Philippa Kaur

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Flash recordingAll field recording and experiments were performed at the paddy field in the Northern Chita Peninsula, Aichi Prefecture, central Japan, in June and July between 2003 and 2016. The ambient temperature at the firefly’s active period was measured using a thermometer. The flashes were recorded with a digital video camera (NV-GS-400, Panasonic, Japan) mounted on a tripod at a height of 30–50 cm from ground and a distance of 1.0–1.5 m away from the specimen. Isolated specimens were selected for recording to exclude the background light from other nontarget specimens. When another specimen appeared near the target specimen, the video recording was cancelled. When a female copulated during video recording in the field, her flashes until 1 min before copulation were regarded as those of a ‘receptive female’. To record the flashes of a ‘mated female’, the female specimens already mated were prepared in aquariums (because virgin and mated females cannot be distinguished in the field): the eggs were obtained from wild female specimens collected one year before at the same field and reared to adults; immediately after emergence the virgin female was confined in a small container with two cultured males for two nights to facilitate copulation. As the parents of the reared specimens were collected from the observation field (same genetic background), the rearing temperature was almost the same as that of the natural field, the emergence period of the cultured specimens overlapped with that of the natural population, the adult body sizes of the reared and natural specimens were indistinguishable, and the flash pattern of the cultured mated females was indistinguishable from that of the wild (potentially) mated females. Thus, we believe that there was no influence of different rearing environments, i.e., the flash behavior of the cultured mated female specimens is expected to be substantially the same as that of wild mated female specimens. To distinguish them from wild (potentially) mated females, the elytra of cultured mated females were marked with colored ink before placing them in the field, and after three days, the flashes of ink-marked specimens were recorded. Of note, we never observed male attraction and copulation in any of the mated females used for field observation; thus, the mated females were unreceptive.Waveform analysisSequential still images were captured from video files at 30 frames per second using VirtualDub (GPL), and then the light intensities in the images were qualified (8-bit linear gray scaling from black to white at 0–255) using ImageJ software. In this study, we defined ‘flash’ as a luminescent waveform from baseline to baseline and ‘flickering’ as fluctuation above baseline in a single flash. The waveforms containing a saturated signal (255, white) were omitted. The waveforms of the maximum signal value lower than 50 were also omitted because of the difficulty in separating signal and noise. Approximately 10–90 waveforms per individual were analyzed; thus, the effect of the occasional interruption of the flash recording by the specimen’s movement and/or vegetation swinging between the specimen and the video camera is statistically ignorable. FD is defined as the time interval between the beginning and the end of a flash (Fig. S1). Flicker intensity (FI) was defined as$${text{FI}} = left{ {begin{array}{*{20}l} {mathop {max }limits_{1 le i le n} left( {frac{{{text{min}}left( {p_{i} ,p_{i + 1} } right) – t_{i} }}{{min left( {p_{i} , p_{i + 1} } right) + t_{i} }}} right)} hfill & {{text{if}} , n ge 1} hfill \ 0 hfill & {{text{if}} , n = 0} hfill \ end{array} } right.$$where p, t, and n denote the peak and the trough (local extrema) in the waveform of a flash and the number of toughs in the flash, respectively (Fig. S1). In total, we measured the FD and FI values of 347, 94, and 355 waveforms from 13 sedentary males, 7 receptive females, and 8 mated females, respectively. We did not consider the flash brightness as a factor because the measured value of the light intensity depends largely on the distance between the light source and the detector; thus, the actual brightness of the lantern cannot be practically measured in the field.e-FireflyFor male attraction experiments, we built an electronic LED device, the e-firefly, to generate patterned flashes with various FDs and FIs using a chip LED (green type, λmax = 568 nm, Everlight Electronics, Taiwan; Figs. S2 and S3) with a microcontroller PIC16F628A (Microchip Technology, USA) (see Figs. S4-S5). An example of the program for the microcontroller is shown in Supplementary Data S1. The brightness was constant in all programs. Flickering frequency ranged between 5–12 Hz, which corresponds to that of sedentary male flashes (approximately 10 Hz)15. To prevent direct access of the attracted specimen to the light source, the chip LED was covered by a steel net painted green (see Fig. S2). For flying male attraction experiments, when the male landed within a 100-mm distance from the e-firefly, we judged the attraction to be a success; otherwise, it was a failure. For sedentary male attraction experiments, the e-firefly was placed 200–300 mm away from the sedentary male. When the approaching male touched the steel net covering the e-firefly, to warrant a positive approach, we measured the time the male remained on the net. If the male did not move away from the net for more than 2 min, we judged the attraction to be a success (strict criterion for judgment); otherwise, it was a failure.Spectral measurementThe luminescence spectra of e-firefly and A. lateralis were measured using a Flame-S spectrophotometer (Ocean Insight, USA). The living A. lateralis specimens were anesthetized on ice and frozen at − 20 °C until use. The lantern started luminescence by thawing at room temperature, and the spectrum was measured during luminescence (within 5 min).Statistical analysisFirst, we considered a discriminant analysis using a logistic regression model that discriminates between receptive females and others in the observational data. We fitted several models with combinations of FD and FI, quadratic terms of FD and FI (FD2, FI2), interaction of FD and FI (FD (times) FI), and temperature (T) as explanatory variables. Based on Akaike’s information criteria (AIC) values and model simplicity, we chose the logistic regression model with FD, FI, FD2 and T as explanatory variables. Let (p)(({varvec{x}})) denote the conditional probability that a flash is from a receptive female given ({varvec{x}}=left(mathrm{FD},mathrm{ FI},mathrm{ T}right)) and (widehat{p})(({varvec{x}})) denote its estimate. The coefficients of the logistic regression model are estimated as follows.
[Model for the observational data with temperature (T)]
$$begin{gathered} {text{log}}frac{{hat{p}}}{{1 – hat{p}}} = begin{array}{*{20}l} { – 32.26 + 69.69 times FD – 43.47 times FI – 76.63 times FD^{2} + 0.87 times T} hfill \ {~quad left( {6.50} right)quad quad left( {15.37} right)quad quad quad left( {8.56} right)quad quad quad quad left( {17.44} right)quad quad quad left( {0.19} right)~~} hfill \ end{array} hfill \ quad {text{AIC: 84}}{text{.75}} hfill \ end{gathered}$$[Model for the observational data without temperature (T)]$$begin{gathered} {text{log}}frac{{hat{p}}}{{1 – hat{p}}} = begin{array}{*{20}l} { – 7.69~ + 47.57 times FD~ – 38.29 times FI~ – 52.86 times FD^{2} ~} hfill \ {~;left( {1.86} right)quad quad left( {9.68} right)quad quad quad left( {7.08} right)quad quad quad quad left( {11.38} right)~~} hfill \ end{array} hfill \ quad {text{AIC: 114}}{text{.89}} hfill \ end{gathered}$$where values in parentheses indicate standard deviations. The same applies hereafter. Temperature (T) is included in the model not because it affects the occurrence of receptive females but because it affects the FD and/or FI of receptive females. The AIC value increased by 30, which is substantial, when temperature was excluded from the model.Figure 2 shows the FD and FI of each flash from receptive females, mated females and males with the discriminant boundaries of receptive females from others for (p=0.5).We next considered a discriminant analysis for the experimental data. Let ({q}^{f}({varvec{x}})) denote the conditional probability that a flying male is attracted to a flash of ({varvec{x}}=left(mathrm{FD},mathrm{ FI},mathrm{ T}right)) and lands, and ({widehat{q}}^{f}({varvec{x}})) denote its estimate. Among several models we fit, the smallest AIC value is attained by the logistic regression model with FD, FI and T as explanatory variables, but the AIC is not much different from the model with FD and FI only.
[Model for flying males with temperature (T)]
$$begin{gathered} {text{log}}frac{{hat{q}^{f} }}{{1 – hat{q}^{f} }} = begin{array}{*{20}l} { – 0.74~~ – 2.42 times FD – 16.82 times FI + 0.31 times T} hfill \ {~;left( {4.01} right)quad quad left( {0.83} right)quad quad quad left( {4.88} right)quad quad quad quad left( {0.20} right)~} hfill \ end{array} hfill \ quad {text{AIC}}:66.96 hfill \ end{gathered}$$

[Model for flying males without temperature (T)]
$$begin{gathered} {text{log}}frac{{hat{q}^{f} }}{{1 – hat{q}^{f} }} = begin{array}{*{20}l} { – 5.36~ – 1.72 times FD – 13.69 times FI} hfill \ {~;left( {1.49} right)quad quad left( {0.63} right)~quad quad left( {4.09} right)~~} hfill \ end{array} hfill \ quad {text{AIC}}:67.61 hfill \ end{gathered}$$
For sedentary males, the model with the smallest AIC value includes all the quadratic terms of FI and FD but not temperature. Let ({q}^{s}({varvec{x}})) denote the conditional probability that a sedentary male is attracted to a flash of ({varvec{x}}=left(mathrm{FD},mathrm{ FI},mathrm{ T}right)) and ({widehat{q}}^{s}left({varvec{x}}right)) denote its estimate. The logistic regression model for ({q}^{s}({varvec{x}})) with the best AIC value is given as follows.
[Model for sedentary males]
$${text{log}}frac{{hat{q}~^{s} }}{{1 – hat{q}~^{s} }} = begin{array}{*{20}l} { – 0.68~ + 7.84 times FD~ + 48.17 times FI – 5.35 times FD^{2} – 166.70 times FI^{2} – 65.67 times FD times FI} hfill \ {;left( {0.97} right)quad quad quad left( {2.99} right)quad quad quad left( {17.74} right)quad quad quad left( {1.74} right)quad quad quad quad left( {72.34} right)quad quad quad quad left( {17.67} right)~} hfill \ end{array}$$
Figure 3 shows successes and failures of attraction of flying males on the left and sedentary males on the right with estimated discriminant boundaries.Let us now estimate probabilities that a flying male is attracted and lands or a sedentary male is attracted to a flash when a flash is from a receptive female or when a flash is either from a sedentary male or mated female. The probability that a flying male is attracted and lands when a flash is from a receptive female is a conditional probability and is expressed as follows.$$begin{aligned} Pleft(left.begin{array}{*{20}c} {text{Flying male}} \ {text{is attracted}} \ end{array} right|begin{array}{*{20}c} {text{Receptive }} \ {{text{female}}} \ end{array} right) & = frac{{Pleft( {begin{array}{*{20}c} {text{Flying male}} \ {text{is attracted}} \ end{array} {text{ and }}begin{array}{*{20}c} {text{Receptive }} \ {{text{female}}} \ end{array} } right) }}{{Pleft( {begin{array}{*{20}c} {{text{Receptive}}} \ {{text{female}}} \ end{array} } right)}}, \ Pleft( {begin{array}{*{20}c} {{text{Receptive}}} \ {{text{female}}} \ end{array} } right) & = mathop int_{Omega } Pleft(left. begin{array}{*{20}c} {{text{Receptive}}} \ {{text{female}}} \ end{array} right|{varvec{x}} right)fleft( {varvec{x}} right)d{varvec{x}} = mathop int_{Omega }pleft( {varvec{x}} right) fleft( {varvec{x}} right)d{varvec{x}} hspace{5mm}{text{and}} \ Pleft( {begin{array}{*{20}c} {text{Flying male}} \ {text{is attracted}} \ end{array} {text{ and }}begin{array}{*{20}c} {text{Receptive }} \ {{text{female}}} \ end{array} } right) & = mathop int_{Omega } Pleft(left. begin{array}{*{20}c} {{text{Receptive}}} \ {{text{female}}} \ end{array} right|varvec{x} right)Pleft(left. begin{array}{*{20}c} {text{Flying male}} \ {text{is attracted}} \ end{array} right|{varvec{x}} right)fleft( {varvec{x}} right)d{varvec{x}} \ & = mathop int_{Omega } pleft( varvec{x} right)q^{f} left( {varvec{x}} right)fleft( {varvec{x}} right)d{varvec{x}}mathbf{.} \ end{aligned}$$Integrals are taken over the domain (Omega) of ({varvec{x}}=(FD, FI, T)) of all females and males, and (f({varvec{x}})) is the joint density function of ({varvec{x}}.) Because (f({varvec{x}})) is unknown, we use the empirical distribution of the observational data, and conditional probabilities given ({varvec{x}}) are replaced with their estimates by logistic regression models. Let ({{varvec{x}}}_{i}=left(F{D}_{i}, F{I}_{i}, {T}_{i}right), i=mathrm{1,2},dots N) denote the (i) th observation in the observational data. The estimates of probabilities are given as follows:$$begin{aligned} hat{P}left( {begin{array}{*{20}c} {{text{Receptive}}} \ {{text{female}}} \ end{array} }right) & = frac{1}{N}mathop sum limits_{i = 1}^{n} hat{p}left( {{varvec{x}}_{i} } right) hspace{15mm} {text{and}} \ hat{P}left( {begin{array}{*{20}c} {text{Flying male}} \ {text{is attracted}} \ end{array} {text{ and }}begin{array}{*{20}c} {text{Receptive }} \ {{text{female}}} \ end{array} } right) & = frac{1}{N}mathop sum limits_{i = 1}^{n} hat{p}left( {{varvec{x}}_{i} } right) hat{q}^{f} left( {{varvec{x}}_{i} } right). \ end{aligned}$$Thus,$$hat{P}left( left. begin{array}{*{20}c} {text{Flying male}} \ {text{is attracted}} \ end{array} right| begin{array}{*{20}c} {text{Receptive }} \ {text{female}} \ end{array} right) = frac{{mathop sum nolimits_{i = 1}^{n} hat{p}left( {{varvec{x}}_{i} } right) hat{q}^{f} left( {{varvec{x}}_{i} } right)}}{{mathop sum nolimits_{i = 1}^{n}hat{p}left(varvec{x}_i right)}}.$$Similarly, we have$$begin{aligned} hat{P}left( left.begin{array}{*{20}c} {text{Flying male}} \ {text{is attracted}} \ end{array}right| {text{Others}} right) & = frac{{mathop sum nolimits_{i = 1}^{n} (1 – hat{p}left( {{varvec{x}}_{i} } right)) hat{q}^{f} left( {{varvec{x}}_{i} } right)}}{{mathop sum nolimits_{i = 1}^{n} (1 – hat{p}left( {{varvec{x}}_{i} } right))}} \ hat{P}left( left. begin{array}{*{20}c} {text{Sedentary male}} \ {text{is attracted}} \ end{array} right| begin{array}{*{20}c} {text{Receptive }} \ {text{female}} \ end{array} right)& = frac{{mathop sum nolimits_{i = 1}^{n} hat{p}left( {{varvec{x}}_{i} } right) hat{q}^{s} left( {{varvec{x}}_{i} } right)}}{{mathop sum nolimits_{i = 1}^{n} hat{p}left( varvec{x}_{i} right)}}hspace{15mm} {text{ and}} \hat{P}left(left. begin{array}{*{20}c} {text{Sedentary male}} \ {text{is attracted}} \ end{array}right| {text{Others}} right) & = frac{{mathop sum nolimits_{i = 1}^{n} left( {1 – hat{p}left( varvec{x}_{i} right)} right) hat{q}^{s} left( {varvec{x}_{i} } right)}}{mathop sum nolimits_{i = 1}^{n} left( {1 – hat{p}left( varvec{x}_{i} right)} right)} . \ end{aligned}$$The estimated probabilities are shown in Table 1.Table 1 Estimated probabilities of a flying male and a sedentary male being attracted to flashes from a receptive female and from others.Full size table More

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The flightless dodo went extinct in the seventeenth century. Biotech company Colossal Biosciences plans to resurrect it.Credit: Hart, F/Bridgeman Images

A biotech company announced an audacious effort to ‘de-extinct’ the dodo last week. The flightless birds vanished from the island of Mauritius — in the Indian Ocean — in the late seventeenth century, and became emblematic of humanity’s negative impacts on the natural world. Could the plan actually work?Colossal Biosciences, based in Dallas, Texas, has landed US$225 million in investment (including funds from the celebrity Paris Hilton) — having previously announced plans to de-extinct thylacines, an Australian marsupial, and create elephants with woolly mammoth traits. But Colossal’s plans depend on huge advances in genome editing, stem-cell biology and animal husbandry, making success far from certain.“It’s incredibly exciting that there’s that kind of money available,” says Thomas Jensen, a cell and molecular reproductive physiologist at Wells College in Aurora, New York. “I’m not sure that the end goal they’re going for is something that’s super feasible in the near future.”Iridescent pigeonsColossal’s plan starts with the dodo’s closest living relative, the iridescent-feathered Nicobar pigeon (Caloenas nicobarica). The company plans to isolate and culture specialized primordial germ cells (PGCs) — which make sperm and egg-producing cells — from developing Nicobars. Colossal’s scientists would edit DNA sequences in the PGCs to match those of dodos using tools such as CRISPR. These gene-edited PGCs would then be inserted into embryos from a surrogate bird species to generate chimeric — those with DNA from both species — animals that make dodo-like egg and sperm. These could potentially produce something resembling a dodo (Raphus cucullatus).To gene-edit Nicobar pigeon PGCs, scientists first need to identify the conditions that allow these cells to flourish in the laboratory, says Jae Yong Han, an avian-reproduction scientist at Seoul National University. Researchers have done this with chickens, but it will take time to identify the appropriate culture conditions that suit other birds’ PGCs.A greater challenge will be determining the genetic changes that could transform Nicobar pigeons into Dodos. A team including Beth Shapiro, a palaeogeneticist at the University of California, Santa Cruz, who is advising Colossal on the dodo project, has sequenced the dodo genome but has not yet published the results. Dodos and Nicobar pigeons shared a common ancestor that lived around 30 million to 50 million years ago, Shapiro’s team reported in 20161. By comparing the nuclear genomes of the two birds, the researchers hope to identify most of the DNA changes that distinguish between them.Insights from ratsTom Gilbert, an evolutionary biologist at the University of Copenhagen, who also advises Colossal, expects the dodo genome to be of high quality — it comes from a museum sample he provided to Shapiro. But he says that finding all the DNA differences between the two birds is not possible. Ancient genomes are cobbled together from short sequences of degraded DNA, and so are filled with unavoidable gaps and errors. And research he published last year comparing the genome of the extinct Christmas Island rat (Rattus macleari) with that of the Norwegian brown rat (Rattus norvegicus)2 suggests that gaps in the dodo genome could lie in the very DNA regions that have changed the most since its lineage split from that of Nicobar pigeons.Even if researchers could identify every genetic difference, introducing the thousands of changes to PGCs would not be simple. “I’m not sure it’s feasible in the near future,” says Jensen, whose team is encountering difficulties making a single genetic change to the genomes of quail.Focusing on only a subset of DNA changes, such as those that alter protein sequences, could slash the number of edits needed. But it’s still not clear that this would yield anything resembling a wild dodo, says Gilbert. “My worry is that Paris Hilton thinks she’s going to get a dodo that looks like a dodo,” he says.A further problem will be the need to find a large bird, such as an emu (Dromaius novaehollandiae), that can act as the surrogate, says Jensen. “Dodo eggs are much, much larger than Nicobar pigeon eggs, you couldn’t grow a dodo inside of a Nicobar egg.”Chicken embryos are fairly receptive to PGCs from other birds, and Jensen’s team has created chimeric chickens that can produce quail sperm — efforts to generate eggs have failed so far. But he thinks it will be far more challenging to transfer PGCs — particularly heavily gene-edited ones — from one wild bird into another.Conservation boon?Colossal chief executive Ben Lamm acknowledges these hurdles, but argues they aren’t dealbreakers. Work towards dodo de-extinction will help with conservation efforts for other birds, he adds. “It will bring a lot of new technologies to the field of bird conservation,” agrees Jensen.Vikash Tatayah, conservation director at the Mauritian Wildlife Foundation in Vacoas-Phoenix, is also enthusiastic about the attention dodo de-extinction could bring to conservation. “It’s something we would like to embrace,” he says.But he points out that the predators that threatened the dodo in the seventeeth century haven’t gone away, whereas most of its habitat has. “You do have to ask,” he says, “if we could have such money, wouldn’t it be better spent on restoring habitat on Mauritius and preventing species from going extinct?” More

This paper describes the complexity of gravitational fractals in terms of global and local dimensions. They are presented in Table 1.Table 1 Global and local dimensions of gravitational fractals and attraction basins.Full size tableThe fractal in hexagonal CPT space, shown in Fig. 1, has a very rich structure, and therefore its characterization by means of fractal dimensions requires two approaches: (1) a global approach treating the fractal as a complex whole and (2) a local approach which allows us to determine the dimension of its fragments which are particularly interesting from a research perspective (see also Table 1). In the subsequent part of the paper, the results obtained are presented and interpreted according to the division in the table.Global dimension of boundaries of gravity attraction basinsTwo types of fractal dimensions have been thus far used in this analysis, i.e., the box and ruler dimensions. Figure 3 shows the distribution of the values of these dimensions determined for the boundaries of attraction as a function of space friction μ.Figure 3Comparison of the variability of the global ruler and box dimensions. Legend: The edge of all attraction basins is a function of the μ coefficient; 1–edges of all basins, 2–entire basins.Full size imageFigure 3 empirically confirms a fact known from chaos theory that whenever a fractal represents full chaos, the ruler dimension may be greater than 2 (Peitgen et al.33, 192–209), whereas the box dimension never exceeds this extreme value. Clearly, for a certain value of μ (in this case μ = 0.19), the numerical values of both types of dimensions are identical.In the bottom part of Fig. 3, line 1 illustrates the variability of the shapes of the attraction basins of individual cities depending on the value of μ, i.e., space resistance. The initially extremely complex shapes of the boundaries are smoothed to take the form of straight lines in the case of a large value of μ (μ = 0.52).In turn, line 2 illustrates not only the boundaries of the attraction basins, but also their internal structure. Clearly, the initially chaotic impacts of individual cities on the agent (μ = 0.005) are gradually smoothed out, so that in the final stage of the process they fully stabilize. This means that each city has a geometrically identical basin of attraction. Hence, if the agent is in the attraction basin of city 1 (purple color), it will always be attracted only by that city. This rule also applies to the other cities. It is obvious that the random process occurring at μ = 0.09 is then replaced by a strictly deterministic one. When chaos becomes complete order (Banaszak et al.15, the numerical values of both types of dimensions appear to stabilize at the level of 1.Global dimension of the boundary of each separate attraction basinFigure 1 also shows the geometric image of the attraction basins of individual cities. They were almost identical, and therefore also the fractal dimensions of the boundaries of these basins must match. The validity of this proposition is confirmed by Fig. 4. Six lines representing the distribution of the fractal dimension of the boundaries of the six basins coincide with almost full accuracy. Further analysis of Fig. 4 allows us to infer the conclusion that there is almost total chaos at the value db = 1.9021 (μ = 0.005). On the other hand, as space resistance increases to the value of μ = 0.22, there is a rapid decrease in the value of the fractal dimension of the boundary of each basin to the level of 1.2628; when μ = 0.34, then db = 1.2382. In that case, the value of the fractal dimension stabilizes, and at μ = 0.46, db = 1.2444 and finally for μ = 0.52, db reaches the value of 1.0412. The icons presented in Fig. 4 in lines 1 and 2 have slightly different structures than the icons in Fig. 3, due to different values of μ in certain cases.Figure 4The box dimension of the edges of the attraction basins depending on the μ coefficient (separately for each attractor). Legend: 1–boundaries of single attraction basins, 2–entire basins.Full size imageThe global dimension of the attraction basin of each city as an irregular geometric figureThe full symmetry of the basins of attraction of individual cities can be disturbed by the shape of the geometric figure on which the deterministic fractal is modeled. Such a situation occurs in the present case. Due to the fact that the fractal in Fig. 1 is formed on the surface of a square, the final basins of attraction of cities 1, 3, 4 and 6 are obviously larger than those of cities 2 and 5. Of course, these differences do not occur when considering the surface inside the hexagon.In Fig. 5, the line marked in black color represents the average value of the fractal dimension of the basins of attraction of individual cities, the value of which is (overline{{d }_{b}}=1.77). It can be seen that at very high values of the fractal dimension in the range (1.750, 1.775), there are db oscillations around this line. This is precisely the effect of modeling the fractal on the surface of the square, rather than the properties of this fractal. Therefore, (overline{{d }_{b}}=1.77) should be regarded as the global dimension of the basin of attraction (of each city) treated as an irregular figure.Figure 5Box dimension of the attraction basins as a geometric irregular figure in the gravitational fractal. Legend: 1-basins of the first city, 2-basins of the second city, 7-basins of all cities.Full size imageLocal dimensions of the boundary of the selected characteristic fragmentsFigure 6 presents fractal dimensions, with the Box and Ruler as functions of μ, and the boundaries of the attraction basins of individual cities occurring in all fragments A, …, E.Figure 6Distribution of the values of fractal dimensions of the boundaries of the attraction basins identified in selected fragments of a fractal; Legend: (A, D)-fragments marked in Fig. 1.Full size imageIt is evident that the structures of Fig. 6 (Box and Ruler) are almost identical. This means that, as has been stated earlier, when describing complex fractal objects, it does not really matter which type of dimension is used.Of interest here is the variability of the structure of both figures along with the increase in the value of the parameter μ. Fragments A, …, E (see Fig. 1) are characterized by high complexity, i.e. the intertwining attraction basins of the individual attractors (cities). This observation is confirmed by the numerical results of both fractal dimensions whose values are in the range (1.68–1.82). To illustrate the spatial complexity of these fragments, and thus their dimensions, by way of example, two fractal fragments are considered below: fragments A and D (see also Fig. 7).Figure 7Box dimension of the edge of each gravitation basin in A and D. Legend: The icons show the variability of the fragments A and D due to the share of the attraction basins of individual cities (3, 4 and 6).Full size imageFigure 6 offers important conclusions concerning the organization of social and economic life in the geographical area surrounding individual cities (attractors).

1.

Out of all the separated fragments, only in fragment A do we find the attraction basins of all the cities intertwined across the entire range of variation μ, i.e. (0.00–0.48). Hence, the graph of fractal dimension (db) (blue line) as a function of μ is continuous, and when the resistance of space is the greatest (μ = 0.48), the fractal dimension d = 1.00. This means that chaos has given way to total order, and fragment A has been symmetrically divided between cities 1 and 6. Hence, there are two colors left, namely red and purple.

2.

A similar situation occurs in the case of fragment D (yellow line), where the attraction basins of individual cities intertwine continuously within the range: 0.00 ≤ μ ≤ 0.46. Beyond the value of 0.46, the entire fragment D is filled with purple: the closest city 1 dominates it.

The research conducted here also confirms the conclusions presented in previous works by Banaszak et al.15,16 concerning the transformation of chaos into spatial order, which means the stabilization of permanent dominance, usually of one attractor (city). Thus, with regard to fragments A and D, in fragment A there is a constant dominance (in half of the area) of cities 1 and 6, from the limit value of μ = 0.24 onward. In the case of fragment D, beginning with the value of μ = 0.36, only city 1 dominates (purple). That is, in the final phase of establishing the order in spatial interactions in the arrangement of areas A and D, the role of the dominant attractor (city) is played by city 1 (purple).Due to the symmetry of Fig. 1, similar effects can be observed in other parts of this fractal, located symmetrically in relation to A, …, E (see Supplementary Material).Figures 1 and 6 confirm the findings, known in the theory of city development, that urban (and other) centers rise in the hierarchy (or their rank decreases), depending on the external and internal factors conditioning their development. In the model used in this study, the parameter μ represents external factors (space resistance). If μ values are low, all cities are attractive from the point of view of spatial interactions and create their own but symmetrical basins of attraction. When the resistance of space increases, one city becomes the dominant center, and its basin of attraction is a uniform compact isotropic surface.However, this is not a simple mechanism, since, as has been demonstrated by simulation experiments described in this paper, within a certain range of μ values, another city (attractor) may dominate the others during chaotic interactions. The dynamic history of urban development confirms this observation, for example, in relation to historical capitals of some countries that have lost their functions as administrative capitals.Local dimension of the boundary of each attraction basin in a selected fragment of a fractalFragments A, …, E (Fig. 1 and the Supplementary Material) consist of mutually intertwined basins of attraction (six cities) whose boundaries with complicated courses have a fractal dimension, e.g. a box dimension.Figure 7(fragment A) shows the distribution of db as a function of μ in this fragment. In the case of total internal chaos, the fractal dimension of the boundaries of the attraction basins of all cities is identical and amounts to 1.9152. A clear differentiation of db is noticeable from μ = 0.1 onward. It should also be noted that orange and blue, red and purple, yellow and green lines mutually coincide. The red–purple line tend towards db = 1 as μ increases. However, orange, blue, yellow and green lines reach a value of db = 0.The fractal dimension db = 1.0 is most closely represented by the blue line (city 2), then the red line (city 6) and the purple line (city 1). Since these lines almost coincide, and the red and purple lines are the last to reach the value db = 1, at μ = 0.48, fragment A is symmetrically covered in red and purple. Therefore, with very high spatial resistance, fragment A is dominated by two cities, namely by 1 and 6.In turn, Fig. 7(fragment D) illustrates the variability of the fractal dimension of boundaries of the attraction basins in this fragment. This dimension depends on the complexity of the mosaic patterns formed in this fragment, with varying μ values. When the values of μ are close to zero, all cities contribute to filling the space of fragment D. When μ = 0.18, city 1 (purple color) falls out of the competition for space, but only up to the value of μ = 0.24, when it starts to compete again with other cities. From the point of view of spatial interactions, in the final phase of this process (μ = 0.44), city 2 (blue) and city 6 (red) dominate to a small extent, because cities 3, 4 and 6, starting from μ = 0.3, do not play any role in fragment D.Figure 7 shows that the value μ = 0.3 is a characteristic point. It is a locus where all the curves representing the attraction basins of individual cities meet. As has already been stated, three of them lose their influence over the space of fragment D.Local dimensions of parts of the attraction basins treated as an irregular geometric figureIn each of the selected fragments A, …, E, some of the boundaries of the attraction basins of individual cities are distributed differently. They create certain holes in the form of irregularly colored mosaic patterns that have a certain fractal dimension. To present its variability, fragments A and D were used again. Figure 8 shows the distribution of db values depending on the value of μ.Figure 8Local dimensions of parts of the attraction basins treated as an irregular geometric figure in (A) and (D). Legend: The icons illustrate the variability of the shape of some of the attraction basins of individual cities in fragment (A) and (D) for cities 3, 4 and 6.Full size imageThe function has several characteristic points. Up to the value of μ = 0.04, attraction basins show a jumble in which no predominant color or shape can be identified. The fractal dimension is then: db = 1.7697. From this value onwards, where μ = 0.042, the interior of fragment A becomes increasingly ordered. With a value of μ = 0.125, the city’s attraction basins 3 and 4 begin to disappear in fragment A. The same happens to the city attraction basins 2 and 5 for the value of μ = 0.24.The final effect of the increase in space resistance (with μ = 0.50) leads to the filling of fragment A with two colors, i.e., purple and red. This means that cities 1 and 6, have won the competition for the space of fragment A. In this case, the fractal dimensions db equal 1.90.Figure 8 presents the variability of the fractal dimension and the effects of the competition for space between cities in fragment D. As is the case in fragment A and all others, i.e. B, C and E (see the Annex with Supplementary Material), the intertwined attraction basins are represented by the area consisting of an endless number of differently colored dots. Hence, up to the value of μ = 0.042, fragment D is dominated by pure spatial chaos that extends over its entire area. It is characterized by the fractal dimension db = 1.7697. This means that with an increase in the value of μ, for the emergence of an irregular shape of a geometric figure, chaos must be accompanied by an increase in the value of the fractal dimension. Its limiting value is number 2. Then, spatial dominance is usually gained by one city and the examined fragment is filled with one color (‘the winner takes it all’).This is precisely the situation in Fig. 8 where city 1 (purple color) has apparently won the competition. Since this color fills area D completely, we find the plausible result db = 2.0. More

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