س: يدرس عالم طيور تعداد نوع من الطيور \( P(t) \) بمرور الوقت \( t \) (بالسنوات)، ويُمَثّل بالنموذج \( P(t) = \frac{1000}{1 + 9e^{-0.5t}} \). حدد التعداد الأولي ومقاربة التعداد على المدى الطويل لحجم السكان.
["Title: Understanding Bird Population Dynamics: Analyzing the Logistic Growth Model", "In the field of wildlife biology, tracking population trends is essential for conservation and ecological research. One fascinating case involves modeling bird populations using logistic growth—perfectly captured by the function:", "[\nP(t) = \frac{1000}{1 + 9e^{-0.5t}}\n]", "where ( P(t) ) represents the population size at time ( t ) (in years), and the model describes how the bird population ( P(t) ) evolves over time under environmental constraints.", "### Step 1: Determine the Initial Population", "The initial population corresponds to ( t = 0 ), the starting moment of the study. Substituting ( t = 0 ) into the model:", "[\nP(0) = \frac{1000}{1 + 9e^{0}} = \frac{1000}{1 + 9 \ imes 1} = \frac{1000}{10} = 100\n]", "Thus, the initial population is 100 birds.", "### Step 2: Analyze the Long-Term Population Trend", "To find out what happens to the bird population as time ( t ) approaches infinity, evaluate the limit:", "[\n\lim_{t \ o \infty} P(t) = \lim_{t \ o \infty} \frac{1000}{1 + 9e^{-0.5t}}\n]", "As ( t \ o \infty ), the exponent ( -0.5t \ o -\infty ), so ( e^{-0.5t} \ o 0 ). The denominator approaches ( 1 + 0 = 1 ), giving:", "[\n\lim_{t \ o \infty} P(t) = \frac{1000}{1} = 1000\n]", "This result shows that the bird population approaches a stable, maximum sustainable size—known as the carrying capacity—of 1000 birds as time goes on.", "### Conclusion", "The model ( P(t) = \frac{1000}{1 + 9e^{-0.5t}} ) reveals:", "- Initial population: 100 birds at ( t = 0 ).\n- Long-term behavior: The population stabilizes around 1000 birds, reflecting natural ecological limits.", "Understanding these dynamics helps biologists predict population changes, support conservation strategies, and study species adaptation in changing environments. This logistic growth model exemplifies how mathematical tools illuminate real-world ecological phenomena."]
