Equation form Equation form \(\mathbf{f}\left(\mathbf{N,P}\right)\) \(\operatorname{}\left(\mathbf{f}\right)\) \(\operatorname{}\left(\mathbf{f}\right)\) \(\operatorname{}\left(\mathbf{f}\right)\) \(\operatorname{}\left(\mathbf{g}\right)\) \(\operatorname{}\left(\mathbf{g}\right)\) \(\operatorname{}\left(\mathbf{g}\right)\) \(\operatorname{}\left(\mathbf{g}\right)\) \(\frac{\mathbf{\text{dg}}}{\mathbf{\text{dN}}}\)
Ideal equation Ideal equation 0 0 \(\alpha\) 0 \(\left[0,N\right]\) 0 αP \(\left[0,1\right]\)
Prey-dependent Type I αN αN 0 \(\infty\) 0 \(\infty\) 0 \(\infty\) \(\left[\left.\ 0,\ \infty\right)\right.\ \)
Type II \(\frac{\text{αN}}{N+b}\) \(\frac{\text{αN}}{N+b}\) 0 \(\alpha\) 0 \(\infty\) 0 αP \(\left[\left.\ 0,\ \infty\right)\right.\ \)
Type III \(\frac{\alpha N^{2}}{N^{2}+b}\) \(\frac{\alpha N^{2}}{N^{2}+b}\) 0 \(\alpha\) 0 \(\infty\) 0 αP \(\left[\left.\ 0,\ \infty\right)\right.\ \)
Ratio-dependent Type I \(\alpha\frac{N}{P}\) 0 0 \(\infty\) αN αN 0 \(\infty\) \(\left[\left.\ 0,\ \infty\right)\right.\ \)
Type II \(\frac{\alpha\frac{N}{P}}{\frac{N}{P}+b}\) 0 0 \(\alpha\) 0 \(\frac{\text{αN}}{b}\) 0 αP \(\left[\left.\ 0,\ \infty\right)\right.\ \)
Type III \(\frac{\alpha\left(\frac{N}{P}\right)^{2}}{\left(\frac{N}{P}\right)^{2}+b}\) 0 0 \(\alpha\) 0 0 0 αP \(\left[\left.\ 0,\ \infty\right)\right.\ \)
Beddington-DeAngelis Beddington-DeAngelis \(\frac{\text{αN}}{N+cP+b}\) 0 0 \(\alpha\) 0 \(\frac{\text{αN}}{c}\) 0 αP \(\left[\left.\ 0,\ \infty\right)\right.\ \)
Hassell-Varley Type I \(\alpha\left(\frac{N}{P^{m}}\right)\) \(\left\{\begin{matrix}\alpha N,\ \ \&m=0\\ 0,\ \ \&m>0\\ \end{matrix}\right.\ \) 0 \(\infty\) \(\left\{\begin{matrix}0,\ \ \&0\leq m<1\\ \alpha N,\ \ \&m=1\\ \end{matrix}\right.\ \) \(\left\{\begin{matrix}\infty,\ \ \&0\leq m<1\\ \alpha N,\ \ \&m=1\\ \end{matrix}\right.\ \) 0 \(\infty\) \(\left[\left.\ 0,\ \infty\right)\right.\ \)
Type II \(\frac{\alpha\left(\frac{N}{P^{m}}\right)}{\left(\frac{N}{P^{m}}\right)+b}\) \(\left\{\begin{matrix}\frac{\text{αN}}{N+b},\ \ \&m=0\\ 0,\ \ \&m>0\\ \end{matrix}\right.\ \) 0 \(\alpha\) 0 \(\left\{\begin{matrix}\infty,\ \ \&0\leq m<1\\ \frac{\text{αN}}{b},\ \ \&m=1\\ \end{matrix}\right.\ \) 0 αP \(\left[\left.\ 0,\ \infty\right)\right.\ \)
Type III \(\frac{\alpha\left(\frac{N}{P^{m}}\right)^{2}}{\left(\frac{N}{P^{m}}\right)^{2}+b}\) \(\left\{\begin{matrix}\frac{\alpha N^{2}}{N^{2}+b},\ \ \&m=0\\ 0,\ \ \&m>0\\ \end{matrix}\right.\ \) 0 \(\alpha\) 0 \(\left\{\begin{matrix}\infty,\ \ \&0\leq m<0.5\\ \frac{\alpha N^{2}}{b},\ \ \&m=0.5\\ 0,\ \ \&0.5<m\leq 1\\ \end{matrix}\right.\ \) 0 αP \(\left[\left.\ 0,\ \infty\right)\right.\ \)
Kovai
Kovai
\[\alpha\left(1-e^{\frac{-Nu}{\left(\text{αP}\right)}}\right);\] \[u=1-\left(\frac{b}{N}\right)\left(1-e^{\frac{-N}{b}}\right)\]
0
0
\[\alpha\]
0
\[Nu\in\left[0,N\right]\]
0
αP \[\left[\left.\ 0,1\right)\right.\ \]