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Exo-civilization Planetary Feedback Equation Derivation

Begin with most general form.

We begin with three coupled equations for the interaction between a exo-civilization population , a resource and which they draw energy from ® and the state of the planetary environment (E).

\frac{dN}{dt} =B_N - D_N + B_{RN}

\frac{dR}{dt} = -D_{NR} + D_{ER}

\frac{dE}{dt} = H_E + D_{NE} + D_{RE}

In the equations above generally B is used as "birth" terms that increase the variable and D is used for "death" terms that decrease the variable (though the meaning depends on the case). Each terms may have dependencies on the other variables - the "phase space" variables (N,R,E) - and other parameters/constraints.

In the above,

B_N is the population natural birth rate.

D_N is the population death rate.

B_{RN} is the additional population birth rate gained from extraction/consumption of resource R.

D_{NR} is the extraction/consumption rate of resource R due to activity of N.

D_{ER} is the modification of extraction/consumption rate of resource R due to changes in planetary environmental state E.

H_E is measure of capacity of planetary environment to return to an pre-civilization equilibrium state E_o. D_{NE} is the forcing of the planetary environment from pre-civilization equilibrium state E_o due to non-resource based population activity.

D_{RE} is the forcing of the planetary environment from pre-civilization equilibrium state E_o due to resource based population activity.

Choosing forms for the terms.

All alpha terms represent either growth rates or measures of the effect of variable i on variable j.

B_N = \alpha_N N.

D_N = \alpha_N N \frac{N}{K(E)}

where the death rate is now dependent on an environmentally dependent carrying capacity K(E).

B_{RN} = \alpha_{RN} \beta_{NR} N R

where \alpha_{RN} is a measure of the additional birth rate gained from extraction/consumption of resource R and \beta_{NR} is the per captia extraction/consumption rate of resource R

D_{NR} = \beta_{NR} N R

D_{ER} = \beta_{NR} N R (\frac{E}{E_c})

where E_c is the critical environmental state beyond which (E>E_c) resource extraction is no longer possible.

H_E = \alpha_E E (1- \frac{E}{E_o})

D_{NE} = \alpha_{EN} N

D_{ER}= \alpha_{ER} \beta_{RN} R N

Finally for the carrying capacity we choose,

K(E) = \frac{E-E_c}{E_c-E_o}

Exo-civilization Planetary Feedback Equations

Our final equations are,

\frac{dN}{dt} =\alpha_N N ( 1 - \frac{N}{\frac{E-E_c}{E_c-E_o}}) - \alpha_{NR} \beta_{R} R N

\frac{dR}{dt} = \beta_{RN} N R (\frac{E}{E_c}-1)

\frac{dE}{dt} = \alpha_E E (1- \frac{E}{E_o}) + \alpha_{EN} N E + \alpha_{ER} \beta_{RN} R N

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Exo-civilization Planetary Feedback Equation Derivation

Exo-civilization Planetary Feedback Equation Derivation

Begin with most general form.

Choosing forms for the terms.

Exo-civilization Planetary Feedback Equations

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