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TableĀ 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 \(x^{*}=\frac{ (1-e^{-q\,dT-\frac{q}{g}(1-e^{-gT})c_{m}-qH} )\ln R^{b}_{0}}{(1-e^{-q(d+\delta )T-\frac{q}{g}(1-e^{-gT})c_{m}-qH}) e^{-q\,dT-\frac{q}{g}(1-e^{-gT})c_{m}-qH}}\) \(R^{b}_{0}\triangleq \frac{b(1-e^{-q\,dT})e^{-q\,dT-\frac{q}{g}(1-e^{-gT})c_{m}-qH}}{1-e^{-q(d+\delta )T-\frac{q}{g}(1-e^{-gT})c_{m}-qH}}\)
\({}\cdot \frac{1}{1-e^{-q\,dT-\frac{q}{g}(1-e^{-gT})c_{m}-qH}}\)
\(y^{*}=\frac{1-e^{-q\delta T}}{1-e^{-q(d+\delta )T-\frac{q}{g}(1-e^{-gT})c_{m}-qH}}\ln R^{b}_{0}\)
Bevertonā€“Holt \(x^{*}=\frac{ (1-e^{-q\,dT-\frac{q}{g}(1-e^{-gT})c_{m}-qH} )\sqrt[n]{\beta (R^{p}_{0}-1)}}{(1-e^{-q(d+\delta )T-\frac{q}{g}(1-e^{-gT})c_{m}-qH}) e^{-q\,dT-\frac{q}{g}(1-e^{-gT})c_{m}-qH}}\) \(R^{p}_{0}\triangleq \frac{p(1-e^{-q\,dT})e^{-q\,dT-\frac{q}{g}(1-e^{-gT})c_{m}-qH}}{\beta (1-e^{-q(d+\delta )T-\frac{q}{g}(1-e^{-gT})c_{m}-qH})}\)
\({}\cdot \frac{1}{1-e^{-q\,dT-\frac{q}{g}(1-e^{-gT})c_{m}-qH}}\)
\(y^{*}=\frac{1-e^{-q\delta T}}{1-e^{-q(d+\delta )T-\frac{q}{g}(1-e^{-gT})c_{m}-qH}}\sqrt[n]{\beta (R^{p}_{0}-1)}\)