Asymptotics in Strong Competition

Abstract

This chapter is devoted to the study of the asymptotic behavior of the following system when κ is sufficiently large.

$$ \left \{ \everymath{\displaystyle} \begin{array}{@{}l@{\quad}l} \frac{\partial u_i}{\partial t}-\frac{\partial^2 u_{i}}{\partial x^2}=a_iu_{i} -u_{i}^2-\kappa u_{i} \sum_{j\neq i} b_{ij}u_{j} & \mbox{in }[0,1]\times(0,+\infty),\\[10pt] u_i=0 &\mbox{on }\{0\}\times(0,+\infty)\cup\{1\}\times(0,+\infty),\\[5pt] u_i=\phi_i & \mbox{on }[0,1]\times\{0\}. \end{array} \right . $$

(6.1)

Here i=1,2,…,M, ϕ _i are given Lipschitz continuous functions on [0,1] such that

$$\phi_i\geq0, \quad \mbox{and if}\quad i\neq j,\ \phi_i \phi_j\equiv0. $$

For simplicity, we assume the coefficients b _ij =1, ∀i≠j. Without this assumption, our proof is still valid with minor changes, due to the special property of dimension 1. In the following, we will point out this whenever necessary. From the discussions in Chap. 4, we know the singular limit of (6.1) as κ→+∞ is a gradient flow (that is, satisfying an energy inequality), with its solution converging to the stationary state as t→+∞. So the natural question arises: does (6.1) for κ large also behave like a gradient system? In this chapter, we will show that with a few assumptions, the answer is yes: for κ large the dynamics of (6.1) is simple. In the first section, we list some non-degeneracy conditions needed in our proof and give the main result of this chapter. The next section is devoted to a coarse convergent property of solutions to (6.1). Finally, in the last section we use the non-degeneracy conditions to finish the proof of the convergence.