A discrete host-parasitoid model with development of pesticide resistance and IPM strategies

Figures & data

Figure 1. The threshold values ${\overline{R}}_0(k,q)$ and numerical simulations of model (4(4) $\begin{array}{r}\left\{\begin{array}{c}\left.\begin{array}{l}{P}_{t+1}=\frac{a{P}_t}{1+b{P}_t}{e}^{-\alpha N_t},\\ N_{t+1}=\beta P_t\left(1-e^{-\alpha N_t}\right)+d{N}_t,\end{array}\right\}t=1,2,\dots ,\\ \left.\begin{array}{l}{\omega }_{qk}=\frac{(1-d_1){\omega }_{q(k-1)}}{1-d_1{\omega }_{q(k-1)}},\\ P_{q{k}^{+}}=(1-d_1{\omega }_{qk})P_{qk},\\ N_{q{k}^{+}}=N_{qk}+{\delta }_k,\end{array}\right\}k=1,2,\dots ,\end{array}\right.\end{array}$(4) ) with constant pulse releasing of natural enemies. The baseline parameter values are as follows: ${\omega }_0=0.99,R_C=0.95,d_1=0.8,q=3,a=1.6,b=0.001,\alpha =2,\beta =0.1,d=0.6$ and ${\delta }_c=0$ (a) The plot of ${\overline{R}}_0(k,q)$ with respect to k and $\delta =0.2$; (b) The time series of the pest population associated with (a); (c) The plot of ${\overline{R}}_0(k,q)$ with respect to k and the releasing constant δ determined by formula (17(17) $\delta ={\delta }_k=\left\{\begin{array}{ll}{\delta }_c,& \text{if}{R}_1(k,q)\le R_C,\\ -\frac{1-d}{\alpha \left(1-d^{(k+1)q}\right)}\ln \left(\frac{R_C}{R_1(k,q)}\right),& \text{if}{R}_1(k,q)>R_C,\end{array}\right.$(17) ); (d) The time series of the pest population associated with (c).

Figure 2. Bifurcation diagrams for model (4(4) $\begin{array}{r}\left\{\begin{array}{c}\left.\begin{array}{l}{P}_{t+1}=\frac{a{P}_t}{1+b{P}_t}{e}^{-\alpha N_t},\\ N_{t+1}=\beta P_t\left(1-e^{-\alpha N_t}\right)+d{N}_t,\end{array}\right\}t=1,2,\dots ,\\ \left.\begin{array}{l}{\omega }_{qk}=\frac{(1-d_1){\omega }_{q(k-1)}}{1-d_1{\omega }_{q(k-1)}},\\ P_{q{k}^{+}}=(1-d_1{\omega }_{qk})P_{qk},\\ N_{q{k}^{+}}=N_{qk}+{\delta }_k,\end{array}\right\}k=1,2,\dots ,\end{array}\right.\end{array}$(4) ) with different bifurcation parameters $a,d,alpha$, and δ. The baseline parameter values are as follows: $d_1=0.8,q=3,a=3.8,b=0.1,delta=0.1,\beta =0.4,d=0.1,\alpha =2$. (a) Bifurcation diagram for the density of the pest population with bifurcation parameter a; (b) Bifurcation diagram for the density of the pest population with bifurcation parameter d; (c) Bifurcation diagram for the density of the pest population with bifurcation parameter α; (d) Bifurcation diagram for the density of the pest population with bifurcation parameter δ.