Gemara

Mishnah: Elul (1) is Rosh Hashanah for Ma'aser Behemah (animals born from one Elul until the next are tithed together);

R. Elazar and R. Shimon say, it is Tishrei 1.

R. Yochanan: Both of them expound the same verse, "Lovshu Charim ha'Tzon va'Amakim Ya'atfu Var Yisro'a'u Af Yashiru". R. Meir (the first Tana) explains that (most) Tzon are clothed, i.e. become pregnant, when the valleys are sprouting with grain, which is in Adar;

(The gestation of Tzon is five months, so) they give birth in Av, therefore Rosh Hashanah is (afterwards,) Elul.

R. Elazar (and R. Shimon) explains that Tzon are clothed when the valleys Yashiru (sing, when grain is housed in straw that rustles in the wind), in Nisan;

They give birth in Elul, therefore Rosh Hashanah is Tishrei.

R. Meir explains that "Yashiru" refers to (the minority of) Tzon that become pregnant late; R. Elazar explains that "Ya'atfu..." refers to Tzon that become pregnant early.

Objection: According to R. Elazar, the verse should first mention "Yashiru", when most Tzon become pregnant!

Rava: Rather, all agree that "Lovshu Charim ha'Tzon..." in Adar. They argue about "Aser Te'aser", which alludes to two Ma'aseros. It equates Ma'aser Behemah to Ma'aser of grain:

R. Meir says that just like Rosh Hashanah for Ma'aser Degen is right after grain is finished (it is fully dried out just before Tishrei), also for Ma'aser Behemah;

R. Elazar says that just like Rosh Hashanah for Ma'aser Degen is Tishrei, also for Ma'aser Behemah.

Bechoros 57b - Mishnah - R. Akiva: There are three Geranos for Ma'aser Behemah (times during the year when one must tithe his animals) - Pras (15 days before) Pesach, Pras Shavu'os, and Pras Sukos;

R. Elazar and R. Shimon say, they are Nisan 1, Sivan 1, and Elul 29.

R. Meir: Elul 1 is Rosh Hashanah for Ma'aser Behemah;

Ben Azai says, animals born in Elul are tithed by themselves.

All animals born between Tishrei 1 and Elul 29 are tithed together (this is like R. Elazar and R. Shimon).

If five animals were born before Rosh Hashanah and five after Rosh Hashanah, they do not join.

Beraisa - Ben Azai: Because Tana'im argue whether Rosh Hashanah is the first of Elul or Tishrei, animals born in Elul are tithed by themselves.

Question: Ben Azai should rule according to whichever is more reasonable!

Surely he understood their reasons, for he considered himself to be much greater than all Chachamim except for the "Kare'ach" (R. Akiva or R. Elazar ben Azaryah)!

Answer (R. Yochanan): The other Tana'im had traditions from Chagai, Zecharyah and Malachi. He did not want to rely on his reasoning against such a tradition.

Rishonim

Rambam (Hilchos Bechoros 7:6): All animals born between Tishrei 1 and Elul 29 join, and we may tithe from one on another. If five were born on Elul 29 and five on (the next day,) Tishrei 1, they do not join.

Ri Korkus: The Rambam rules that Tishrei 1 is Rosh Hashanah, because this is the opinion of R. Elazar and R. Shimon against R. Meir. Also, the Stam Mishnah (Bechoros 57b) is like them, and the Mishnayos there focus on Ma'aser Behemah, so it is more authoritative than the Mishnah in Rosh Hashanah that is like R. Meir. Also, the Geranos of R. Akiva correspond to their opinion.

Lechem Mishneh: The Rambam rules like R. Elazar and R. Shimon because they are Rabim against R. Meir. Also, the argument in the Mishnah (Bechoros 57b) is followed by a Stam Mishnah like them. Therefore, this overrides the Stam Mishnah in Rosh Hashanah. Meir. However, the SMaG (Aseh 212, p.224 Sof Amud 3) rules that Elul 1 is Rosh Hashanah, like the Mishnah in Rosh Hashanah.

Me'iri (Reish Rosh Hashanah DH veha''Mishnah): Rosh Hashanah for Ma'aser Behemah does not depend on the Geranos. The Mishnah says that animals born before and after a Goren join, but those born before and after Rosh Hashanah do not join. The summer Goren is Elul 29, but Rosh Hashanah is Elul 1. The Rambam rules that Rosh Hashanah is Tishrei 1, because in Bechoros this is a Stam Mishnah following an argument. However, it seems that the Halachah is Elul 1, in order to comply with our Mishnah of 'four Roshei Hashanah'.

Note: The Gemara said that Ben Azai holds that the summer Goren is Av 29, 'in accordance with his opinion that Elulim are tithed by themselves.' The Me'iri holds that fixing the Goren to be Elul 29 does not indicate that Rosh Hashanah is Tishrei. Rather, it is in order that animals will be available for Sukos. (R. Akiva fixes every Goren 15 days before the festival.) The Ri Korkus said that R. Akiva supports the Pesak that Tishrei is Rosh Hashanah. He must hold that it is unreasonable to include new and old animals in a Goren. The Gemara did not specify R. Meir's opinion about Geranos. The Ri Korkus must hold that he agrees with Ben Azai that the summer Goren is Av 29. According to the Me'iri, presumably he holds like the majority opinion that it is Elul 29.

The Me'iri and Lechem Mishneh call the Mishnah in Bechoros 'an argument followed by a Stam.' The Ri Korkus does not mention this, nor do SMaG and Ritva (Rosh Hashanah 2a DH Echad), who rule like R. Meir. It seems that they do not consider it 'an argument followed by a Stam' because it is all in the same Mishnah, or because the argument is not explicit. Before the opinion of R. Elazar and R. Shimon is brought Stam, R. Meir's opinion is explicitly stated, Ben Azai said that that Elulim are tithed by themselves; the Gemara explains that this is because he is unsure when is Rosh Hashanah. Also, R. Akiva, R. Elazar and R. Shimon said that the summer Goren is Elul 29. According to the Me'iri and Ritva, this is no indication of when is Rosh Hashanah.

Rambam (Hilchos Ma'aser Behemah 6:2): Chachamim decreed not to separate Ma'aser Behemah after the Churban (Bechoros 53a). The concern is lest one eat it Tam and be Chayav Kares for Shechutei Chutz.