##### Model Description

This is an ecological-economic model that endogenizes discontinuous change between states of fire-driven, semi-arid rangelands which may exist in varying degrees as grassland and woodlands depending partly on the given soil conditions. More sandy soils will result in woodlands, whereas clay soils will more likely result in grasslands. One of the critical features of this model is the existence and impact of fires on these different types of rangelands. Grasslands may build up biomass and create a fireload that once ignited may destroy grass and woodlands. The natural recovery rates of these different rangelands are important in determining the system dynamics. Without human control through the use of livestock, the equilibrium of the grasslands is determined by their intrinsic growth rates, competition coefficients, as well as their susceptibility to the effects of fire. Accordingly, the model also considers the behavior of fire and herbivores and the interaction between these elements in order to determine optimally controlled management parameters. The time horizon of management planning can heavily influence the pathways and conditions of the system. Specifically, by controlling the offtake of livestock that grazes on the grasslands, planners may influence the interaction between the grasslands and woodlands to optimal paths which depend not only on natural parameters but also on the time horizons considered and discount rates.

$x_{t+1} - x_t = \alpha_t x_t (1 - \gamma x_t/g_t) - u_t \\$ |

This equation describes the growth of the biomass of the livestock, x, as a function of the logistic growth determined by alpha, the intrinsic growth rate of the the livestock, gamma the rate of depreciation of grasses due to grazing and g_t as the amount of grass at time t. Here u_t is the control variable which is the herd offtake at time t which determines the utility towards users of the entire social ecological system. |

$g_{t+1}-g_{t} = \beta_t g_t (1 - c_{gg} g_t/g_{max} - c_{wg}w_t/w_{max}) - \gamma_t x_t - b(g_t) g_t\\$ |

This equation describes the growth of the grass biomass. The first term on the RHS describes the logistic growth of the grass which is a function of beta_t, which is the rate of regeneration of the grass at time t, taking into account the competition between grasses and woods. This is then decreased by the grazing of the livestock, gamma x, and by fires if the grass biomass hits a flashpoint, b(g_t) g_t. |

$w_{t+1} - w{t} = \phi_t w_t (1 - c_{ww} w_t/w_{max} - c_{gw} g_t/g_{max}) - f(g_t)w_t \\$ |

This equation described the growth of the woods biomass. The first term on the RHS describes the logistic growth of the woods which is a function of phi_t, the rate of regeneration of woods at time t, taking into account the intraspecies competition between grasses and woods. This is then decreased by the possible fire effects if grass biomass exceeds the flashpoint with the function f(g_t) w_t. |

$b(g_t)= \frac{\sigma g_t[|(\frac{(g_t-\gamma x_t)}{g_f} - 1)| + (\frac{(g_t - \gamma_t x_t)}{g_f}-1)]}{2(\frac{g_t-\gamma_tx_t}{g_f}-1)}\\$ |

This describes the fire function for the grasslands. If the grass biomass is greater than the flashpoint, g_f, sigma percent of the grasslands are destroyed. |

$f(g_t)= \frac{\overline{\omega} w_t[|(\frac{(g_t-\gamma x_t)}{g_f} - 1)| + (\frac{(g_t - \gamma_t x_t)}{g_f}-1)]}{2(\frac{g_t-\gamma_tx_t}{g_f}-1)}$ |

The fire function for woody biomass. If the grass biomass is greater than the flashpoint, g_f, omega percent of the woodlands are destroyed. |

# Wood and Grassland Difference Equation # based on Biodivesity, resilience... by Charles Perrings and Brian Walker # note- not a functioning XPPAUT program as not all equations have been rendered # in the program par alpha=1, beta=1, gamma=1, u=01 x'= x*(alpha*(1-gamma*x/g)- u +1) init x=1 # set method to discrete @ meth=discrete @ total=100 @ ylo=-.1,yhi=1,xhi=100 done

Madler S, Arizona State University.

Biodiversity, resilience and the control of ecological-economic systems: the case of fire-driven rangelands. Ecological Economics. 22:73-83.

. 1997.