I just thought I'd put this out there for comment.
Semelparity is when a species only reproduces once in a lifetime. Usually semelparous species die soon after giving birth. Yinrih, however, are both semelparous and long-lived thanks to a high degree of parental care conferring a survival advantage.
Yinrih live on average around 724 Terran years (let's say 700 to make the math easier). A yinrih pup reaches maturity at around 53 Terran years, but let's assume because of social factors they don't reproduce until around the age of 100 on average.
The average litter size is 1.5 times the number of contributing parents. Yinrih can have between 2 to 12 biological parents. This group of parents is called a childermoot. Let's say that once you account for childhood death, infertility, conscious decision not to have pups, etc, the number of pups in a litter that go on to become parents is about 1.2 times the size of the childermoot.
Let's say at year zero there are 50000 newly hatched yinrih. They reproduce at year 100, giving a total population of 50000*1.2 + 50000 = 60000 This trend continues until year 700. The first cohort of 50000 yinrih dies, so from here out we subtract the population older than 700. The population reaches 3 billion (my threshold for achieving orbital flight) at around the year 5750.
population table
| Terran Year | Population | Cohort Size |
|---|---|---|
| 0 | 50,000.00 | |
| 100 | 110,000.00 | 60,000.00 |
| 200 | 182,000.00 | 72,000.00 |
| 300 | 268,400.00 | 86,400.00 |
| 400 | 372,080.00 | 103,680.00 |
| 500 | 496,496.00 | 124,416.00 |
| 600 | 645,795.20 | 149,299.20 |
| 700 | 774,954.24 | 179,159.04 |
| 800 | 879,945.09 | 214,990.85 |
| 900 | 955,934.11 | 257,989.02 |
| 1000 | 997,120.93 | 309,586.82 |
| 1100 | 996,545.11 | 371,504.19 |
| 1200 | 945,854.13 | 445,805.02 |
| 1300 | 835,024.96 | 534,966.03 |
| 1400 | 702,029.95 | 641,959.23 |
| 1500 | 592,435.94 | 770,351.08 |
| 1600 | 560,923.13 | 924,421.29 |
| 1700 | 673,107.76 | 1,109,305.55 |
| 1800 | 1,007,729.31 | 1,331,166.66 |
| 1900 | 1,659,275.17 | 1,597,400.00 |
| 2000 | 2,741,130.21 | 1,916,880.00 |
| 2100 | 4,339,356.25 | 2,300,256.00 |
| 2200 | 6,507,227.50 | 2,760,307.19 |
| 2300 | 9,258,673.00 | 3,312,368.63 |
| 2400 | 12,560,407.60 | 3,974,842.36 |
| 2500 | 16,322,489.12 | 4,769,810.83 |
| 2600 | 20,386,986.95 | 5,723,773.00 |
| 2700 | 24,514,384.34 | 6,868,527.60 |
| 2800 | 28,417,261.20 | 8,242,233.12 |
| 2900 | 31,800,713.44 | 9,890,679.74 |
| 3000 | 34,410,856.13 | 11,868,815.69 |
| 3100 | 36,093,027.36 | 14,242,578.83 |
| 3200 | 36,861,632.83 | 17,091,094.59 |
| 3300 | 36,983,959.40 | 20,509,313.51 |
| 3400 | 37,080,751.28 | 24,611,176.21 |
| 3500 | 38,196,901.53 | 29,533,411.46 |
| 3600 | 41,836,281.84 | 35,440,093.75 |
| 3700 | 49,953,538.20 | 42,528,112.50 |
| 3800 | 64,894,245.84 | 51,033,735.00 |
| 3900 | 89,273,095.01 | 61,240,482.00 |
| 4000 | 125,777,714.02 | 73,488,578.40 |
| 4100 | 176,883,256.82 | 88,186,294.08 |
| 4200 | 244,509,908.18 | 105,823,552.89 |
| 4300 | 329,661,889.82 | 126,988,263.47 |
| 4400 | 432,094,267.78 | 152,385,916.17 |
| 4500 | 550,063,121.34 | 182,863,099.40 |
| 4600 | 680,225,745.61 | 219,435,719.28 |
| 4700 | 817,770,894.73 | 263,322,863.14 |
| 4800 | 956,875,073.67 | 315,987,435.76 |
| 4900 | 1,091,550,088.41 | 379,184,922.92 |
| 5000 | 1,216,910,106.09 | 455,021,907.50 |
| 5100 | 1,330,842,127.31 | 546,026,289.00 |
| 5200 | 1,436,010,552.77 | 655,231,546.80 |
| 5300 | 1,542,062,663.32 | 786,277,856.16 |
| 5400 | 1,667,825,195.99 | 943,533,427.39 |
| 5500 | 1,843,190,235.18 | 1,132,240,112.87 |
| 5600 | 2,110,328,282.22 | 1,358,688,135.44 |
| 5700 | 2,523,843,938.67 | 1,630,425,762.53 |
| 5800 | 3,149,512,726.40 | 1,956,510,915.04 |
It is likely I'm making some mistakes. I've assumed that the yinrih's exotic reproductive strategy doesn't effect the population. We can assume that a human has two parents, but this is not the case for yinrih.
Addendum:
I tried a slightly different strategy. Same starting numbers, but this time I got the present population by summing the cohorts younger than 700. The good news is I came up with a similar result, with the population reaching my 3 billion threshold around 5250.
Table for method 2
| Terran Year | Population | Cohort size |
|---|---|---|
| 0 | 50000 | 50,000.00 |
| 100 | 110,000.00 | 60,000.00 |
| 200 | 182,000.00 | 72,000.00 |
| 300 | 268,400.00 | 86,400.00 |
| 400 | 372,080.00 | 103,680.00 |
| 500 | 496,496.00 | 124,416.00 |
| 600 | 645,795.20 | 149,299.20 |
| 700 | 774,954.24 | 179,159.04 |
| 800 | 929,945.09 | 214,990.85 |
| 900 | 1,115,934.11 | 257,989.02 |
| 1000 | 1,339,120.93 | 309,586.82 |
| 1100 | 1,606,945.11 | 371,504.19 |
| 1200 | 1,928,334.13 | 445,805.02 |
| 1300 | 2,314,000.96 | 534,966.03 |
| 1400 | 2,776,801.15 | 641,959.23 |
| 1500 | 3,332,161.38 | 770,351.08 |
| 1600 | 3,998,593.66 | 924,421.29 |
| 1700 | 4,798,312.39 | 1,109,305.55 |
| 1800 | 5,757,974.87 | 1,331,166.66 |
| 1900 | 6,909,569.85 | 1,597,400.00 |
| 2000 | 8,291,483.82 | 1,916,880.00 |
| 2100 | 9,949,780.58 | 2,300,256.00 |
| 2200 | 11,939,736.69 | 2,760,307.19 |
| 2300 | 14,327,684.03 | 3,312,368.63 |
| 2400 | 17,193,220.84 | 3,974,842.36 |
| 2500 | 20,631,865.01 | 4,769,810.83 |
| 2600 | 24,758,238.01 | 5,723,773.00 |
| 2700 | 29,709,885.61 | 6,868,527.60 |
| 2800 | 35,651,862.74 | 8,242,233.12 |
| 2900 | 42,782,235.28 | 9,890,679.74 |
| 3000 | 51,338,682.34 | 11,868,815.69 |
| 3100 | 61,606,418.81 | 14,242,578.83 |
| 3200 | 73,927,702.57 | 17,091,094.59 |
| 3300 | 88,713,243.08 | 20,509,313.51 |
| 3400 | 106,455,891.70 | 24,611,176.21 |
| 3500 | 127,747,070.04 | 29,533,411.46 |
| 3600 | 153,296,484.05 | 35,440,093.75 |
| 3700 | 183,955,780.85 | 42,528,112.50 |
| 3800 | 220,746,937.03 | 51,033,735.00 |
| 3900 | 264,896,324.43 | 61,240,482.00 |
| 4000 | 317,875,589.32 | 73,488,578.40 |
| 4100 | 381,450,707.18 | 88,186,294.08 |
| 4200 | 457,740,848.62 | 105,823,552.89 |
| 4300 | 549,289,018.34 | 126,988,263.47 |
| 4400 | 659,146,822.01 | 152,385,916.17 |
| 4500 | 790,976,186.41 | 182,863,099.40 |
| 4600 | 949,171,423.69 | 219,435,719.28 |
| 4700 | 1,139,005,708.43 | 263,322,863.14 |
| 4800 | 1,366,806,850.11 | 315,987,435.76 |
| 4900 | 1,640,168,220.14 | 379,184,922.92 |
| 5000 | 1,968,201,864.17 | 455,021,907.50 |
| 5100 | 2,361,842,237.00 | 546,026,289.00 |
| 5200 | 2,834,210,684.40 | 655,231,546.80 |
| 5300 | 3,401,052,821.28 | 786,277,856.16 |