Table 1 Deterministic and stochastic models used for fitting all combinations of deterministic and stochastic models.
From: Shape and rate of movement of the invasion front of Xylella fastidiosa spp. pauca in Puglia
Deterministic model | Function | ||
|---|---|---|---|
Negative exponential | \(f_{1} \left( x \right) = a \cdot {\text{exp}}\left( { - rx} \right)\) | ||
Logistic | \(f_{2} \left( x \right) = \frac{1}{{1 + {\text{exp}}\left( {r\left( {x - x_{50} } \right)} \right)}}\) | ||
CNE* | \(f_{3} \left( x \right) = { }\left\{ {\begin{array}{ll} {1 \; \mid \; x < x_{100} ,} \\ {\exp \left( { - r\left( {x - \left( {x_{100} + ct} \right)} \right)} \right) \; \mid \; x \ge x_{100} .} \\ \end{array} } \right.{ }\) | ||
Stochastic model | function | Mean | Variance |
|---|---|---|---|
Binomial distribution | \(g_{1} \left( {x,N,p} \right) = \left( {\begin{array}{*{20}c} N \\ x \\ \end{array} } \right)p^{x} \left( {1 - p} \right)^{N - x}\) | \(Np\) | \(Np\left( {1 - p} \right)\) |
Beta-binomial distribution | \(g_{2} \left( {x,N,p,\theta } \right) = \frac{\Gamma \left( \theta \right)}{{\Gamma \left( {p\theta } \right)\Gamma \left( {\left( {1 - p} \right)\theta } \right)}}\frac{N!}{{x!\left( {N - x} \right)!}}\frac{{\Gamma \left( {x + p\theta } \right)\Gamma \left( {N - x + \left( {1 - p} \right)\theta } \right)}}{{\Gamma \left( {N + \theta } \right)}}\) | \(Np\) | \(Np\left( {1 - p} \right)\left( {1 + { }\frac{N - 1}{{\theta + 1}}} \right)\) |
