Posts

2016-02-20: Apple VS FBI

2016-02-19: More Zika may be better than less

2016-02-17: Dependent Non-Commuting Random Variable Systems

2016-01-14: Life at the multifurcation

2015-09-28: AI ain't that smart

2015-06-24: MathEpi citation tree

2015-03-31: Too much STEM is bad

2015-03-24: Dawn of the CRISPR age

2015-02-12: A Comment on How Biased Dispersal can Preclude Competitive Exclusion

2015-02-09: Hamilton's selfish-herd paradox

2015-02-08: Risks and values of microparasite research

2014-11-10: Vaccine mandates and bioethics

2014-10-18: Ebola, travel, president

2014-10-17: Ebola comments

2014-10-12: Ebola numbers

2014-09-23: More stochastic than?

2014-08-17: Feynman's missing method for third-orders?

2014-07-31: CIA spies even on congress

2014-07-16: Rehm on vaccines

2014-06-21: Kurtosis, 4th order diffusion, and wave speed

2014-06-20: Random dispersal speeds invasions

2014-05-06: Preservation of information asymetry in Academia

2014-04-16: Dual numbers are really just calculus infinitessimals

2014-04-14: More on fairer markets

2014-03-18: It's a mad mad mad mad prisoner's dilemma

2014-03-05: Integration techniques: Fourier--Laplace Commutation

2014-02-25: Fiber-bundles for root-polishing in two dimensions

2014-02-17: Is life a simulation or a dream?

2014-01-30: PSU should be infosocialist

2014-01-12: The dark house of math

2014-01-11: Inconsistencies hinder pylab adoption

2013-12-24: Cuvier and the birth of extinction

2013-12-17: Risk Resonance

2013-12-15: The cult of the Levy flight

2013-12-09: 2013 Flu Shots at PSU

2013-12-02: Amazon sucker-punches 60 minutes

2013-11-26: Zombies are REAL, Dr. Tyson!

2013-11-22: Crying wolf over synthetic biology?

2013-11-21: Tilting Drake's Equation

2013-11-18: Why $1^\infty != 1$

2013-11-15: Adobe leaks of PSU data + NSA success accounting

2013-11-14: 60 Minutes misreport on Benghazi

2013-11-11: Making fairer trading markets

2013-11-10: L'Hopital's Rule for Multidimensional Systems

2013-11-09: Using infinitessimals in vector calculus

2013-11-08: Functional Calculus

2013-11-03: Elementary mathematical theory of the health poverty trap

2013-11-02: Proof of the area of a circle using elementary methods

Random dispersal speeds invasions

There's been a controversy in theoretical ecology over invasions for a number of years now. One question is how does the randomness (aka stochasticity) of dispersal events alter the speed of the invasion.

Consider a density-independent invasion process where the population density \(n_t(x)\) is a solution of the integrodifference equation

\[ n_{t+1}(x) = \mathscr{R}_0 \int_{-\infty}^{\infty} k(x-y) n_t(y) \mathrm{d} y\]

and the dispersal kernel is a shifted Gauss distribution

\[ k(x) = \frac{1}{\sqrt{2\pi} \sigma} e^{- \frac{\left(x- u\right)^{2}}{2 \sigma^{2}}}.\]

This kernel converges to a delta-function as the standard deviation \(\sigma\) vanishes, which corresponds to the case of fixed deterministic dispersal. So we can use it to see how stochasticity alters invasion speed. The dispersal kernel's moment generating function

\[ M[k](s) = e^{ s^2 \sigma^{2}/2 + s u } \]

Using standard methods for density-independent branching processes, we can determine the wave-speed as a function of wave-number...

\[ c(s) = \frac{s \sigma^{2}}{2} + u + \frac{1}{s} \log{\left(\mathscr{R}_{0} \right)} \]

The minimum wave speed is then

\[ c^* = \sigma \sqrt{2 \log{\left (\mathscr{R}_{0} \right )}} + u \]

So increased dispersal stochasticity accelerates the waves compared to deterministic dispersal (which is the delta-function).