Bifurcation Diagrams Stability of Spatial Patterns

Some authors used subsolutions, supersolutions, and comparison theorems to analyze the stability of the nontrivial steady state pix) [257]. For the Fisher equation, Skellam s linear stability analysis about p(x) can be used [414]. If we set p x, t) = p x) + hp(x, t) and consider the linearization of (9.1) about p(x), we obtain 

Q(p) = p. Since p(x) 0 and 5i does not change sign on (0, L), it follows that pLi and all of the other eigenvalues are positive which guarantees the stability of the 

This conclusion is not quite as straightforward for other kinetic terms. For example, if F(p) = p(l - p) p -a),then Q(p) = p (2p —1—a), which does not have a definite sign on [0, L]. If (2(p) 0 on [0, L], the nonuniform steady state is stable. If 2(P) 0 on [0, L], the nonuniform steady state is unstable. If Q( ) changes sign on [0, L], no conclusion can be drawn and one has to resort to other tools. The bifurcation diagram that emerges for the logistic case is that of a simple forward or supercritical bifurcation at as illustrated in Fig. 9.2. Below the trivial steady state is stable and the population dies out. Above L, the nonuniform steady state is stable, the trivial state is unstable, and the population persists or survives. 

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