Intermediates, Catalysts, Persistence, and Boundary Steady States

For dynamical systems arising from chemical reaction networks, persistence is the property that each species concentration remains positively bounded away from zero, as long as species concentrations were all positive in the beginning. We describe two graphical procedures for simplifying reaction networks without breaking known necessary or sufficient conditions for persistence, by iteratively removing so-called intermediates and catalysts from the network. The procedures are easy to apply and, in many cases, lead to highly simplified network structures, such as monomolecular networks. For specific classes of reaction networks, we show that these conditions for persistence are equivalent to one another. Furthermore, they can also be characterized by easily checkable strong connectivity properties of a related graph. In particular, this is the case for (conservative) monomolecular networks, as well as cascades of a large class of post-translational modification systems (of which the MAPK cascade and the $n$-site futile cycle are prominent examples). Since one of the aforementioned sufficient conditions for persistence precludes the existence of boundary steady states, our method also provides a graphical tool to check for that.Comment: The main result was made more general through a slightly different approach. Accepted for publication in the Journal of Mathematical Biolog

Intermediates and Generic Convergence to Equilibria

Known graphical conditions for the generic or global convergence to equilibria of the dynamical system arising from a reaction network are shown to be invariant under the so-called successive removal of intermediates, a systematic procedure to simplify the network, making the graphical conditions easier to check.Comment: Added theorem 1 and corrected an error in the proof of theorem

Simetria, compacidade e multiplicidade de soluções para um problema elíptico semilinear em Rn.

Dissertação (mestrado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2008.Mostramos que o problema elíptico semilinear ( − u + b(|x|)u = f(|x| , u) u E C2(RN) , onde b : [0,∞) → R é uma função contínua limitada inferiormente por uma constante positiva e f : [0,∞)×R → R é uma função contíınua satisfazendo certas condições de crescimento (subcrítico e superquadrático) e convexidade, possui soluções radiais com qualquer quantidade finita prescrita de nós para N > ou = 2. Também mostramos que, se a hipótese de convexidade for substituíıda pela hipóotese de que f é não-decrescente e íımpar com respeito à variável u, entáo o problema possui ao menos uma solução não-radial para N = 4 ou N > ou = 6. A falta de compacidade em domíınios ilimitados é superada com a restrição a subespaços de funções invariantes pela ação de subgrupos do grupo O(N) das transformações lineares ortogonais de RN e os objetivos são alcançados combinando-se o Teorema do Passo da Montanha e o Princípio da Criticalidade Simétrica. Para a obtenção das soluções radiais nodais, aplicamos o método de Nehari de concatenação de soluções positivas e negativas em regiões anulares vizinhas. ____________________________________________________________________________________________________________ ABSTRACTWe show that the semilinear elliptic problem - ∆u + b(|x|)u = f(|x|,u) u Є C2(RN), where b : [0, ∞) → R is a continuous function bounded below by a positive constant and e f : [0, ∞) x R → R is a continuous function for which certain growth (subcritic and superquadratic) and convexity conditions hold, has radial solutions with a any prescribed finite number of nodes, for N ≥ 2 . We also show that if the convexity hypothesis is replaced by f nondecreasing and odd with respect to the variable u, then the problem still has at least one nonradial solution, for N =4 or N ≥ 6. The lack of compactness is overcome by the restriction to subspaces of functions invariant under the action of subgroups of the group O(N) of the orthogonal linear transformations of RN, and the results are achieved through a combination of the Mountain Pass Theorem and the Principle of Symmetric Criticality. Nodal radial solutions are constructed following the method of Nehari of piecing together positive and negative solutions on alternating annuli