Changes between Version 22 and Version 23 of u/afrank

Timestamp:

08/25/16 12:28:00 (8 years ago)

Author:

Adam Frank

Comment:

—

u/afrank

@@ -5 +5 @@
 
 = Exo-civilization Planetary Feedback Equation Derivation =
+
+We begin with three coupled equations for the interaction between a eco-civilization population (N), a resource and which they draw energy from (R) and the state of the planetary environment (E).
+
+$\frac{dN}{dt} =\alpha_N N - \beta_N N - \alpha_{NR} \beta_{RN}(R) N$
+
+$\frac{dR}{dt} = -\beta_{RN}(N) R + \beta_{ER}(E) R $
+
+$\frac{dE}{dt} =  \gamma_E(E) E + \alpha_{NE}(N)  E  + \alpha_{RE}(R) $
+
+
+In the equations above $\alpha$ is used for "birth" terms and $\beta$ is used for "death" terms.
+
+In the above,
+
+$\alpha_N$ is the per captia population birth rate.
+
+$\beta_N$  is the per captia population death rate.
+
+$\alpha_{NR}$ is the per captia additional birth rate gained from extraction/consumption of resource $R$.
+
+$\beta_{RN}(R)$ is the per captia extraction/consumption rate of resource $R$.
+
+$\beta_{NR}(N)$ is the per unit extraction/consumption rate of resource $R$.
+
+$\beta_{ER}(E)$ is the "per unit" reduction in resource extraction due to changes in planetary environmental state.
+
+$\gamma_E(E)$ is measure of capacity of planetary environment to return to an pre-civilization equilibrium state ($E_o$).
+ 
+$\alpha_{NE}(N)$ is per captia forcing of environment from pre-civilization equilibrium state ($E_o$).
+
+$\alpha_{NE}(N)$ is per unit forcing of environment from pre-civilization equilibrium state ($E_o$) based on resource extraction.
+
+== Now we propose models for terms.
+
+== (1) Population
+Let the percaptja death rate be dependent on an environmentally dependent carrying capacity $K(E)$
+
+$\beta_N = \alpha_N \frac{N}{K(E)} $
+
+Let the per captia additional birth rate gained from extraction/consumption of resource $R$ be,
+
+$\beta_{R}(R) = \beta_R R$
+
+Thus we our population ODE now becomes a form of the competition predator-prey models
+
+  
+$\frac{dN}{dt} =\alpha_N N ( 1 - \frac{N}{K(E)}) - \alpha_{NR} \beta_{R} R N$
+
+In what follows we will use
+
+$K(E) = \frac{E-E_c}{E_c-E_o}$
+
+== (2) Resource
+
+Let the per captia and per unit extraction/consumption rate of resource $R$ be the same 
+
+$\beta_{RN} = \beta_{NR}$
+
+and let the  "per unit" reduction in resource extraction due to changes in planetary environmental state be
+
+$\beta_{RE} = \beta_{RN} N R (\frac{E}{E_c})$
+
+where $E_c$ is a "critical" state of the system above which no resource can be extracted.  
+
+
+Thus our resource equation becomes.
+
+$\frac{dR}{dt} = \beta_{RN} N R (\frac{E}{E_c}-1)$
+
+
+== (3) Environment
+
+Let $\gamma_E$ give a form that acts like logistic growth/decline
+
+$\gamma_E(E) = \alpha_E(1- \frac{E}{E_o})$
+
+Choose a simple form for per captia environmental driving and resource driving,
+
+$ \alpha_{NE}(N) = \alpha_{EN} N$
+
+$\alpha_{ER}(R) = \alpha_{ER} \beta_{RN} R N$
+
+We also choose a simple form of the carrying capacity which lowers the capacity as E approaches the critical value
+
+Thus our Environment equation becomes
+
+$\frac{dE}{dt} =  \alpha_E E (1- \frac{E}{E_o}) +  \alpha_{EN} N  E  + \alpha_{ER} \beta_{RN} R N$
+
+= 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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