Table 1 Description of the model equations, and interpretation of the system state variables and parameters.
From: Long-term transients help explain regime shifts in consumer-renewable resource systems
Model equations | ||||
Population dynamics: | \(\frac{{{\mathrm{d}}P}}{{{\mathrm{d}}t}} = rP\left( {1 - \frac{P}{{qR}}} \right)\) | |||
Renewable resource dynamics: | \(\frac{{{\mathrm{d}}R}}{{{\mathrm{d}}t}} = f_{\mathrm{i}}(R) - hP\) | |||
Resource growth (baseline model): | \(f_1\left( R \right) = cR\left( {1 - \frac{R}{{K_{\max }}}} \right)\) | |||
Resource growth (extended model): | \(f_2\left( R \right) = cs^ \ast R\left( {1 - \frac{R}{{K_{\max }}}} \right)\left( {\frac{R}{{K_{\min }}} - 1} \right)\) | |||
Interpretation of symbols | ||||
Symbol | Interpretation | Unit | Value | References |
r | Relative population growth rate | Year−1 | 0.0044 | |
q | Per capita dependency on the renewable resource | Individuals # resource units−1 | 1 | |
h | Per capita exploitation rate | Individual−1 year−1 | Varied | This study |
c | Relative resource growth rate | Year−1 | Varied | This study |
Kmax | Carrying capacity of the resource | # Resource units | 70,000 | |
Kmin | Critical resource level | # Resource units | \(\frac{{K_{\max }}}{{10}}\) | e.g., ref. 35 |
s* | Scaling factor setting \(\max \left( {f_2\left( R \right)} \right) \equiv \max \left( {f_1\left( R \right)} \right)\) | Unitless | For given \(K_{\max },\;K_{\min }\): ≈ 0.198 | e.g., ref. 58; Supplementary Methods 2 |
P | Population size | Individuals | State variable | - |
R | Resource availability | # Resource units | State variable | - |
