In professorbeautiful/LevinsLoops: Building dynamic simulation for Levins loop models.

Using the package LevinsLoops

The moving equilibrium ideas of Richard Levins are based on differential equation dynamic system models for the evolution of a vector of values X = X(t). A local equilibrium point is a solution to $\partial(X)/\partial(t) = AX = 0$. Not all equilibria are stable. Conditions of stability at equibrium points are given by Routh-Hurwitz. systems can be analyzed for stability at the IpmNet site.

At a stable equilibrium, when parameters of the system change, the equilibrium point moves as well. Suppose the parameters change as $A = A(s)$ for some $s$ changing so slowly relative to $t$ that the system tracks the moving equilibrium closeky. The solution is obtained by the total derivative $$0 = d/ds(0) = \frac{d}{ds} (A(s)X_{eq}(s)) = \frac{\partial{A}}{\partial{s}}X_{eq}(s) + A(s) \frac{\partial{X_{eq}(s)}}{\partial{s}}.$$ Solving for the derivative, $$\frac{\partial{X_{eq}(s)}}{\partial{s}} = (-1) \times A(s)^{-1} \times \frac{\partial{A}}{\partial{s}} \times X_{eq}(s).$$

The Dynamic Model Explorer

The app LoopModelExplorer makes available the qualitative predictions of Richard Levins's qualitative approach to dynamical systems.

The dropdown list on the upper right will select a pre-coded model. The first model is from Fig 2 Levins & Schultz 1996. The loaded model string is

R -o R Ro->H H o->x Ho->y y-oy R -< R R>->H H >->x H>->y y>-y

which is represented by the following diagram, created in the app window: creates the following graph.

library(LevinsLoops)
library(LoopAnalyst)
library(DOT)
invisible(makeCMplot(stringToCM(modelStringList[1]), size=4))

Convergence to the equilibrium point""

library(LevinsLoops)
dynamSim(stringToCM(modelStringList[5]), initial = c(1000,0,0,0,0), Tmax = 100)

professorbeautiful/LevinsLoops documentation built on May 26, 2019, 8:33 a.m.