2014. doi: 10.1007/978-3-662-43696-7.  Google Scholar

[66] K. Taira, Analytic Semigroups and Semilinear Initial-Boundary Value Problems, 2$^{nd}$ edition, London Mathematical Society Lecture Note Series, 434. Cambridge University Press, Cambridge, 2016.  doi: 10.1017/CBO9781316729755.  Google Scholar
[67]

K. Taira, The hypoelliptic Robin problem for quasilinear elliptic equations, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 1601-1618.  doi: 10.3934/dcdss.2020091.  Google Scholar

[68]

K. Taira, Dirichlet problems with discontinuous coefficients and Feller semigroups, Rend. Circ. Mat. Palermo, 69 (2020), 287-323.  doi: 10.1007/s12215-019-00404-5.  Google Scholar

[69]

K. Taira, Boundary Value Problems and Markov Processes: Functional Analysis Methods for Markov Processes, doi: 10.1007/978-3-030-48788-1.  Google Scholar

[70]

K. Taira, Logistic Neumann problems with discontinuous coefficients, Ann. Univ. Ferrara Sez. VII Sci. Mat., 66 (2020), 409-485.  doi: 10.1007/s11565-020-00350-6.  Google Scholar

[71]

K. Taira, Semilinear degenerate elliptic boundary value problems via the Semenov approximation, Rend. Circ. Mat. Palermo, 70 (2021), 1305-1388.  doi: 10.1007/s12215-020-00560-z.  Google Scholar

[72]

K. Taira and K. Umezu, Bifurcation for nonlinear elliptic boundary value problems II, Tokyo J. Math., 19 (1996), 387-396.  doi: 10.3836/tjm/1270042527.  Google Scholar

[73]

H. Triebel, Theory of Function Spaces, Monographs in Mathematics, Vol. 78, Birkhäuser-Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.  Google Scholar

[74]

H. Triebel, Theory of Function Spaces II, Monographs in Mathematics, Vol. 84, Birkhäuser-Verlag, Basel, 1992 doi: 10.1007/978-3-0346-0419-2.  Google Scholar

[75] G. M. Troianiello, Elliptic Differential Equations and Obstacle Problems, The University Series in Mathematics. Plenum Press, New York, 1987.  doi: 10.1007/978-1-4899-3614-1.  Google Scholar
[76] J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987.  doi: 10.1017/CBO9781139171755.  Google Scholar
[77]

K. Yosida, Functional Analysis, Classics in Mathematics. Springer-Verlag, Berlin, 1995. doi: 10.1007/978-3-642-61859-8.  Google Scholar

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd edition, Pure and Applied Mathematics, Vol. 140, Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar

[2]

G. A. Afrouzi and K. J. Brown, On principal eigenvalues for boundary value problems with indefinite weight and Robin boundary conditions, Proc. Amer. Math. Soc., 127 (1999), 125-130.  doi: 10.1090/S0002-9939-99-04561-X.  Google Scholar

[3]

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-709.  doi: 10.1137/1018114.  Google Scholar

[4]

H. Amann, Nonlinear elliptic equations with nonlinear boundary conditions, North-Holland Math. Studies, 21 (1976), 43-63.   Google Scholar

[5] A. Ambrosetti and A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge Studies in Advanced Mathematics, No. 104, Cambridge University Press, Cambridge, 2007.  doi: 10.1017/CBO9780511618260.  Google Scholar
[6] A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis, Cambridge Studies in Advanced Mathematics, No. 34, Cambridge University Press, Cambridge, 1995.   Google Scholar
[7]

J. Bergh and J. Löfström, Interpolation Spaces, An Introduction, Springer-Verlag, Berlin Heidelberg New York, 1976.  Google Scholar

[8]

J.-M. Bony, Principe du maximum dans les espaces de Sobolev, C. R. Acad. Sc. Paris, 265 (1967), 333-336.   Google Scholar

[9]

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

[10]

R. F. Brown, A Topological Introduction to Nonlinear Analysis, 3$^{rd}$ edition, Springer, Cham, 2014. doi: 10.1007/978-3-319-11794-2.  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]

K.-C. Chang, Methods in Nonlinear Analysis, Springer Monogr. Math., Springer-Verlag, Berlin, 2005.  Google Scholar

[13] I. Chavel, Eigenvalues in Riemannian Geometry, Pure and Applied Mathematics, 115. Academic Press, Inc., Orlando, FL, 1984.   Google Scholar
[14]

J. Chazarain and A. Piriou, Introduction à La Théorie Des Équations Aux Dérivées Partielles Linéaires, Gauthier-Villars, Paris, 1981.  Google Scholar

[15]

S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[16]

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

[17]

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180.  doi: 10.1007/BF00282325.  Google Scholar

[18]

E. N. Dancer, Global solution branches for positive mappings, Arch. Rational Mech. Anal., 52 (1973), 181-192.  doi: 10.1007/BF00282326.  Google Scholar

[19]

D. G. de Figueiredo, Positive solutions of semilinear elliptic problems, Differential Equations, 957 (1982), 34-87.   Google Scholar

[20]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar

[21]

P. Drábek and J. Milota, Methods of Nonlinear Analysis, Applications to Differential Equations, 2$^{nd}$ edition, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser/Springer Basel AG, Basel, 2013. doi: 10.1007/978-3-0348-0387-8.  Google Scholar

[22]

W. H. Fleming, A selection-migration model in population genetics, J. Math. Biol., 2 (1975), 219-233.  doi: 10.1007/BF00277151.  Google Scholar

[23]

J. M. FraileP. Koch MedinaJ. López-Gómez and S. Merino, Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation, J. Differential Equations, 127 (1996), 295-319.  doi: 10.1006/jdeq.1996.0071.  Google Scholar

[24]

J. García-MeliánR. Gómez-ReñascoJ. López-Gómez 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. Rational Mech. Anal., 145 (1998), 261-289.  doi: 10.1007/s002050050130.  Google Scholar

[25]

J. García-MeliánJ. D. Rossi and J. C. Sabina de Lis, Existence and uniqueness of positive solutions to elliptic problems with sublinear mixed boundary conditions, Commun. Contemp. Math., 11 (2009), 585-613.  doi: 10.1142/S0219199709003508.  Google Scholar

[26]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001.  Google Scholar

[27]

I. C. Gohberg and M. G. Kreĭn, The basic propositions on defect numbers, root numbers and indices of linear operators, Uspehi Mat. Nauk., 12 (1957), 43–118; English translation: Amer. Math. Soc. Transl., 13 (1960), 185–264. doi: 10.1090/trans2/013/08.  Google Scholar

[28]

P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematical Series, 247, Longman Scientific & Technical, Harlow, 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]

L. Hörmander, The Analysis of Linear Partial Differential Operators III, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 274. Springer-Verlag, Berlin, 1994.  Google Scholar

[31]

M. A. Krasnosel'skii, Positive Solutions of Operator Equations, P. Noordhoff, Groningen, 1964. Google Scholar

[32]

M. G. Kreĭn and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Amer. Math. Soc. Transl., 1950 (1950), 128 pp.  Google Scholar

[33]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, Rhode Island 1968.  Google Scholar

[34] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Equations of Elliptic Type, Nauka, Moscow, 1964 (Russian), English translation; Academic Press, New York London, 1968.   Google Scholar
[35]

J. López-Gómez, Metasolutions: Malthus versus Verhulst in population dynamics, A dream of Volterra, Elsevier/North-Holland, Amsterdam, Stationary Partial Differential Equations, 2 (2005), 211-309.  doi: 10.1016/S1874-5733(05)80012-9.  Google Scholar

[36]

J. López-Gómez, Linear Second Order Elliptic Operators, World Scientific Publishing, Co. Pte. Ltd., Hackensack, NJ, 2013. doi: 10.1142/8664.  Google Scholar

[37]

J. López-Gómez and J. C. Sabina de Lis, First variations of principal eigenvalues with respect to the domain and point-wise growth of positive solutions for problems where bifurcation from infinity occurs, J. Differential Equations, 148 (1998), 47-64.  doi: 10.1006/jdeq.1998.3456.  Google Scholar

[38]

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

[39]

J. Moser, A new proof of de Giorgi's theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math., 13 (1960), 457-468.  doi: 10.1002/cpa.3160130308.  Google Scholar

[40]

L. Nirenberg, Topics in Nonlinear Functional Analysis, New York University, Courant Institute of Mathematical Sciences, New York; revised reprint of the 1974 original, Courant Lecture Notes in Mathematics, No, 6, American Mathematical Society, Providence, Rhode Island, 2001. doi: 10.1090/cln/006.  Google Scholar

[41]

T.-C. 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), 503-527.  doi: 10.2307/2154124.  Google Scholar

[42] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York London, 1992.   Google Scholar
[43]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Corrected reprint of the 1967 original. Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5.  Google Scholar

[44]

P. H. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math., 3 (1973), 161-202.  doi: 10.1216/RMJ-1973-3-2-161.  Google Scholar

[45]

R. Redlinger, Über die $C^{2}$-Kompaktheit der Bahn von Lösungen semilinearer parabolischer systeme, Proc. Roy. Soc. Edinburgh, 93 (1982/83), 99-103.  doi: 10.1017/S0308210500031693.  Google Scholar

[46] M. Reed and B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators, Academic Press, New York San Francisco London, 1978.   Google Scholar
[47]

T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, De Gruyter Series in Nonlinear Analysis and Applications, 3. Walter de Gruyter & Co., Berlin, 1996. doi: 10.1515/9783110812411.  Google Scholar

[48]

D. H. Sattinger, Topics in Stability and Bifurcation Theory, Lecture Notes in Mathematics, No. 309, Springer-Verlag, New York Heidelberg Berlin, 1973.  Google Scholar

[49]

J. C. Saut and B. Scheurer, Remarks on a non linear equation arising in population genetics, Comm. Partial Differential Equations, 3 (1978), 907-931.  doi: 10.1080/03605307808820080.  Google Scholar

[50]

M. Schechter, Principles of Functional Analysis, 2$^{nd}$ edition, Graduate Studies in Mathematics, 36. American Mathematical Society, Providence, RI, 2002. doi: 10.1090/gsm/036.  Google Scholar

[51] N. N. Semenov, Chemical Kinetics and Chain Reactions, Clarendon Press, Oxford, 1935.   Google Scholar
[52]

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

[53]

S. Senn and P. Hess, On positive solutions of a linear elliptic eigenvalue problem with Neumann boundary conditions, Math. Ann., 258 (1981/82), 459-470.  doi: 10.1007/BF01453979.  Google Scholar

[54]

K. Taira, The Yamabe problem and nonlinear boundary value problems, J. Differential Equations, 122 (1995), 316-372.  doi: 10.1006/jdeq.1995.1151.  Google Scholar

[55]

K. Taira, Bifurcation for nonlinear elliptic boundary value problems I, Collect. Math., 47 (1996), 207-229.   Google Scholar

[56]

K. Taira, Boundary value problems for elliptic integro-differential operators, Math. Z., 222 (1996), 305-327.  doi: 10.1007/BF02621868.  Google Scholar

[57]

K. Taira, Introduction to semilinear elliptic boundary value problems, Taiwanese J. Math., 2 (1998), 127-172.  doi: 10.11650/twjm/1500406929.  Google Scholar

[58]

K. Taira, Positive solutions of diffusive logistic equations, Taiwanese J. Math., 5 (2001), 117-140.  doi: 10.11650/twjm/1500574891.  Google Scholar

[59]

K. Taira, Diffusive logistic equations in population dynamics}, Adv. Differential Equations, 7 (2002), 237-256.   Google Scholar

[60]

K. Taira, Introduction to diffusive logistic equations in population dynamics, Korean J. Comput. Appl. Math., 9 (2002), 289-347.  doi: 10.1007/BF03021545.  Google Scholar

[61]

K. Taira, Logistic Dirichlet problems with discontinuous coefficients, J. Math. Pures Appl., 82 (2003), 1137-1190.  doi: 10.1016/S0021-7824(03)00058-8.  Google Scholar

[62]

K. Taira, Diffusive logistic equations with degenerate boundary conditions, Mediterr. J. Math., 1 (2004), 315-365.  doi: 10.1007/s00009-004-0018-2.  Google Scholar

[63]

K. Taira, Degenerate elliptic eigenvalue problems with indefinite weights, Mediterr. J. Math., 5 (2008), 133-162.  doi: 10.1007/s00009-008-0140-7.  Google Scholar

[64]

K. Taira, Degenerate elliptic boundary value problems with asymmetric nonlinearity, J. Math. Soc. Japan, 62 (2010), 431-465.   Google Scholar

[65]

K. Taira, Semigroups, Boundary Value Problems and Markov Processes, 2$^{nd}$ edition, Springer Monographs in Mathematics, Springer, Heidelberg, 2014. doi: 10.1007/978-3-662-43696-7.  Google Scholar

[66] K. Taira, Analytic Semigroups and Semilinear Initial-Boundary Value Problems, 2$^{nd}$ edition, London Mathematical Society Lecture Note Series, 434. Cambridge University Press, Cambridge, 2016.  doi: 10.1017/CBO9781316729755.  Google Scholar
[67]

K. Taira, The hypoelliptic Robin problem for quasilinear elliptic equations, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 1601-1618.  doi: 10.3934/dcdss.2020091.  Google Scholar

[68]

K. Taira, Dirichlet problems with discontinuous coefficients and Feller semigroups, Rend. Circ. Mat. Palermo, 69 (2020), 287-323.  doi: 10.1007/s12215-019-00404-5.  Google Scholar

[69]

K. Taira, Boundary Value Problems and Markov Processes: Functional Analysis Methods for Markov Processes, doi: 10.1007/978-3-030-48788-1.  Google Scholar

[70]

K. Taira, Logistic Neumann problems with discontinuous coefficients, Ann. Univ. Ferrara Sez. VII Sci. Mat., 66 (2020), 409-485.  doi: 10.1007/s11565-020-00350-6.  Google Scholar

[71]

K. Taira, Semilinear degenerate elliptic boundary value problems via the Semenov approximation, Rend. Circ. Mat. Palermo, 70 (2021), 1305-1388.  doi: 10.1007/s12215-020-00560-z.  Google Scholar

[72]

K. Taira and K. Umezu, Bifurcation for nonlinear elliptic boundary value problems II, Tokyo J. Math., 19 (1996), 387-396.  doi: 10.3836/tjm/1270042527.  Google Scholar

[73]

H. Triebel, Theory of Function Spaces, Monographs in Mathematics, Vol. 78, Birkhäuser-Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.  Google Scholar

[74]

H. Triebel, Theory of Function Spaces II, Monographs in Mathematics, Vol. 84, Birkhäuser-Verlag, Basel, 1992 doi: 10.1007/978-3-0346-0419-2.  Google Scholar

[75] G. M. Troianiello, Elliptic Differential Equations and Obstacle Problems, The University Series in Mathematics. Plenum Press, New York, 1987.  doi: 10.1007/978-1-4899-3614-1.  Google Scholar
[76] J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987.  doi: 10.1017/CBO9781139171755.  Google Scholar
[77]

K. Yosida, Functional Analysis, Classics in Mathematics. Springer-Verlag, Berlin, 1995. doi: 10.1007/978-3-642-61859-8.  Google Scholar

Figure 1.  The bounded domain $ D $ and the unit outward normal $ \mathbf{n} $ to $ \partial D $
Figure 2.  The boundary portion $ M $ is deadly and its complement $ {\partial D} \setminus M $ is a barrier
Figure 3.  The structural condition (Z.1) on the function $ h(x) $
Figure 4.  The bifurcation diagram of Theorem 1.5
Figure 5.  The bifurcation diagram of Theorem 1.5: Malthus versus Verhulst
Figure 6.  The positive solution curve (16) for $ \lambda > \lambda_{1}(m) $ under condition (Z.3) via the Semenov approximation
Figure 7.  The bifurcation diagram of Remark 1.4 under condition (Z.3) (Verhulst theory)
Figure 8.  Conditions (b) and (d) in Theorem 3.3
Figure 9.  The bifurcation curves $ \varGamma_{1} $ and $ \varGamma_{2} $ of the nonlinear equation (20) in Theorem 3.3
Figure 10.  The point $ \left(1/ \mathop{\mathrm{spr}}(B), 0\right) $ is a bifurcation point of the nonlinear equation (21) to the trivial solution in Theorem 3.4
Figure 11.  The mapping properties of the resolvent $ R_{c} $ in the spaces $ C(\overline{D}) $, $ W^{2,p}(D) $ and $ C^{1}_{B}(\overline{D}) $
Figure 12.  The mapping properties of the resolvent $ R_{c} = \left(- \varDelta + c(x)\right)^{-1} $ in the spaces $ C(\overline{D}) $, $ C_{e}(\overline{D}) $ and $ C^{1}_{B}(\overline{D}) $
Figure 13.  The first eigenvalues $ \mu_{1}(\lambda) = \gamma_{1}(\lambda) - \lambda $, $ \mu_{1}(0) = \gamma_{1} $ and $ \mu_{1}\left(\lambda_{1}(m)\right) = 0 $
Figure 14.  The first eigenvalues $ \mu_{D}(\lambda) $, $ \mu_{N}(\lambda) $ and $ \mu_{1}(\lambda) $ in the case $ \int_{D}m(x)\,dx < 0 $
Figure 15.  The first eigenvalues $ \mu_{D}(\lambda) $, $ \mu_{N}(\lambda) $ and $ \mu_{1}(\lambda) $ in the case $ \int_{D}m(x)\,dx = 0 $
Figure 16.  The first eigenvalues $ \mu_{D}(\lambda) $, $ \mu_{N}(\lambda) $ and $ \mu_{1}(\lambda) $ in the case $ \int_{D}m(x)\,dx > 0 $
Figure 17.  A flowchart of proof of Theorem 1.5, part (i)
Figure 18.  A flowchart of proof of Lemma 7.2
Figure 19.  The set of solutions of the semilinear problem (1) consists of a pitchfork near $ \lambda = \lambda_{1}(m) $
Figure 20.  The critical value $ \overline{\lambda}(h) $ of the positive bifurcation solution curve $ \mathcal{C} = \{(\lambda, u(\lambda))\} $
Figure 21.  The mapping properties of the resolvent $ R_{\lambda} = (\lambda I - \varDelta)^{-1} $ in the spaces $ C(\overline{D}) $, $ C_{e}(\overline{D}) $ and $ C^{1}_{B}(\overline{D}) $
Figure 22.  A positive bifurcation solution curve $ (\lambda, u(\lambda)) $ of the nonlinear operator equation $ u = H(\lambda,u) $ can be continued beyond the point $ (\lambda^{\ast}, u^{\ast}) $ via the implicit function theorem (Theorem 3.1)
Figure 23.  The mapping properties of the negative Laplacian $ - \varDelta $ and the resolvent $ R_{0} = \left(- \varDelta\right)^{-1} $
Figure 24.  A flowchart of proof of Theorem 1.5, part (ii)
Figure 25.  The bifurcation diagram of Theorem 8.1 (the Dirichlet case)
Figure 26.  The bifurcation diagram of Theorem 8.2 in the case $ \int_{D}m(x)\,dx < 0 $ and $ \nu_{1}(m) > 0 $ (the Neumann case)
Figure 27.  The bifurcation diagram of Theorem 8.2 in the case $ \int_{D}m(x)\,dx = 0 $ and $ \nu_{1}(m) = 0 $ (the Neumann case)
Figure 28.  The bifurcation diagram of Theorem 8.2 in the case $ \int_{D}m(x)\,dx > 0 $ and $ \nu_{1}(m) < 0 $ (the Neumann case)
Figure 30.  The bifurcation diagrams of Theorem 8.3 in the case $ \int_{D}m(x)\,dx < 0 $ and $ \nu_{1}(m) > 0 $
Figure 31.  The bifurcation diagrams of Theorem 8.3 in the case $ \int_{D}m(x)\,dx = 0 $ and $ \nu_{1}(m) = 0 $
Figure 32.  The bifurcation diagrams of Theorem 8.3 in the case $ \int_{D}m(x)\,dx > 0 $ and $ \nu_{1}(m) < 0 $
Figure 29.  The open subset $ D^{+} $ with boundary $ \partial D^{+} $
Table 1.  A biological meaning of each term
Term Biological interpretation
D Terrain
x Location of the terrain
u(x) Population density of a species inhabiting the terrain
A member of the population moves about the terrain via the type of random walks occurring in Brownian motion
$\frac{1}{\lambda }$ Rate of diffusive dispersal
m(x) Intrinsic growth rate
h(x) Coefficient of intraspecific competition
Term Biological interpretation
D Terrain
x Location of the terrain
u(x) Population density of a species inhabiting the terrain
A member of the population moves about the terrain via the type of random walks occurring in Brownian motion
$\frac{1}{\lambda }$ Rate of diffusive dispersal
m(x) Intrinsic growth rate
h(x) Coefficient of intraspecific competition
Table 2.  A biological meaning of boundary conditions
Boundary Condition Biological interpretation
Dirichlet case
(a(x') ≡ 0, b(x') ≡ 1)
Completely hostile (deadly) exterior
Neumann case
(a(x') ≡ 1, b(x') ≡ 0)
Barrier
Robin or mixed-type case
(a(x') + b(x') > 0)
Hostile but not completely deadly exterior
Boundary Condition Biological interpretation
Dirichlet case
(a(x') ≡ 0, b(x') ≡ 1)
Completely hostile (deadly) exterior
Neumann case
(a(x') ≡ 1, b(x') ≡ 0)
Barrier
Robin or mixed-type case
(a(x') + b(x') > 0)
Hostile but not completely deadly exterior
Table 3.  An overview of theorems for eigenvalue problems with indefinite weights
Problems Conditions Theorems
(6)
(mixed type case)
(M.1) (H.1), (H.2) Theorem 1.3 for λ1(m)
(55)
(Dirichlet case)
(M.1) Theorem 6.1 for γ1(m)
(58)
(Neumann case)
(M.2) Theorem 6.2 for ν1(m)
(56), (59), (61) (M.1), (M.2)
(H.1), (H.2)
Theorem 6.3
for µD(λ), µN(λ), µ1(λ)
Problems Conditions Theorems
(6)
(mixed type case)
(M.1) (H.1), (H.2) Theorem 1.3 for λ1(m)
(55)
(Dirichlet case)
(M.1) Theorem 6.1 for γ1(m)
(58)
(Neumann case)
(M.2) Theorem 6.2 for ν1(m)
(56), (59), (61) (M.1), (M.2)
(H.1), (H.2)
Theorem 6.3
for µD(λ), µN(λ), µ1(λ)
Table 4.  An overview of existence theorems for diffusive logistic problems
Problems Conditions Theorems
(1)
(mixed type case)
(M.1)
(Z.1), (Z.2)
(H.1), (H.2)
Theorem 1.5 for u(λ)
(91)
(Dirichlet case)
(M.1)
(Z.1), (Z.2)
Theorem 8.1 for v(λ)
(93)
(Neumann case)
(M.2)
(Z.1), (Z.2)
Theorem 8.2 for w(λ)
(1), (91), (93) (M.1), (M.2)
(Z.1), (Z.2)
(H.1), (H.2)
Theorem 8.3 for v(λ), w(λ), u(λ)
(1)
(mixed type case)
(M.1), (Z.3)
(H.1), (H.2)
Theorem 1.6 for u(λ)
Problems Conditions Theorems
(1)
(mixed type case)
(M.1)
(Z.1), (Z.2)
(H.1), (H.2)
Theorem 1.5 for u(λ)
(91)
(Dirichlet case)
(M.1)
(Z.1), (Z.2)
Theorem 8.1 for v(λ)
(93)
(Neumann case)
(M.2)
(Z.1), (Z.2)
Theorem 8.2 for w(λ)
(1), (91), (93) (M.1), (M.2)
(Z.1), (Z.2)
(H.1), (H.2)
Theorem 8.3 for v(λ), w(λ), u(λ)
(1)
(mixed type case)
(M.1), (Z.3)
(H.1), (H.2)
Theorem 1.6 for u(λ)
[1]

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

[2]

Santiago Cano-Casanova. Bifurcation to positive solutions in BVPs of logistic type with nonlinear indefinite mixed boundary conditions. Conference Publications, 2013, 2013 (special) : 95-104. doi: 10.3934/proc.2013.2013.95

[3]

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

[4]

W. E. Fitzgibbon, M.E. Parrott, Glenn Webb. Diffusive epidemic models with spatial and age dependent heterogeneity. Discrete & Continuous Dynamical Systems, 1995, 1 (1) : 35-57. doi: 10.3934/dcds.1995.1.35

[5]

Xiaoyan Zhang, Yuxiang Zhang. Spatial dynamics of a reaction-diffusion cholera model with spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2625-2640. doi: 10.3934/dcdsb.2018124

[6]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825

[7]

Michael E. Filippakis, Donal O'Regan, Nikolaos S. Papageorgiou. Positive solutions and bifurcation phenomena for nonlinear elliptic equations of logistic type: The superdiffusive case. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1507-1527. doi: 10.3934/cpaa.2010.9.1507

[8]

Tzung-shin Yeh. S-shaped and broken s-shaped bifurcation curves for a multiparameter diffusive logistic problem with holling type-Ⅲ functional response. Communications on Pure & Applied Analysis, 2017, 16 (2) : 645-670. doi: 10.3934/cpaa.2017032

[9]

Bastian Gebauer, Nuutti Hyvönen. Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem. Inverse Problems & Imaging, 2008, 2 (3) : 355-372. doi: 10.3934/ipi.2008.2.355

[10]

Stephen Pankavich, Christian Parkinson. Mathematical analysis of an in-host model of viral dynamics with spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1237-1257. doi: 10.3934/dcdsb.2016.21.1237

[11]

Cong He, Hongjun Yu. Large time behavior of the solution to the Landau Equation with specular reflective boundary condition. Kinetic & Related Models, 2013, 6 (3) : 601-623. doi: 10.3934/krm.2013.6.601

[12]

Alain Hertzog, Antoine Mondoloni. Existence of a weak solution for a quasilinear wave equation with boundary condition. Communications on Pure & Applied Analysis, 2002, 1 (2) : 191-219. doi: 10.3934/cpaa.2002.1.191

[13]

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

[14]

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

[15]

Lin Wang, James Watmough, Fang Yu. Bifurcation analysis and transient spatio-temporal dynamics for a diffusive plant-herbivore system with Dirichlet boundary conditions. Mathematical Biosciences & Engineering, 2015, 12 (4) : 699-715. doi: 10.3934/mbe.2015.12.699

[16]

Jian-Wen Sun, Wan-Tong Li, Zhi-Cheng Wang. A nonlocal dispersal logistic equation with spatial degeneracy. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 3217-3238. doi: 10.3934/dcds.2015.35.3217

[17]

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

[18]

Muhammad I. Mustafa. On the control of the wave equation by memory-type boundary condition. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 1179-1192. doi: 10.3934/dcds.2015.35.1179

[19]

Jesús Ildefonso Díaz, L. Tello. On a climate model with a dynamic nonlinear diffusive boundary condition. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 253-262. doi: 10.3934/dcdss.2008.1.253

[20]

Chris Cosner, Andrew L. Nevai. Spatial population dynamics in a producer-scrounger model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1591-1607. doi: 10.3934/dcdsb.2015.20.1591

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (58)
  • HTML views (31)
  • Cited by (0)

Other articles
by authors

[Back to Top]