Question

Oscillating growth rates Some species have growth rates that oscillate with an (approximately) constant period P. Consider the growth rate function N'(t) = r+A sin 2πt/P, where A and r are constants with units of individuals/yr, and t is measured in years. A species becomes extinct if its population ever reaches 0 after t=0.

c. Suppose P=10, A=50, and r=5. If the initial population is N(0)=10, does the population ever become extinct? Explain.

Accepted Answer

Identify the given growth rate function: $$N'(t) = r + A \sin\left(\frac{2\pi t}{P}\right)$$, where $$P=10$$, $$A=50$$, and $$r=5. $$Write the differential equation for the population $$N(t)$$: $$\frac{dN}{dt} = 5 + 50 \sin\left(\frac{2\pi t}{10}\right). $$Integrate the growth rate function to find the population function $$N(t)$$, using the initial condition $$N(0) = 10. $$This gives:

$$N(t) = \int \left(5 + 50 \sin\left(\frac{2\pi t}{10}\right)\right) dt + C. $$Perform the integration step-by-step:  
- The integral of $$5$$ with respect to $$t is$$ $$5t.  
- $$The integral of $$50 \sin\left(\frac{2\pi t}{10}\right)$$ with respect to $$t$$ involves using the substitution $$u = \frac{2\pi t}{10}$$, so the integral becomes $$-\frac{50 \cdot 10}{2\pi} \cos\left(\frac{2\pi t}{10}\right)$$ plus a constant. Apply the initial condition $$N(0) = 10 to $$solve for the constant of integration $$C$$, then analyze the resulting function $$N(t) to $$determine if it ever reaches zero for $$t \u003e 0. $$This involves checking the minimum values of $$N(t)$$ over time.

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Key Concepts

Differential Equations and Population Growth

Oscillatory Functions and Periodicity

Extinction Condition and Population Thresholds