AZ �TVEN�VES WRIGHT-SEJT�S T�RT�NETE
Szerz�: R�ST Gergely, PhD hallgat�
T�mavezet�: Dr. KRISZTIN Tibor egyetemi tan�r
Int�zm�ny: Szegedi Tudom�nyegyetem, Term�szettudom�nyi Kar, Bolyai Int�zet,
Alkalmazott �s Numerikus Matematika Tansz�k

Az y'(t) = - ay(t -1)[1+y(t)] �gynevezett k�sleltetett logisztikus differenci�legyenletet el�sz�r Hutchinson haszn�lta �kol�giai modellk�nt a negyvenes �vekben. Az�ta az egyenlet sz�mtalan v�ltozat�t alkalmazt�k a popul�ci�dinamika ter�let�n. Ugyanez az egyenlet n�h�ny �v m�lva felt�nt Lord Cherwell munk�iban, megh�kkent� m�don a pr�msz�mok eloszl�s�val kapcsolatos vizsg�latok sor�n. Wright 1955-ben bizony�totta h�res stabilit�si t�tel�t, amely szerint ha a < 3/2, akkor minden megold�s a 0 egyens�lyi helyzethez konverg�l. Ismert, hogy a=Pi/2 -n�l Hopf-bifurk�ci� t�rt�nik �s megjelenik egy periodikus megold�s. A k�rd�s az, mi t�rt�nik a k�t �rt�k k�z�tt. Wright azt sejtette, hogy ebben az intervallumban is igaz a 0 megold�s glob�lis attraktivit�sa. Wright m�dszereivel hosszas sz�mol�s ut�n az eredm�ny kiterjeszthet� a < 37/24 -ig, a Pi/2-ig azonban senkinek sem siker�lt eljutnia. Gopalsamy 1986-ban k�z�lt egy bizony�t�st a sejt�sre a Math. Proc. Cambridge Phil. Soc. centen�riumi �nnepi sz�m�ban, k�s�bb azonban kider�lt, hogy a bizony�t�s hib�s, �gy a sejt�s tov�bbra is nyitott maradt. �jkelet� eredm�nyeink alapj�n �gy t�nik, siker�lhet megoldani ezt az �tven�ves nevezetes probl�m�t.

Kulcsszavak: logisztikus modell, k�sleltetett visszacsatol�s, funkcion�l-differenci�legyenlet, glob�lis stabilit�s

THE STORY OF THE FIFTY-YEAR-OLD WRIGHT'S CONJECTURE
Author: Gergely R�ST, PhD student
Supervisor: Tibor KRISZTIN, university professor
Institute: University of Szeged, Faculty of Science, Bolyai Institute,
Department of Applied Mathematics

The delayed logistic differential equation y'(t) = - ay(t -1)[1+y(t)]was first studied by Hutchinson as an ecological model in the fourties. Since then different versions of this equation were applied to describe population dynamics. Some years later the same equation appeared in the works of Lord Cherwell, mysteriously related to the distribution of primes. Wright proved his celebrated stability theorem in 1955, namely that if a < 3/2, then every solution tends to the equilibrium 0. It is well-known that a=Pi/2 is a Hopf-bifurcation point and a periodic solution appears. What can we expect between these two values? Wright conjectured that in this whole interval the solution 0 is globally attractive. By the methods of Wright, the result can be extended at the cost of considerable elaboration to a < 37/24, but no one could reach Pi/2. Gopalsamy published a proof to the conjecture in the centennial volume of Math. Proc. Cambridge Phil. Soc. in 1986, but later it turned out that the proof is not correct, so the conjecture remained open. Due to our recent results, it seems to be that there is a chance to solve this fifty-year-old notable conjecture.

Keywords: logistic model, delayed feedback, functional differential equations, global stability