An ichthyologist observes that the population \( P(t) \) of a certain fish species in the Great Barrier Reef declines exponentially according to \( P(t) = P_0 e^{-0.05t} \), where \( t \) is in years. How many years will it take for the population to decrease to 25% of its original size?

["How Long Until a Fish Population Drops to 25% of Its Original Size?\nAn Exponential Decay Model of Coral Reef Fish Populations", "Understanding how marine species populations change over time is crucial for conservation efforts, especially in vulnerable ecosystems like the Great Barrier Reef. A recent observation by ichthyologists reveals that a certain fish population declines exponentially, governed by the equation:", "[\nP(t) = P_0 e^{-0.05t}\n]", "where ( P(t) ) is the population at time ( t ) (in years), ( P_0 ) is the initial population, and the decay rate is 0.05 per year.", "But how long will it take for this population to fall to just 25% of its original size? Let’s solve this step by step.", "---", "### The Exponential Decay Formula Explained", "The standard exponential decay model is:", "[\nP(t) = P_0 e^{-kt}\n]", "In this case, ( k = 0.05 , \ ext{year}^{-1} ). To find when the population reaches 25% of ( P_0 ), we set:", "[\nP(t) = 0.25 P_0\n]", "Substitute into the decay equation:", "[\n0.25 P_0 = P_0 e^{-0.05t}\n]", "---", "### Solving for Time ( t )", "Divide both sides by ( P_0 ) (assuming ( P_0 <br/>\neq 0 )):", "[\n0.25 = e^{-0.05t}\n]", "Take the natural logarithm (ln) of both sides:", "[\n\ln(0.25) = \ln\left(e^{-0.05t}\right)\n]", "Using the logarithmic identity ( \ln(e^x) = x ), this simplifies to:", "[\n\ln(0.25) = -0.05t\n]", "Now solve for ( t ):", "[\nt = \frac{\ln(0.25)}{-0.05}\n]", "Calculate ( \ln(0.25) ):", "[\n\ln(0.25) = \ln\left(\frac{1}{4}\right) = -\ln(4) \approx -1.3863\n]", "Now substitute:", "[\nt = \frac{-1.3863}{-0.05} = \frac{1.3863}{0.05} \approx 27.73\n]", "---", "### Final Answer: Approximately 27.7 Years", "It will take about 27.7 years for the fish population in the Great Barrier Reef to decline to 25% of its original size under the observed exponential decay model.", "---", "### Conservation Implications", "This slow but steady decline underscores the fragility of reef ecosystems amid climate change and habitat loss. Monitoring these trends helps scientists and managers develop timely interventions to protect biodiversity and support reef resilience.", "By tracking exponential population trends, we gain vital insight into how quickly species respond to environmental pressures—and the urgent need for sustainable action.", "---", "Keywords: exponential decay, ichthyology, Great Barrier Reef, fish population model, ( P(t) = P_0 e^{-0.05t} ), population decline, marine conservation, 25% population, ( t = time ), ecological modeling, decay rate 0.05 per year."]