Theory and Modern Applications
Table 1 Description of variables and parameters used in system (1) and their non-dimensionalizing transformations. All variables and parameters are assumed to be nonnegative
From: Optimal control problem arises from illegal poaching of southern white rhino mathematical model
Var. or par. | Description | Unit | Non-dimensional var. or par. |
|---|---|---|---|
\(N_{i}\) | total number of rhinos in class i | rhino | \(x_{i}=\frac{N_{i}}{K}\) |
M | total number of illegal hunters | human | \(y=\frac{\beta ^{*} M}{r}\) |
t | time | year | τ = rt |
K | carrying capacity of rhinos | rhino | – |
r | intrinsic growth rate of rhinos | \(\frac{1}{\text{time}}\) | – |
\(\alpha ^{*}\) | transition rate from juvenile rhinos to adult rhinos ready to be hunt | \(\frac{1}{\text{time}}\) | \(\alpha =\frac{\alpha ^{*}}{r}\) |
\(\beta ^{*}\) | success rate of hunters rhino | \(\frac{1}{\text{human}\times \text{time}}\) | – |
p | proportion of surviving rhinos after being hunted | – | p∈[0,1] |
\(\delta ^{*}\) | transition rate due to the regrowth of rhino horn | \(\frac{1}{\text{time}}\) | \(\delta =\frac{\delta ^{*}}{r}\) |
\(\gamma ^{*}\) | conversion coefficient on the number of rhinos that have been killed by hunters | \(\frac{\text{human}}{\text{rhino}}\) | \(q=\frac{\gamma ^{*} \beta ^{*} K}{r}\) |
\(\mu ^{*}\) | natural death rate of rhinos | \(\frac{1}{\text{time}}\) | \(\mu =\frac{\mu ^{*}}{r}\) |
\(\xi ^{*}\) | natural drop-out rate of hunter due to being sufficiently aware to stop hunting | \(\frac{1}{\text{time}}\) | \(\xi =\frac{\xi ^{*}}{r}\) |
\(h^{*}\) | hunter arrest rate by the government | \(\frac{1}{\text{time}}\) | \(h=\frac{h^{*}}{r}\) |