27.2 Exercises

Exercise 27.1 (Inspired by Logan and Wolesensky (2009)) Let \(R(t)\) denote the rainfall at a location at time \(t\), which is a random process. Assume that probability of the change in rainfall from day \(t\) to day \(t+\Delta t\) is the following:

change probability
\(\Delta R = \rho\) \(\lambda \Delta t\)
\(\Delta R = 0\) \(1- \lambda \Delta t\)

The stochastic differential equation generated by this process is \(dR = \lambda \rho \; dt + \sqrt{\lambda \rho^{2}} \; dW(t)\).

  1. What is the Fokker-Planck partial differential equation for the probability distribution \(f(R,t)\)?
  2. What is a formula that solves the Fokker-Planck partial differential equation?
  3. Make some representative plots of the solution as it evolves over time
 

Exercise 27.2 A particle is moving in a gravitational field but still allowed to diffuse randomly. In this case the stochastic differential equation is \(dx = -g \; dt + \sqrt{D} \; dW(t)\).

  1. What is the Fokker-Planck partial differential equation for the probability distribution \(f(x,t)\)?
  2. Based on the work done in this section, what is the equation for the probability distribution \(f(x,t)\)?
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Exercise 27.3 Consider the stochastic differential equation \(\displaystyle dS = \left( 1 - S \right) + \sigma dW(t)\), where \(\sigma\) controls the amount of stochastic noise.

  1. First let \(\sigma = 0\) so the equation is entirely deterministic. Classify the stability of the equilibrium solutions for this differential equation.
  2. Still let \(\sigma = 0\). Apply separation of variables to solve this differential equation.
  3. Now let \(\sigma = 0.1\). Do 100 realizations of this stochastic process, with initial condition \(S(0)=0.5\). What do you notice?
  4. Now try different values of \(\sigma\) larger and smaller than 0.1. What do you notice?
  5. What is the Fokker-Planck partial differential equation for the probability distribution \(f(S,t)\)?

 

Exercise 27.4 (Inspired by Logan and Wolesensky (2009)) Consider the differential equation \(\displaystyle x' = \lambda x - c \mu x^{2}\), which is similar to a logistic differential equation. The per capita rate equation for this differential equation is \(\displaystyle \frac{x'}{x} = \lambda - c \mu x\).

  1. Assume there is noise to this per capita rate, i.e. \(\displaystyle \frac{x'}{x} \rightarrow \displaystyle \frac{x'}{x} + \mbox{ Noise}\). With this revised equation, what are the deterministic and stochastic parts?
  2. What is the Fokker-Planck partial differential equation for the probability distribution \(f(x,t)\)?
  3. The assumes that \(f_{t} = 0\), so \(f(x,t) \rightarrow f(x)\). Through direct verification, show that \(f(x)=Dx^{\alpha-1} e^{-\beta x}\) is a solution to the steady-state distribution, where \(\alpha = 2 \lambda - 1\) and \(\beta = 2c\mu\).
  4. With \(\lambda = c = \mu = D = 1\), make a graph of \(f(x)\) and graph it below:

 

Exercise 27.5 (Inspired by Gardiner (2004)) A type of chemical reaction is \(X + A \leftrightarrow 2X\), where \(A\) acts like an enzyme. The stochastic differential equation that describes this scenario is:

\[\begin{equation} dX = \left( A X - X^{2} \right) \; dt + \left( AX + X^{2} \right) dW(t) \end{equation}\]

  1. What is the Fokker-Planck equation for this stochastic differential equation?
  2. The steady-state distribution for this process is \(p(X) = e^{-2X}(A+X)^{4A-1}X^{-1}\). With \(A=1\) make a plot of this distribution.

Exercise 27.6 Models of cell membranes take account for the energy needed for ions and other materials to cross the cell membrane, usually expressed as a membrane potential \(U(x)\), where \(x\) is the current position of a particle distance. The probability \(p\) of the particle being at position \(x\) at time \(t\) is given by the Fokker-Planck equation:

\[\begin{equation} v \frac{\partial p}{\partial t} = \frac{\partial}{\partial x} \left( U'(x) p \right) + k T \frac{\partial^{2} p}{\partial x^{2}}, \end{equation}\]

where \(k\) is Boltzmann’s constant and \(T\) is the temperature.

  1. Write the Fokker-Planck equation in steady state. Show that a solution for the steady-state equation is \(p(x)=C e^{-\frac{U(x)}{kT}}\)
  2. If the steady-state distribution is normal, what is function will \(U(x)\) be?

References

Gardiner, C W. 2004. Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences. 3rd ed. springer.
Logan, J. David, and William Wolesensky. 2009. Mathematical Methods in Biology. 1st ed. Hoboken, N.J: Wiley.