# zbMATH — the first resource for mathematics

Coexistence theorems of steady states for predator-prey interacting systems. (English) Zbl 0655.35021
The author gives necessary and sufficient conditions for the existence of positive solutions to the following system: $\Delta u+uM(u,v)=0,\quad d\Delta v+v(g(u)-m(v))=0,\quad (u,v)|_{\partial \Omega}=(0,0),\quad \Omega \subset {\mathbb{R}}\quad n,$ where M satisfies the so-called prey growth rate conditions, g and m are increasing functions satisfying $$g(0)-m(v)<0$$ for v larger than some constant C. The paper includes many well-known systems and results as special cases, and some interesting examples are given.
Reviewer: C.F.Wang

##### MSC:
 35J60 Nonlinear elliptic equations 92D25 Population dynamics (general) 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35A05 General existence and uniqueness theorems (PDE) (MSC2000)
##### Keywords:
existence; positive solutions; prey growth rate
Full Text:
##### References:
 [1] N. D. Alikakos, \?^\? bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations 4 (1979), no. 8, 827 – 868. · Zbl 0421.35009 · doi:10.1080/03605307908820113 · doi.org [2] Herbert Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), no. 4, 620 – 709. · Zbl 0345.47044 · doi:10.1137/1018114 · doi.org [3] Henri Berestycki, Le nombre de solutions de certains problèmes semi-linéaires elliptiques, J. Funct. Anal. 40 (1981), no. 1, 1 – 29 (French, with English summary). · Zbl 0452.35038 · doi:10.1016/0022-1236(81)90069-0 · doi.org [4] H. Berestycki and P.-L. Lions, Some applications of the method of super and subsolutions, Bifurcation and nonlinear eigenvalue problems (Proc., Session, Univ. Paris XIII, Villetaneuse, 1978) Lecture Notes in Math., vol. 782, Springer, Berlin, 1980, pp. 16 – 41. [5] J. Blat and K. J. Brown, Bifurcation of steady-state solutions in predator-prey and competition systems, Proc. Roy. Soc. Edinburgh Sect. A 97 (1984), 21 – 34. · Zbl 0554.92012 · doi:10.1017/S0308210500031802 · doi.org [6] E. D. Conway, Diffusion and predator-prey interaction: pattern in closed systems, Partial differential equations and dynamical systems, Res. Notes in Math., vol. 101, Pitman, Boston, MA, 1984, pp. 85 – 133. · Zbl 0554.92011 [7] -, Diffusion and the predator-prey interaction: steady states with flux at the boundary, Contemporary Math., vol. 17, Amer. Math. Soc., Providence, R. I., 1983, pp. 215-234. · Zbl 0538.35042 [8] 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. · Zbl 0505.35047 · doi:10.1016/S0196-8858(82)80009-2 · doi.org [9] Edward D. Conway and Joel A. Smoller, Diffusion and the predator-prey interaction, SIAM J. Appl. Math. 33 (1977), no. 4, 673 – 686. · Zbl 0368.35021 · doi:10.1137/0133047 · doi.org [10] Michael G. Crandall and Paul H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal. 52 (1973), 161 – 180. · Zbl 0275.47044 · doi:10.1007/BF00282325 · doi.org [11] 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. · Zbl 0512.47045 · doi:10.1016/0022-247X(83)90098-7 · doi.org [12] E. N. Dancer, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc. 284 (1984), no. 2, 729 – 743. · Zbl 0524.35056 [13] Jerome A. Goldstein, Semigroups of linear operators and applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. · Zbl 0592.47034 [14] 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. · Zbl 0606.35034 · doi:10.1017/S0308210500026391 · doi.org [15] A. Leung, Monotone schemes for semilinear elliptic systems related to ecology, Math. Methods Appl. Sci. 4 (1982), no. 2, 272 – 285. · Zbl 0493.35044 · doi:10.1002/mma.1670040118 · doi.org [16] P. de Mottoni and F. Rothe, Convergence to homogeneous equilibrium state for generalized Volterra-Lotka systems with diffusion, SIAM J. Appl. Math. 37 (1979), no. 3, 648 – 663. · Zbl 0425.35055 · doi:10.1137/0137048 · doi.org [17] C. V. Pao, On nonlinear reaction-diffusion systems, J. Math. Anal. Appl. 87 (1982), no. 1, 165 – 198. · Zbl 0488.35043 · doi:10.1016/0022-247X(82)90160-3 · doi.org [18] Joel Smoller, Shock waves and reaction-diffusion equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 258, Springer-Verlag, New York-Berlin, 1983. · Zbl 0508.35002 [19] J. Smoller, A. Tromba, and A. Wasserman, Nondegenerate solutions of boundary value problems, Nonlinear Anal. 4 (1980), no. 2, 207 – 216. · Zbl 0429.34024 · doi:10.1016/0362-546X(80)90049-8 · doi.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.