Using the Hawk-Dove Model and Ordinary Differential Equation

Systems to Study Asian Carp InvasionYvonne Feng and Kelly Pham

OutlineBackgroundMotivationIntroduction to our modelsDifferent Invasion ProblemsLimitations of our modelsFuture Work

Background Native habitat: ChinaProlific (spawns rapidly)Eats planktonEats approximately 6.6-11.3% of their body

weight

Invasion ProblemsAsian carp introduced to US in 1970’sMigrated to Mississippi RiverCompetes with native species for food50% of total catch in 2008Currently 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 ModelHawk-Dove as basic modelRepresent it as an ODE system

(normalized)

Choose V = 2 and C = 4

Diffusion- Reaction ModelDivide river into n cells and add spatial

component

Formula: ∂w/∂t = F(w) + D∆ww 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

Davenport

Initial Conditions (Carp) : w0 =(0.2, 0.1, 0)

La Crosse

Saint Louis

Carp Native Fish

Carp -1 2

Native Fish

0 1

Pop

ula

tion

Fra

ctio

n o

f A

sian

C

arp

s

Time Step(Chosen automatically by matlab)

Cell # (each cell represent a spot in the

river)

Plot of Asian Carps Population in Cell r at Time t

Modeling the ImplementationsElectric Fence

Change diagonal entry of coefficient matrix to 0.000001

Targeted RemovalAdd matrix to payoff to matrix A for the

cells where targeted removal is happening

ProblemsAsian 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 DovesTotal biomass in each spot is conservedThe carrying capacity of the river is

constantFish dispersal is independent of

temperature, amount of food, flow

Problem: Prevent Future InvasionAsian Carps are introduced in cell #1-3(ex. Cell 1: 025, Cell2: 0.1, Cell3: 0.05)

Electric Fence: 16 million dollars eachTargeted 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:

Cell 1

Cell 2

Cell 3

Cell 4

Cell 5

Cell 6

Cell 7

Cell 8

Cell 9

Cell 10

0

0.1

0.2

0.3

0.4

0.5

0.6

No Treat-mentFence be-tween Cell #3 and 4

Fish Cell 4 - 7

Pop

ula

tion

Fra

cti

on

of

Asi

an

C

arp

Final Population Fraction of Asian Carps

DiscussionIf 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 10More Money efficient than

implementing Electric Fence

Problem: During InvasionRandom Asian Carps Initial

Population FractionsResources: 2 sets of targeted fishingAverage Invasion Index: Average of

the sum of Asian Carps Population after targeted fishing over 20 iterations

#1 Group of Targeted Fishing in Cell#

#1 G

rou

p o

f Targ

ete

d F

ish

ing

in

C

ell

#Average Invasion Index of 20 random Asian Carps Initial Conditions

DiscussionPutting 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 WorkAdd 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?