Dynamics of Delay Equations, Theory and Applications - Abstract
Kiss, Gabor
author
Gábor Kiss
and Gergely Röst
?ffiltextsuperscriptaUniversity of Szeged, Bolyai Institute, Aradi vértanúk tere 1., 6720 Szeged, Hungary
?eceived
addressBolyai Institute
University of Szeged
Aradi vértanúk tere 1.
6720 Szeged, Hungary
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In citegurney1980nicholson,
beginequationlabeleq:NB
x'(t)=-gamma x(t) px(t-tau)e^-alpha x(t-tau), quad p,alpha,gamma,tau>0
endequation
is proposed to model oscillations in it lucilia cuprina populations.
In this model, $frac1alpha$ is the size of the population at which it achieves maximal reproduction success, $gamma$ is the adult death-rate, and parameter $p$ is the maximal egg-production rate.
?ing a global Hopf-bifurcation theorem from citeMR1451617, existence of periodic solutions to eqrefeq:NB is established in citeMR2112957.
The existence of the global attractor for eqrefeq:NB is shown in citeMR2352875 where bounds of the attractor are given, as well.
Recently, citeMR3090069 considers mortality during maturation in eqrefeq:NB leading to
beginequation*labeleq:NBm
x'(t)=-gamma x(t) pe^-deltataux(t-tau)e^-alpha x(t-tau),
endequation*
where, $delta>0$ is the immature death-rate.
Dynamical properties of both the trivial $x=0$ and the positive equilibrium $x^*=frac1alphaleft(lnfracpgamma-deltatau right)$ are analysed in great details.
We report on the effect of considering two maturation sites with different maturation times.
That is, we consider
beginequation*labeleq:NBgm2
x'=-gamma x pleft ( qe^-delta_1 tau_1f(x(t-tau_1)) (1-q)e^-delta_2 tau_2f(x(t-tau_2))right).
endequation*
Our main interest is to investigate how the stability and oscillatory
properties of the blowflies equation are affected by the heterogeneity
of the maturation sites.
beginthebibliography1
bibitemgurney1980nicholson
WSC Gurney, SP Blythe, and RM Nisbet.
newblock Nicholson's blowflies revisited.
newblock em Nature, 287:17--21, 1980.
bibitemMR2352875
G Röst and J Wu.
newblock Domain-decomposition method for the global dynamics of delay
differential equations with unimodal feedback.
newblock em Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.,
463(2086):2655--2669, 2007.
bibitemMR3090069
H Shu, L Wang, and J Wu.
newblock Global dynamics of Nicholson's blowflies equation revisited: onset
and termination of nonlinear oscillations.
newblock em J. Differential Equations, 255(9):2565--2586, 2013.
endthebibliography
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