Local stability implies global stability for the 2-dimensional Ricker map

Consider the difference equation $x_{k+1}=x_k e^{\alpha-x_{n-d}}$ where $\alpha$ is a positive parameter and d is a non-negative integer. The case d = 0 was introduced by W.E. Ricker in 1954. For the delayed version d >= 1 of the equation S. Levin and R. May conjectured in 1976 that local stability of the nontrivial equilibrium implies its global stability. Based on rigorous, computer aided calculations and analytical tools, we prove the conjecture for d = 1.Comment: for associated C++ program, mathematica worksheet and output, see http://www.math.u-szeged.hu/~krisztin/ricke

Differenciálegyenletek kvalitatív elmélete alkalmazásokkal = Qualitative theory of differential equations with applications

Differenciálegyenletek kvalitatív elméletében végeztünk kutatásokat. Az elméleti eredményeket is fontos alkalmazások motiválták. Emellett alkalmazásokkal is foglalkoztunk. A főbb eredmények: másodrendű nem-autonóm differenciálegyenletek megoldásainak aszimptotikus vizsgálatára dolgoztunk ki új módszereket; bizonyos funkcionál-differenciálegyenletekre új típusú attraktorok szerkezetét írtuk le; járványterjedési jelenségek vizsgálatára differenciálegyenletes modelleket adtunk meg, és azok kvalitatív tulajdonságait leírva a járványok terjedéséről fontos információkat kaptunk. | We studied the qualitative theory of differential equations. The theoretical results were motivated by important applications. In addition we considered applications, too. Some of the main results: we developed new methods to study the asymptotic behaviour of solutions of second order nonautonomous differential equations; we described the structure of new type of attractors for certain functional differenctial equations; different epidemic models were developed to describe the spread of infectious diseases, and we studied the qualitative properties of these models to get important information about the diseases

A note on dissipativity and permanence of delay difference equations

We give sufficient conditions on the uniform boundedness and permanence of non-autonomous multiple delay difference equations of the form $$x_{k+1}=x_k f_k(x_{k-d},\dots,x_{k-1},x_k),$$ where $f_k\colon D \subseteq (0,\infty)^{d+1}\to (0,\infty)$. Moreover, we construct a positively invariant absorbing set of the phase space, which implies also the existence of the global (pullback) attractor if the right-hand side is continuous. The results are applicable for a wide range of single species discrete time population dynamical models, such as (non-autonomous) models by Ricker, Pielou or Clark

Oscillation criteria for linear difference equations with several variable delays

We obtain new sufficient criteria for the oscillation of all solutions of linear delay difference equations with several (variable) finite delays. Our results relax numerous well-known limes inferior-type oscillation criteria from the literature by letting the limes inferior be replaced by the limes superior under some additional assumptions related to slow variation. On the other hand, our findings generalize an oscillation criterion recently given for the case of a constant, single delay