Coexistence theorems of steady states for predator-prey interacting systems

Li, Lige

doi:10.1090/S0002-9947-1988-0920151-1

Coexistence theorems of steady states for predator-prey interacting systems
HTML articles powered by AMS MathViewer

by Lige Li PDF

Trans. Amer. Math. Soc. 305 (1988), 143-166 Request permission

Abstract:

In this paper we give necessary and sufficient conditions for the existence of positive solutions of steady states for predator-prey systems under Dirichlet boundary conditions on $\Omega \Subset {{\mathbf {R}}^n}$. We show that the positive coexistence of predatorprey densities is completely determined by the "marginal density," the unique density of prey or predator while the other one is absent, i.e. the $({u_0}, 0)$ or $(0, {\nu _0})$. More specifically, the situation of coexistence is determined by the spectral behavior of certain operators related to these marginal densities and is also completely determined by the stability properties of these marginal densities. The main results are Theorems 1 and 4.2.

References

N. D. Alikakos, $L^{p}$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations 4 (1979), no. 8, 827–868. MR 537465, DOI 10.1080/03605307908820113
Herbert Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), no. 4, 620–709. MR 415432, DOI 10.1137/1018114
Henri Berestycki, Le nombre de solutions de certains problèmes semi-linéaires elliptiques, J. Functional Analysis 40 (1981), no. 1, 1–29 (French, with English summary). MR 607588, DOI 10.1016/0022-1236(81)90069-0
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. MR 572249
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. MR 751174, DOI 10.1017/S0308210500031802
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. MR 759745

Diffusion and the predator-prey interaction

steady states with flux at the boundary

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
Edward D. Conway and Joel A. Smoller, Diffusion and the predator-prey interaction, SIAM J. Appl. Math. 33 (1977), no. 4, 673–686. MR 492877, DOI 10.1137/0133047
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
Jerome A. Goldstein, Semigroups of linear operators and applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. MR 790497
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
A. Leung, Monotone schemes for semilinear elliptic systems related to ecology, Math. Methods Appl. Sci. 4 (1982), no. 2, 272–285. MR 659042, DOI 10.1002/mma.1670040118
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. MR 549146, DOI 10.1137/0137048
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
Joel Smoller, Shock waves and reaction-diffusion equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York-Berlin, 1983. MR 688146
J. Smoller, A. Tromba, and A. Wasserman, Nondegenerate solutions of boundary value problems, Nonlinear Anal. 4 (1980), no. 2, 207–216. MR 563804, DOI 10.1016/0362-546X(80)90049-8