# zbMATH — the first resource for mathematics

Uniqueness and nonuniqueness of coexistence states in the Lotka-Volterra competition model. (English) Zbl 0829.92015
In the last decade, the reaction-diffusion system $u_t = k_1 \Delta u + u(a - bu - cv), \quad v_t = k_2 \Delta v + v(d - eu - fv), \tag{1}$ has received intensive study in various directions. The motivation for this study comes from ecology and biology, and system (1) is usually referred to as the Lotka-Volterra competition model. Here $$u$$ and $$v$$ denote the population densities of two species in certain competing environments. The positive constants $$k_1, k_2$$ in (1) are the diffusion rates of these species from high density regions to low density ones; while $$a,d,b,c,e,f$$ are in general functions of $$(x,t)$$, and $$a,d$$ represent the birth rates; $$b$$ and $$f$$ account for the rates of self-limitation, and $$c$$ and $$e$$ account for the rates of competition. System (1) is usually considered in a smooth bounded domain $$\Omega$$ and accompanied with certain homogeneous boundary conditions – Dirichlet or Neumann or the third type, depending on the external environment. With this interpretation, we see that only nonnegative solutions of (1) are of real interest.
To make the main idea more transparent, we only consider the simplest model of (1) where $$k_1 = k_2 = 1$$ and $$a = d,b,c,e,f$$ are positive constants. We are basically concerned with the so-called coexistence states, i.e., the nonnegative time-independent solution $$(u,v)$$ of which both $$u$$ and $$v$$ are not identically zero.

##### MSC:
 92D25 Population dynamics (general) 35K57 Reaction-diffusion equations 92D40 Ecology 35Q92 PDEs in connection with biology, chemistry and other natural sciences
Full Text:
##### References:
 [1] Blat, Proc. Roy. Soc. Edinburgh Sec. A 97 pp 21– (1984) · Zbl 0554.92012 · doi:10.1017/S0308210500031802 [2] Blat, SIAM J. Math. Anal. 17 pp 1339– (1986) [3] Cantrell, Houston J. Math. 15 pp 341– (1989) [4] Cantrell, SIAM J. Appl. Math. 53 pp 219– (1993) [5] Neurobiology, Shanghai Medical University Press, 1989. [6] Cosner, SIAM J. Appl. Math. 44 pp 1112– (1984) [7] Crandall, J. Funct. Anal. 8 pp 321– (1971) [8] Dancer, J. Math. Anal. Appl. 91 pp 131– (1983) [9] Dancer, Trans. Amer. Math. Soc. 284 pp 729– (1984) [10] Dancer, J. Differential Equations 60 pp 236– (1985) [11] De Figueiredo, Comm. Partial Differential Equations 17 pp 339– (1992) [12] Mathematical Models in Biology, Random House, New York, 1988. [13] Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics, No. 28, Springer-Verlag, Berlin, 1979. · doi:10.1007/978-3-642-93111-6 [14] and , Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer-Verlag, Berlin, New York, 1983. · Zbl 0361.35003 · doi:10.1007/978-3-642-61798-0 [15] Kahane, Funkcial. Ekvac. 35 pp 51– (1992) [16] , and , Linear and Quasilinear Equations of Parabolic Type, AMS, Providence, 1968. [17] Systems of Nonlinear Partial Differential Equations: Applications to Biology and Engineering, Kluwer Academic Publishers, Dordrecht, Boston, 1989. · doi:10.1007/978-94-015-3937-1 [18] Li, J. Math. Anal. Appl. 138 pp 537– (1989) [19] Li, Differential Integral Equations 4 pp 817– (1991) [20] Lopez-Gomez, C. R. Acad. Sci. Paris Sér. I Math. 313 pp 933– (1991) [21] Manes, Boll. Un. Mat. Ital. 7 pp 285– (1973) [22] Pao, J. Math. Anal. Appl. 83 pp 54– (1981) [23] Pao, Nonlinear Anal. 6 pp 1163– (1982) [24] Rabinowitz, J. Funct. Anal. 7 pp 487– (1971) [25] Mathematical Ideas in Biology, Cambridge University Press, London, New York, 1968. · doi:10.1017/CBO9780511565144 [26] and , Nonlinear problems in nuclear reactor analysis, pp. 298–307 in: Nonlinear Problems in the Physical Sciences and Biology, , and , eds., Lecture Notes in Mathematics No. 322, Springer-Verlag, Berlin, New York, 1973. [27] Uhlenbeck, Amer. J. Math. 98 pp 1059– (1976)
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.