×

zbMATH — the first resource for mathematics

Global positive coexistence of a nonlinear elliptic biological interacting model. (English) Zbl 0698.92022
The author studies an elliptic predator-prey system of the Dirichlet problem of the type \[ (1) \Delta u+u M(u,v)=0,\;d\Delta v+v(g(u)- m(v))=0,\;(u,v)|_{\partial \Omega}=(0,0). \] Under some assumptions on \(M\), \(g\) and \(m\), and providing that the domain \(\Omega\) is large, he gives a necessary and sufficient condition for the coexistence of positive solutions to (1). This necessary and sufficient condition is equivalent to the condition that the corresponding o.d.e. system \[ du/dt=u M(u,v),\;dv/dt=v(g(u)-m(v)) \] has positive equilibrium \({\tilde u}>0\), \({\tilde v}>0\). So, when \(\Omega\) is large, this condition does not depend on the shape of \(\Omega\) and is algebraically computable. Also, some stability properties of the positive solutions to (1) are studied.
Reviewer: A.Canada

MSC:
92D25 Population dynamics (general)
35J65 Nonlinear boundary value problems for linear elliptic equations
35B99 Qualitative properties of solutions to partial differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Amann, H., Fixed point equation and nonlinear eigenvalue problem in ordered Banach spaces, SIAM rev., 18, 4, 620-709, (1976) · Zbl 0345.47044
[2] Brown, K.J., Nontrivial solutions of predator-prey systems with small diffusion, Nonlin. anal. theoret. meth. appl., 11, 6, 685-689, (1987) · Zbl 0631.92014
[3] Brown, K.J.; Lin, S.S., On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight function, J. math. anal. appl., 76, 112-120, (1980) · Zbl 0437.35058
[4] Brown, P., Decay to uniform states in ecological interactions, SIAM J. appl. math., 38, 22-37, (1980) · Zbl 0511.92019
[5] Conway, E.D.; Gardner, R.; Smoller, J., Stability and bifurcation of steady-state for predator-prey equations, Adv. appl. math., 3, 288-334, (1982) · Zbl 0505.35047
[6] Dancer, E.N., On positive solution of some pairs of differential equations II, J. differ. eq., 60, 2, 236-258, (1985) · Zbl 0549.35024
[7] DeMottoni, P.; Rothe, F., Convergence to homogeneous equilibrium states for generalized Volterra-Lotka systems, SIAM J. appl. math., 648-663, (1979) · Zbl 0425.35055
[8] Gidas, B.; Ni, W.M.; Nirenberg, L., Symmetry and related properties via the maximum principle, Commun. math. phys., 68, 209-243, (1979) · Zbl 0425.35020
[9] Korman, P.; Leung, A.W., A general monotone scheme for elliptic systems with applications to ecological models, Proc. roy. soc. Edinburgh, 102A, 315-325, (1986) · Zbl 0606.35034
[10] Li, L., Coexistence theorems of steady states for predator-prey interacting systems, Trans. am. math. soc., 305, 143-166, (1988) · Zbl 0655.35021
[11] Li, L.; Lloyd, M.R., A numerical behavior of positive solutions to elliptic predator-prey system over large regions, () · Zbl 0722.35036
[12] Pazy, A., Semigroups of linear operators and applications to partial differential equations, (1983), Springer-Verlag New York · Zbl 0516.47023
[13] Varmus, H.E., Retroviruses, (), 411-503
[14] Casten, R.G.; Holland, C.J., Stability properties of solutions of reaction-diffusion equations, SIAM, J. appl. math., 33, 353-364, (1977) · Zbl 0372.35044
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.