Population Dynamics SDE

Following Burgman et al. (1993, pp. 55-57), due to random impacts such as droughts, birth and death rates are modeled as functions of stochastic processes that are in turn defined by the solutions of their governing SDE's as follows. Let ${\bf W}_t \equiv (W_t^{(B)},W_t^{(f)},W_t^{(r)},W_t^{(N)})'$be a vector of three independent Wiener processes. The solution to the SDE $dX_t = -(\alpha_f + \beta_f^2X_t)(1 - X_t^2)dt + \beta_f(1 - X_t^2)dW_t^{(f)}$ is

$$X_t = \frac{(1 + X_0)\exp(-2\alpha_ft+2\beta_fW_t^{(f)}) + X_0 - 1} {(1 + X_0)\exp(-2\alpha_ft+2\beta_fW_t^{(f)}) + 1 - X_0}$$ (3)

(Kloeden and Platen 1995, p. 124) and is bounded between -1 and 1 when $X_0 \in (-1,\: 1)$.Let $a_f(X_t) \equiv -(\alpha_f + \beta_f^2X_t)(1 - X_t^2)$ (drift function), $b_f(X_t) \equiv \beta_f(1 - X_t^2)$ (diffusion function), and $f_t = U(X_t) \equiv (1 + X_t) / 2$.The distribution of f_t at t is the solution to the SDE

(see Kloeden and Platen 1995, pp. 108-109). This SDE was chosen because its solution is bounded between and 1 making f_t a well-defined birth rate. A similar development leads to the SDE $dr_t = -.5(\alpha_r + \beta_r^2(2r_t - 1))(1 - (2r_t - 1)^2)dt + .5\beta_r(1 - (2r_t - 1)^2)dW_t^{(r)}$for the death rate.

The tendency of more females to have litters within protected areas (see Gros (1998)) is represented by having the parameter $\alpha_f$ be conditional on the region's status. Similarly, to represent the effect of poaching and pest hunting on r_t, $\alpha_r$ is conditional on H_t (see Figure 2). The variability of the sample paths of f_t and r_t are controlled by the parameters $\beta_f$ and $\beta_r$, respectively. Although the SDE's for f_t and r_t are not derived from biological theory, their use allows birth and death rates to be modeled as bounded, temporal stochastic processes with parameters that can represent different temporal trends (with $\alpha_.$), and different amounts of variability (with $\beta_.$).

All other unmodeled effects (such as migration and/or parameter values that are age-dependent) that could influence the within-region cheetah count differential (dN_t) are represented by the derivative of a Wiener process. This is accomplished by converting (9) to an SDE:  

$$dN_t = \left[ f_t(1 - P^{cN_t})N_t - r_tN_t - (f_t - r_t)\frac{N_t^2}{K_t} \right]dt + \beta_N dW_t^{(N)}$$ (4)

where P, c, N₀, and $\beta_N$ are fixed parameters, and K_t is a deterministic function of the B_t temporal stochastic process.

Equation (11) does not have an analytical solution, and hence the joint distribution of $(B_t, \: K_t, \: f_t,\: r_t,\: N_t)'$is numerically approximated by generating a large number of sample paths of the vector SDE with the Explicit Order 1.0 Strong Scheme described in Kloeden and Platen (1995, p. 376). Note that if B_t, K_t, f_t, and r_t are fixed, the distribution of N_t is the solution of an SDE with constant coefficients. The solution that is found when B_t, K_t, f_t, and r_t are temporal stochastic processes corresponds to finding the joint probability distribution of $(B_t, \: K_t, \: f_t,\: r_t,\: N_t)'$ for given values of t, q, and m.

The influence diagram displays the intended causal relationship: K_t, f_t, and r_t are causing N_t (using the probabilistic definition of causality given in Section 2.2.2). Defining the ecosystem model as an influence diagram then, is critical to conveying causal structure to all stakeholders. Such rigorous statements about causality cannot be deduced from examination of the system of SDE's alone (see Pearl (1995)).

The final output variable, D_t measures the fraction of a region's area over which cheetah have been detected. Let ra be a region's surface area and d= N_t/ra, i.e., the density of cheetah in the region. An observation on D_t can be computed from maps of cheetah presence/absence by district. This is done by dividing the sum of all areas of districts in the region on which cheetah have been detected by ra. The influence diagram models D_t as a deterministic function of N_t and ra as follows. Let $\xi$ be the minimum cheetah density that results in a cheetah detection report. Let $\rho$ be a cheetah density above which cheetah are certain to be reported. Then D_t = 0 if $d < \xi$, $=(d-\xi)/(\rho - \xi)$ if $d \in (\xi,\: \rho)$,and =1 if $d \gt \rho$.Note that it is possible for N_t to be positive but D_t to be zero, i.e., $\xi$ can be interpreted as the minimum density detection limit.

The loss node is a deterministic function of D_t. This loss function is arrived at through public debate and reflects a composite of the ecological and economic values that have been expressed by all stakeholders. For purposes of illustration, the loss function used here represents the values of: (a) high loss if cheetah go extinct, and (b) bounded economic loss of livestock predation. These values are operationalized in the asymmetric loss function: $L = 10^4I_{\{D_t<.15\}}(D_t) + 1I_{\{.15 < D_t < .5\}}(D_t) + 5I_{\{.5 < D_t < .8\}}(D_t) + 10I_{\{.8 < D_t\}}(D_t)$.The value of m that minimizes the expected value of this loss node would be chosen for implementation.