Resilience in reaction-diffusion systems

Reaction-diffusion systems with zero-flux Neumann boundariesare widely used to model various kinds of interaction in, forexample, the scientific fields of ecology, biology, chemistry,medicine and industry. The physical systems within these fieldsare often known to be (conditionally or unconditionally) resilientwith respect to shocks, disturbances or catastrophies in theimmediate environment. In order to be good mathematical modelsof such situations the reaction-diffusion systems must havethe same resilient or asymptotic behaviour as that of the physicalsituation. Three fundamentally different kinds of reaction termsare usually distinguished according to the entry signs of thereaction Jacobian: mutualism, mixed (predator-prey) interactionand competition. The asymptotic stability (in the Poincarésense) of mutualistic systems has already been studied extensively,but the results cannot be generalized (globally) to the othertwo fundamental types, which are not order-preserving. A partial(local) generalization is, however given here for these twotypes, involving simple Jacobian inequalities and knowledge(often prompted by the underlying physical situation) of invariantsets in solution space. The return time of resilient systemsand the approach rate of asymptotically stable solutions arealso estimated.     

On the similarity solutions for free convection in a saturated porous medium adjacent to impermeable horizontal surfaces

The similarity solutions for free convection about a horizontal surface in a saturated porous medium when the wall temperature is prescribed as a power law of distance from the origin, i.e. proportional tox ^m, are considered. Solutions valid form1 are obtained. Numerical solutions are obtained which reveal that there is a valuem ₀ such that a solution of the similarity equations is possible only form>m ₀. It is shown thatm ₀=–2/5 and the nature of the singularity is discussed and an asymptotic expansion is obtained.

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Ähnlichkeitslösungen für eine freie Konvektion in einem gesättigten porösen Medium, das an eine undurchlässige, horizontale Oberfläche grenzt

Zusammenfassung Die Ähnlichkeitslösungen sind für eine freie Konvektion über eine horizontale Oberfläche in einem gesättigten porösen Medium in Betracht gezogen worden, wenn die Wandtemperatur beschrieben ist als eine Potenzfunktion des Abstandes vom Ursprung, z. B. proportional zux ^m.Gültige Lösungen sind fürm1 bestimmt. Numerische Lösungen sind bestimmt worden, die offenbaren, daß es einen Wertm ₀ geben muß, so daß eine Lösung von ähnlichen Lösungen nur fürm>m ₀ möglich ist. Es wird gezeigt, daßm ₀=–2/5 und daß die Beschaffenheit der Singularität besprochen und die asymptotische Ausdehnung bestimmt worden ist.

     

The boundary-layer flow past a suddenly heated vertical surface in a saturated porous medium

The boundary-layer flow generated on an impermeable vertical surface in a saturated porous medium is considered in the case when wall heating at a rate proportional tox is switched on at timet=0, (x measures distance along the wall and is a constant). The similarity equations which hold in the limit of larget are discussed and are shown to have a solution only for >–1. The behaviour of the solution as –1 and as is obtained. Numerical solutions of the initial value problem are then obtained for a range of values of . A direct numerical integration is possible for 1, while an iterative procedure is required for <1, with the numerical scheme becoming unstable for =–0.5.

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Grenzschichtströmung an einer plötzlich aufgeheizten vertikalen Fläche, in einem gesättigten porösen Medium

Zusammenfassung Es wird die an einer undurchlässigen, vertikalen Fläche hervorgerufene Grenzschichtströmung im Falle eines Einschalten der Heizung beit=0 betrachtet. Die Stärke der Wandheizung is proportional zux , wobeix die Koordinate längs der Wand ist und eine Konstante. Die Ähnlichkeitsgleichungen werden für den Bereich von großen Zeitent besprochen und es wird gezeigt, daß eine Lösung nur für >–1 vorliegt. Es wird das Verhalten der Lösungen für –1 und erhalten. Numerische Lösungen für die Anfangsbedingungen des Problems werden für eine Reihe von -Werten errechnet. Eine direkte numerische Integration ist für 1 möglich, während für <1 eine Iteration erforderlich ist, wobei das numerische Verhalten für =–0.5 instabil wird.

     

The effects of a complexing agent on travelling waves in autocatalytic systems with applied electric fields

The effects of electric fields on the reaction fronts that arisein a system governed by an autocatalytic reaction and a complexationreaction between the autocatalyst and a complexing agent areconsidered. The complexation reaction is assumed to be fastrelative to the autocatalytic reaction and the equations forthis limit are derived. The corresponding travelling waves arediscussed, the case of quadratic autocatalysis being treatedin detail. The existence of minimum speed waves is examined,being dependent on the ratio of diffusion coefficients D, theconcentration S₀ and equilibrium constant K of the complexationreaction as well as the electric field strength E. It is seenthat, for some parameter values, minimum speed waves have negativeautocatalayst concentrations, and waves which have the lowestspeed consistent with non-negative concentrations are also obtained.Numerical integrations of the initial-value problem are performedfor representative parameter values. These show the developmentof the appropriate travelling wave (when it exists) as the largetime behaviour of the system, and, in cases where no travellingwave exists, the numerical integrations show the electrophoreticseparation of substrate and autocatalyst.