Representation of Hunting Pressure and Carrying Capacity

H_t takes on the values infrequent and frequent and is influenced by m, R_t and U.

A region's cheetah carrying capacity is a deterministic function of herbivore biomass: $K_t \equiv \mbox{nearest integer}( \beta_{K_t}^{(0)} + \beta_{K_t}^{(1)}B_t)$.Herbivore count is not the central focus of this cheetah viability influence diagram. Therefore, a single birth-death model for the meta-population size of cheetah-prey herbivores at time t is used as a first approximation to the population dynamics of the prey populations. Let k₀ be the herbivore carrying capacity at time t₀. The SDE for B_t with random carrying capacity is $dB(t) = B(t)(k_0 - B(t))dt + B(t)\sigma dW_t^{(B)}$ where dW_t^(B) is a zero mean, unit variance white noise process. With zero diffusion (no white noise component), this is the population dynamics model given by Wells et al. (1998) assuming a high population density and low probability of unsuccessful mating (see Section 3.1.5, below). B_t is influenced by the nodes t, C, and m. U could also be an influence but was not used here due to the increased complexity.

For $t \rightarrow \infty$,the distribution of B(t) approaches that of a gamma random variable with scale parameter $\beta = 2/\sigma^2$ and shape parameter $\alpha = 2k_0/\sigma^2$(Prajneshu 1980). For Kenya however, data on herbivore numbers through time is available. For this reason, the limiting distribution is not used. Instead, this logistic growth SDE is added to the system of SDE's (see below) and this system of four equations is solved numerically.