Skip to main content

Theory and Modern Applications

Table 1 The positive equilibrium and the threshold of model (3.2) and (3.3)

From: A pest control model with birth pulse and residual and delay effects of pesticides

Birth function

Equilibrium

\(R_{0}\)

Ricker

\(x^{*}=\frac{ (1-e^{-q\,dT-\frac{q}{g}(1-e^{-gT})c_{m}-qH} )\ln R^{b}_{0}}{(1-e^{-q(d+\delta )T-\frac{q}{g}(1-e^{-gT})c_{m}-qH}) e^{-q\,dT-\frac{q}{g}(1-e^{-gT})c_{m}-qH}}\)

\(R^{b}_{0}\triangleq \frac{b(1-e^{-q\,dT})e^{-q\,dT-\frac{q}{g}(1-e^{-gT})c_{m}-qH}}{1-e^{-q(d+\delta )T-\frac{q}{g}(1-e^{-gT})c_{m}-qH}}\)

\({}\cdot \frac{1}{1-e^{-q\,dT-\frac{q}{g}(1-e^{-gT})c_{m}-qH}}\)

\(y^{*}=\frac{1-e^{-q\delta T}}{1-e^{-q(d+\delta )T-\frac{q}{g}(1-e^{-gT})c_{m}-qH}}\ln R^{b}_{0}\)

Beverton–Holt

\(x^{*}=\frac{ (1-e^{-q\,dT-\frac{q}{g}(1-e^{-gT})c_{m}-qH} )\sqrt[n]{\beta (R^{p}_{0}-1)}}{(1-e^{-q(d+\delta )T-\frac{q}{g}(1-e^{-gT})c_{m}-qH}) e^{-q\,dT-\frac{q}{g}(1-e^{-gT})c_{m}-qH}}\)

\(R^{p}_{0}\triangleq \frac{p(1-e^{-q\,dT})e^{-q\,dT-\frac{q}{g}(1-e^{-gT})c_{m}-qH}}{\beta (1-e^{-q(d+\delta )T-\frac{q}{g}(1-e^{-gT})c_{m}-qH})}\)

\({}\cdot \frac{1}{1-e^{-q\,dT-\frac{q}{g}(1-e^{-gT})c_{m}-qH}}\)

\(y^{*}=\frac{1-e^{-q\delta T}}{1-e^{-q(d+\delta )T-\frac{q}{g}(1-e^{-gT})c_{m}-qH}}\sqrt[n]{\beta (R^{p}_{0}-1)}\)