Step Known Derived from known
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.