Caveat: this is my first question here, it is quite interdisciplinary, but I hope to be in the correct place to ask. I am a user of Mathematics Stack Exchange since some years ago, and this question is related with some questions there (here, here whose general formula is discussed here and here).

Context: I am preparing a mathematical paper regarding a new family of dynamical systems (if you are not familiar with the concept, simplifying the idea it is a mathematical formula in which starting from a initial value, once applied to the formula the resulting value is again applied to the formula, and so on, finally the values are plotted and eventually a -sometimes interesting- pattern emerges) whose attractors (plotted patterns) in the present case seem to have unexpected pareidolic properties.

Basically some of the patterns generated by these systems show similarities with some structures of invertebrate life forms, specially insects, marine jellyfish, and zooplancton and also due to the patterns of the accumulation of points, also with life forms presenting bioluminescence properties.

For each interesting pattern so far I have tried to find the closest life form example, to compare both the model and the life form patterns.

So my target is including in the paper the closest life form similar to each mathematical pattern. Initially it is just a pareidolic coincidence, but it might be interesting if the mathematical formula can resemble models of some organic structures.

These are the ones I have been able to gather, both the model and the closest life form I found. The pictures I am using at the right side of and below the images are just for the sake of completeness (they belong to their respective owners, I do not own them, if there is any problem I will remove them, so just please let me know). The formula can be verified at the MSE links I have added at the beginning of the question and the Python code to generate them is in this link (please feel free to use it and modify it). The questions are after the examples (click to enlarge):

  1. Patterns similar to thorax and abdomen of Bembicini wasp, head and body of Turritopsis dohrnii (inmortal jellyfish) and Tardigrade limbs:

enter image description here

  1. Patters similar to Drain fly:

enter image description here

  1. Patterns similar to Acherontia atropos (thorax and abdomen and main proportions of body, specially the patterns near the head are interesting):

enter image description here

  1. Patterns similar to Ctenophora (down left) and Mnemiopsis_leidyi (down right):

enter image description here

I would like to ask the following questions:

  1. Are there better examples of life forms for the models I am showing above (1,2,3,4)? If somebody could provide examples or specific names would be very appreciated.

  2. Are there papers or references in the Biology field regarding mathematical models of this kind?

  3. In the case I would like to add images from internet in the paper, I know that in the case of Wikipedia there is a standard way to add the reference in the bibliography. Apart from Wikipedia, are there other sources of images which I could use without problems in a paper (thinking about the kind of life forms that are the target of the patterns)? Thanks in advance.

  • $\begingroup$ I have tried to add the tag "mathematical-models" but it is converted into "theoretical-biology". Tried to correct it but it seems I can not do it for some reason. $\endgroup$ – iadvd May 11 '18 at 8:46
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    $\begingroup$ I think those tags have been merged (with theoretical-biology as the main one), which is probably why you cannot change to it. $\endgroup$ – fileunderwater May 11 '18 at 18:13
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    $\begingroup$ The question has nothing to do with mathematical modelling or theoretical biology anyway (nor does it have much to do with the other tags). $\endgroup$ – Remi.b May 11 '18 at 18:52
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    $\begingroup$ The question is entirely based upon a personal visual appreciation. I am hence voting to close as opinion-based. $\endgroup$ – Remi.b May 11 '18 at 18:52
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    $\begingroup$ @Remi.b I definitely think it should be open, since Mathematical modelling of biological patterns (using semi-mechanistic models or phenemenological models) is a valid topic in Mathematical biology. The background to this particular Q could be stronger though (eg links to the research field). $\endgroup$ – fileunderwater May 11 '18 at 19:12

For the second subquestion (papers or references of similar models), it could be useful for you to look at the books by J.D Murray for inspiration; Mathematical Biology, vol I & II. A pdf of volume 1 can be found here, and the second volume here. Even if they are a complete of years old (last edition from 2002 I think) they provide a really nice background to the field of Mathematical biology, and he has quite many examples on pattern formation of different kinds. I don’t remember if dynamics systems are included per se though.

However, several chapters include reaction diffusion systems and wave models, to model e.g. coat/fur patterns and wound healing. It is also packed with references, so you should be able to find lots of classic citations that might be useful.

Some chapters you might find interesting:

  • Reaction Diffusion, Chemotaxis and Non-local Mechanisms
  • Travelling Waves in Reaction Diffusion Systems with Weak Diffusion: Analytical Techniques and Results
  • Animal Coat Patterns and Other Practical Applications of Reaction Diffusion Mechanisms
  • Mechanical Theory for Generating Pattern and Form in Development
  • Bacterial Patterns and Chemotaxis

I've mostly looked at population modeling in his books, so I'm not very familier with these topics myself.

  • $\begingroup$ thanks for the references! indeed the classical reaction difussion was included in my readings regarding the characterization of these systems. This is very useful. $\endgroup$ – iadvd May 11 '18 at 22:53
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    $\begingroup$ Good. Sorry for not being able to help more with the specifics. There are probably also more recent books that might be more relevant, but Murray provides a nice overview for those that come from a mathematical background. Since it requires quite alot of mathematics it is probably rather inaccessible for many with a biological background. $\endgroup$ – fileunderwater May 14 '18 at 14:19
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    $\begingroup$ hi again, I managed to publish recently an article at Springer Nature CS regarding this matter, thank you for taking time to guide me to the important bibliography of J.D.Murray. Hoping you the best, cheers! link.springer.com/article/10.1007/… $\endgroup$ – iadvd Dec 12 '19 at 3:58
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    $\begingroup$ @iadvd Great news and I'm glad to hear. I will download and look at the paper later on. Fun to hear that Bio-SE input has made its way into a published paper as well even if it was rather vague in this case). I guess it would be really cool if we eventuelly started to find mentions/references in papers to Bio-SE Qs (or Stack Exchange in general), but we are probably not there yet. $\endgroup$ – fileunderwater Dec 12 '19 at 9:26

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