W.H. Ruan

Asymptotic behavior and positive solutions of a chemical reaction diffusion system

Abstract This paper is concerned with some qualitative analysis for a coupled system of three reaction diffusion equations which arises from certain chemical reactions first discovered by Belousov and Zhabotinskii. The analysis includes the existence of a bounded global time-dependent solutions, the stability and instability of the zero solution, and the existence and nonexistence of a positive steady-state solution, including a global attractor of the system. The global existence and stability problem is determined by the method of upper and lower solutions, and the existence of a positive steady-state solution is based on the fixed point index and bifurcation theory. This analysis leads to a necessary and sufficient condition for the existence and nonexistence of a positive steady-state solution in relation to the various physical parameters of the system.

Convergence to constant states in a population genetic model with diffusion

Summary In this final section we briefly summarize the results according to the four cases described in the introduction. Case 1 (Heterozygote intermediate case). Kaa ≤ KAa ≤ KAA and Kaa Case 2 (Heterozygote superior case). Kaa ≤ KAA Case 3 (Heterozygote inferior case). KAa Case 4 (Identical carrying capacities). Kaa = KAa = KAA. There are infinitely many equilibria: p e [0, 1], N = N*, where N* is the common value of KAA, KAa and Kaa. The N-component of each regular solution converges uniformly to N* as t → ∞ (theorem 3.1). The ω-limit set set of each regular solution contains only constant equilibria (the remark after theorem 3.1). From lemma 3.2 we see that in all the cases an equilibrium is unstable if it is a local minimum point of the functionand is asymptotically stable if it is a local maximum point of .