A model which I'm imagining could at a very concrete and highly unrealistic state begin with a species splitting into three branches. Each of these three branches would then split into three branches, and so on. I doubt it would be too difficult to show or at least understand that this should inevitably lead to a fractal.
In order to arrange some realistic aspect to the model, I would then allow some or all of the three branches to not form. These instances would model extinction of species in a way. Instead of forming, then dying out, the branch just doesn't form. The rate at which this occurs would depend on how many branches are currently able to form (once a branch has divided or failed to branch at all, it would no longer be counted), then a random number of branches that would form will not. I would think this sort of model would retain fractal behavior, but certainly will not look as "nice" as without extinctions.
Another way to make the model more realistic is to allow a variable number of branches, say between 0 and 5 or so, or whatever current estimates on speciation might suggest. This part may destroy the fractal like behavior, but as I have seen in Barnsley's fern, this variable extension of the phylogeny may be fine.
I'm sure there are plenty of other ideas which may be implemented, but I think what I have listed should be reasonable for a simple model. The real question would then be, how closely does an averaging of many simulations of the model reflect the phylogeny of today's species within all higher levels of taxonomy?
Another thing I've been thinking about is linear versus nonlinear systems. Primarily, I've been thinking about why some nonlinear systems cannot be linearized, especially globally. Locally, around fixed points, nonlinear systems may be linearized by evaluating the Jacobian matrix at the fixed points. However, what restricts us from taking the nonlinear terms and calling them a new term by a change of variables? Certainly, most, if not all, cases will result in a system with more dimensions (each independent variable gets its own dimension). However, if nonlinear terms are linearly independent among the linear terms, then could a new system be generated in order to make the system easier to solve? Perhaps this is all bogus because the nonlinear terms may be formed by the linear terms, which would then be a case of ALL nonlinear terms being linearly dependent which would then disallow one to make a change of variables of the nonlinear terms in order to present a new dimension to the problem.
Another idea related to this is to explore what is needed in order to execute linearization of a system- as in constructing the Jacobian matrix evaluated at the fixed points of a system. Following along with the steps indicated by Strogatz in his book, Nonlinear Dynamics and Chaos, I found (and am assuming) that the only requirements needed for a two dimensional system with arbitrary coupling are that the transformation functions used in the change of variables need to be differentiable and have an inverse which is also differentiable.
For example, let the derivatives of x and y be x' = f(x,y) and y' = g(x,y), and let the change of variables, u and v, be u = F(x,x*) and v = G(y,y*), where x* and y* are fixed points. Rewriting functions for x and y gives: x = Finv(u,x*) and y = Ginv(v,y*), where Finv and Ginv are the inverse functions for F and G. Differentiating u and v gives: u' = F' + u' F and v' = G' + v' G. Rearranging for Finv and Ginv, then differentiating gives: x' = Finv' + u' Finv = Finv' + u' x and y' = Ginv' + v' Ginv = Ginv' + v' y. By substitution of these new x' and y' equations with the originals gives: f = Finv' + u' x and g = Ginv' + v' y. Solving these for u' and v' and then expanding them as Taylor series evaluated at the fixed points should give appropriate Jacobian matrices from which an analysis of the eigen values will determine what sort of behavior can be expected.
No comments:
Post a Comment