Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Diffusive logistic equation with constant yield harvesting, I: Steady States

Authors: Shobha Oruganti, Junping Shi and Ratnasingham Shivaji
Journal: Trans. Amer. Math. Soc. 354 (2002), 3601-3619
MSC (2000): Primary 35J65; Secondary 35J25, 35B32, 92D25
Published electronically: May 7, 2002
MathSciNet review: 1911513
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider a reaction-diffusion equation which models the constant yield harvesting to a spatially heterogeneous population which satisfies a logistic growth. We prove the existence, uniqueness and stability of the maximal steady state solutions under certain conditions, and we also classify all steady state solutions under more restricted conditions. Exact global bifurcation diagrams are obtained in the latter case. Our method is a combination of comparison arguments and bifurcation theory.

References [Enhancements On Off] (What's this?)

  • [AB] Afrouzi, G. A.; Brown, K. J., On a diffusive logistic equation. J. Math. Anal. Appl., 225, (1998), no. 1, 326-339. MR 99g:35130
  • [ACS] Ali, Ismael; Castro, Alfonso; Shivaji, R., Uniqueness and stability of nonnegative solutions for semipositone problems in a ball. Proc. Amer. Math. Soc., 117, (1993), no. 3, 775-782. MR 93d:35048
  • [ABC] Ambrosetti, Antonio; Brézis, Haim; Cerami, Giovanna, Combined effects of concave and convex nonlinearities in some elliptic problems. J. Funct. Anal., 122, (1994), no. 2, 519-543. MR 95g:35059
  • [BC] Brauer, Fred; Castillo-Chávez, Carlos, Mathematical models in population biology and epidemiology. Texts in Applied Mathematics, 40. Springer-Verlag, New York, (2001). CMP 1 822 695
  • [BK] Brezis, H.; Kamin, S, Sublinear elliptic equations in R$^n$, Manu. Math., 74, (1992), 87-106. MR 93f:35062
  • [CMS] Castro, Alfonso; Maya, C.; Shivaji, R., Nonlinear eigenvalue problems with semipositone structure. Elec. Jour. Differential Equations, Conf. 5., (2000), 33-49. CMP 1 799 043
  • [CS] Castro, Alfonso; Shivaji, R., Positive solutions for a concave semipositone Dirichlet problem. Nonlinear Anal., 31, (1998), no. 1-2, 91-98. MR 98j:35061
  • [CC1] Cantrell, Robert Stephen; Cosner, Chris, Diffusive logistic equations with indefinite weights: population models in disrupted environments. Proc. Roy. Soc. Edinburgh Sect. A, 112, (1989), no. 3-4, 293-318. MR 91b:92015
  • [CC2] Cantrell, R. S.; Cosner, C., The effects of spatial heterogeneity in population dynamics. J. Math. Biol., 29, (1991), no. 4, 315-338. MR 92b:92049
  • [Cl] Clark, Colin W., Mathematical Bioeconomics, The Optimal Management of Renewable Resources, John Wiley & Sons, Inc. New York (1990). MR 91c:90037
  • [CP] Clément, Ph.; Peletier, L. A., An anti-maximum principle for second-order elliptic operators. J. Differential Equations, 34, (1979), no. 2, 218-229. MR 83c:35034
  • [CR1] Crandall, Michael G.; Rabinowitz, Paul H, Bifurcation from simple eigenvalues. J. Functional Analysis, 8, (1971), 321-340. MR 44:5836
  • [CR2] Crandall, Michael G.; Rabinowitz, Paul H., Bifurcation, perturbation of simple eigenvalues and linearized stability. Arch. Rational Mech. Anal., 52, (1973), 161-180. MR 49:5962
  • [F] Fisher, R.A., The wave of advance of advantageous genes. Ann. Eugenics, 7, (1937), 353-369.
  • [He] Henry, Daniel, Geometric theory of semilinear parabolic equations. Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, (1981). MR 83j:35084
  • [Ho] Holling, C.S., The components of predation as revealed by a study of small-mammal predation on the European pine sawfly. Canadian Entomologist, 91, (1959), 294-320.
  • [KPP] Kolmogoroff, A., Petrovsky, I, Piscounoff, N, Study of the diffusion equation with growth of the quantity of matter and its application to a biological problem. (French) Moscow Univ. Bull. Math. 1, (1937), 1-25.
  • [KS] Korman, Philip; Shi, Junping, New exact multiplicity results with an application to a population model, Proceedings of Royal Society of Edinburgh Sect. A, 131, (2001), 1167-1182. CMP 1 862 448
  • [M] Murray, J. D., Mathematical biology. Second edition. Biomathematics, 19., Springer-Verlag, Berlin, (1993). MR 94j:92002
  • [OS] Ouyang Tiancheng; Shi, Junping, Exact multiplicity of positive solutions for a class of semilinear problems: II. Jour. Differential Equations, 158, 94-151, no. 1, (1999). MR 2001b:35117
  • [Sa] Sattinger, D. H., Monotone methods in nonlinear elliptic and parabolic boundary value problems. Indiana Univ. Math. J., 21, (1971/72), 979-1000. MR 45:8969
  • [S] Shi, Junping, Persistence and bifurcation of degenerate solutions. Jour. Func. Anal., 169, (1999), no. 2, 494-531. MR 2001h:47115
  • [SS] Shi, Junping; Shivaji, Ratnasingham, Global bifurcation for concave semipositon problems, Recent Advances in Evolution Equations, (2002). (to appear)
  • [SY] Shi, Junping; Yao, Miaoxin, On a singular nonlinear semilinear elliptic problem. Proc. Roy. Soc. Edingbergh Sect. A, 128, (1998), no. 6, 1389-1401. MR 99j:35070
  • [Y] Yosida, Kosaku, Functional analysis. Fifth edition. Grundlehren der Mathematischen Wissenschaften, Band 123. Springer-Verlag, Berlin-New York, (1978). MR 58:17765

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35J65, 35J25, 35B32, 92D25

Retrieve articles in all journals with MSC (2000): 35J65, 35J25, 35B32, 92D25

Additional Information

Shobha Oruganti
Affiliation: Department of Mathematics, Mississippi State University, Mississippi State, Mississippi 39762

Junping Shi
Affiliation: Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187 and Department of Mathematics, Harbin Normal University, Harbin, Heilongjiang, P. R. China 150080

Ratnasingham Shivaji
Affiliation: Department of Mathematics, Mississippi State University, Mississippi State, Mississippi 39762

Keywords: Diffusive logistic equation, harvesting, steady states, comparison methods, bifurcation
Received by editor(s): September 5, 2001
Received by editor(s) in revised form: October 15, 2001
Published electronically: May 7, 2002
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society