Some uniqueness and exact multiplicity results for a predator-prey model

Du, Yihong; Lou, Yuan

doi:10.1090/S0002-9947-97-01842-4

Some uniqueness and exact multiplicity results for a predator-prey model
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by Yihong Du and Yuan Lou PDF

Trans. Amer. Math. Soc. 349 (1997), 2443-2475 Request permission

Abstract:

In this paper, we consider positive solutions of a predator-prey model with diffusion and under homogeneous Dirichlet boundary conditions. It turns out that a certain parameter $m$ in this model plays a very important role. A good understanding of the existence, stability and number of positive solutions is gained when $m$ is large. In particular, we obtain various results on the exact number of positive solutions. Our results for large $m$ reveal interesting contrast with that for the well-studied case $m=0$, i.e., the classical Lotka-Volterra predator-prey model.

References

K. J. Brown, Nontrivial solutions of predator-prey systems with small diffusion, Nonlinear Anal. 11 (1987), no. 6, 685–689. MR 893773, DOI 10.1016/0362-546X(87)90035-6
J. Blat and K. J. Brown, Global bifurcation of positive solutions in some systems of elliptic equations, SIAM J. Math. Anal. 17 (1986), no. 6, 1339–1353. MR 860917, DOI 10.1137/0517094
A. Casal, J. C. Eilbeck, and J. López-Gómez, Existence and uniqueness of coexistence states for a predator-prey model with diffusion, Differential Integral Equations 7 (1994), no. 2, 411–439. MR 1255897
Shui Nee Chow and Jack K. Hale, Methods of bifurcation theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 251, Springer-Verlag, New York-Berlin, 1982. MR 660633, DOI 10.1007/978-1-4613-8159-4
E. Conway, R. Gardner, and J. Smoller, Stability and bifurcation of steady-state solutions for predator-prey equations, Adv. in Appl. Math. 3 (1982), no. 3, 288–334. MR 673245, DOI 10.1016/S0196-8858(82)80009-2
Donald S. Cohen and Theodore W. Laetsch, Nonlinear boundary value problems suggested by chemical reactor theory, J. Differential Equations 7 (1970), 217–226. MR 259356, DOI 10.1016/0022-0396(70)90106-3
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
E. N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl. 91 (1983), no. 1, 131–151. MR 688538, DOI 10.1016/0022-247X(83)90098-7
E. N. Dancer, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc. 284 (1984), no. 2, 729–743. MR 743741, DOI 10.1090/S0002-9947-1984-0743741-4
E. N. Dancer, On positive solutions of some pairs of differential equations. II, J. Differential Equations 60 (1985), no. 2, 236–258. MR 810554, DOI 10.1016/0022-0396(85)90115-9
E. N. Dancer, Counterexamples to some conjectures on the number of solutions of nonlinear equations, Math. Ann. 272 (1985), no. 3, 421–440. MR 799671, DOI 10.1007/BF01455568
E. N. Dancer, On the number of positive solutions of weakly nonlinear elliptic equations when a parameter is large, Proc. London Math. Soc. (3) 53 (1986), no. 3, 429–452. MR 868453, DOI 10.1112/plms/s3-53.3.429
E. N. Dancer, On uniqueness and stability for solutions of singularly perturbed predator-prey type equations with diffusion, J. Differential Equations 102 (1993), no. 1, 1–32. MR 1209974, DOI 10.1006/jdeq.1993.1019
E. N. Dancer, Multiple fixed points of positive mappings, J. Reine Angew. Math. 371 (1986), 46–66. MR 859319, DOI 10.1515/crll.1986.371.46
E. N. Dancer and Yi Hong Du, Competing species equations with diffusion, large interactions, and jumping nonlinearities, J. Differential Equations 114 (1994), no. 2, 434–475. MR 1303035, DOI 10.1006/jdeq.1994.1156
Y. Du, Uniqueness, multiplicity and stability for positive solutions of a pair of reaction-diffusion equations, Proc. Roy. Soc. Edinburgh 126A (1996), 777-809.
Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244, DOI 10.1007/BFb0089647
Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
Philip Korman and Anthony W. Leung, A general monotone scheme for elliptic systems with applications to ecological models, Proc. Roy. Soc. Edinburgh Sect. A 102 (1986), no. 3-4, 315–325. MR 852364, DOI 10.1017/S0308210500026391
Nela Lakoš, Existence of steady-state solutions for a one-predator–two-prey system, SIAM J. Math. Anal. 21 (1990), no. 3, 647–659. MR 1046793, DOI 10.1137/0521034
Lige Li, Coexistence theorems of steady states for predator-prey interacting systems, Trans. Amer. Math. Soc. 305 (1988), no. 1, 143–166. MR 920151, DOI 10.1090/S0002-9947-1988-0920151-1
Lige Li, On the uniqueness and ordering of steady states of predator-prey systems, Proc. Roy. Soc. Edinburgh Sect. A 110 (1988), no. 3-4, 295–303. MR 974745, DOI 10.1017/S0308210500022289
J. López-Gómez and R. Pardo, Existence and uniqueness of coexistence states for the predator-prey model with diffusion: the scalar case, Differential Integral Equations 6 (1993), no. 5, 1025–1031. MR 1230477
C. V. Pao, On nonlinear reaction-diffusion systems, J. Math. Anal. Appl. 87 (1982), no. 1, 165–198. MR 653613, DOI 10.1016/0022-247X(82)90160-3
C. V. Pao, Nonlinear parabolic and elliptic equations, Plenum Press, New York, 1992. MR 1212084
Paul H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis 7 (1971), 487–513. MR 0301587, DOI 10.1016/0022-1236(71)90030-9
Yoshio Yamada, Stability of steady states for prey-predator diffusion equations with homogeneous Dirichlet conditions, SIAM J. Math. Anal. 21 (1990), no. 2, 327–345. MR 1038895, DOI 10.1137/0521018
Q.X. Ye and Z.Y. Li, Introduction to Reaction-Diffusion Equations (in Chinese), Science Press, Beijing, 1990.