1. An Agent-Based Mathematical
Model about Carp Aggregation*
Chao Wu
Department of Computer Science and Engineering
University of Tennessee at Chattanooga
*This project is sponsored by NSF with proposal #1111542 and #1240734
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2. Outline
Introduction
Derivation of Agent-based Mathematical
Model
Experimental Result
Further Work
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3. Mathematical Modeling of Carp
Aggregation -- Background
 Asian carp （ invaders ):
 Originally introduced into Great-Lake areas to
control weed and parasite growth in aquatic farms.
 However,
 Threat native fish （ out-compete for food and
space ）
 Lower water quality （ kill off sensitive organisms ）
 Current status:
 Dominate Mississippi River, spread northward up
the Mississippi River and its tributaries （ even to
Minnesota ） .
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4.  In order to effectively control the growth of carp
 Invent barrier technologies
 Fishing
 Chemical poison, …
 We need to understand and forecast the collective
behavior of Asian carp: Aggregation
Mathematical Modeling of Carp
Aggregation -- Motivation
Data acquisition Forecast of carp aggregation
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5. Mathematical Modeling of Carp
Aggregation--Primitive assumption
 Aggregation is:
 a random and spontaneous behavior
 a gradual process. (originated from the inter-carp
collision)
 happen at perceptible distance
 Collision is always caused by the interaction
between neighboring carp
 an analog van der Walls forces (Molecular
Dynamics).
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6.  Carp aggregation consists of numerical solution of the Newton’s
second law.
mi and Xi indicate the mass and coordinates of i-th carp respectively. The force indicates the
influence from other carp and external environment; it is derived from the global potential energy .
 The global potential energy is divided into pair-wise potential energy(or two-body energy). In
simplicity, the externally applied potential energy and three-body(or higher-order) interaction are
ignored.
Mathematical Modeling of Carp
Aggregation: Fundamentals of MD
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7.  In this work, the pair-wise interaction Uij is defined using modified van der Waals forces,
where the corresponding potential energy function Uij between carp-i and carp- j is defined
by the Formula. Rs,Rhand Rk are illustrated in Figure. rij indicates the distance between two
neighboring carp. σ and ε are constant coefficient for analog van der Waals forces. It should
be remarked that the moving orientation, water flow velocity and blind zone is not
considered in the formulation of Formula.
Mathematical Modeling of Carp Aggregation:
Potential Energy and Four Characteristic Zones
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8.  Figure 3(a) shows the potential energy incurred by the pair-wise interaction
between two neighboring carp. As a comparison, the Lennard-Jones potential energy
is also shown in Figure 3(b).
Mathematical Modeling of Carp
Aggregation – Potential Energy
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9. Mathematical Modeling of Carp
Aggregation: Force
Derived from the potential energy, the interaction
force can be defined as follows:
In reality, the repulsion zone, parallel orientation zone and
attraction zone are not constant but dependent on many factors.
(not show in the slides) 9

10. Mathematical Modeling of Carp
Aggregation: Force
Figure 4(a) shows the inter-carp force field. As a comparison, the force field
derived from Lennard-Jones potential energy is given in Figure 4(b).
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11. Mathematical Modeling of Carp
Aggregation – Blind Zone
Pair-wise potential energy and force between carp
with blind-zone :
vi is the velocity of i-th carp. βmax is the maximal
perceptible angle, 0<βmax ≤π.βij indicates the angle
between vi and rij , it is defined by the following
formula:
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12. Mathematical Modeling of Carp Aggregation
– Potential Energy with Blind Zone
Taking into account of blind-zone, the inter-carp
potential energy is defined as:
where
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13. Mathematical Modeling of Carp
Aggregation – Force with Blind-Zone
the corresponding inter-carp force would be:
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14. Mathematical Modeling of Carp
Aggregation – Numerical Method
By introducing the momentum of carp, the
trajectory status of carp can be obtained using the
following Verlet algorithm:
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15. The simulation steps are given below:
1.Initialize the position and velocity of carp
2.Get the interaction force of individual carp
3.Update the acceleration of individual carp
4.Update the position and velocity of individual
carp. Go back to Step 2.
Mathematical Modeling of Carp
Aggregation
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16.  The simulation parameters:
 1. The maximum perceptible angle ( 2π/3)
 2. 700 X 700 canvas (The fish that hit the wall
will be bounced back).
 3. The number of fish is 20, 50, 100 respectively
 4. The constant coefficient σ is determined by
repel zone radius, while ε is adjusted according to
certain zone(In this case, it is 0.05 for repel zone
and 50000 for attraction zone).
Simulation Results
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17.  Fish(n = 20):
Simulation Result of Carp Aggregation
1 2 3
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18.  Fish(n = 50):
Simulation Result of Carp Aggregation
4 5 6
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19.  Fish(n = 100):
Simulation Result of Carp Aggregation
7 8 9
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20.  Fish(n = 100):
Simulation Result of Carp Aggregation
10 11 12
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21.  Fish(n = 100):
Simulation Result of Carp Aggregation
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22.  Fish(n = 100):
Energy change of Carp Aggregation
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23.  A more complex mathematical model:
 Other factors: lateral line, fish mass
 Scale up fish number: calculation problem
 Complicated Environment
 Macro-cell strategy: divide the global physics
domain into multiple overlapping/non-
overlapping cell (elements)
 Analyze each cell (element) using data-mining
method (e.g., NN, regression, HMM)
 Cell of interest will be analyzed using modeling and
simulation
 Validation of mathematical model under the
collaboration of University of Minnesota.
Future Work
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24. DATA MINING FOR MACRO-CELL
STRATEGY – CLASSIFICATION AND
OUTLIER
Cell Kinetic
energy
Carp
density
temperature Oxygen
density
Aggregation
occurs?
1 18112 50 60 1.93 no
2 8007 120 65 1.77 no
3 12000 70 70 2.11 no
4 6893 150 55 1.90 yes
5 21332 90 40 1.88 no
6 9231 130 80 1.95 yes
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