The observant blogger may notice a few changes on my blog. First and foremost, I have switched from Blogger to WordPress. Why did I abandon Blogger? Because in WordPress I can do this…
To simpify the basic Lotka-Volterra model,
the number of parameters are reduced by defining
,
and rescaling time
. The rescaled model becomes
With
and
, this further simplifies to
where
,
. This two-parameter formulation fully characterizes the behavior of the original system. This system has three equilibria, two resulting from the extinction of the predator or of both species,
and
, and one coexistence equilibrium at
. Clearly, the latter only exists if
. The stability of these equilibria is determined by the Jacobian matrix
obtained by linearizing about an equilibrium
:
![\mathbf{J} = \left[ \begin{array}{cc} 1-2x^*-y^* & -x^* \\ \beta(1-\alpha) & \beta(x^*-\alpha) \end{array} \right]](http://s0.wp.com/latex.php?latex=%5Cmathbf%7BJ%7D+%3D+%5Cleft%5B+%5Cbegin%7Barray%7D%7Bcc%7D+1-2x%5E%2A-y%5E%2A+%26+-x%5E%2A+%5C%5C+%5Cbeta%281-%5Calpha%29+%26+%5Cbeta%28x%5E%2A-%5Calpha%29+%5Cend%7Barray%7D+%5Cright%5D&bg=ffffff&fg=333333&s=0 "\mathbf{J} = \left[ \begin{array}{cc} 1-2x^*-y^* & -x^* \\ \beta(1-\alpha) & \beta(x^*-\alpha) \end{array} \right]")
Now, that felt really good! Expect more of that in future posts. Because my research is largely mathematically oriented it is only logical that my blogging software has good math support. The math support in WordPress is not perfect but light years ahead of any other blogging software that I know of (e.g. Blogger and MovableType). I have to admit that my migration was largely inspired by Terry Tao’s blog.
Yet another change on my blog is that I now have a dedicated domain, www.pineda-krch.com, that takes you directly to the blog.
hey mario,
we’re going to be rolling out blogs for OpenWetWare users soon using wordpress, and I’d be interested to know what plugins/tweaks (if any) you recommend for science bloggers. here’s where the blogs will live eventually (there’s just an example up there now):
http://openwetware.org/blog/
thanks a bunch,
jason
Jason,
that’s great news. I wish I’d known about it a week ago
I am glad you guys choose WordPress. It was not an easy decision to migrate but what swayed me was the
integration that WP provides. I think this particular functionality will be really useful on the OWW blog.
If the blog will be integrated with the wiki part of OWW it would be neat if one could use wiki formatting in blog posts and comments (I am not familiar with a plugin that does this, though). A few other plugins that I would be interested in are “Subscribe to Comments plugin” and syntax highlighting for code (particularly one supporting R). Being able to preview comments would also be useful.
Who knows, if the OWW WP blog will have all sorts of extra nifty functionality (which I cannot implement on my WordPress.com site) maybe we are looking at yet another migration.
All the best and see you on the OWW blog,
Mario