Approximate probabilistic cellular automata for the dynamics of single-species populations under discrete logisticlike growth with and without weak Allee effects

Abstract

We investigate one-dimensional elementary probabilistic cellular automata (PCA) whose dynamics in first-order mean-field approximation yields discrete logisticlike growth models for a single-species unstructured population with nonoverlapping generations. Beginning with a general six-parameter model, we find constraints on the transition probabilities of the PCA that guarantee that the ensuing approximations make sense in terms of population dynamics and classify the valid combinations thereof. Several possible models display a negative cubic term that can be interpreted as a weak Allee factor. We also investigate the conditions under which a one-parameter PCA derived from the more general six-parameter model can generate valid population growth dynamics. Numerical simulations illustrate the behavior of some of the PCA found.

A probabilistic cellular automata model for the dynamics of a population driven by logistic growth and weak Allee effect

“…Recently, all six-parameter left-right symmetric elementary (two-state) PCA that recover the logistic map with or without weak Allee effect from their first-order mean field approx imation have been identified [31], including their one-parameter counterparts. The one-parameter PCA found, however, have a particular strucure: they are all mixed (or 'diploid') PCA pA-qB with A and B even and A + B = 254.…”

“…PCA p254-q72 does not fit into such PCA, because the transition probabilities φ(1 | 011) = φ(1 | 110) = 1 (see table 1), making bits A 3 = B 3 = 1 and A 6 = B 6 = 1 simultaneously; it thus belongs to another set of mixed PCA that yields logistic-like maps like (8) including the weak Allee effect in the first-order mean field approximation and displays an extinction-survival phase transition. Preliminary results indicate that mixed PCA displaying extinction-survival phase transitions abound, and some effort is currently being made to spot patterns in the mess [31][32][33]. Of particular interest to us is the subclass of mixed PCA that yields cubic maps like (8) in first-order mean field approximation, together with the analysis of the maps themselves.…”

“…The PCA in table 1 is an example of a mixed PCA, also known as 'diploid' cellular automata [28][29][30][31][32][33]. In a mixed PCA, two or more deterministic CA rules are combined such that sometimes one rule is applied, some other times another rule is applied.…”

“…The single-cell mean field approximation thus predict a first-order phase transition for PCA p254-q72 at p * mf = p 0 = (8 − 2 √ 5)/11 0.321. The mean field maps ( 5) and ( 6) can be compared with a discrete-time model for the dynamics of populations under weak Allee effects that reads [24,25,31]…”

“…It is remarkable that the rationale presented in section 3 for the construction a model for a population of individuals simultaneously under the stresses of overcrowding and loneliness recovers a logistic-like map including the weak Allee effect. Although the mean field equation for the dynamics of a one-dimensional PCA of symmetric radius r = 1 is almost always a cubic map and the analysis of cubic maps from PCA can be found in the literature, (i) not every choice of transition probabilities ensue a cubic map with the correct sign, as detailed in [31], and (ii) the connexion between mixed PCA, cubic maps and Allee effects seems to be novel and relevant.…”