Model Description
This model tells a story of resilience of a rangeland system in Australia. Anderies et al. (2002) provides the following overview of the model. "We developed a stylized mathematical model to explore the effects of physical, ecological, and economic factors on the resilience of a managed fire-driven rangeland system. Depending on grazing pressure, the model exhibits one of three distinct configurations: a fire-dominated, grazing-dominated, or shrub-dominated rangeland system. Transaction costs and costs due to shrub invasion, via their effect on grazing decisions, strongly influence which stable configuration is occupied. This, in turn, determines the resilience of the rangeland system. These results are used to establish conditions under which management for profit is consistent with the maintenance of resilience".
Reference
Anderies, J. M., Janssen, M. A., & Walker, B. H. (2002). Grazing Management, Resilience, and the Dynamics of a Fire-driven Rangeland System. Ecosystems, 5(1), 23–44. doi:10.1007/s10021-001-0053-9.
Default Dynamics
(Please describe default dynamics for this model.)
$\dot{c}=r_cs-\delta_cc$ |
| Change in the biomass of crowns of grasses |
$\dot{s}=c(a_c+r_ss)(1-\frac{s}{s*}-\alpha_{ws}({\frac{w}{w*}})^{\beta})-\gamma_gs-\gamma_{Is}If(s;a_s,b_s)$ |
| Change in shoot biomass of grasses |
$\dot{w}=r_ww(1-\frac{w}{w*})-\gamma_{Iw}wf(I,a_w,b_w)$ |
| Change in woody shrub biomass |
$\dot{I}=I(r_If(s;a_I,b_I)-\delta_I)$ |
| Fire dynamics |
#DEFINITION OF PARAMETERS================================== #rc = crown intrinsic growth rate,deltac= crown decay rate #sstar = maximum shoot biomass, #wstar = maximum shrub biomass, rhats=ratio of shoot growth rate to crown decay #aws =competitive effect of woody shrubs on grass, beta=nonlinearity of the #shrub competitive effect of woody shrubs on grass, gammahatg is the ratio #of grazing rate to crown decay rate, rhati= the ratio of fire index growth rate #to crown decay rate, ai controls fuel load at which fire flares up #bi controls the sharpness of the fuel load threshold #deltahati is the minimum fuel load required to maintain a fire #rhatw is the ratio of shrug growth rate to crown decay #gammaiw is the ratio of the effect of fire on shrubs to crown decay rate #aw is the ratio of shrug growth rate to crown decay #bw is the sharpness of shrub consumption threshold par rc=1, deltac=1, sstar=1, wstar=1 par rhats=3, ahatc=0.1, alphaws=0.5, beta=3, gammahatg=0.1,rhati=60, par ai=0.5, bi=3, deltahati=0.1, rhatw=0.1, gammahatiw=1, aw=1, bw=8, par bs=1, as=0.1 init w=0.6,s=0.25,c=0.6, i=0.1 #Parameters for crown and shoot dynamics #I did not know how to derive gammais so made it = aw #Table 1 in the paper says "see aw" for the variable gammais rs=rhats ac=ahatc*rs gammag=gammahatg rw=rhatw*deltac/wstar ri=rhati*deltac deltai=deltahati*ri gammaiw=gammahatiw*deltac gammais=aw e(s,ai,bi)=s^bi/(ai^bi+s^bi) f(s,as,bs)=s^bs/(ai^bs+s^bs) g(i,aw,bw)=i^bw/(aw^bw+I^bw) #DYNAMICS======================================================== #FIRE DYNAMICS i'=i*(ri*e(s,ai,bi)-deltai) #CHANGE IN CROWN BIOMASS OVER TIME c'=rc*s-deltac*c #THE CHANGE IN SHOOT BIOMASS OVER TIME s'=c*(ac+rs*s)*(1-(s/sstar)-alphaws*(w/wstar)^beta)-gammag*s-gammais*I*f(s,as,bs) #THE CHANGE IN WOODY SHRUBS OVER TIME w'=rw*w*(1-(w/wstar))-gammaiw*w*g(i,aw,bw)
Barnett A, Arizona State University.
Bozicevic M, Arizona State University.
