Before the perturbation |
Before the perturbation |
Before the perturbation |
1. In the steady state past, the static structure function is derived
from static state variables; this step generates Eq. 3.
|
\(\left\{X_{\text{past}}\right\};\)
\(\left\{\frac{dX_{\text{past}}}{dt}\right\}=0;\text{\ \ }\)
\(\lambda_{j,\text{past}}=0\) \((j=3,4,5)\)
|
\(\lambda_{1,\text{past}};\ \ \lambda_{2,\text{past}};\ R_{\text{past}}\)
|
Initializing the system right after the perturbation is imposed
at t = 0 |
Initializing the system right after the perturbation
is imposed at t = 0 |
Initializing the system right after the
perturbation is imposed at t = 0 |
2. A perturbation is now imposed, expressed by a change in one or more
of the parameters, {c}, in the transition functions,
k({X},{c}). The initial time derivatives of the
state variables are calculated from Eqs. 21-23. |
\(\left\{X_{0}\right\}=\left\{X_{\text{past}}\right\};\ {\lambda_{j,0}=\lambda}_{j,\text{past}};\)
R0 = Rpast; perturbed
transition functions\(\ \)
|
\(\{\frac{dX_{0}}{dt}\}\) \(\neq\)
0 |
After the system is initialized to the Perturbation |
After the system is initialized to the Perturbation |
After the system is initialized to the
Perturbation |
3. The state variables are updated using their time derivatives using
Eq. 20. |
\(\left\{X_{0}\right\};\ \left\{\frac{dX_{0}}{dt}\right\}\) |
\(\left\{X_{1}\right\}\) |
4. The transition functions are updated by substituting updated state
variables. |
{k({X0},{c0})} |
{k({X1},{c1})} |
5. The time derivatives of the state variables are up-dated using Eqs.
21-23. |
\(\left\{k\left(\left\{X_{1}\right\},\left\{c_{1}\right\}\right)\right\};\ R_{0}\) |
\(\left\{\frac{dX_{1}}{dt}\right\}\ \) |
6. The structure function is updated from the constraints derived above
using Eqs. 14-18. |
\(\left\{X_{1}\right\};\{\frac{dX_{1}}{\text{dt}}\}\);
{k({X1},{c1})} |
\({\lambda_{1,1},\ ..,\lambda_{5,1},\ R}_{1}\) |
Subsequent steps repeat steps 3-6. With each update of the
structure function, the updated effects of the perturbation on abundance
and metabolic rate distributions can be derived. |
Subsequent
steps repeat steps 3-6. With each update of the structure function, the
updated effects of the perturbation on abundance and metabolic rate
distributions can be derived. |
Subsequent steps repeat steps
3-6. With each update of the structure function, the updated effects of
the perturbation on abundance and metabolic rate distributions can be
derived. |