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Understanding Population Dynamics Using Partial Differential Equations
Serena Wang, Gargi Mishra, Caledonia Wilson, Michelle Serrano
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Pierre-Francois Verhulst came up with the logistic model
Thomas Malthus Pierre-Francois Verhulst
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Logistic Model
y’ = ky(A-y)
k: the population growth constant
A: the carrying capacity of the environment
General solution:
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What are Partial Differential Equations?
A partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives.
http://www.its.caltech.edu/~sparikh/ma142.html
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Time Evolution of a Spatial Profile Curve
Our Model:
● Modelling population over a thin strip of habitat ● Want to model population as a function of location and time along
this strip● Let u(t,x) denote the size of a population located at a location x for a
particular time t
● Then our initial conditions take the form u(0,x)=u0(x)
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The Initial Profile Curve, u0(x)
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Changing the Logistic Equation
In our new model, the carrying capacity, A, now depends on location, x
Location --> Location -->
Pop
ulat
ion
--->
Pop
ulat
ion
--->
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Why use partial differential equations?
● Allows us to conceptualize how the initial profile curve changes with time: “the time evolution of the profile curve”
● Breaking down problem like this allows us to model migration and dispersion: we focused on migration
How u(t,x) changes with time when we let x be constant
How the curve u0(x) is modified over time
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Mathematica Plot: Time Evolution of the Profile Curve
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Stability of Equilibria
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Migrating Populations
c: “speed” of migrationx: Locationt: timeu(t,x): function of time and location
Initial Condition:
PREVIOUS EXAMPLE
NEW EXAMPLE
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Migrating Populations: A General Solution for the PDE
Integrating both sides of this equation with respect to t:
Gives:
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Derivation of the Equation
Take partial derivative of:
To Get:
x
ut=0 t=1
-c*t
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Mathematica Plot: A randomly migrating population
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Case Study: Seasonal Migration
- In our model, we use migration of the gray whales to simulate the behavior of the function.
- The migration of gray whales is seasonal, ranging from the Southern Baja peninsula near the Tropic of Cancer to the Chukchi Sea north of the Arctic Circle.
- To model the periodic nature of the whales’ migration, we can translate the initial graph using a periodic function to obtain
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Periodic Migration
Now, u(t,x) satisfies the periodic migration equation
Where u0(x) acts as the initial profile curve. The parameter ‘c’ still
governs the speed of the migration dependent on the time of
year.
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Mathematica Plot: Movement of Curve
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References
Kerckhove, Michael. "From Population Dynamics to Partial Differential Equations." The Mathematica Journal 14 (2012). doi:10.3888/tmj.14-9.
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Thank You! Questions?