Using the HawkDove Model and Ordinary Differential Equation

Using the Hawk-Dove Model and Ordinary Differential Equation Systems to Study Asian Carp Invasion 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 2 n 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 2 n x 2 n matrix that contains the Laplacian matrix and the diagonal matrix of diffusion coefficients

La Crosse Davenport Initial Conditions (Carp) : w 0 =(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 Tim e. S tep (Ch ose n a uto ma tica lly by ma tlab ) Cel h ce c a e ( l# ll re nt a prese spo riv e h t t in er)

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, Cell 2: 0. 1, Cell 3: 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(Cell 4 – Cell 10)

Population Fraction of Asian Carp Results 0. 6 Beginning of Invasion: Final Population Fraction of Asian Carps 0. 5 No Treatment 0. 4 0. 3 0. 2 Fence between Cell #3 and 4 0. 1 Fish Cell 4 7 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 ü Least Population of Asian Carps that invade cell #4 to 10 üMore 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?