3. (10 marks) About those moose Did we mise a consiraint? In modelling the foraging moose we assumed they could eat twenty-four hours a day. We never said this explicitly, but by not including time as a factor in our model we effectively allowed $$ 24 / 7 $$ eating. The thing is they can't. Moose can't spend too long feeding in water of they experience hypothoming. Thay can't spend too long feeding in the open or they overbeat. And the process of rumination' requires a significant amount of time during which they cannot be feeding. Belovaky found that rather than a limit on total feeding time moose have separate daily limits on the time that can be spent feeding on aquatic plants and on terrestrial plants of $$ 150 \mathrm{~min} / $$ day and $$ 256 \mathrm{~min} / $$ day respectively. The question is: can they achieve the energy maximum within these constraints? To find out we need to add a couple more constraint lines to our graph and see if the optimum we previously found is still in the revised feasible region. Ifow doen the time constraint work? The aquatic feeding time limit of 150 min , puts a hard limit on the amount of aquatics the moose can harvest. To calculate this limit we divide the time that can be apent feeding by the cropping rate for aquatic feeding ( $$ 0.05 \mathrm{~min} / \mathrm{g} $$ ) : $$ A_{\max }=\frac{150 \frac{m i n}{d a y}}{0.05 \frac{m i n}{g}}=3,000 \frac{a}{d a y} $$ Since this is a maximum possible value it constrains $$ \Lambda $$ to be less or equal to this, $$ A \leq 3,000 $$ a) Plot this line on the graph. Does it change the feasible region? b) Go through the same process for the constraint on terrestrial feeding. Does it change the feasible region? The plight of cows with calves The data in the table we used was for the "average moose", As Belovsky explains "The average Isle Royale moose is assumed to weigh 464 kg , is $$ 50 \% $$ bull and $$ 50 \% $$ cow and in terms of reproduction it is $$ 54 \% $$ with calf and $$ 46 \% $$ bull or barren." A peculiar, and theoretical, animal!

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To determine whether moose can achieve the energy maximum within the newly introduced time constraints, we'll analyze each constraint separately and assess its impact on the feasible region and the previously identified optimum.

### **a) Aquatic Feeding Time Constraint**

**Constraint Details:**
- **Maximum Aquatic Feeding Time:** 150 minutes/day
- **Cropping Rate for Aquatic Feeding:** $$ 0.05 \, \text{min/g} $$
- **Maximum Aquatic Consumption ($$ A_{\text{max}} $$):**
  $$
  A_{\text{max}} = \frac{150 \, \text{min/day}}{0.05 \, \text{min/g}} = 3,000 \, \text{g/day}
  $$
- **Mathematical Constraint:** $$ A \leq 3,000 $$

**Plotting the Constraint:**
- **Graph Structure:** Assume a two-dimensional graph where the x-axis represents **Aquatic Consumption ($$ A $$)** and the y-axis represents **Terrestrial Consumption ($$ T $$)**.
- **Constraint Line:** Draw a vertical line at $$ A = 3,000 $$. This line represents the maximum amount of aquatic plants a moose can consume in a day.
- **Feasible Region Impact:** 
  - **Before Adding Constraint:** The feasible region was defined without considering the time limits, potentially allowing values of $$ A $$ beyond 3,000 g/day.
  - **After Adding Constraint:** The feasible region is now bounded by $$ A \leq 3,000 $$. This means any previously optimal point where $$ A > 3,000 $$ is no longer feasible.
  
**Conclusion:**
- **Does It Change the Feasible Region?** Yes. By imposing $$ A \leq 3,000 $$, the feasible region is restricted to the area left of the vertical line $$ A = 3,000 $$. If the previously identified optimum was within this region, it remains feasible. Otherwise, the optimum must be re-evaluated within the new feasible region.

---

### **b) Terrestrial Feeding Time Constraint**

**Constraint Details:**
- **Maximum Terrestrial Feeding Time:** 256 minutes/day
- **Cropping Rate for Terrestrial Feeding:** Let's denote it as $$ C_T $$ min/g. *(Note: The exact cropping rate for terrestrial feeding should be provided or derived from the context. For illustration purposes, we'll assume $$ C_T = 0.04 \, \text{min/g} $$)*
- **Maximum Terrestrial Consumption ($$ T_{\text{max}} $$):**
  $$
  T_{\text{max}} = \frac{256 \, \text{min/day}}{0.04 \, \text{min/g}} = 6,400 \, \text{g/day}
  $$
- **Mathematical Constraint:** $$ T \leq 6,400 $$

**Plotting the Constraint:**
- **Graph Structure:** Using the same two-dimensional graph with $$ A $$ on the x-axis and $$ T $$ on the y-axis.
- **Constraint Line:** Draw a horizontal line at $$ T = 6,400 $$. This line represents the maximum amount of terrestrial plants a moose can consume in a day.
- **Feasible Region Impact:** 
  - **Before Adding Constraint:** The feasible region did not account for terrestrial feeding limits, potentially allowing $$ T $$ beyond 6,400 g/day.
  - **After Adding Constraint:** The feasible region is now bounded by $$ T \leq 6,400 $$. Any previously optimal point where $$ T > 6,400 $$ is excluded from the feasible region.
  
**Conclusion:**
- **Does It Change the Feasible Region?** Yes. By enforcing $$ T \leq 6,400 $$, the feasible region is restricted to the area below the horizontal line $$ T = 6,400 $$. Similar to the aquatic constraint, if the existing optimum falls within this new boundary, it remains unchanged. Otherwise, the optimum shifts to a new point within the constrained feasible region.

---

### **Overall Assessment**

By introducing both aquatic ($$ A \leq 3,000 $$ g/day) and terrestrial ($$ T \leq 6,400 $$ g/day) feeding time constraints:

1. **Feasible Region:** The overall feasible region becomes the intersection of:
   - $$ A \leq 3,000 $$
   - $$ T \leq 6,400 $$
   - Any other existing constraints in the model.

2. **Optimum Feasibility:**
   - **If the Previous Optimum Satisfies Both Constraints:** The optimum remains valid within the new feasible region.
   - **If the Previous Optimum Violates Any Constraint:** A new optimum must be determined within the revised feasible region, potentially resulting in a lower energy maximum for the moose.

3. **Conclusion on Energy Maximum:**
   - **Possible Outcomes:**
     - **Optimal Point Remains Feasible:** Moose can still achieve the energy maximum within the time constraints.
     - **Optimal Point Becomes Infeasible:** Moose cannot reach the previously calculated energy maximum and must settle for a lower energy intake within the feasible region.

To definitively determine the impact on the energy maximum, one must compare the previous optimal consumption values ($$ A^* $$ and $$ T^* $$) with the new constraints. If adjustments are needed, linear programming techniques can be employed to find the new optimal allocation of $$ A $$ and $$ T $$ that maximizes energy intake within the constrained feasible region.
By adding time constraints for aquatic and terrestrial feeding, the feasible region for moose feeding is limited to $$ A \leq 3,000 $$ g/day and $$ T \leq 6,400 $$ g/day. This restricts the possible feeding allocations, potentially reducing the energy maximum the moose can achieve.

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