Neutral Army Growth

Neutral monster armies in Heroes IV grow over time. I did some experiments to figure out the growth rate.

Data

turn	50 peasants	100	1000
1	50	100	1000
2	50	101	1005
3	51	102	1010
4	52	104	1015
5	52	105	1020
6	53	107	1025
7	54	108	1030
8	55	110	1035
9	55	111	1041
10	56	113	1046
11	57	115	1051
12	58	116	1056
13	59	118	1061
14	60	120	1066
15	60	121	1071
16	61	123	1077
17	62	125	1082
18	63	127	1087
19	64	128	1092
20	65	130	1097
21	66	132	1102
22	67	134	1107
23	68	136	1113
24	69	138	1118
25	70	140	1123

turn	50 catapults	100	1000
1	50	100	1000
2	50	100	1000
3	50	100	1000
4	51	101	1001
5	51	101	1001
6	52	102	1002
7	52	102	1002
8	52	102	1002
9	53	103	1003
10	53	103	1003

Hypothesis

The growth rate seems to be about 1.4% per day, rounded to the nearest whole number of creatures. That is, given an initial number of creatures C_i, the number of creatures C in the army on day D is given by the formula

C = round(C_i * 1.014 ^ (D - 1))

However, there seems to be a cap on the growth rate. For peasants, it is about 5.125 creatures per day. For catapults, it seems to be around 1/3 of a creature per day. I don't know how the cap is determined.

Hypothesis: the growth rate cap is 5.125 * sqrt(S) PEONS per day, where S is the strength of an individual creature in PEONs. Note that catapults have an individual strength of 200 PEONs.