scholarly journals Dynamics of a discrete-time pioneer–climax model

2021 ◽  
Author(s):  
Nora M. Gilbertson ◽  
Mark Kot
Keyword(s):  
Discrete Time ◽  
Global Bifurcation ◽  
Relative Strength ◽  
Homoclinic Bifurcation ◽  
Invariant Circle ◽  
Ecological Implications ◽  
Zero Growth ◽  
Sudden Transition ◽  
Coexistence State

AbstractWe present a simple mathematical model for the dynamics of a successional pioneer–climax system using difference equations. Each population is subject to inter- and intraspecific competition; population growth is dependent on the combined densities of both species. Nine different geometric cases, corresponding to different orientations of the zero-growth isoclines, are possible for this system. We fully characterize the long-term dynamics of the model for each of the nine cases, uncovering diverse sets of potential behaviors. Competitive exclusion of the pioneer species and of the climax species are both possible depending on the relative strength of competition. Stable coexistence of both species may also occur; in two cases, a coexistence state is destabilized through a Neimark–Sacker bifurcation, and an attracting invariant circle is born. The invariant circle eventually disappears into thin air in a heteroclinic or homoclinic bifurcation, leading to the sudden transition of the system to an exclusion state. Neither global bifurcation has been observed in a discrete-time pioneer–climax model before. The homoclinic bifurcation is novel to all pioneer–climax models. We conclude by discussing the ecological implications of our results.

Dynamics of a Discrete-time Pioneer–climax Model

2021 ◽  
Author(s):  
Nora Gilbertson ◽  
Mark Kot
Keyword(s):  
Discrete Time ◽  
Global Bifurcation ◽  
Relative Strength ◽  
Homoclinic Bifurcation ◽  
Invariant Circle ◽  
Ecological Implications ◽  
Zero Growth ◽  
Sudden Transition ◽  
Coexistence State

Abstract We present a simple mathematical model for the dynamics of a successional pioneer–climax system using difference equations. Each population is subject to inter- and intraspecific competition; population density growth is dependent on the combined densities of both species. Nine different geometric cases, corresponding to different orientations of the zero-growth isoclines, are possible for this system. We fully characterize the long-term dynamics of the model for each of the nine cases, uncovering diverse sets of potential behaviors. Competitive exclusion of the pioneer species and exclusion of the climax species are both possible depending on the relative strength of competition. Stable coexistence of both species may also occur; in two cases, a coexistence state is destabilized through a Neimark–Sacker bifurcation and an attracting invariant circle is born. The invariant circle eventually disappears into thin air in a heteroclinic or homoclinic bifurcation, leading to the sudden transition of the system to an exclusion state. Neither global bifurcation has been observed in a discrete-time pioneer–climax model before. The homoclinic bifurcation is novel to all pioneer–climax models. We conclude by discussing the ecological implications of our results.

ROBUST STABILITY, STABILISATION AND H-INFINITY CONTROL FOR PREMIUM-RESERVE MODELS IN A MARKOVIAN REGIME SWITCHING DISCRETE-TIME FRAMEWORK

2016 ◽  
Vol 46 (3) ◽  
pp. 747-778 ◽  
Author(s):  
Lin Yang ◽  
Athanasios A. Pantelous ◽  
Hirbod Assa
Keyword(s):  
Discrete Time ◽  
Regime Switching ◽  
Structural Changes ◽  
Linear Matrix ◽  
H Infinity Control ◽  
R Process ◽  
The Stability ◽  
Markovian Regime Switching ◽  
Markovian Regime

AbstractThe premium pricing process and the medium- and long-term stability of the reserve policy under conditions of uncertainty present very challenging issues in relation to the insurance world. Over the last two decades, applications of Markovian regime switching models to finance and macroeconomics have received strong attention from researchers, and particularly market practitioners. However, relatively little research has so far been carried out in relation to insurance. This paper attempts to consider how a linear Markovian regime switching system in discrete-time could be applied to model the medium- and long-term reserves and the premiums (abbreviated here as the P-R process) for an insurer. Some recently developed techniques from linear robust control theory are applied to explore the stability, stabilisation and robust H∞-control of a P-R system, and the potential effects of abrupt structural changes in the economic fundamentals, as well as the insurer's strategy over a finite time period. Sufficient linear matrix inequality conditions are derived for solving the proposed sub-problems. Finally, a numerical example is presented to illustrate the applicability of the theoretical results.

On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states

1966 ◽  
Vol 3 (02) ◽  
pp. 403-434 ◽  
Author(s):  
E. Seneta ◽  
D. Vere-Jones
Keyword(s):  
Random Walk ◽  
Markov Chains ◽  
Recent Work ◽  
Discrete Time ◽  
Branching Process ◽  
Stationary Distributions

Distributions appropriate to the description of long-term behaviour within an irreducible class of discrete-time denumerably infinite Markov chains are considered. The first four sections are concerned with general reslts, extending recent work on this subject. In Section 5 these are applied to the branching process, and give refinements of several well-known results. The last section deals with the semi-infinite random walk with an absorbing barrier at the origin.

ONE-DIMENSIONAL CHAOS IN IMPULSED LINEAR OSCILLATING CIRCUITS

1993 ◽  
Vol 03 (04) ◽  
pp. 921-941 ◽  
Author(s):  
LAURA GARDINI ◽  
RENZO LUPINI
Keyword(s):  
Asymptotic Behavior ◽  
Fixed Points ◽  
Critical Point ◽  
Critical Points ◽  
Phase Plane ◽  
Global Bifurcation ◽  
Homoclinic Bifurcation ◽  
Combined Effect ◽  
Chaotic Oscillations ◽  
One Dimensional

The dynamics of a damped linear oscillating circuit subject to impulses is represented by a one-dimensional endomorphism (or noninvertible map) π: ℝ → ℝ. The asymptotic behavior of orbits in the phase-plane is characterized in terms of critical points and point singularities of π (fixed points or cycles). Their combined effect, that is, the merging of a critical point into a repelling cycle, causes a global bifurcation or a homoclinic bifurcation, with transition to chaotic oscillations.

scholarly journals Market Share Delegation in a Bertrand Duopoly: Synchronisation and Multistability

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
Luciano Fanti ◽  
Luca Gori ◽  
Cristiana Mammana ◽  
Elisabetta Michetti
Keyword(s):  
Market Share ◽  
Discrete Time ◽  
Price Competition ◽  
Initial Conditions ◽  
Two Dimensional ◽  
Discrete Time System ◽  
Multiple Attractors ◽  
Time System ◽  
Market Share Delegation

This paper tackles the issue of local and global analyses of a duopoly game with price competition and market share delegation. The dynamics of the economy is characterised by a differentiable two-dimensional discrete time system. The paper stresses the importance of complementarity between products as a source of synchronisation in the long term, in contrast to the case of their substitutability. This means that when products are complements, players may coordinate their behaviour even if initial conditions are different. In addition, there exist multiple attractors so that even starting with similar conditions may end up generating very different dynamic patterns.

scholarly journals Transient Dynamics of the Lorenz System with a Parameter Drift

2021 ◽  
Vol 31 (02) ◽  
pp. 2150029
Author(s):  
Julia Cantisán ◽  
Jesús M. Seoane ◽  
Miguel A. F. Sanjuán
Keyword(s):  
Lorenz System ◽  
Scaling Law ◽  
Transient State ◽  
Rate Of Change ◽  
Transient Dynamics ◽  
Change Rate ◽  
The Lorenz System ◽  
Sudden Transition ◽  
The Relationship

Nonautonomous dynamical systems help us to understand the implications of real systems which are in contact with their environment as it actually occurs in nature. Here, we focus on systems where a parameter changes with time at small but non-negligible rates before settling at a stable value, by using the Lorenz system for illustration. This kind of systems commonly show a long-term transient dynamics previous to a sudden transition to a steady state. This can be explained by the crossing of a bifurcation in the associated frozen-in system. We surprisingly uncover a scaling law relating the duration of the transient to the rate of change of the parameter for a case where a chaotic attractor is involved. Additionally, we analyze the viability of recovering the transient dynamics by reversing the parameter to its original value, as an alternative to the control theory for systems with parameter drifts. We obtain the relationship between the paramater change rate and the number of trajectories that tip back to the initial attractor corresponding to the transient state.

scholarly journals The influence of mixing on the stratospheric age of air changes in the 21st century

2019 ◽  
Vol 19 (2) ◽  
pp. 921-940 ◽  
Author(s):  
Roland Eichinger ◽  
Simone Dietmüller ◽  
Hella Garny ◽  
Petr Šácha ◽  
Thomas Birner ◽  
...  
Keyword(s):  
21St Century ◽  
Climate Models ◽  
Climate Model ◽  
Phase 1 ◽  
Relative Strength ◽  
Mixing Efficiency ◽  
Climate Projections ◽  
Model Studies ◽  
State Potential

Abstract. Climate models consistently predict an acceleration of the Brewer–Dobson circulation (BDC) due to climate change in the 21st century. However, the strength of this acceleration varies considerably among individual models, which constitutes a notable source of uncertainty for future climate projections. To shed more light upon the magnitude of this uncertainty and on its causes, we analyse the stratospheric mean age of air (AoA) of 10 climate projection simulations from the Chemistry-Climate Model Initiative phase 1 (CCMI-I), covering the period between 1960 and 2100. In agreement with previous multi-model studies, we find a large model spread in the magnitude of the AoA trend over the simulation period. Differences between future and past AoA are found to be predominantly due to differences in mixing (reduced aging by mixing and recirculation) rather than differences in residual mean transport. We furthermore analyse the mixing efficiency, a measure of the relative strength of mixing for given residual mean transport, which was previously hypothesised to be a model constant. Here, the mixing efficiency is found to vary not only across models, but also over time in all models. Changes in mixing efficiency are shown to be closely related to changes in AoA and quantified to roughly contribute 10 % to the long-term AoA decrease over the 21st century. Additionally, mixing efficiency variations are shown to considerably enhance model spread in AoA changes. To understand these mixing efficiency variations, we also present a consistent dynamical framework based on diffusive closure, which highlights the role of basic state potential vorticity gradients in controlling mixing efficiency and therefore aging by mixing.

Classified Discrete-Time Markov Chains

2015 ◽  
pp. 87-115
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Discrete Time ◽  
Formal Definition ◽  
Markov Chain Models ◽  
Definition Of ◽  
Markovian Systems

The main focus of this chapter is on the formalization of classified DTMCs. The chapter begins by presenting the formalization of some foundational notions of classified states, which are categorized based on reachability, periodicity, or absorbing features. Then, these results along with the formal definition of a DTMC, presented in the previous chapter, are used to formalize classified Markov chains, such as aperiodic and irreducible DTMCs. Based on these concepts, some long-term properties are verified for the purpose of formally checking the correctness of the functions of Markovian systems or analyzing the performance of Markov chain models.

Denmark

2019 ◽  
pp. 192-209 ◽  
Author(s):  
Jørgen Goul Andersen
Keyword(s):  
Private Consumption ◽  
Social Protection ◽  
Social Citizenship ◽  
Public Employees ◽  
Tax Reforms ◽  
Public Consumption ◽  
Zero Growth ◽  
Catch Up ◽  
The Government

Jørgen Goul Andersen shows that Denmark entered the crisis with a strong economy, but also with a housing and credit bubble beginning to burst. This contributed to constraining private consumption for a long period. When the government switched in May 2010 to zero growth in public consumption and to significant cuts in social protection, this left exports as the only possible driver of growth. It took ten years for private consumption per capita to catch up with 2007 levels, while the number of public employees declined by 5 per cent from 2010 to 2016. Welfare and tax reforms since 2008 have contributed to an erosion of social citizenship, accompanied by strong rhetoric questioning the deservingness of people in a vulnerable position. Whereas the weakest groups are at risk of being excluded from social protection, the long-term decline in welfare services may ‘crowd in’ private welfare for the upper middle classes.

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