College

**Calculus I with Applications**

In Week 7, we modeled the temperature of Kodiak, Alaska, ignoring the effects of climate change. To include climate change in our model, we model the temperature with:

[tex]\[ T(t) = \frac{5}{9} \ln (t+100)\left[3 \sin \left(\frac{1}{2} t+5\right)+10\right] - \frac{160}{9} \][/tex]

where [tex]\( t \)[/tex] is given in months after December 2019, and [tex]\( T \)[/tex] is the mean water temperature in degrees Celsius [tex]\((^{\circ} C)\)[/tex] of the location where the eggs were left to hatch. We then compose [tex]\( T(t) \)[/tex] with the model:

[tex]\[ H(T) = \frac{e^{6.727}}{T+2.394} \][/tex]

where [tex]\( H \)[/tex] is the number of days it took the eggs to hatch after spawning (and [tex]\( T \)[/tex] as above). Let:

[tex]\[ f(t) = (H \circ T)(t) \][/tex]

1. (a) Suppose we are interested in comparing [tex]\( f \)[/tex] to the first year of data. What is the appropriate domain to restrict [tex]\( f \)[/tex] to?

(b) Suppose that we are interested in comparing data specifically for Alaskan Coho salmon in the first year, which spawn from July to November. What is the appropriate domain to restrict [tex]\( f \)[/tex] to?

(c) Suppose that we are interested in comparing data specifically for Alaskan Coho salmon in the first two years, which spawn from July to November. What is the appropriate domain to restrict [tex]\( f \)[/tex] to?

2. After an initial comparison of the data with [tex]\( f \)[/tex] for the first two years, we decide that [tex]\( f \)[/tex] fits our observations closely enough to warrant further investigation. Next, we want to check if [tex]\( f \)[/tex] matches the extremal behaviors observed in the data (in other words, the shortest and longest hatching periods).

Using a computer program, the critical values of [tex]\( f \)[/tex] are computed (up to four significant figures) as:

[tex]\[ t = 5.743, 11.97, 18.31, 24.54, 30.87, 37.11 \ldots \][/tex]

(a) In Alaska, daily temperatures typically increase from January to June, and daily temperatures typically decrease from June to January. Determine whether [tex]\( t = 5.743 \)[/tex] and [tex]\( t = 11.97 \)[/tex] are local minimums or local maximums.

(b) Below is a table of the critical points and their corresponding outputs:

[tex]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
\text{Months Since 2020} & 62.25 & 68.56 & 74.81 & 81.13 & 87.38 & 93.69 \\
\hline
\text{Temperature} \left(^{\circ} C \right) & 2.013 & 19.25 & 2.303 & 19.77 & 2.573 & 20.26 \\
\hline
\text{Days until Hatch} & 189.4 & 38.56 & 177.7 & 37.66 & 168.0 & 36.85 \\
\hline
\end{array}
\][/tex]

[tex]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
\text{Months Since 2020} & 99.95 & 106.3 & 112.5 & 118.8 & 125.1 & 131.4 \\
\hline
\text{Temperature} (^{\circ} C) & 2.826 & 20.71 & 3.063 & 21.14 & 3.286 & 21.54 \\
\hline
\text{Days until Hatch} & 159.9 & 36.13 & 153.0 & 35.47 & 146.9 & 34.87 \\
\hline
\end{array}
\][/tex]

What is the day, month, and year for [tex]\( t = 62.25 \)[/tex]?

(c) Recall from 1(b) and 1(c) that Alaskan Coho salmon spawn from July to November. If we are investigating Alaskan Coho salmon, which critical points in the table have an interpretation in the model [tex]\( f \)[/tex]?

(d) According to the California Department of Fish and Wildlife, when the mean water temperature is [tex]\(10.5^{\circ} C\)[/tex], California Coho salmon eggs hatch in 38 days. Is [tex]\( f \)[/tex] an effective model to study California Coho salmon in 2025?

**References**

Sparks, M. M., Falke, J. A., Quinn, T. P., Adkison, M. D., Schindler, D. E., Bartz, K., Young, D., & Westley, P. A. H. (2019). Influences of spawning timing, water temperature, and climatic warming on early life history phenology in western Alaska Sockeye Salmon. Canadian Journal.

Answer :

Let's walk through the solution step-by-step:

1. Domain Restriction (1st Year and Spawning Seasons):

- (a) First Year: Since [tex]\( t \)[/tex] is in months after December 2019, the first year would be from [tex]\( t = 0 \)[/tex] to [tex]\( t = 12 \)[/tex]. Therefore, the appropriate domain to restrict [tex]\( f \)[/tex] to is [tex]\( [0, 12] \)[/tex].

- (b) Alaskan Coho salmon (July-November, First Year): These months correspond to [tex]\( t = 7 \)[/tex] to [tex]\( t = 11 \)[/tex] assuming January is March 2020. Hence, the domain is [tex]\( [7, 11] \)[/tex].

- (c) Alaskan Coho salmon (July-November, First Two Years): This includes two spawning periods:
- First Year: [tex]\( t = 7 \)[/tex] to [tex]\( t = 11 \)[/tex]
- Second Year: [tex]\( t = 19 \)[/tex] to [tex]\( t = 23 \)[/tex] (One year later)
- Therefore, the domain is the union: [tex]\( [7, 11] \cup [19, 23] \)[/tex].

2. Local Minimums and Maximums (Critical Values Analysis):

- Increasing and Decreasing Temperatures:
- In Alaska, temperatures typically rise from January to June and fall from July to January.
- For the given critical values, [tex]\( t = 5.743 \)[/tex] is in the first half of the cycle (March), suggesting it's likely a minimum.
- [tex]\( t = 11.97 \)[/tex], occurring around July, when temperatures tend to decrease again, is likely a maximum.

3. Date Calculation for [tex]\( t = 62.25 \)[/tex]:

- Months Since December 2019:
- [tex]\( t = 62.25 \)[/tex] months after December 2019 corresponds to June 2025.
- To get the specific day, consider 0.25 of a month (approximately 7 days assuming 30 days per month).
- Therefore, the date would be June 7, 2025.

4. Interpretation for Alaskan Coho Salmon Spawning:

- Relevant Critical Points: Since Coho salmon spawn from July to November, we look for critical values within those periods:
- [tex]\( t = 68.56 \)[/tex] and [tex]\( t = 74.81 \)[/tex] appear during the July-November timespan two years after December 2019.

5. Model Suitability for California Coho Salmon:

- Comparison with California Data:
- The model [tex]\( f \)[/tex] was based on specific Alaskan conditions, and using it for California Coho salmon might not be suitable.
- In California, at a mean water temperature of [tex]\( 10.5^\circ C \)[/tex], eggs hatch in 38 days, which doesn't directly align with the model's predictions for Alaska.

In conclusion, our analysis respects the local seasonal patterns and critical times for Alaskan Coho spawned hatching and compares them critically to another region—California, establishing the contextual accuracy and limitations of the given model.