Abstract
The paper studies families of positive solution curves for non-autonomous two-point problems
depending on two positive parameters \(\lambda\) and \(\mu\). We regard \(\lambda\) as a primary parameter, giving us the solution curves, while the secondary parameter \(\mu\) allows for evolution of these curves. We give conditions under which the solution curves do not intersect, and the maximum value of solutions provides a global parameter. Our primary application is to constant yield harvesting for diffusive logistic equation. We implement numerical computations of the solution curves, using continuation in a global parameter, a technique that we developed in (Korman in Nonlinear Anal. 93:226, 2013).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Allgower, E.L., Georg, K.: Numerical Continuation Methods. An Introduction. Springer Series in Computational Mathematics, vol. 13. Springer, Berlin (1990)
Crandall, M.G., Rabinowitz, P.H.: Bifurcation, perturbation of simple eigenvalues and linearized stability. Arch. Ration. Mech. Anal. 52, 161–180 (1973)
Costa, D.G., Drábek, P., Tehrani, H.: Positive solutions to semilinear elliptic equations with logistic type nonlinearities and constant yield harvesting in \(R^{n}\). Commun. Partial Differ. Equ. 33, 1597–1610 (2008)
Gidas, B., Ni, W.-M., Nirenberg, L.: Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68, 209–243 (1979)
Girão, P., Tehrani, H.: Positive solutions to logistic type equations with harvesting. J. Differ. Equ. 247(2), 574–595 (2009)
Hastings, S.P., McLeod, J.B.: Classical Methods in Ordinary Differential Equations. With Applications to Boundary Value Problems. Graduate Studies in Mathematics, vol. 129. Am. Math. Soc., Providence (2012)
Hung, K.C., Wang, S.H.: Classification and evolution of bifurcation curves for a multiparameter p-Laplacian Dirichlet problem. Nonlinear Anal. 74(11), 3589–3598 (2011)
Hung, K.C., Wang, S.H.: Bifurcation diagrams of a p-Laplacian Dirichlet problem with Allee effect and an application to a diffusive logistic equation with predation. J. Math. Anal. Appl. 375(1), 294–309 (2011)
Korman, P.: Symmetry of positive solutions for elliptic problems in one dimension. Appl. Anal. 58(3–4), 351–365 (1995)
Korman, P.: Global Solution Curves for Semilinear Elliptic Equations. World Scientific, Hackensack (2012)
Korman, P.: Exact multiplicity and numerical computation of solutions for two classes of non-autonomous problems with concave-convex nonlinearities. Nonlinear Anal. 93, 226–235 (2013)
Korman, P., Li, Y., Ouyang, T.: Exact multiplicity results for boundary-value problems with nonlinearities generalising cubic. Proc. R. Soc. Edinb., Sect. A 126A, 599–616 (1996)
Nirenberg, L.: Topics in Nonlinear Functional Analysis. Courant Institute Lecture Notes. Am. Math. Soc., Providence (1974)
Oruganti, S., Shi, J., Shivaji, R.: Diffusive logistic equation with constant yield harvesting. I. Steady states. Transl. Am. Math. Soc. 354(9), 3601–3619 (2002)
Ouyang, T., Shi, J.: Exact multiplicity of positive solutions for a class of semilinear problems, II. J. Differ. Equ. 158(1), 94–151 (1999)
Shi, J.: A radially symmetric anti-maximum principle and applications to fishery management models. Electron. J. Differ. Equ. 27 (2004), 13 pp. (electronic)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Korman, P. Families of Solution Curves for Some Non-autonomous Problems. Acta Appl Math 143, 165–178 (2016). https://doi.org/10.1007/s10440-015-0033-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10440-015-0033-2