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The phrase "arising independently" is correct and usable in written English.
It can be used to describe something that emerges or develops on its own, without external influence or assistance.
Example: "The new species was found to be arising independently in the isolated ecosystem."
Alternatives: "emerging autonomously" or "developing independently".
Exact(18)
In the same way, the cichlids have evolved pairs of matching species, arising independently in separate lakes in Africa.
This observation strongly indicated that the 1555A>G mutation arising independently in different mtDNA haplogroup backgrounds among eastern Asian populations.
Windt (2010; see also Metzinger 2013b) argues that in certain dreams, self-experience is reduced to pure spatiotemporal-self-location arising independently of the phenomenology of embodied selfhood, suggesting that neither body ownership nor the sense of agency are strictly necessary for self-consciousness.
Scientists are divided over the likelihood of life arising independently on Mars, or on other planets in our galaxy.
Regulated expression of this primordial RAG-1/RAG-2 complex in lymphocytes that could then co-evolve gradually with subsequent "slave elements" arising independently in T-cell receptor and immunoglobulin genes.
That sequencing of the C/L/A translocation indeed identified the same base substitution suggests that these two translocations are derived from each other instead of arising independently from separate CAN1 invasions.
Similar(42)
Electric organs appear to have arisen independently in several fishes.
This is a pity, because the evolution of eyes, which have arisen independently in diverse animal groups, is highly contentious.
"The fact that they have arisen independently in different countries suggests this is in response to the vaccine," said Lan.
Instead, anthropologists, historians, and archeologists tell us that state governments have arisen independently under one of two sets of circumstances.
Agriculture arose independently in at least three regions: South America, Mesoamerica, and eastern North America.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com