Diffusive logistic equation with constant yield harvesting, I: Steady States
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- by Shobha Oruganti, Junping Shi and Ratnasingham Shivaji PDF
- Trans. Amer. Math. Soc. 354 (2002), 3601-3619 Request permission
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
- G. A. Afrouzi and K. J. Brown, On a diffusive logistic equation, J. Math. Anal. Appl. 225 (1998), no. 1, 326–339. MR 1639260, DOI 10.1006/jmaa.1998.6044
- Ismael Ali, Alfonso Castro, and R. Shivaji, Uniqueness and stability of nonnegative solutions for semipositone problems in a ball, Proc. Amer. Math. Soc. 117 (1993), no. 3, 775–782. MR 1116249, DOI 10.1090/S0002-9939-1993-1116249-5
- Antonio Ambrosetti, Haïm Brezis, and Giovanna Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994), no. 2, 519–543. MR 1276168, DOI 10.1006/jfan.1994.1078
- Brauer, Fred; Castillo-Chávez, Carlos, Mathematical models in population biology and epidemiology. Texts in Applied Mathematics, 40. Springer-Verlag, New York, (2001).
- Haïm Brezis and Shoshana Kamin, Sublinear elliptic equations in $\textbf {R}^n$, Manuscripta Math. 74 (1992), no. 1, 87–106. MR 1141779, DOI 10.1007/BF02567660
- Castro, Alfonso; Maya, C.; Shivaji, R., Nonlinear eigenvalue problems with semipositone structure. Elec. Jour. Differential Equations, Conf. 5., (2000), 33–49.
- Alfonso Castro and R. Shivaji, Positive solutions for a concave semipositone Dirichlet problem, Nonlinear Anal. 31 (1998), no. 1-2, 91–98. MR 1487532, DOI 10.1016/S0362-546X(96)00189-7
- Robert Stephen Cantrell and Chris Cosner, 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 1014659, DOI 10.1017/S030821050001876X
- R. S. Cantrell and C. Cosner, The effects of spatial heterogeneity in population dynamics, J. Math. Biol. 29 (1991), no. 4, 315–338. MR 1105497, DOI 10.1007/BF00167155
- Colin W. Clark, Mathematical bioeconomics, 2nd ed., Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1990. The optimal management of renewable resources; With a contribution by Gordon Munro; A Wiley-Interscience Publication. MR 1044994
- Ph. Clément and L. A. Peletier, An anti-maximum principle for second-order elliptic operators, J. Differential Equations 34 (1979), no. 2, 218–229. MR 550042, DOI 10.1016/0022-0396(79)90006-8
- Michael G. Crandall and Paul H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis 8 (1971), 321–340. MR 0288640, DOI 10.1016/0022-1236(71)90015-2
- Michael G. Crandall and Paul H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal. 52 (1973), 161–180. MR 341212, DOI 10.1007/BF00282325
- Fisher, R.A., The wave of advance of advantageous genes. Ann. Eugenics, 7, (1937), 353–369.
- Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244
- 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.
- 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.
- 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.
- J. D. Murray, Mathematical biology, 2nd ed., Biomathematics, vol. 19, Springer-Verlag, Berlin, 1993. MR 1239892, DOI 10.1007/b98869
- Tiancheng Ouyang and Junping Shi, Exact multiplicity of positive solutions for a class of semilinear problem. II, J. Differential Equations 158 (1999), no. 1, 94–151. MR 1721723, DOI 10.1016/S0022-0396(99)80020-5
- D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J. 21 (1971/72), 979–1000. MR 299921, DOI 10.1512/iumj.1972.21.21079
- Junping Shi, Persistence and bifurcation of degenerate solutions, J. Funct. Anal. 169 (1999), no. 2, 494–531. MR 1730558, DOI 10.1006/jfan.1999.3483
- Shi, Junping; Shivaji, Ratnasingham, Global bifurcation for concave semipositon problems, Recent Advances in Evolution Equations, (2002). (to appear)
- Junping Shi and Miaoxin Yao, On a singular nonlinear semilinear elliptic problem, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), no. 6, 1389–1401. MR 1663988, DOI 10.1017/S0308210500027384
- Kôsaku Yosida, Functional analysis, 5th ed., Grundlehren der Mathematischen Wissenschaften, Band 123, Springer-Verlag, Berlin-New York, 1978. MR 0500055
Additional Information
- Shobha Oruganti
- Affiliation: Department of Mathematics, Mississippi State University, Mississippi State, Mississippi 39762
- Email: so1@ra.msstate.edu
- 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
- MR Author ID: 616436
- ORCID: 0000-0003-2521-9378
- Email: shij@math.wm.edu
- Ratnasingham Shivaji
- Affiliation: Department of Mathematics, Mississippi State University, Mississippi State, Mississippi 39762
- MR Author ID: 160980
- Email: shivaji@ra.msstate.edu
- Received by editor(s): September 5, 2001
- Received by editor(s) in revised form: October 15, 2001
- Published electronically: May 7, 2002
- © Copyright 2002 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 354 (2002), 3601-3619
- MSC (2000): Primary 35J65; Secondary 35J25, 35B32, 92D25
- DOI: https://doi.org/10.1090/S0002-9947-02-03005-2
- MathSciNet review: 1911513