Local Stability in 3D Discrete Dynamical Systems: Application to a Ricker Competition Model

Luís, Rafael; Rodrigues, Elias

doi:10.1155/2017/6186354

Cited by 15 publications

(8 citation statements)

References 26 publications

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“…The demonstrated stability analysis can be applied to other forms of two-species continuous-time and discrete-time population models in returning more complex dynamical systems, such as controlling species, functional responses, time delay [27,28] and the Allee effect. Our work can be extended to study the dynamics of three or more species systems [29] and to understand the stabilising and destabilising factors before obtaining the model outcomes.…”

Section: Discussionmentioning

confidence: 99%

Stability Switching in Lotka-Volterra and Ricker-Type Predator-Prey Systems with Arbitrary Step Size

Don

Burrage

Helmstedt

et al. 2023

Axioms

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Dynamical properties of numerically approximated discrete systems may become inconsistent with those of the corresponding continuous-time system. We present a qualitative analysis of the dynamical properties of two-species Lotka-Volterra and Ricker-type predator-prey systems under discrete and continuous settings. By creating an arbitrary time discretisation, we obtain stability conditions that preserve the characteristics of continuous-time models and their numerically approximated systems. Here, we show that even small changes to some of the model parameters may alter the system dynamics unless an appropriate time discretisation is chosen to return similar dynamical behaviour to what is observed in the corresponding continuous-time system. We also found similar dynamical properties of the Ricker-type predator-prey systems under certain conditions. Our results demonstrate the need for preliminary analysis to identify which dynamical properties of approximated discretised systems agree or disagree with the corresponding continuous-time systems.

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Section: Discussionmentioning

confidence: 99%

Stability Switching in Lotka-Volterra and Ricker-Type Predator-Prey Systems with Arbitrary Step Size

Don

Burrage

Helmstedt

et al. 2023

Axioms

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“…The critical point y 0 is called a stable critical point if for every ò > 0 ∃a δ such that if ψ(t) is any solution of ( )  f = y y satisfying ∥ψ(t 0 ) − y 0 ∥ < δ, then the solution ψ(t) exists ∀t t 0 and it satisfies ∥ψ(t) − y 0 ∥ < ò∀t t 0 . The critical point y 0 is called asymptotically stable if ∃a δ such that if ψ(t) is any solution of  y=f(y) satisfying ∥ψ(t 0 ) − y 0 ∥ < δ then ( ) y ¥ t lim t =y 0 [61]. The difference between the stable and asymptotically stable critical point is that near an asymptotically critical point all trajectories reach at that point, but near a stable critical point all trajectories make a circle near at that point [62].…”

Section: Observational Fit Of the Model Parameters By Using Hubble Da...mentioning

confidence: 99%

Dynamical system method of viscous fluid in f(T) gravity theory

Samaddar,

Sanasam

2024

Phys. Scr.

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In this paper, using the recently introduced f(T) gravity framework, we have analyzed the viscous fluid cosmological model in the FLRW cosmological model by assuming a specific form of the bulk viscosity coefficient as follows: ς=ς0+ς1H+ς2(\dot H/H) and a non-linear f(T) model, particularly f(T) = T − α√−T, where α is the model parameter. The bulk viscosity coefficients are estimated by using the Hubble 31 data set. Using the phase space technique, we examine the asymptotic behavior of our cosmological bulk viscous model.We found one stable critical point. Phase space analysis and geometrical interpretations are given. We discover that according to our model, the Universe evolved from a matter-dominated decelerated phase (a past attractor) to a stable de-Sitter accelerated epoch (afuture attractor). The evolution of the EoS parameter shows the acceleration phase of the cosmic expansion, whereas the negative behavior of viscosity-induced pressure indicates the accelerated expansion of the universe. Additionally, the cosmic matter-energy density exhibits positive behavior. We look into how statefinder parameters behave for our model. We discover that the trajectory of our model starts at the essence and ends at the CDM region. We use the Om diagnostic test, which reveals that our modeldisplays quintessence behavior. The energy condition criteria are analyzed, and we discover that SEC has been violated, whereas WEC, NEC, and DEC satisfy the positivity criteria. Our f(T) cosmological model, with the influence of bulk viscosity, successfully describes the expansion history of the universe and provides a good fit to recent observational data.

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“…Notice that, these 20 parameters are assumed to be positive. For a complete study in local dynamics of this model in dimensions 1, 2 and 3 we refer the papers [10,11].…”

Section: Ricker Competition Model Of 4 Speciesmentioning

confidence: 99%