Study of a logistic equation with local and non-local reaction terms

Manuel Delgado, Giovany M. Figueiredo, Marcos T.O. Pimenta, Antonio Suárez



We examine a logistic equation with local and non-local reaction terms both for time dependent and steady-state problems. Mainly, we use bifurcation and monotonicity methods to prove the existence of positive solutions for the steady-state equation and sub-supersolution method for the long time behavior for the time dependent problem. The results depend strongly on the size and sign of the parameters on the local and non-local terms.


Logistic equation; local and non-local terms; bifurcation methods

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W. Allegretto and A. Barabanova, Existence of positive solutions of semilinear elliptic equations with nonlocal terms, Funkcial. Ekvac. 40 (1997), 395–409.

M. G. Crandall and P.H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), 321–340.

F.J.S.A. Corrêa, M. Delgado and A. Suárez, Some nonlinear heterogeneous problems with nonlocal reaction term, Adv. Differential Equations 16 (2011), 623–641.

F.J.S.A. Corrêa, M. Delgado and A. Suárez, Some non-local population models with non-linear diffusion, Math. Comput. Modelling 54 (2011), 2293–2305.

F.A. Davidson and N. Dodds, Existence of positive solutions due to nonlocal interactions in a class of nonlinear boundary value problems, Methods Appl. Anal. 14 (2007), 15–27.

M. Delgado, J. López-Gómez and A. Suárez, On the symbiotic Lotka–Volterra model with diffusion and transport effects, J. Differential Equations 160 (2000), 175–262.

P. Freitas, Nonlocal reaction-diffusion equations, Differential equations with applications to biology (Halifax, NS, 1997), 187–204, Fields Inst. Commun., 21, Amer. Math. Soc., Providence, RI, 1999.

J. Furter and M. Grinfeld, Local vs. nonlocal interactions in population dynamics, J. Math. Biol. 27 (1989), 65–80.

J. Garcı́a-Melián, Multiplicity of positive solutions to boundary blow-up elliptic problems with signchanging weights, J. Funct. Anal. 261 (2011), 1775–1798.

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 (1981), 883–901.

S. B. Hsu, J. López-Gómez, L. Mei and M. Molina-Meyer, A nonlocal problem from conservation biology, SIAM J. Math. Anal. 46 (2014), 435–459.

J. López-Gómez, On the structure and stability of the set of solutions of a nonlocal problem modeling Ohmic heating, J. Dynam. Differential Equations 10 (1998), 537–566.

J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, Research Notes in Mathe matical Series 426, Chapman & Hall / CRC, Florida 2001.

J. López-Gómez, Varying bifurcation diagrams of positive solutions for a class of indefinite super linear boundary value problems, Trans. Amer. Math. Soc. 352 (2000), 1825–1858.

J. López-Gómez, Global existence versus blow-up in superlinear indefinite parabolic problems, Sci. Math. Jpn. 61 (2005), 493–516.

J. López-Gómez, Linear Second Order Elliptic Operators, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013.

J. López-Gómez and P. Quittner, Complete and energy blow-up in indefinite superlinear parabolic problems, Crete Contin. Dyn. Syst. 14 (2006), 169–186.

J. López-Gómez, A. Tellini and F. Zanolin, High multiplicity and complexity of the bifurcation diagrams of large solutions for a class of superlinear indefinite problems, Commun. Pure Appl. Anal. 13 (2014), 1–73.

P. Quittner, and P. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and steady States, Birkhäuser Advanced Texts: Basel Textbooks, Verlag, Basel, 2007.

P. Quittner, and P. Souplet, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.

P. Rouchon, Boundedness of global solutions of nonlinear diffusion equation with localized reaction term, Differential Integral Equations 16 (2003), 1083–1092.


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