How were Arabia, South India and North Africa able to resist the Indo Europeans

I don't know. Possibly There are three different reasons for the three different areas.

And I would suggest that possibly those three regions might have resisted the invasions by Indo European speaking hordes that you imagine because those invasions didn't happen in those regions. People need reasons of some sort to decide to invade some place. On the other hand nobody needs a reason to not make the decision to invade some place. If the spread of Indo European languages was caused by invasions by Indo European speakers then that was a massive series of invasions. And maybe those who believe in those hypothetical prehistoric invasions should be more impressed by the scale of them than wonder why they weren't even more widespread.

India was probably just too big, they concentrated their dominance in the north. The middle east at the time was dominated by the Mesopotamian states and Egypt, which prevented the Indo-Europeans from expanding (their are several tribes which may have been IE that invaded the middle east in the ancient period) into Arabia and North Africa. Persia was conquered because much of it was undeveloped, except Elam, which was mostly in the south.

If you examine such migrations (and that is a probably teh more appropriate word than invasion) they are nearly all across the same latitude.

The reason is simple. The animals, crops and technology of the immigrant is appropriate to that latitude and less so further north or south.

What if the main way those IE languages spread was not by massive invasion, or at least not by great conquests?
What should convince us those "conquests" was the reason and not other events and processes?
Spread of skills and trade or barter and religious practises and technologies are othef possibillities.
That does not neccesarily imply absence of violence but only it may have been ( I guess so) had other elements.

Theseus said:

The Indo Europeans conquered the major part of Europe, Persia, Central Asia and Northern parts of South Asia and were able to change the native languages to an Indo European language during the ancient period. But why did the Indo European languages not spread to Arabia, South India and North Africa despite of the close proximity

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Some did .
Hicksos conquered Egypt if not wrong .

Theseus said:

Unfortunately languages spread through invasions many times in the past. And a small elite was able to change the native language of the conquered regions.
North Africa was fully arabized after the Arab expansion. The Roman conquest of Western Europe led to the spread of the Roman script and Latin languages to the extent that some Western European languages have almost 50% Latin words. A small group of Central Asian warlords were able to transform Anatolia from Christian Greek to Muslim Turkish. Today China is a homogenous country because a small group of Han people were able to conquer whole China and impose their language upon the native people

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You seems to be right about most of what You write about spread
of languages known from written sources.
That does not mean ir is the only possible way.
And there are as far as I know very little to support any idea
there were large scale conquests in Europe by "greay kings"
in those prehistoric times.
There could have been hundreds of "microconquests", but what reasons
do we have to prefer such a hypothesis over other possibillities?

Theseus said:

No this does not make much sense as Sri Lanka has a very different latitude than Western European countries yet both are mostly IE speaking regions. On the other hand North Africa has a similar latitude to Iran yet North Africa is not IE but Iran is IE

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This seems to me to be thinking about the issue in completely the wrong way. The arrival of Indo-European languages in, for example, Greece, is not connected in any way with the arrival of Indo-European languages in Sri Lanka. The two events are separated by more than a millenium (possibly much more); and the people carrying these languages were not 'Indo-Europeans', if by this we mean speakers of a common language with a shared culture. Both spoke languages believed to be descended from a common ancestor, but long, long before. They were not involved in some common invasion or population movement.

The question makes no more sense than asking why Austronesian languages are commonly spoken in Los Angeles and Antananarivo, but not in Seoul. It's entirely the wrong level of analysis.

The Indo Europeans did make inroads and conquests of North Africa multiple times:
First: the bronze age Hyksos, Hittites, and Sea Peoples.
Second: The Greeks and the Romans, who successfully conquered the region, from Egypt to Morocco, and converted it to Indo European.

I don't know the situation with India. Others could tell you better.

Arabia is an easier situation, the place was very difficult to invade, and I would guess that the Arabs were quite powerful on their own lands.

I don't know, but could no water and a barren landscape have anything to do with it?
The Nile delta and the Tigris and Euphrates valley were population population centers but Arabia and most of North Africa wasn't even mapped until the early 20th century.

• Arabia wasn't a very inhabitable place. It was mostly desert and sparsely populated. It's climate wasn't suitable for living and thus it was full of locally governed tribal nomadic societies. That's the reason why the Achaemenids, Romans, and other large empires never sought out to conquer Arabia. Furthermore, the Levant and Iraq hadn't become Indo-Europeanized. These were the frontier regions of Southwest Asia and they would have needed to get colonized first for the IE people to get into the interior of Southwest Asia. Ancient Levantines and Iraqis most likely resisted becoming IEs because they had their own far more advanced civilizations and were a power in their own right for most of history. North Africa and West Asia were conquered by Indo-Europeans in the past but the living conditions in North Africa weren't suitable enough for a full-scale colonization.

• North Africa wasn't Indo-Europeanized because the IEs had to colonize/get past West Asia first to get into North Africa, something that never happened. If West Asia got Indo-Europeanized, then there's a good chance that the same would have happened in North Africa.

• South India (India as a whole to a lesser extent) is a place full of forests, swamps, and some mountains & rivers. The main geographical feature that separates South India from North India are two main mountain ranges: the Vindhyas and Satapuras. It's actually an interesting feat that Maharashtra (and Goa) managed to get IE-ized considering that it is south of the above mentioned mountain ranges and thus geographically in South India. But as we all know, there are multiple factors involved in ethnolinguistic colonization. Besides geography, empires play a large role in spreading language and colonizing. I suspect that the Satavahanas and Western Kshatrapas had a major role to play in Maharashtra becoming IE. Also, India is a vast place with a huge population density. Even today, there are significant pockets of Austro-Asiatic and Dravidian-speakers in Central India and Eastern India.

Theseus said:

The Indo Europeans conquered the major part of Europe, Persia, Central Asia and Northern parts of South Asia and were able to change the native languages to an Indo European language during the ancient period. But why did the Indo European languages not spread to Arabia, South India and North Africa despite of the close proximity

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If we go to Indian history, we don't know any such attempt made to change the language of Southern kingdoms.

Flesh said:

• Arabia wasn't a very inhabitable place. It was mostly desert and sparsely populated. It's climate wasn't suitable for living and thus it was full of locally governed tribal nomadic societies. That's the reason why the Achaemenids, Romans, and other large empires never sought out to conquer Arabia. Furthermore, the Levant and Iraq hadn't become Indo-Europeanized. These were the frontier regions of Southwest Asia and they would have needed to get colonized first for the IE people to get into the interior of Southwest Asia. Ancient Levantines and Iraqis most likely resisted becoming IEs because they had their own far more advanced civilizations and were a power in their own right for most of history. North Africa and West Asia were conquered by Indo-Europeans in the past but the living conditions in North Africa weren't suitable enough for a full-scale colonization.

• North Africa wasn't Indo-Europeanized because the IEs had to colonize/get past West Asia first to get into North Africa, something that never happened. If West Asia got Indo-Europeanized, then there's a good chance that the same would have happened in North Africa.

• South India (India as a whole to a lesser extent) is a place full of forests, swamps, and some mountains & rivers. The main geographical feature that separates South India from North India are two main mountain ranges: the Vindhyas and Satapuras. It's actually an interesting feat that Maharashtra (and Goa) managed to get IE-ized considering that it is south of the above mentioned mountain ranges and thus geographically in South India. But as we all know, there are multiple factors involved in ethnolinguistic colonization. Besides geography, empires play a large role in spreading language and colonizing. I suspect that the Satavahanas and Western Kshatrapas had a major role to play in Maharashtra becoming IE. Also, India is a vast place with a huge population density. Even today, there are significant pockets of Austro-Asiatic and Dravidian-speakers in Central India and Eastern India.

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Satavahana and Andhra are synonym of each other. Telugu people also refer themselves as Andhra that's why their state is referred as Andhra Pradesh aka Land of the Andhras/Telugus.

Vindya ranges is not a linguistic division but rather a division of Indo-Gangetic plains and Deccan plateau, Madhya Pradesh, Chattisgarh, Maharashtra, Goa all are IE speaking.

Devdas said:

Satavahana and Andhra are synonym of each other. Telugu people also refer themselves as Andhra that's why their state is referred as Andhra Pradesh aka Land of the Andhras/Telugus.

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Okay. I thought it was an Indo-Aryan state. My bad. So what do you think was mostly responsible for the IE-ization of Maharashtra?

Vindya ranges is not a linguistic division but rather a division of Indo-Gangetic plains and Deccan plateau, Madhya Pradesh, Chattisgarh, Maharashtra, Goa all are IE speaking.

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It's a physical barrier still. Physical barriers limit linguistic contact. I mentioned the Vindhyas as the region that differentiates North India from South India. It isn't an impenetrable barrier, but still a barrier. I think if it weren't there, then North Indians would have an easier time travelling down south. And more land could have become IE-ized, I don't know for sure though.

Theseus said:

The Indo Europeans conquered the major part of Europe, Persia, Central Asia and Northern parts of South Asia and were able to change the native languages to an Indo European language during the ancient period. But why did the Indo European languages not spread to Arabia, South India and North Africa despite of the close proximity

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The reason for not invading North Africa was Egypt , if not wrong Hittite had a constant clash with them and Hycksos conquered Egypt , enstablished a dinasty and founded a new capital in Avaris ... Then where expelled , but further west the tribes were more primitive and uninteresting for conquest, as for Arabia it was just a big desert. The crescent fertile was instead conquered by Mitanni , Hurartians etc.