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Smoothly vanishing density in the contact process by an interplay of disorder and long-distance dispersal
by Róbert Juhász, István A. Kovács
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Róbert Juhász |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202405_00018v1 (pdf) |
| Date submitted: | 2024-05-14 10:28 |
| Submitted by: | Juhász, Róbert |
| Submitted to: | SciPost Physics Core |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
Realistic modeling of ecological population dynamics requires spatially explicit descriptions that can take into account spatial heterogeneity as well as long-distance dispersal. Here, we present Monte Carlo simulations and numerical renormalization group results for the paradigmatic model, the contact process, in the combined presence of these factors in both one and two-dimensional systems. Our results confirm our analytic arguments stating that the density vanishes smoothly at the extinction threshold, in a way characteristic of infinite-order transitions. This extremely smooth vanishing of the global density entails an enhanced exposure of the population to extinction events. At the same time, a reverse order parameter, the local persistence displays a discontinuity characteristic of mixed-order transitions, as it approaches a non-universal critical value algebraically with an exponent $\beta_p'<1$.
Current status:
Reports on this Submission
Strengths
1) The work tackles an interesting problem: understanding the combined effect of quenched disorder and long-range dispersal on the dynamical transition in a prototypical lattice model of population dynamics, specifically the Contact Process.
2) The numerical analysis is performed with appropriate methods, which are compared with each others, and the results are presented rather clearly.
Weaknesses
This study provides a numerical verification of results that are anticipated from the authors' previous work. Specifically, the 1D behavior of the stationary density described in Eq. (2) and the local persistence in Eq. (4) are consistent with the SDRG arguments presented in Ref. [30]. Additionally, the qualitative similarity between the 1D and the 2D behavior was suggested by the arguments in Ref. [28]. The numerical results on the exponent characterizing the local persistence in 2D, for which there is no theoretical prediction as far as I understand, are not fully conclusive.
Report
SUMMARY. In this work the Authors analyze a variation of the Contact Process, a lattice model for population dynamics, in which (1) populated (active) sites become empty (inactive) with a random site-dependent rate, and (2) the dispersal is long range, i.e. the probability of populating a site at distance R decays as a power of R. This model displays a transition between an extinction phase, where the stationary density of populated sites is zero, and a fluctuating/active phase, where the density is finite.
The authors conduct a numerical analysis of this model using both Monte Carlo simulations and Strong Disorder Renormalization Group (SDRG) methods, in one and two dimensions. They compute the stationary density (order parameter of the transition) and the local persistence (the probability that a given site is never activated). In the 1D case, their analysis corroborates existing predictions based on SDRG arguments, confirming the infinite-order vanishing of the stationary density at the extinction transition, (Eq. 2), and the mixed-order transition exhibited by local persistence, (Eq. 4). The numerical results further suggest that these behaviors extend to the 2D case.
ASSESSMENT. This work provides a consistent numerical verification of theoretical predictions in 1D and extends these insights to 2D, contributing to the understanding of the effects of quenched disorder and long-range dispersal on the Contact Process. The results are quite clearly presented. Although most of the results verified in this paper have been theoretically derived by the authors in previous works, this paper provides a meaningful and due numerical validation. I therefore recommend publication in SciPost Physics Core.
Requested changes
I think that the presentation of certain results could be slightly improved, to make the manuscript more self-consistent. My suggestions in this direction are reported below.
(i) The Authors stress that the infinite-order behavior in Eq. (2) is a consequence of the combined presence of quenched randomness in the local rates and long-range dispersal, while the two ingredients taken separately lead to an algebraic vanishing of the density at the extinction transition. Do the Authors have some insight on the mechanism behind this?
(ii) The simplified, analytically tractable SDRG analysis of Ref. [31] is mentioned in several points of the manuscript: I would suggest to briefly describe what are the assumptions behind that simplified treatment, in order to make the manuscript self-contained. I would do the same regarding the numerical algorithm to perform SDRG mentioned on p.3 and introduced in Ref. [39].
(iii) What justifies the choice of c, the dilution parameter, used in the Monte Carlo numerics?
(iv) Could the Authors comment on what are the expectations in the case of truly long-range dispersal, i.e. when the exponent alpha is smaller than the dimensionality d?
(v) In the discussion, the Authors mention the model with an environmental gradient as a perspective, and refer to comparison with satellite image data; I would add some reference or clarify this context: which type of data should one think of, which one hopes to describe in terms of a contact process? How should the environmental gradient be implemented?
Recommendation
Ask for minor revision