chap09


Modeling and Simulation in Python

Chapter 9

Copyright 2017 Allen Downey

License: Creative Commons Attribution 4.0 International


In [1]:
# Configure Jupyter to display the assigned value after an assignment
%config InteractiveShell.ast_node_interactivity='last_expr_or_assign'

# import everything from SymPy.
from sympy import *

# Set up Jupyter notebook to display math.
init_printing()

The following displays SymPy expressions and provides the option of showing results in LaTeX format.


In [2]:
from sympy.printing import latex

def show(expr, show_latex=False):
    """Display a SymPy expression.
    
    expr: SymPy expression
    show_latex: boolean
    """
    if show_latex:
        print(latex(expr))
    return expr

Analysis with SymPy

Create a symbol for time.


In [3]:
t = symbols('t')

If you combine symbols and numbers, you get symbolic expressions.


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expr = t + 1

The result is an Add object, which just represents the sum without trying to compute it.


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type(expr)

subs can be used to replace a symbol with a number, which allows the addition to proceed.


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expr.subs(t, 2)

f is a special class of symbol that represents a function.


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f = Function('f')

The type of f is UndefinedFunction


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type(f)

SymPy understands that f(t) means f evaluated at t, but it doesn't try to evaluate it yet.


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f(t)

diff returns a Derivative object that represents the time derivative of f


In [10]:
dfdt = diff(f(t), t)

In [11]:
type(dfdt)

We need a symbol for alpha


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alpha = symbols('alpha')

Now we can write the differential equation for proportional growth.


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eq1 = Eq(dfdt, alpha*f(t))

And use dsolve to solve it. The result is the general solution.


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solution_eq = dsolve(eq1)

We can tell it's a general solution because it contains an unspecified constant, C1.

In this example, finding the particular solution is easy: we just replace C1 with p_0


In [15]:
C1, p_0 = symbols('C1 p_0')

In [16]:
particular = solution_eq.subs(C1, p_0)

In the next example, we have to work a little harder to find the particular solution.

Solving the quadratic growth equation

We'll use the (r, K) parameterization, so we'll need two more symbols:


In [17]:
r, K = symbols('r K')

Now we can write the differential equation.


In [18]:
eq2 = Eq(diff(f(t), t), r * f(t) * (1 - f(t)/K))

And solve it.


In [19]:
solution_eq = dsolve(eq2)

The result, solution_eq, contains rhs, which is the right-hand side of the solution.


In [20]:
general = solution_eq.rhs

We can evaluate the right-hand side at $t=0$


In [21]:
at_0 = general.subs(t, 0)

Now we want to find the value of C1 that makes f(0) = p_0.

So we'll create the equation at_0 = p_0 and solve for C1. Because this is just an algebraic identity, not a differential equation, we use solve, not dsolve.

The result from solve is a list of solutions. In this case, we have reason to expect only one solution, but we still get a list, so we have to use the bracket operator, [0], to select the first one.


In [22]:
solutions = solve(Eq(at_0, p_0), C1)
type(solutions), len(solutions)

In [23]:
value_of_C1 = solutions[0]

Now in the general solution, we want to replace C1 with the value of C1 we just figured out.


In [24]:
particular = general.subs(C1, value_of_C1)

The result is complicated, but SymPy provides a method that tries to simplify it.


In [25]:
particular = simplify(particular)

Often simplicity is in the eye of the beholder, but that's about as simple as this expression gets.

Just to double-check, we can evaluate it at t=0 and confirm that we get p_0


In [26]:
particular.subs(t, 0)

This solution is called the logistic function.

In some places you'll see it written in a different form:

$f(t) = \frac{K}{1 + A e^{-rt}}$

where $A = (K - p_0) / p_0$.

We can use SymPy to confirm that these two forms are equivalent. First we represent the alternative version of the logistic function:


In [27]:
A = (K - p_0) / p_0

In [28]:
logistic = K / (1 + A * exp(-r*t))

To see whether two expressions are equivalent, we can check whether their difference simplifies to 0.


In [29]:
simplify(particular - logistic)

This test only works one way: if SymPy says the difference reduces to 0, the expressions are definitely equivalent (and not just numerically close).

But if SymPy can't find a way to simplify the result to 0, that doesn't necessarily mean there isn't one. Testing whether two expressions are equivalent is a surprisingly hard problem; in fact, there is no algorithm that can solve it in general.

Exercises

Exercise: Solve the quadratic growth equation using the alternative parameterization

$\frac{df(t)}{dt} = \alpha f(t) + \beta f^2(t) $


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Exercise: Use WolframAlpha to solve the quadratic growth model, using either or both forms of parameterization:

df(t) / dt = alpha f(t) + beta f(t)^2

or

df(t) / dt = r f(t) (1 - f(t)/K)

Find the general solution and also the particular solution where f(0) = p_0.


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