Travelling fronts in a food-limited population model with time delay

2002 ◽  
Vol 132 (1) ◽  
pp. 75-89 ◽  
Author(s):  
S. A. Gourley ◽  
M. A. J. Chaplain
Keyword(s):  
Population Model ◽  
Time Delays ◽  
Front Solutions ◽  
Heteroclinic Connection ◽  
Non Local ◽  
Spatial Terms ◽  
Travelling Fronts ◽  
Manifold Theory ◽  
And Diffusion ◽  
Existence Question

In this paper we study travelling front solutions of a certain food-limited population model incorporating time-delays and diffusion. Special attention is paid to the modelling of the time delays to incorporate associated non-local spatial terms which account for the drift of individuals to their present position from their possible positions at previous times. For a particular class of delay kernels, existence of travelling front solutions connecting the two spatially uniform steady states is established for sufficiently small delays. The approach is to reformulate the problem as an existence question for a heteroclinic connection in R4. The problem is then tackled using dynamical systems techniques, in particular, Fenichel's invariant manifold theory. For larger delays, numerical simulations reveal changes in the front's profile which develops a prominent hump.

scholarly journals Transmission and diffusion: Linguistic change in the regional French of Béarn

2015 ◽  
Vol 26 (3) ◽  
pp. 327-352
Author(s):  
DAMIEN MOONEY
Keyword(s):  
Individual Change ◽  
Vowel Quality ◽  
Linguistic Change ◽  
Local Varieties ◽  
Transmission Process ◽  
Non Local ◽  
Diffusion Interface ◽  
Successive Generations ◽  
And Diffusion ◽  
Shed Light

ABSTRACTThis article examines the seemingly dichotomous linguistic processes of transmission and diffusion (Labov, 2007) in the regional variety of French spoken in Béarn, southwestern France. Using a sociophonetic apparent time methodology, an analysis of nasal vowel quality provides evidence for the advancement of linguistic changes from below taking place between successive generations during the transmission process, as well as for change from above taking place in the variety as a result of exposure to diffusing non-local varieties of French. The results address Labov's (2007) assertion that it is rare to investigate incremental changes occurring from below in European dialectological studies and shed light on the transmission–diffusion interface by showing the adoption of an individual change from above to instigate a faithfully-transmitted counterclockwise chain shift in the regional French nasal vowel system.

scholarly journals Permanence of dispersal population model with time delays

2006 ◽  
Vol 192 (2) ◽  
pp. 417-430 ◽  
Author(s):  
Yasuhiro Takeuchi ◽  
Jing’an Cui ◽  
Rinko Miyazaki ◽  
Yasuhisa Saito
Keyword(s):  
Population Model ◽  
Time Delays

Coupled non-local periodic parabolic systems with time delays

2008 ◽  
Vol 87 (4) ◽  
pp. 479-495 ◽  
Author(s):  
Rong-Nian Wang ◽  
Ti-Jun Xiao ◽  
Jin Liang
Keyword(s):  
Time Delays ◽  
Parabolic Systems ◽  
Non Local

Stability and Hopf bifurcation analysis for Nicholson's blowflies equation with non-local delay

2012 ◽  
Vol 23 (6) ◽  
pp. 777-796 ◽  
Author(s):  
RUI HU ◽  
YUAN YUAN
Keyword(s):  
Hopf Bifurcation ◽  
Upper And Lower Solutions ◽  
Laboratory Data ◽  
Steady State Solution ◽  
Steady States ◽  
Nicholson’S Blowflies Equation ◽  
Non Local ◽  
The Stability ◽  
Manifold Theory ◽  
Theoretical Results

We consider a diffusive Nicholson's blowflies equation with non-local delay and study the stability of the uniform steady states and the possible Hopf bifurcation. By using the upper- and lower solutions method, the global stability of constant steady states is obtained. We also discuss the local stability via analysis of the characteristic equation. Moreover, for a special kernel, the occurrence of Hopf bifurcation near the steady state solution and the stability of bifurcated periodic solutions are given via the centre manifold theory. Based on laboratory data and our theoretical results, we address the influence of various types of vaccinations in controlling the outbreak of blowflies.

ON DIMENSION OF NON-LOCAL BIFURCATIONAL PROBLEMS

1996 ◽  
Vol 06 (05) ◽  
pp. 919-948 ◽  
Author(s):  
D. TURAEV
Keyword(s):  
Lyapunov Exponents ◽  
Center Manifold ◽  
Critical Dimension ◽  
Lyapunov Dimension ◽  
Stable And Unstable Manifolds ◽  
Local Bifurcations ◽  
Non Local ◽  
Unstable Manifolds ◽  
The Difference ◽  
Manifold Theory

An analogue of the center manifold theory is proposed for non-local bifurcations of homo- and heteroclinic contours. In contrast with the local bifurcation theory it is shown that the dimension of non-local bifurcational problems is determined by the three different integers: the geometrical dimension dg which is equal to the dimension of a non-local analogue of the center manifold, the critical dimension dc which is equal to the difference between the dimension of phase space and the sum of dimensions of leaves of associated strong-stable and strong-unstable foliations, and the Lyapunov dimension dL which is equal to the maximal possible number of zero Lyapunov exponents for the orbits arising at the bifurcation. For a wide class of bifurcational problems (the so-called semi-local bifurcations) these three values are shown to be effectively computed. For the orbits arising at the bifurcations, effective restrictions for the maximal and minimal numbers of positive and negative Lyapunov exponents (correspondingly, for the maximal and minimal possible dimensions of the stable and unstable manifolds) are obtained, involving the values dc and dL. A connection with the problem of hyperchaos is discussed.

scholarly journals A generalized fractional-order elastodynamic theory for non-local attenuating media

2020 ◽  
Vol 476 (2238) ◽  
pp. 20200200 ◽  
Author(s):  
Sansit Patnaik ◽  
Fabio Semperlotti
Keyword(s):  
Fractional Order ◽  
Elastic Waves ◽  
Energy Transport ◽  
Transport Processes ◽  
Non Local ◽  
The Many ◽  
Propagation Of Elastic Waves ◽  
And Diffusion ◽  
Fully Coupled ◽  
Theoretical Results

This study presents a generalized elastodynamic theory, based on fractional-order operators, capable of modelling the propagation of elastic waves in non-local attenuating solids and across complex non-local interfaces. Classical elastodynamics cannot capture hybrid field transport processes that are characterized by simultaneous propagation and diffusion. The proposed continuum mechanics formulation, which combines fractional operators in both time and space, offers unparalleled capabilities to predict the most diverse combinations of multiscale, non-local, dissipative and attenuating elastic energy transport mechanisms. Despite the many features of this theory and the broad range of applications, this work focuses on the behaviour and modelling capabilities of the space-fractional term and on its effect on the elastodynamics of solids. We also derive a generalized fractional-order version of Snell’s Law of refraction and of the corresponding Fresnel’s coefficients. This formulation allows predicting the behaviour of fully coupled elastic waves interacting with non-local interfaces. The theoretical results are validated via direct numerical simulations.

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