A unifying approach to discrete single-species populations models

Sandefur, James T.

doi:10.3934/dcdsb.2017194

Cited by 5 publications

(3 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For brain dynamics, the collective behavior of a group of neurons can be represented as a node with a particular dynamics along time [ 3 – 6 ]. In general, the mathematical way to describe and characterize dynamics is by (ordinary or partial) differential equations (continuous time) [ 7 ] or difference equations (discrete time) [ 8 ]. Global models on brain dynamics are grounded on anatomical structural networks built under parcelling of the brain surface [ 9 – 11 ].…”

Section: Introductionmentioning

confidence: 99%

Informational structures: A dynamical system approach for integrated information

et al. 2018

View full text Add to dashboard Cite

Integrated Information Theory (IIT) has become nowadays the most sensible general theory of consciousness. In addition to very important statements, it opens the door for an abstract (mathematical) formulation of the theory. Given a mechanism in a particular state, IIT identifies a conscious experience with a conceptual structure, an informational object which exists, is composed of identified parts, is informative, integrated and maximally irreducible. This paper introduces a space-time continuous version of the concept of integrated information. To this aim, a graph and a dynamical systems treatment is used to define, for a given mechanism in a state for which a dynamics is settled, an Informational Structure, which is associated to the global attractor at each time of the system. By definition, the informational structure determines all the past and future behavior of the system, possesses an informational nature and, moreover, enriches all the points of the phase space with cause-effect power by means of its associated Informational Field. A detailed description of its inner structure by invariants and connections between them allows to associate a transition probability matrix to each informational structure and to develop a measure for the level of integrated information of the system.

show abstract

Section: Introductionmentioning

confidence: 99%

Informational structures: A dynamical system approach for integrated information

et al. 2018

View full text Add to dashboard Cite

show abstract

“…Briefly, in the Leslie model, a (female) animal population is divided into different age classes, typically three, with separate transition rates to the next generation. Whereas the Leslie model and other models for population dynamics typically treat time as a continuous variable, we account for the special proliferation mechanism with discrete size steps at discrete time points and model the system as a discrete dynamical system [ 21 ]. This means that we determine the population state at discrete points in time t, defining distinct generations in accordance with the MacDonald–Pfitzer rule.…”

Section: Methodsmentioning

confidence: 99%

Is the diatom sex clock a clock?

Fuhrmann‐Lieker

Kubetschek

Ziebarth

et al. 2021

J. R. Soc. Interface.

View full text Add to dashboard Cite

The unique life cycle of diatoms with continuous decreasing and restoration of the cell size leads to periodic fluctuations in cell size distribution and has been regarded as a multi-annual clock. To understand the long-term behaviour of a population analytically, generic mathematical models are investigated algebraically and numerically for their capability to describe periodic oscillations. Whereas the generally accepted simple concepts for the proliferation dynamics do not sustain oscillating behaviour owing to broadening of the size distribution, simulations show that a proposed limited lifetime of a newly synthesized cell wall slows down the relaxation towards a time-invariant equilibrium state to the order of a hundred thousand generations. In combination with seasonal perturbation events, the proliferation scheme with limited lifetime is able to explain long-lasting rhythms that are characteristic for diatom population dynamics. The life cycle thus resembles a pendulum clock that has to be wound up from time to time by seasonal perturbations rather than an oscillator represented by a limit cycle.

show abstract

“…where N t is the population density in year t, F(.) is a growth function that describes the ecological mechanisms and processes that underlie the growth dynamics (Sandefur, 2018), and λ (r, r ) is the dispersal kernel which gives the probability distribution of the event that an individual located before dispersal at position r = (x , y ) moves after dispersal to the position r = (x, y) over a dispersal domain Ω (Lewis et al, 2006;Lutscher, 2019). Here, our focus is on the rate of spread in the population which depends on the properties of the dispersal kernel.…”

Section: Equivalence Between Dispersal Kernelsmentioning

confidence: 99%

Equivalence between short- and long-distance dispersal in individual animal movement

Ahmed,

Petrovskii

et al. 2024

Preprint

View full text Add to dashboard Cite

Random walks (RW) provide a useful modelling framework for the movement of animals at an individual level. If the RW is uncorrelated and unbiased such that the direction of movement is completely random, the dispersal is characterised by the statistical properties of the probability distribution of step lengths, or the dispersal kernel. Whether an individual exhibits short-or long-distance dispersal can be distinguished by the rate of asymptotic decay in the end-tail of the distribution of step-lengths. If the decay is exponential or faster, referred to as a thin-tail, then the step length variance is finite -as occurs in Brownian motion. On the other hand, inverse power-law step length distributions have a heavy end-tail with slower decay, resulting in an infinite step length variance, which is the hallmark of a Lévy walk. Although different approaches to relate these different dispersal mechanisms have been used, they are ad hoc and sub-optimal. We provide a more robust method by ensuring that the survival probability, that is the probability of occurrence of steps longer than a fixed characteristic step length is the same for both distributions. Moreover, we derive an optimal value for the survival probability by minimising the L2-distance between the dispersal kernels. By computing the optimal probability for movement paths with commonly used thin- and heavy-tailed step length distributions, we form equivalence between short-and long-distance dispersal of animals in different spatial dimensions. We also demonstrate how our findings can be applied to ecological scenarios, to more accurately relate dispersal mechanisms within a modelling framework for spatio-temporal population dynamics.

show abstract

A unifying approach to discrete single-species populations models

Cited by 5 publications

References 13 publications

Informational structures: A dynamical system approach for integrated information

Informational structures: A dynamical system approach for integrated information

Is the diatom sex clock a clock?

Equivalence between short- and long-distance dispersal in individual animal movement

Contact Info

Product

Resources

About