**Population dynamics model. **The following population dynamics model was applied to reconstruct the initial dugong population size in 1894 from fishery statistics between 1894 and 1914:

*N*t-1 = *N*t (1 + *r *–* r N*t / *K*) - *C*t ,

where *r* is the intrinsic rate of population increase, *N*t is the population size in year *t*, *K* is the carrying capacity, and *C*t is the number of individuals removed from the waters near the Ryukyu Islands in year *t*. The carrying capacity (*K*) in 1893 was sufficient to sustain the initial population of dugongs at that time (*N*1894). The intrinsic rate of population increase (*r*) was given as 1%, 2%, 4%, or 5% within a range of natural one.

**Stochastic model.** We applied a stochastic model to estimate the number of individuals in 1979 from stranding and bycatch record with a binomial distribution because the number of reported deaths was very small.

*N*t-1 = *N*t - *C*t ＋ β [*N*t - Ct, r],

where *C*t is the number of deaths in year *t*, β [*N*t - Ct,* r*] is a binomial number, and Prob [ β [*N, r*] = x] = NCx rx (1 - *r*)N-x. As it is a stochastic model, the number of individuals in 2004 varied among each run; the lower limit was *N*1979, by which <50% of the runs reached *N*2004 < 6 individuals.

**Population viability analysis.** We conducted a population viability analysis (PVA) to evaluate the impact of bycatch on the population decline between 1997 and 2019 based on the stranding records. We denoted fecundity as *f*, the survival rate until 1 year old as *s*0, the annual survival rate after 1 year old as *s*, the age at maturity as *a*m, and the physiological longevity as *A.* We assumed that the fecundity *f* was 1/3 if at least one adult male existed in the population; the sex ratio at birth was 1:1 on average; the age at maturity *a*m was 8 years of age23, and the physiological longevity *A* was 73 years6. We ignored environmental stochasticity because no mass deaths caused by infectious diseases or changes in survival or mortality rates due to environmental fluctuations have not been recorded during this period. We also ignored density effects because the carrying capacity of the location was sufficiently greater than the initial population size, and our goal was to investigate the possibility of population recovery after a decrease in population using a population dynamics model and estimate the natural growth rate during this period. The detailed extinction risk depends on age structure.

According to the life history parameters, except the physiological longevity compiled by (ref. 23), the annual survival probability of an *a* year-old individual is *s* for *a* = 1, 2, …, 72; for *a* = 0, and 0 for *a* = 73; the reproductive probability of an adult female >8 years old is 2*f*. As the number of years for a population to become extinct or recover depends on age composition, age-specific survival, and reproductive rates, we obtain the population growth rate by the maximum eigenvalue of the following Leslie matrix, **L**={*L*ij} (*i *= 1,...20, *j *= 1,...,20) as:

*L*i1 = *s*0*f */ 2 for *i* ³* a*m, *L*i+1,*i* = *s* for *i *= 1,…, 72, and *L*ij = 0 otherwise.

The population growth rate l was 2.3%, 0.3%, -0.4%, and -0.7% if (*s*, *s*0) = (0.95, 0.8), (0.93, 0.8), (0.92, 0.8), and (0.95, 0.4), respectively.

We assumed that the sex of each individual in 1979 was randomly sampled by the 1:1 sex ratio, and its age was randomly sampled by the stable age structure that is given by the eigenvector of the Leslie matrix with the maximum eigenvalue. The probability that *N*1997 £ 11 is >5% if *N*1979 £ 16,* N*1979 £ 17, and *N*1979 £ 23, under the scenarios (*s*, *f*) = (0.95, 0.33), (0.97, 0.14), (0.95, 0.17), respectively. We assumed that the number of individuals at age 1 year in year *t *+ 1, denoted by *N*1,*t*+1, is determined by the binomial distribution:

where *N*f represents the number of adult females in year *t*. We assumed that no twins were born. We assumed that the probability that an individual with age x survived in the next year is s if *x *= 1 or s0 if *x *= 0. We also assumed that *C*t individuals who died by bycatch were randomly chosen from any sex and age because the age of individuals caught by bycatch is rarely known. We do not know the sex of some individuals.

**Data availability**

All data in this study are included in this article and its supplementary data files.