User:Carl Boettiger/Notebook/Stochastic Population Dynamics/2010/05/14

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Structured Population Dynamics

Beetle Model

Macroscopic equations:

\dot E &= f_E(E,L,A) = -\mu_e E - c_{le} E L - c_{ae} E A - a_e E + b A \\
\dot L &= f_L(E, L) =  -\mu_L L + a_e E - a_L L \\
\dot P &=  f_P(L,P) = -\mu_P P - a_p P + a_L L\\
\dot A &= f_A(P,A) = -\mu_A A + a_p P

Has the corresponding variance-covariance dynamics

&\frac{d}{dt} \sigma_E^2 =  2\partial_E f_E \sigma_E^2 + g_E + 2\textrm{Cov}(E,L) \partial_L f_E + 2\textrm{Cov}(E,A) \partial_A f_E  \\
&\frac{d}{dt} \sigma_L^2 =  2\partial_L f_L \sigma_L^2 + g_L + 2\textrm{Cov}(E,L) \partial_E f_L \\
&\frac{d}{dt} \sigma_P^2 =  2\partial_P f_P \sigma_P^2 + g_P + 2\textrm{Cov}(L,P) \partial_L f_P   \\
&\frac{d}{dt} \sigma_A^2 =  2\partial_A f_A \sigma_A^2 + g_A + 2\textrm{Cov}(P,A) \partial_P f_A      \\
&\frac{d}{dt} \textrm{Cov}(E,L) = \left( \partial_E f_E + \partial_L f_L \right) \textrm{Cov}(E,L) + \partial_L f_E \sigma_L^2 + \partial_E f_L \sigma_E^2 \\
&\frac{d}{dt} \textrm{Cov}(E,A) = \left( \partial_E f_E + \partial_A f_A \right) \textrm{Cov}(E,A) + \partial_A f_E \sigma_A^2 + \partial_E f_A \sigma_E^2 \\
&\frac{d}{dt} \textrm{Cov}(L,P) = \left( \partial_P f_P + \partial_L f_L \right) \textrm{Cov}(L,P) + \partial_L f_P \sigma_L^2 + \partial_P f_L \sigma_P^2 \\
&\frac{d}{dt} \textrm{Cov}(A,P) = \left( \partial_P f_P + \partial_A f_A \right) \textrm{Cov}(A,P) + \partial_A f_P \sigma_A^2 + \partial_P f_A \sigma_P^2 

Where g_i is the second jump moment, which is a function of the state (E, L, P, A) just as f_i is. (In this case it will correspond to the sum of all birth and death terms).

General Form & Algorithm

Consider the dynamics are given by

 \dot X_i = f_i(\vec X)

and define variance-covariance matrix M and the Jacobian matrix of f as J. Then the dynamics of the diagonal elements (variance terms) are written as

 \dot \sigma_i^2 = 2 g_i + 2\sum_k J_{ik} M_{ik} \partial_k f_i

While the dynamics of the off-diagonal elements (covariance terms) are given by

 \frac{d}{dt} \textrm{Cov(i,j)} = \dot M_{ij} = \partial_i f_i M_{ji} + \partial_j f_j M_{ij} + \partial_i f_j M_{ii} + \partial_j f_i M_{jj}

Track current implementation of the algorithm here.

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