Yvonne Feng and Kelly Pham
Outline
 Background
 Motivation
 Introduction to our models
 Different Invasion Problems
 Limitations of our models
 Future Work
Background
 Native habitat: China
 Prolific (spawns rapidly)
 Eats plankton
 Eats approximately 6.6-11.3% of their body weight
Invasion Problems
 Asian carp introduced to US in 1970’s
 Migrated to Mississippi River
 Competes with native species for food
 50% of total catch in 2008
 Currently threatening the Great Lakes
Why Research This?
 To study and understand the interaction
between the native and invasive species
 To study the speed of the invasion with aims
to identify parameters to slow down or to
stop the invasion
Game Theory Model
 Hawk-Dove as basic model
 Represent it as an ODE system
(normalized)
 Choose V = 2 and C = 4
Diffusion- Reaction Model
 Divide river into n cells and add spatial component
 Formula: ∂w/∂t =
F(w) + D∆w
 w is the 2n x 1 vector that represents the population
fractions in each cell
 F is the change of population fractions over time in each
cell (our ODE model)
 D∆ is the 2n x 2n matrix that contains the Laplacian
matrix and the diagonal matrix of diffusion coefficients
La Crosse
Davenport
Initial Conditions
(Carp) : w0 =(0.2, 0.1, 0)
Saint
Louis
Carp
Native
Fish
Carp
-1
2
Native
Fish
0
1
Population Fraction of Asian Carps
Plot of Asian Carps Population in Cell r at Time t
Modeling the Implementations
 Electric Fence
 Change diagonal entry of coefficient matrix to
0.000001
 Targeted Removal
 Add matrix to payoff to matrix A for the cells
where targeted removal is happening
Problems
Asian Carps are introduced in
certain spots in the river
Asian Carps heavily invade the
entire river
Assumptions
 Fish in each spot is either an Asian carp or a native
fish
 All carps act like Hawks; all native fish act like
Doves
 Total biomass in each spot is conserved
 The carrying capacity of the river is constant
 Fish dispersal is independent of temperature,
amount of food, flow
Problem: Prevent Future Invasion
 Asian Carps are introduced in cell #1-3
 (ex. Cell 1: 025, Cell2: 0.1, Cell3: 0.05)
 Electric Fence: 16 million dollars each
 Targeted Fishing: 2 million dollars each set
 Goal: Find the best fishing strategy to prevent
Asian Carps from invading into other
areas(Cell4 – Cell 10)
Results
Beginning of Invasion:
Population Fraction of Asian Carp
Final Population Fraction of Asian Carps
0.6
0.5
0.4
No
Treatment
0.3
0.2
Fence
between
Cell #3 and
4
0.1
0
Cell 1 Cell 2 Cell 3 Cell 4 Cell 5 Cell 6 Cell 7 Cell 8 Cell 9 Cell 10
Discussion
 If the Targeted Fishing is as good as our
assumption, with the given initial Asian
Carps Population Fractions:
 Fishing Strategy:Cell#4-7
 Least Population of Asian Carps that invade
cell #4 to 10
More Money efficient than implementing
Electric Fence
Problem: During Invasion
 Random Asian Carps Initial Population
Fractions
 Resources: 2 sets of targeted fishing
 Average Invasion Index: Average of the sum
of Asian Carps Population after targeted
fishing over 20 iterations
#1 Group of Targeted Fishing in Cell#
Average Invasion Index of 20 random Asian Carps Initial Conditions
#1 Group of Targeted Fishing in Cell#
Discussion
 Putting all of the targeted fishing groups in
one cell is a bad strategy
 With the current 20 random initial Asian
Carps population iterations, and given two
groups of targeted fishing:
results suggest that placing the two fishing
groups in separate cells between the center
and end of the invasion domain is a good
strategy
Limitations
 Native and invasive fish interactions are most
likely more complicated than represented in
the Hawk-Dove mode
 Most likely, there will be a change in biomass
 In addition to fish dispersal, fish also exhibit
active movement towards food sources and
favorable environmental conditions
Future Work
 Add a Retaliator to our Hawk-Dove model
 Incorporate a term for active movement of
fish
 Reassess results for later time points
Thank you!
Any Questions?