A herpetologist studying the conservation of endangered reptiles and amphibians in Southeast Asia models the population growth of a rare species using the equation \( P(t) = P_0 e^{rt} \), where \( P_0 \) is the initial population, \( r \) is the growth rate, and \( t \) is time in years. Given \( P_0 = 100 \), \( r = 0.05 \), and \( P(t) = 200 \), solve for \( t \).

["### Using Exponential Growth Models to Support Endangered Reptile Conservation in Southeast Asia", "Conserving endangered reptiles and amphibians in Southeast Asia requires precise scientific modeling to guide effective conservation strategies. One critical tool is population dynamics modeling, particularly using exponential growth equations that estimate how quickly threatened species can recover under favorable conditions. A leading herpetologist studying a rare amphibian species is applying the equation ( P(t) = P_0 e^{rt} ) to inform conservation efforts, where:", "- ( P_0 = 100 ) (initial population),\n- ( r = 0.05 ) (annual growth rate),\n- ( P(t) = 200 ) (target population),\n- ( t ) is time in years.", "Understanding how long it will take for this species to double in numbers under current conditions helps prioritize habitat protection, anti-poaching measures, and breeding programs.", "To determine ( t ), we substitute known values into the equation:", "[\n200 = 100 e^{0.05t}\n]", "Divide both sides by 100:", "[\n2 = e^{0.05t}\n]", "To solve for ( t ), take the natural logarithm (ln) of both sides:", "[\n\ln(2) = \ln(e^{0.05t}) = 0.05t\n]", "Thus,", "[\nt = \frac{\ln(2)}{0.05}\n]", "Using ( \ln(2) \approx 0.6931 ), we calculate:", "[\nt \approx \frac{0.6931}{0.05} = 13.862\n]", "Therefore, the model predicts it will take approximately 13.86 years for the population to grow from 100 to 200 individuals under a sustained 5% annual growth rate.", "This result underscores both the progress possible through biological recovery and the urgency to maintain or accelerate growth rates—especially given that habitat loss and climate change continue to threaten these fragile species. Conservationists use such precise modeling not only to set realistic recovery timelines but also to secure funding, engage local communities, and enforce protective policies grounded in science.", "For herpetologists and conservation planners, molecular tracking, environmental monitoring, and population modeling like ( P(t) = P_0 e^{rt} ) are indispensable tools. By combining field data with mathematical projections, they develop actionable plans to secure a future for Southeast Asia’s unique reptiles and amphibians.", "---", "关键词: 生存繁殖模型, 指数增长公式, 両栖动物保护, 东南亚旅游保护, 生物多样性恢复, 爱德里生物学家, 野生种群动态", "Stay updated on conservation science and support efforts to protect endangered reptiles and amphibians in Southeast Asia."]