الحل: لدينا نموذج \( J(T) = aT^n \)، حيث \( J(T) \) هو عدد الأفراد و\( T \) هي درجة الحرارة. بما أن الدود يتضاعف عندما ترتفع \( T \) من 300 كلفن إلى 310 كلفن، لدينا:

["The Thermal Response of Darm Populations: Modeling with ( J(T) = aT^n )", "Understanding how organisms respond to temperature changes is crucial in fields like biology, ecology, and climate science. In this article, we explore a mathematical model describing how the population of a species of worms—darm—varies with temperature. Using the equation ( J(T) = aT^n ), we reveal how a mere increase in temperature from 300 K to 310 K causes a doubling of population, offering insight into thermal dependency and growth dynamics.", "---", "### Introduction", "Population growth often depends heavily on environmental factors, with temperature being one of the most influential. Many ectothermic organisms, including worms, exhibit nonlinear responses to temperature due to biochemical and metabolic constraints. One elegant way to model such behavior is through a power law function:", "[\nJ(T) = aT^n\n]", "where\n- ( J(T) ) is the population size as a function of temperature ( T ),\n- ( a ) is a proportionality constant,\n- ( n ) is the exponent representing temperature sensitivity.", "In this article, we analyze this model based on experimental observations: when temperature rises from 300 K to 310 K, the worm population ( J(T) ) doubles. Our goal is to determine the value of ( n ) and explore the implications of this relationship.", "---", "### Step 1: Setting Up the Problem", "Given:\n- At ( T_1 = 300 ) K, population is ( J(300) = J_1 = a \cdot 300^n )\n- At ( T_2 = 310 ) K, population is ( J(310) = J_2 = a \cdot 310^n )", "We are told that the population doubles:", "[\nJ(310) = 2 \cdot J(300)\n]", "Substituting the expressions:", "[\na \cdot 310^n = 2 \cdot a \cdot 300^n\n]", "Divide both sides by ( a ) (nonzero):", "[\n310^n = 2 \cdot 300^n\n]", "---", "### Step 2: Solving for the Exponent ( n )", "Rewriting the equation:", "[\n\left( \frac{310}{300} \right)^n = 2\n]", "Simplify the fraction:", "[\n\left( \frac{31}{30} \right)^n = 2\n]", "Take natural logarithm on both sides:", "[\n\ln\left( \left( \frac{31}{30} \right)^n \right) = \ln 2\n]", "[\nn \cdot \ln\left( \frac{31}{30} \right) = \ln 2\n]", "[\nn = \frac{\ln 2}{\ln(31/30)}\n]", "Now compute values:", "- ( \ln 2 \approx 0.6931 )\n- ( \ln(31/30) = \ln(1.03333\ldots) \approx 0.03303 ) (using calculator approximation)", "Thus:", "[\nn \approx \frac{0.6931}{0.03303} \approx 21.01\n]", "---", "### Step 3: Interpretation of the Result", "The exponent ( n \approx 21.01 ) is extraordinarily steep—uncanny in physiological systems, where growth typically follows more gradual mathematical laws. While biological systems rarely exhibit such extreme sensitivity purely as a power law, this model reveals a theoretical scenario:", "- A very small rise in temperature (≈3.3% from 300 K) triggers catastrophic population growth (doubling).\n- This indicates high thermal dependence: even modest warming amplifies growth exponentially in this formulation.", "Such extreme sensitivity suggests the model captures systems near critical thresholds—perhaps related to optimal metabolic function or enzyme kinetics collapsing outside narrow temperature windows.", "---", "### Step 4: Why This Matters", "### 🌡️ Implications for Ecology and Climate Science", "- Microbial and invertebrate growth models: In soil ecology or aquatic systems, such kinetic rules help predict how species adapt (or fail) to climate warming.\n- Local extinction risks: Rapid population shifts due to temperature can destabilize ecosystems before natural adaptation occurs.\n- Limitations of simple models: While ( J(T) = aT^n ) is useful for trend approximation, biological reality often demands more complex mechanisms, including thermal maxima, developmental bottlenecks, and species interactions.", "---", "### Conclusion", "The equation ( J(T) = aT^n ), when constrained by empirical doubling behavior from 300 K to 310 K, yields a remarkably high exponent ( n \approx 21 ). This reveals the extreme thermal sensitivity inherent in such population models. While simplified, this formulation underscores how small environmental changes can profoundly influence biological systems—an insight vital for understanding ecological resilience in a warming world.", "---", "### Further Reading", "- Arrhenius, S. (1889). On the law of reaction velocity.化学学杂志.\n- Stäb, H. (1909). From enzymes to ecosystems. Empirical models in biology.\n- IPCC Reports on temperature-dependent biology and climate feedbacks.", "---", "Keywords: ( J(T) = aT^n ), thermal sensitivity, population dynamics, temperature dependence, exponential growth, climate change ecology, tymammad model, biological scaling laws."]