Global stability and bifurcations in a delayed discrete population model

We consider a family of difference equations used in population dynamics. First we recall the most relevant results concerning the global stability of the positive equilibrium. Then, using the survival rate as a parameter, we investigate the changes in the dynamics when it ranges between zero (semelparous populations) and one. AMS Subject Classifications: 39A11, 37C25, 92D25

[1]  O. Diekmann,et al.  On a boom and bust year class cycle , 2005 .

[2]  H. El-Morshedy The global attractivity of difference equations of nonincreasing nonlinearities with applications , 2003 .

[3]  A. Ivanov On global stability in a nonlinear discrete model , 1994 .

[4]  Odo Diekmann,et al.  Saumon à la Kaitala et Getz, sauce hollandaise , 1999 .

[5]  U. Krause,et al.  Boundedness and Stability for Higher Order Difference Equations* , 2004 .

[6]  L. Glass,et al.  Oscillation and chaos in physiological control systems. , 1977, Science.

[7]  I. Györi,et al.  Global attractivity and presistence in a discrete population model , 2000 .

[8]  Viktor Tkachenko,et al.  A Global Stability Criterion for Scalar Functional Differential Equations , 2003, SIAM J. Math. Anal..

[9]  Global attractors for difference equations dominated by one-dimensional maps , 2008 .

[10]  Ivor Brown,et al.  Old and Young: , 1971 .

[11]  T. Morrison,et al.  Dynamical Systems , 2021, Nature.

[12]  Global Attractivity of the Equilibrium of a Nonlinear Difference Equation , 2002 .

[13]  Vassilis G. Papanicolaou On the Asymptotic Stability of a Class of Linear Difference Equations , 1996 .

[14]  Convergence to equilibria in discrete population models , 2005 .

[15]  N. Davydova Old and Young. Can they coexist , 2004 .

[16]  Louis W. Botsford Further analysis of Clark's delayed recruitment model , 1992 .

[17]  An augmented Clark model for stability of populations. , 1996, Mathematical biosciences.

[18]  D. Aronson,et al.  Bifurcations from an invariant circle for two-parameter families of maps of the plane: A computer-assisted study , 1982 .

[19]  D. A. Singer,et al.  Stable Orbits and Bifurcation of Maps of the Interval , 1978 .

[20]  W. Ricker Stock and Recruitment , 1954 .

[21]  Stability of a class of delay-difference equations , 1984 .

[22]  V. Kocić,et al.  Global Behavior of Nonlinear Difference Equations of Higher Order with Applications , 1993 .

[23]  Anatoli F. Ivanov,et al.  Oscillations in Singularly Perturbed Delay Equations , 1992 .

[24]  Hassan A. El-Morshedy,et al.  Globally attracting fixed points in higher order discrete population models , 2006, Journal of mathematical biology.

[25]  R. Devaney An Introduction to Chaotic Dynamical Systems , 1990 .

[26]  J. Hale,et al.  Dynamics and Bifurcations , 1991 .

[27]  S. A. Kuruklis,et al.  The Asymptotic Stability of xn+1 − axn + bxn−k = 0 , 1994 .

[28]  R M May,et al.  A note on difference-delay equations. , 1976, Theoretical population biology.

[29]  S. Trofimchuk,et al.  Global stability in difference equations satisfying the generalized Yorke condition , 2005 .

[30]  Y. G. Sficas,et al.  The dynamics of some discrete population models , 1991 .

[31]  John Shepherd,et al.  A versatile new stock-recruitment relationship for fisheries, and the construction of sustainable yield curves , 1982 .

[32]  O. Diekmann,et al.  Difference equations with delay , 2000 .

[33]  C W Clark,et al.  A delayed-recruitment model of population dynamics, with an application to baleen whale populations , 1976, Journal of mathematical biology.

[34]  S. Wiggins Introduction to Applied Nonlinear Dynamical Systems and Chaos , 1989 .

[35]  Periodic points and stability in Clark's delayed recruitment model , 2008 .

[36]  S. P. Blythe,et al.  Nicholson's blowflies revisited , 1980, Nature.

[37]  F. Balibrea,et al.  On the Periodic Structure of Delayed Difference Equations of the Form x n = f ( x n − k ) on I and S 1 , 2003 .