| 10 | | $\frac{dN}{dt} =\alpha_N N - \beta_N N - \alpha_{NR} \beta_{RN}(R) N$ |
| 11 | | |
| 12 | | $\frac{dR}{dt} = -\beta_{RN}(N) R + \beta_{ER}(E) R $ |
| 13 | | |
| 14 | | $\frac{dE}{dt} = \gamma_E(E) E + \alpha_{NE}(N) E + \alpha_{RE}(R) $ |
| 15 | | |
| 16 | | |
| 17 | | In the equations above $\alpha$ is used for "birth" terms and $\beta$ is used for "death" terms. |
| | 13 | $\frac{dN}{dt} =B_N - D_N + B_{RN}$ |
| | 14 | |
| | 15 | $\frac{dR}{dt} = -D_{NR} + D_{ER} $ |
| | 16 | |
| | 17 | $\frac{dE}{dt} = H_E + D_{NE} + D_{RE} $ |
| | 18 | |
| | 19 | |
| | 20 | In the equations above generally B is used "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 there terms may have dependencies on the other variables (the "phase space" variables (N,R,E) and other parameters/constraints. |
| 21 | | $\alpha_N$ is the per captia population birth rate. |
| 22 | | |
| 23 | | $\beta_N$ is the per captia population death rate. |
| 24 | | |
| 25 | | $\alpha_{NR}$ is the per captia additional birth rate gained from extraction/consumption of resource $R$. |
| 26 | | |
| 27 | | $\beta_{RN}(R)$ is the per captia extraction/consumption rate of resource $R$. |
| 28 | | |
| 29 | | $\beta_{NR}(N)$ is the per unit extraction/consumption rate of resource $R$. |
| 30 | | |
| 31 | | $\beta_{ER}(E)$ is the "per unit" reduction in resource extraction due to changes in planetary environmental state. |
| 32 | | |
| 33 | | $\gamma_E(E)$ is measure of capacity of planetary environment to return to an pre-civilization equilibrium state ($E_o$). |
| | 24 | $B_N$ is the population natural birth rate. |
| | 25 | |
| | 26 | $D_N$ is the population death rate. |
| | 27 | |
| | 28 | $B_{RN}$ is the additional population birth rate gained from extraction/consumption of resource $R$. |
| | 29 | |
| | 30 | $D_{NR}$ is the extraction/consumption rate of resource $R$ due to activity of $N$. |
| | 31 | |
| | 32 | $D_{ER}$ is the modification of extraction/consumption rate of resource $R$ due to changes in planetary environmental state $E$. |
| | 33 | |
| | 34 | $H_E$ is measure of capacity of planetary environment to return to an pre-civilization equilibrium state ($E_o$). |
| 35 | | $\alpha_{NE}(N)$ is per captia forcing of environment from pre-civilization equilibrium state ($E_o$). |
| 36 | | |
| 37 | | $\alpha_{NE}(N)$ is per unit forcing of environment from pre-civilization equilibrium state ($E_o$) based on resource extraction. |
| 38 | | |
| 39 | | == Now we propose models for terms. |
| 40 | | |
| 41 | | == (1) Population |
| 42 | | Let the percaptja death rate be dependent on an environmentally dependent carrying capacity $K(E)$ |
| 43 | | |
| 44 | | $\beta_N = \alpha_N \frac{N}{K(E)} $ |
| 45 | | |
| 46 | | Let the per captia additional birth rate gained from extraction/consumption of resource $R$ be, |
| 47 | | |
| 48 | | $\beta_{R}(R) = \beta_R R$ |
| 49 | | |
| 50 | | Thus we our population ODE now becomes a form of the competition predator-prey models |
| 51 | | |
| 52 | | |
| 53 | | $\frac{dN}{dt} =\alpha_N N ( 1 - \frac{N}{K(E)}) - \alpha_{NR} \beta_{R} R N$ |
| 54 | | |
| 55 | | In what follows we will use |
| | 36 | $D_{NE}$ is the forcing of the planetary environment from pre-civilization equilibrium state ($E_o$) due to non-resource based population activity. |
| | 37 | |
| | 38 | $D_{RE}$ is the forcing of the planetary environment from pre-civilization equilibrium state ($E_o$) due to resource based population activity. |
| | 39 | |
| | 40 | |
| | 41 | == Choosing forms for the terms. |
| | 42 | |
| | 43 | All alpha terms represent either growth rates or measures of the effect of variable $i$ on variable "j". |
| | 44 | |
| | 45 | |
| | 46 | $B_N = \alpha_N N$. |
| | 47 | |
| | 48 | $D_N = \alpha_N N \frac{N}{K(E)}$ |
| | 49 | |
| | 50 | where the death rate is now dependent on an environmentally dependent carrying capacity K(E). |
| | 51 | |
| | 52 | $B_{RN} = \alpha_{RN} \beta_{NR} N R$ |
| | 53 | |
| | 54 | where $\alpha_{RN}$ is the per captia additional birth rate gained from extraction/consumption of resource R and $\beta_{NR}$ is the per captia extraction/consumption rate of resource R |
| | 55 | |
| | 56 | $D_{NR} = \beta_{NR} N R$ |
| | 57 | |
| | 58 | $D_{ER} = \beta_{NR} N R (\frac{E}{E_c})$ |
| | 59 | |
| | 60 | where $E_c$ is the critical environmental state beyond which $E>E_c$ resource extraction is no longer possible. |
| | 61 | |
| | 62 | $H_E = \alpha_E E (1- \frac{E}{E_o})$ |
| | 63 | |
| | 64 | $D_{NE} = \alpha_{EN} N$ |
| | 65 | |
| | 66 | $D_{ER}= \alpha_{ER} \beta_{RN} R N$ |
| | 67 | |
| | 68 | Finally for the carrying capacity we choose, |
| 59 | | == (2) Resource |
| 60 | | |
| 61 | | Let the per captia and per unit extraction/consumption rate of resource $R$ be the same |
| 62 | | |
| 63 | | $\beta_{RN} = \beta_{NR}$ |
| 64 | | |
| 65 | | and let the "per unit" reduction in resource extraction due to changes in planetary environmental state be |
| 66 | | |
| 67 | | $\beta_{RE} = \beta_{RN} N R (\frac{E}{E_c})$ |
| 68 | | |
| 69 | | where $E_c$ is a "critical" state of the system above which no resource can be extracted. |
| 70 | | |
| 71 | | |
| 72 | | Thus our resource equation becomes. |
| 73 | | |
| 74 | | $\frac{dR}{dt} = \beta_{RN} N R (\frac{E}{E_c}-1)$ |
| 75 | | |
| 76 | | |
| 77 | | == (3) Environment |
| 78 | | |
| 79 | | Let $\gamma_E$ give a form that acts like logistic growth/decline |
| 80 | | |
| 81 | | $\gamma_E(E) = \alpha_E(1- \frac{E}{E_o})$ |
| 82 | | |
| 83 | | Choose a simple form for per captia environmental driving and resource driving, |
| 84 | | |
| 85 | | $ \alpha_{NE}(N) = \alpha_{EN} N$ |
| 86 | | |
| 87 | | $\alpha_{ER}(R) = \alpha_{ER} \beta_{RN} R N$ |
| 88 | | |
| 89 | | We also choose a simple form of the carrying capacity which lowers the capacity as E approaches the critical value |
| 90 | | |
| 91 | | Thus our Environment equation becomes |
| 92 | | |
| 93 | | $\frac{dE}{dt} = \alpha_E E (1- \frac{E}{E_o}) + \alpha_{EN} N E + \alpha_{ER} \beta_{RN} R N$ |
