##### Model Description

The model of Tsembaga agro-ecology explores the coupled dynamics involving population growth, renewable resource base, resource consumption by humans, and the self-regulating effect of cultural ritual. The model demonstrates that the cultural ritual of Tsembaga (Kaiko) can stabilize the Tsembaga population and its resource level. This is achieved by attenuating wildly fluctuating limit cycles of population and resource levels down to desirable small-amplitude cycles. Anderies (1998) describes the model as the following.

*"In his classic work,Pigs for the Ancestors, Roy Rappaport proposed that the ritual cycle of the Tsembaga was a mechanism to regulate human population growth and prevent the degradation of the Tsembaga ecosystem. Rappaport provided detailed ethnographic and ecological information to support his claim, but many aspects of Rappaport's model were subsequently criticised. Several simulation models of the Tsembaga ecosystem were constructed to test Rappaport's hypothesis (Shantzis & Behrens, 1973; Foin & Davis, 1984) and evaluate possible alternatives (e.g. Foin & Davis, 1987). The basic conclusions were that it was possible to develop models supporting Rappaport's hypothesis but they were extremely sensitive to parameter choices, and other simpler population control mechanisms might be more likely (Buchbinder, 1977; Foin & Davis, 1987).*

*In this paper, a much simpler dynamical system model for a slash-and-burn agricultural system is developed and applied to the Tsembaga system. By analysing the structure of the model for different physical and socioeconomic conditions, sources of instability and possible stabilising mechanisms are identified. The model indicates that behavioral plasticity (ability to modify behavior over a wide range of behavioral options, quickly and easily) is a fundamental source of instability which is strong enough to nullify more direct stabilising influences such as malnutrition and disease. This suggests that the only possible mechanism to counter to this fundamentally destabilising force may be cultural, i.e. the ritual cycle. Finally, a condition is outlined for which the ritual cycle will produce (local) stability".*

##### Default Dynamics

(Please describe default dynamics for this model.)

$\Large \frac{dx_{1}}{dt}=\left ( b_{n}-a\cdot exp\left ( -\alpha \frac{k\left ( c_{2}x_{1}A_{c} \right )^{a_{1}}\left ( x_{2} \right )^{a_{2}}}{x_{1}} \right ) -wx_{1} \right )x_{1}$ |

The rate of change in Tsembaga population density as a function of net natural birth rate, net death rate due to food availability, and warfare from conflicts. |

$\Large \frac{dx_{2}}{dt}=x_{2}n_{r}(1-x_{2})-\beta k(c_{2}x_{1}A_{c})^{a_{1}}(x_{2})^{a_{2}}$ |

The rate of change in renewable natural capital as a function of logistic growth and food production. |

$\Large \frac{dx_{3}}{dt}=(r-h)x_{3}$ |

The rate of change pig population as a function of intrinsic growth rate and harvest by humans. |

$\Large \frac{dc_{2}}{dt}=\lambda _{c_{2}}\left ( d_{f}- \frac{k\left ( c_{2}x_{1}A_{c} \right )^{a_{1}}\left ( x_{2} \right )^{a_{2}}}{x_{1}} \right )(c_{2}^{max}-c_{2})$ |

The rate of change in fraction of population devoting 1 man year of energy as a function of minimum per capita food demand and per capita food intake. |

$\Large \frac{dh}{dt}=\tau (x_{3}/x_{1}-g(h))$ |

The rate of change in pig harvest as a function of the ratio of pig to human population. |

$\Large \frac{dw}{dt}=\tau (x_{1}/x_{3}-\gamma g(w)+\delta )$ |

The rate of change in human deaths due to warfare as a function of the ratio of human to pig population. |

#Tsembaga ecosystem model with production functions # #define some auxiliary quantities #aux foodprod=foodrat #aux deathrate=ro-growth #aux pigpep=x3/x1 #aux peppig=x1/x3 #physical system specification #define physical parameters #al=death/food response #b=death/med response par al=5.7,nr=0.2,rp=0.2 par ro=0.02,df=4.9,a1=0.7,a2=0.3 par beta=0.06,x2min=0,cut=0.05 #define sundry functions g(dd)=(dd-2)/(cut^(3-dd)) h(dd)=(3-dd)/(cut^(2-dd)) f(x,c,b)=if(x<c)then(g(b)*(x^3)+h(b)*(x^2))else(x^b) pcd(k,x1,x2,a1,a2)=k*f(x1,cut,a1)*f(x2-x2min,cut,a2) d(x,b,a)=a*exp(-b*x) #define some hidden variables k=1.219/((0.8^(a2))*(18.15^a1)) pfg=pcd(k,364*w*x1,x2,a1,a2) #foodrat=(pfg-x3)/x1 foodrat=(pfg)/x1 growth=ro-d(foodrat,al,df) #define right hand sides par wd=0 dx1/dt=(growth-wd*x1)*x1 dx2/dt=x2*nr*(1-x2)-beta*pfg #dx3/dt=(rp-hvst)*x3 #cultural system specification #define cultural parameters par wmax=0.25,lw=0.5,pk=3,tau=5,hk=1,scale=1,shft=1 #functions pone(x)=-562.5*x**2+ 33.75*x+0.7 ptwo(x)=562.5*x**2 -1091.25*x+529.6 line(x)=-0.918350703*x+1.234175351 gt(x)=if(x<0.030816312)then(pone(x))else(if(x<0.96918369)then(line(x))else(ptwo(x))) ppone(x)=-1200*x**2+120*x pptwo(x)=-ppone(x-0.9)+5.6 pline(x)=-.333436*x+3.01678 gpt(x)=if(x<0.050139)then(ppone(x))else(if(x<0.94986)then(pline(x))else(pptwo(x))) #hidden variables #hvst=pk*ko*ko*ko #wd=hk*b*b*b #define right hand sides #par w=0.14 dw/dt=lw*(1-foodrat)*(wmax-w) #dko/dt=tau*(x3/x1-gt(ko)) #db/dt=tau*(x1/(x3)-(scale*gpt(b)+shft)) #initial condtion @ xp=x2,yp=x1 @ bounds=1000 @ total=4000 @ dt=0.1 done

Yu JHD, Arizona State University.

Bozicevic M, Arizona State University.

Culture and human agro-ecosystem dynamics: the Tsembaga of New Guinea. Journal of Theoretical Biology. 192:515-530.

. 1998.