118890142
An Application of the Generalized Morse Index to Travelling Wave Solutions of a Competitive Reaction-Diffusion Model.
Abstract : The existence of travelling wave solutions of a diffusion reaction-system is studied via the generalized Morse index of isolated invariant sets. This index theory is analogous to degree theory, and the method of proof follows lines familiar from the latter theory. The equations in question are 'deformed' to a 'standard' system where the index can be easily computed, and the existence theorem follows from the 'non-triviality' of the index. The index theory has been described in other papers; here the main job is to construct 'isolating neighborhoods' which are analogous (in the degree theory) to open sets with no critical points on the boundary. Some novel means of location such neighborhoods are described.
120064138
THE SET OF POSITIVE SOLUTIONS OF SEMILINEAR EQUATIONS IN LARGE BALLS
The exact number of positive solutions of Δ u + f ( u ) = 0 on finite balls in ℝ N is determined. The assumptions about f(u) are similar to those imposed by Serrin and the second author in a previous study of uniqueness of the positive solution when the spatial domain is all of ℝ N (see [7, 8]). For finite balls of sufficiently large radius it is shown here that there are exactly two positive and, hence, radial solutions. To this end, we first prove the linear nondegeneracy of the positive solution of ℝ N . This is obtained by applying the technique of monotone separation of graphs [7] to the linearised equations. Somewhat sharper estimates are required here (see Part I, Section 2).
43895421
The Existence of Periodic Travelling Waves for Singularly Perturbed Predator-Prey Equations via the Conley Index
This paper is concerned with the existence of periodic travelling wave solutions of reactiondiffusion equations, which arise as models of predator-prey interactions in mathematical ecology. The relevant equations are somewhat similar to the Fitz-Hugh-Nagumo equations, and our results apply to this latter system as well. In studying the Fitz-Hugh-Nagumo equations, Conley ([ 3 I), initiated a new topological approach to this problem. His method depends on the construction of an isolating neighborhood (cf. Section 2A) about a “singular” periodic solution. Associated with such a neighborhood is a topological invariant, the Conley Index, which, when nontrivial, implies the existence of a smooth solution near the singular one. The new aspects of the present discussion are in the method used to construct the isolating neighborhood (in IFi4), and in the computation of the index. With regard to the isolating neighborhood, our construction is global in that a parameter taken to be small in Conley’s equations, is now allowed to assume large values. In order to compute the index, we “continue” our equations to those studied by Conley. Actually, we go one step further and continue Conley’s equations to the Van der Pol equations crossed with a pair of linear equations admitting a repelling critical point. The index for this latter system is easily computed and turns out to be non-trivial. It follows from the invariance of the index under continuation (21, that the desired
153458691
Rationing, bargaining, and voting in 2-sided markets
This paper analyzes the temporary equilibrium of 2-sided markets from the point of view of bargaining theory. The minimum rule, quantity traded equal the minimum of demand and supply, is shown to hold under both the Nash bargain and the λ-transfer value. Moreover, both concepts imply that rationing is uniform on the long side of the market. The paper then considers disequilibrium price dynamics. Under majoritarian price dynamics, nex period's price is the majority rule price of this period's active traders. The sequence of temporary equilibria according to the majoritarian price dynamics is shown to converge to Walrasian equilibrium.