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Using the Hawk-Dove Model and Ordinary Differential Equation Systems to Study Asian Carp Invasion

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## Using the Hawk-Dove Model and Ordinary Differential Equation Systems to Study Asian Carp Invasion

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**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 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**Initial Conditions (Carp) : w0 =(0.2, 0.1, 0)**La Crosse Davenport Saint Louis**Plot of Asian Carps Population in Cell r at Time t**Population Fraction of Asian Carps Cell # (each cell represent a spot in the river) Time Step(Chosen automatically by matlab)**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**Final Population Fraction of Asian Carps Beginning of Invasion: Population Fraction of Asian Carp**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**Average Invasion Index of 20 random Asian Carps Initial**Conditions #1 Group of Targeted Fishing in Cell# #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