Previous |  Up |  Next

# Article

Full entry | PDF   (0.2 MB)
Keywords:
predator-prey model; coexistence state
Summary:
The existence of a positive solution for the generalized predator-prey model for two species $$\begin{gathered} \Delta u + u(a + g(u,v)) = 0\quad \mbox {in}\ \Omega ,\\ \Delta v + v(d + h(u,v)) = 0\quad \mbox {in} \ \Omega ,\\ u = v = 0\quad \mbox {on}\ \partial \Omega , \end{gathered}$$ are investigated. The techniques used in the paper are the elliptic theory, upper-lower solutions, maximum principles and spectrum estimates. The arguments also rely on some detailed properties of the solution of logistic equations.
References:
[1] Cantrell, R., Cosner, C.: On the uniqueness and stability of positive solutions in the Lotka-Volterra competition model with diffusion. Houston J. Math. 15 (1989), 341-361. MR 1032394 | Zbl 0721.92025
[2] Cosner, C., Lazer, C.: Stable coexistence states in the Volterra-Lotka competition model with diffusion. Siam J. Appl. Math. 44 (1984), 1112-1132. DOI 10.1137/0144080 | MR 0766192 | Zbl 0562.92012
[3] Conway, E., Gardner, R., Smoller, J.: Stability and bifurcation of steady state solutions for predator-prey equations. Adv. in Appl. Math. 3 (1982), 288-344. DOI 10.1016/S0196-8858(82)80009-2 | MR 0673245 | Zbl 0505.35047
[4] Dancer, E. N.: On positive solutions of some pair of differential equations. Trans. Am. Math. Soc. 284 (1984), 729-743. DOI 10.1090/S0002-9947-1984-0743741-4 | MR 0743741
[5] Dancer, E. N.: On positive solutions of some pair of differential equations II. J. Differ. Equations. 60 (1985), 236-258. DOI 10.1016/0022-0396(85)90115-9 | MR 0810554
[6] Kang, J. H.: A cooperative biological model with combined self-limitation and cooperation terms. J. Comput. Math. Optimization 4 (2008), 113-126. MR 2433652
[7] Kang, J. H., Lee, J. H.: Steady state coexistence solutions of reaction-diffusion competition models. Czech. Math. J. 56 (2006), 1165-1183. DOI 10.1007/s10587-006-0086-5 | MR 2280801 | Zbl 1164.35351
[8] Lou, Y.: Necessary and sufficient condition for the existence of positive solutions of certain cooperative system. Nonlinear Analysis, Theory, Methods and Applications 26 (1996), 1079-1095. MR 1375651 | Zbl 0856.35038
[9] Dunninger, Dennis: Lecture Note of Applied Analysis. Department of Mathematics, Michigan State University.
[10] Li, L., Logan, R.: Positive solutions to general elliptic competition models. Differ. Integral Equations 4 (1991), 817-834. MR 1108062 | Zbl 0751.35014
[11] Mingxin, Wang, Zhengyuan, Li, Qixiao, Ye: The existence of positive solutions for semilinear elliptic systems. Acta Sci. Nat. Univ. Pekin. 28 36-49 (1992).
[12] Li, Zhengyuan, Mottoni, P. De: Bifurcation for some systems of cooperative and predator-prey type. J. Partial Differ. Equations 25-36 (1992). MR 1177527 | Zbl 0769.35026

Partner of