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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
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