Qualitative Analysis of a Nonautonomous Beverton--Holt Ricker Model

Hüls, Thorsten; Pötzsche, Christian

doi:10.1137/140955434

Cited by 15 publications

(7 citation statements)

References 32 publications

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“…Note that each f i ∈ F , so T admits a carrying simplex Σ. Denote the set of all maps (46) by Fig. 8 for the orbit simulation.…”

Section: 3mentioning

confidence: 99%

“…Note that f i enjoys the properties: f i (z, r) > 1 for z < r, f i (z, r) = 1 for z = r, and f i (z, r) < 1 for z > r. The discrete-time system induced by map (2) is often used in the modeling of n-species in competition; see [38,58,20,21,70,60,73,14,54,46]. The variable x i is the density of species i and f i is its growth function (or fitness function).…”

mentioning

confidence: 99%

“…The discrete-time system induced by map ( 2) is often used in the modeling of n-species in competition; see [38,58,20,21,70,60,73,14,54,46]. The variable x i is the density of species i and f i is its growth function (or fitness function).…”

mentioning

confidence: 99%

“…Note that each f i ∈ F , so T admits a carrying simplex Σ. Denote the set of all maps (46) by ) for all r 2 > 0. When r 2 ≈ 0.032889, DT [r2] (p) has a pair of complex conjugate eigenvalues with modulus 1 which do not equal ±1, ±i, (−1 ± √ 3i)/2.…”

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confidence: 99%

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Permanence and universal classification of discrete-time competitive systems via the carrying simplex

Gyllenberg¹,

Jiang

Niu³

et al. 2020

Discrete &Amp; Continuous Dynamical Systems - A

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We study the permanence and impermanence for discrete-time Kolmogorov systems admitting a carrying simplex. Sufficient conditions to guarantee permanence and impermanence are provided based on the existence of a carrying simplex. Particularly, for low-dimensional systems, permanence and impermanence can be determined by boundary fixed points. For a class of competitive systems whose fixed points are determined by linear equations, there always exists a carrying simplex. We provide a universal classification via the equivalence relation relative to local dynamics of boundary fixed points for the three-dimensional systems by the index formula on the carrying simplex. There are a total of 33 stable equivalence classes which are described in terms of inequalities on parameters, and we present the phase portraits on their carrying simplices. Moreover, every orbit converges to some fixed point in classes 1 − 25 and 33; there is always a heteroclinic cycle in class 27; Neimark-Sacker bifurcations may occur in classes 26 − 31 but cannot occur in class 32. Based on our permanence criteria and the equivalence classification, we obtain the specific conditions on parameters for permanence and impermanence. Only systems in classes 29, 31, 33 and those in class 27 with a repelling heteroclinic cycle are permanent. Applications to discrete population models including the Leslie-Gower models, Atkinson-Allen models and Ricker models are given.Received xxxx 20xx; revised xxxx 20xx.

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“…Note that each f i ∈ F , so T admits a carrying simplex Σ. Denote the set of all maps (46) by Fig. 8 for the orbit simulation.…”

Section: 3mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Permanence and universal classification of discrete-time competitive systems via the carrying simplex

Gyllenberg¹,

Jiang

Niu³

et al. 2020

Discrete &Amp; Continuous Dynamical Systems - A

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“…An alternative path could be to study directly the system (3.5) by means of some ad-hoc procedures. Examples of such studies are [11], [17] and [2].…”

Section: The Qualitative Study Of the Ode Modelmentioning

confidence: 99%

From ODE to Open Markov Chains, via SDE: an application to models for infections in individuals and populations

Esquível

Patrício

Guerreiro

2020

Computational and Mathematical Biophysics

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We present a methodology to connect an ordinary differential equation (ODE) model of interacting entities at the individual level, to an open Markov chain (OMC) model of a population of such individuals, via a stochastic differential equation (SDE) intermediate model. The ODE model here presented is formulated as a dynamic change between two regimes; one regime is of mean reverting type and the other is of inverse logistic type. For the general purpose of defining an OMC model for a population of individuals, we associate an Ito processes, in the form of SDE to ODE system of equations, by means of the addition of Gaussian noise terms which may be thought to model non essential characteristics of the phenomena with small and undifferentiated influences. The next step consists on discretizing the SDE and using the discretized trajectories computed by simulation to define transitions of a finite valued Markov chain; for that, the state space of the Ito processes is partitioned according to some rule. For the example proposed for illustration, the state space of the ODE system referred – corresponding to a model of a viral infection – is partitioned into six infection classes determined by some of the critical points of the ODE system; we detail the evolution of some infected population in these infection classes.

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