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Solving Volterra-Lotka systems with diffusion by monotone iteration. (English) Zbl 0735.35060
An abstract theorem concerning a monotone iteration method in a partially ordered linear space is presented first. Then the method is applied to a stationary Volterra-Lotka system in a convex domain $$\Omega$$: $\alpha\Delta u+au-bu^ 2+cuv=0,\;\beta\Delta v+dv-ev^ 2+fuv=0 \hbox{ in } \Omega \hbox{ and } u=v=0 \hbox{ on } \partial\Omega,$ with already known subsolutions $$\b{u}_ 0$$, $$\b{v}_ 0$$ and supersolutions $$\bar u_ 0$$, $$\bar v_ 0$$ as an initial step. As a result, solutions $$u^*,v^*\in C^ 2(\Omega)\cap C(\bar\Omega)$$ are obtained in order- intervals $$[\b{u}_ 0,\bar u_ 0]$$ and $$[\b{v}_ 0,\bar v_ 0]$$ respectively under appropriate assumptions on the positive constants $$\alpha$$, $$\beta$$, $$a$$, $$b$$, $$d$$, $$e$$ and the real ones $$c$$, $$f$$. An extended predator-prey system is also considered. One numerical example is given.
##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 92D25 Population dynamics (general)
MGOO
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##### References:
 [1] Blat, J., Brown, K. J.: Bifurcation of steady-state solutions in predator-prey and competition systems. Proc. Roy. Soc. Edinb. 97A, 21-34 (1984) · Zbl 0554.92012 [2] Dancer, E. N.: On positive solutions of some pairs of differential equations. Trans. Am. Math. Soc. 284, 729-743 (1984) · Zbl 0524.35056 · doi:10.1090/S0002-9947-1984-0743741-4 [3] Förster, H., Witsch, K.: Multigrid software for the solution of elliptic problems on rectangular domains: MG00 (Release 1). In: Hackbusch, W., Trottenberg, U. (eds.) (Lect. Notes Math. vol. 960, pp. 427-460). Berlin Heidelberg New York: Springer 1982 [4] Förster, H., Witsch, K.: On efficient multigrid software for elliptic problems on rectangular domains. Math. Comp. Simul. 23, 293-298 (1981) · doi:10.1016/0378-4754(81)90087-2 [5] Goh, B. S.: Management and analysis of biological populations. Amsterdam Oxford New York: Elsevier 1980 [6] Hadeler, K. P., Rothe, F., Vogt, H.: Stationary solutions of reaction-diffusion equations. Math. Meth. Appl. Sci. 1, 418-431 (1979) · Zbl 0424.35047 · doi:10.1002/mma.1670010307 [7] Huy, C. U., McKenna, P. J., Walter, W.: Finite-difference approximations to the Dirichlet problem for elliptic systems. Numer Math. 49, 227-237 (1986) · Zbl 0602.65068 · doi:10.1007/BF01389626 [8] Korman, P., Leung, A. W.: A general monotone scheme for elliptic systems with applications to ecological models. Proc. Roy. Soc. Edinb. 102A, 315-325 (1986) · Zbl 0606.35034 [9] Korman, P., Leung, A. W.: On the existence and uniqueness of positive steady states in the Volterra-Lotka ecological models with diffusion. Appl. Anal. 26, 145-160 (1987) · Zbl 0639.35026 · doi:10.1080/00036818708839706 [10] Krasnoselski, M.: Positive solutions of operator equations. Groningen: Noordhoff 1964 [11] Leung, A.: Limiting behaviour for a prey-predator model with diffusion and crowding effects. J. Math. Biol. 6, 87-93 (1978) · Zbl 0386.92011 · doi:10.1007/BF02478520 [12] McKenna, P. J., Walter, W.: On the Dirichlet problem for elliptic systems. Appl. Anal. 21, 207-224 (1986) · Zbl 0593.35042 · doi:10.1080/00036818608839592 [13] Metzler, W.: Dynamische Systeme in der Ökologie. Stuttgart: Teubner 1987 · Zbl 0631.92022 [14] Ortega, J. M., Rheinboldt, W. C.: Iterative solutions of nonlinear equations in several variables. New York: Academic Press 1970 · Zbl 0241.65046 [15] Potra, F. A.: Newton-like methods with monotone convergence for solving nonlinear operator equations. Nonlinear Anal. Theory Methods Appl. 11, 697-717 (1987) · Zbl 0633.65050 · doi:10.1016/0362-546X(87)90037-X [16] Potra, F. A., Rheinboldt, W. C.: On the monotone convergence of Newton’s method. Computing 36, 81-90 (1986) · Zbl 0572.65034 · doi:10.1007/BF02238194 [17] Rothe, F.: Convergence to the equilibrium state in the Volterra-Lotka diffusion equations. J. Math. Biol. 3, 319-324 (1976) · Zbl 0355.92013 [18] Southwood, T. R. E.: Ecological methods, 2nd edn. New York: Wiley 1978 [19] Voller, R. L.: Monotonieeigenschaften von Abbildungen and Einschlie?ung von Lösungen nichtlinearer Operatorgleichungen. Habilitationsschrift, Düsseldorf 1989 · Zbl 0734.47044 [20] Voss, H.: Ein neues Verfahren zur Einschlie?ung der Losungen von Operatorgleichungen. Z. Angew. Math. Mech. 56, 218-219 (1976) · Zbl 0341.65040 · doi:10.1002/zamm.19760560509
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