1. Predator-Prey System Whales, Krill, and Fishermen

2. Outline Introduction Constructing a model for growth rate
Sanity Check
Model for fishing efforts
Analyzing the Model
Effects of Fishing
Maximum Sustainable Yield
Total Fishing Yield
Limitations with our models
Conclusion

3. Introduction
Baleen whales use their baleen plates to filter feed krill.
They overeat for almost half a year, then fast for breeding season.
The first goal of this presentation is to formulate a model which will help us understand how whales and krill interact and the effect the fisherman has on both populations.
The next goal is to analyze different fishing efforts for the whales and krill and to find a maximal sustainable fishing yield.

4. The Growth Rate Model Krill Population Growth Rate
𝑑 𝑁 1 𝑑𝑡 = 𝑟 1 𝑁 1 1− 𝑁 1 𝑘 −𝑐 𝑁 1 𝑁 2 − 𝑟 1 𝐹 1 𝑁 1
Whale Population Growth Rate
𝑑 𝑁 2 𝑑𝑡 = 𝑟 2 𝑁 2 1− 𝑁 2 (𝛼 𝑁 1 ) − 𝑟 2 𝐹 2 𝑁 2
Key
Natural Population Growth Rate
Carrying Capacity
Number of Krill Eaten By Whales
Number of Species Harvested From Fishing

5. 𝛂 𝛂𝒌 Variable/Parameter Description Dimension N1 N2 r1 r2 k c F1 F2 Y1
Krill population
[krill] N2
Whale population
[whales] r1
Rate of krill population growth
1/[time] r2
Rate of whale population growth k
Carrying capacity of krill 𝛂
Number of whales sustained per one krill
[whales]/[krill] 𝛂𝒌
Carrying capacity of whales
([whales]/[krill])[krill] c
Rate of whales consuming krill F1
Fishing effort for the krill
[time][krill] F2
Fishing effort for the whales
[time][whales] Y1
Krill yield
[krill]/[time] Y2
Whale yield
[whales]/[time]

6. Dimensional Analysis X1 – Proportional Population of the Krill
𝑋 ­ 1 = 𝑁 1 𝑘 = [𝑘𝑟𝑖𝑙𝑙] [𝑘𝑟𝑖𝑙𝑙] =[1]
X2 – Proportional Population of the Whales
𝑋 ­ 2 = 𝑁 2 𝛼𝑘 = [𝑤ℎ𝑎𝑙𝑒𝑠] 𝑤ℎ𝑎𝑙𝑒𝑠 ∗[𝑘𝑟𝑖𝑙𝑙] [𝑘𝑟𝑖𝑙𝑙] =[1]
Krill population at time t over the carrying capacity of krill.
Whale population at time t over the carrying capacity of whales.

7. The Fishing effort Model
Krill Proportional Population Growth Rate
𝑑𝑋 ­ 1 𝑑𝑡 = 𝑟 1 𝑋 1 (1− 𝐹 1 − 𝑋 1 −𝑣 𝑋 2 )
Whale Proportional Population Growth Rate
𝑑 𝑋 2 𝑑𝑡 = 𝑟 2 𝑋 2 (1− 𝐹 2 − 𝑋 2 𝑋 1 )

8. Constants
𝑣= 𝑐𝛼𝑘 𝑟 1 ⟹ 𝑐= 1 𝑤ℎ𝑎𝑙𝑒 ∗[𝑡𝑖𝑚𝑒] 𝑟 1 = 1 [𝑡𝑖𝑚𝑒] 𝛼= 𝑤ℎ𝑎𝑙𝑒𝑠 𝑘𝑟𝑖𝑙𝑙 𝑘=[𝑘𝑟𝑖𝑙𝑙]
𝑣= 1 𝑤ℎ𝑎𝑙𝑒𝑠 ∗[𝑡𝑖𝑚𝑒] ∗ 𝑤ℎ𝑎𝑙𝑒𝑠 𝑘𝑟𝑖𝑙𝑙 𝑘𝑟𝑖𝑙𝑙 𝑡𝑖𝑚𝑒 = 1
Rate of whales consuming krill over the krill population growth rate.

9. 𝑣 Variable/Parameter Description Dimension X1
Proportional krill population
[1] X2
Proportional whale population r1
Rate of krill population growth
1/[time] r2
Rate of whale population growth F1
Fishing effort for the krill
[time][krill] F2
Fishing effort for the whales
[time][whales] Y1
Krill yield
[krill]/[time] Y2
Whale yield
[whales]/[time] 𝑣
𝑐𝛼𝑘 𝑟 1 ≝ Rate of whales consuming
krill over krill population growth rate

10. Equilibrium Points and Stability
𝑋 1 ∗ = 1− 𝐹 1 1+𝑣(1− 𝐹 2 ) 𝑋 2 ∗ = (1− 𝐹 1 )(1− 𝐹 2 ) 1+𝑣(1− 𝐹 2 )
Stable Node at ( 𝑋 1 ∗ , 𝑋 2 ∗ ) and at (0,0)
As the whale and krill populations approach the equilibrium point they will stay there unless fishing efforts change.

11. Effects of Fishing
If F1 > 1: 𝑑 𝑋 1 𝑑𝑡 = 𝑟 1 𝑋 1 1− 𝐹 1 − 𝑋 1 −𝑣 𝑋 2 < − 𝑟 1 𝑋 1 𝑋 1 +𝑣 𝑋 2 If F2 > 1: 𝑑 𝑋 2 𝑑𝑡 = 𝑟 2 𝑋 2 (1− 𝐹 2 − 𝑋 2 𝑋 1 )< − 𝑟 2 𝑋 2 2 𝑋 1
In this case it is best to leave the whales and krill alone before we harvest them to extinction!

12. Maximum Sustainable krill Yield
𝑌 1 ∗ = 𝑟 1 𝐹 1 𝑁 1 ∗ = 𝑟 1 𝐹 1 𝑘 𝑋 1 ∗ = 𝑟 1 𝐹 1 𝑘(1− 𝐹 1 ) 1+𝑣(1− 𝐹 2 ) For a fixed F2 𝑑 𝑌 1 ∗ 𝑑 𝐹 1 𝑟 1 𝐹 2 1− 𝐹 1 1+𝑣 1− 𝐹 2 =0 => 𝐹 1 = is the maximum sustainable yield for the krill.
Recall our original krill equilibrium point:
𝑋 1 ∗ = 1− 𝐹 1 1+𝑣(1− 𝐹 2 )
And if we pretend there are no fishermen
F1=F2=0
𝑋 1 ∗ = 1 1+𝑣
If 𝑣=1 then 𝑋 1 ∗ =0.5
Meaning the whales are consuming the krill at the point of maximum sustainable yield naturally.

13. Maximum Sustainable Whale Yield
𝑌 2 ∗ = 𝑟 2 𝐹 2 𝑁 2 ∗ = 𝑟 2 𝐹 2 𝛼𝑘 𝑋 2 ∗ = 𝑟 2 𝐹 2 𝛼𝑘(1− 𝐹 1 )(1− 𝐹 2 ) 1+𝑣(1− 𝐹 2 ) For a fixed F1 𝑑 𝑌 2 ∗ 𝑑 𝐹 2 𝑟 2 𝐹 2 𝛼𝑘 1− 𝐹 1 (1− 𝐹 2 ) 1+𝑣(1− 𝐹 2 ) =0 ⇒ 1+𝑣 + 𝐹 2 −2−2𝑣 +𝑣 𝐹 2 2 =0 ⇒ 𝐹 2 = 1+𝑣 − 1+𝑣 𝑣 If
𝑣=0 ⇒ 𝐹 2 =0.5
𝑣=0.001⇒ 𝐹 2 =0.5001
𝑣=1⇒ 𝐹 2 =0.586
𝑣=5⇒ 𝐹 2 =0.710
If the rate at which whales consume krill increases so does the fishing effort for whales.

14. Total Fishing Yield
If 𝛽 is large:
𝑌≅ 𝑘 𝑟 1 (1− 𝐹 1 )(𝛽 𝐹 2 1− 𝐹 2 ) 1+𝑣(1− 𝐹 2 )
𝑌 is maximized by letting 𝐹 1 =0.
The total yield is maximized by not harvesting any krill, only whales.
If 𝛽 is small:
𝑌≅ 𝑘 𝑟 1 (1− 𝐹 1 ) 𝐹 1 1+𝑣(1− 𝐹 2 )
𝑌 is maximized by making 𝐹 2 as large as possible.
The total yield is maximized by harvesting as many whales as possible.
𝑌= 𝑌 1 ∗ + 𝛾 𝑌 2 ∗ = 𝑘 𝑟 1 (1− 𝐹 1 )( 𝐹 1 +𝛽 𝐹 2 1− 𝐹 2 ) 1+𝑣(1− 𝐹 2 )
𝛾 Is a constant representing the relative value of the whale and krill
𝛽=𝛾𝛼 𝑟 2 𝑟 1

15. Limitations Rise of the constants Necessary to measure 𝛽 and 𝑣
Extensive measurements is required for both species growth rate.
Measuring species interaction
Rate at with the whales eat krill

16. Conclusion
Fishing strategies that involve harvesting both species will happen when 𝛽 is not extremely larger or small. Here, we will find the most realistic fishing model relating to the population growth rate of the krill and whales.
The model depicted gives an accurate description of the growth rate of the species, where the growth rate depends on:
Natural population growth rate
Carrying capacity
Fishing effort

17. THANK YOU!