IPM strategies to a discrete switching predator-prey model induced by a mate-finding Allee effect

Figures & data

Figure 1. EIL: the lowest population density that will cause economic damage. ET: population density at which control measures should be invoked to prevent an increasing pest population from reaching EIL.

Figure 2. The two real branches $W(0,z)$ and $W(-1,z)$ of Lambert W function.

Figure 3. Bifurcation diagram for the existence of regular equilibria of system (5(5) $\dot{Z}(t)=\{\begin{array}{ll}{F}_{S_1}(Z),& Z\in S_1,\\ F_{S_2}(Z),& Z\in S_2,\end{array}$(5) ) with respect to r and ET, parameters are $a=1.5,\theta =7.5,q=0.25$.

Figure 4. Bifurcation diagram for system (5(5) $\dot{Z}(t)=\{\begin{array}{ll}{F}_{S_1}(Z),& Z\in S_1,\\ F_{S_2}(Z),& Z\in S_2,\end{array}$(5) ) with respect to r. All other parameters as follows: $a=2,\theta =4,q=0.05,ET=0.45$ and $(H_0,P_0)=(0.5,0.4)$.

Figure 5. Phase-plan of system (5(5) $\dot{Z}(t)=\{\begin{array}{ll}{F}_{S_1}(Z),& Z\in S_1,\\ F_{S_2}(Z),& Z\in S_2,\end{array}$(5) ) with different r. [A] r = 2.18; [B] r = 2.213; [C] r = 2.4; [D] r = 2.65. The other parameters are identical to those in Figure 4.

Figure 6. Bifurcation diagram for system (5(5) $\dot{Z}(t)=\{\begin{array}{ll}{F}_{S_1}(Z),& Z\in S_1,\\ F_{S_2}(Z),& Z\in S_2,\end{array}$(5) ) with respect to q. All other parameters as follows: $a=1.68,ET=0.72,r=2.58,(H_0,P_0)=(0.1,0.1)$, and [A] $\theta =9.5$; [B] $\theta =5$; [C] $\theta =1$.

Figure 7. Bifurcation diagram for system (5(5) $\dot{Z}(t)=\{\begin{array}{ll}{F}_{S_1}(Z),& Z\in S_1,\\ F_{S_2}(Z),& Z\in S_2,\end{array}$(5) ) with respect to θ. All other parameters as follows: $a=2,q=0.8,ET=0.8,r=2.29,(H_0,P_0)=(0.3,0.2)$, and [A] r = 2.13; [B] r = 2.

Figure 8. Switching effect of system (5(5) $\dot{Z}(t)=\{\begin{array}{ll}{F}_{S_1}(Z),& Z\in S_1,\\ F_{S_2}(Z),& Z\in S_2,\end{array}$(5) ) under different initial densities. Parameters are $a=2,r=1.9,\theta =2,q=0.1,ET=0.65$.

Figure 9. Pest outbreak frequency depends on initial density $(H_0,P_0)$ of system (5(5) $\dot{Z}(t)=\{\begin{array}{ll}{F}_{S_1}(Z),& Z\in S_1,\\ F_{S_2}(Z),& Z\in S_2,\end{array}$(5) ). The parameters are fixed as $a=1.4,\theta =6,q=0.05,ET=0.65,r=2.2$.

Figure 10. The coexisting attractors of system (5(5) $\dot{Z}(t)=\{\begin{array}{ll}{F}_{S_1}(Z),& Z\in S_1,\\ F_{S_2}(Z),& Z\in S_2,\end{array}$(5) ) with different initial values. Parameters are $a=2,\theta =4,q=0.05,ET=0.45,r=2.3$, and [A] $(H_0,P_0)=(0.6,0.4)$; [B] $(H_0,P_0)=(0.1,0.1)$.

Figure 11. Basin of attraction of two attractors shown in Fig. 10 with $H\in [0.3,0.67]$ and $P\in [0.2,0.8]$. The white and black points are attracted to the attractors shown in Figure 10 from left to right.

Figure 12. Attractors' switch-like behavior of system (5(5) $\dot{Z}(t)=\{\begin{array}{ll}{F}_{S_1}(Z),& Z\in S_1,\\ F_{S_2}(Z),& Z\in S_2,\end{array}$(5) ) with $r_t=r+\sigma u$ has random perturbation as each 90 generations. Parameters are: $a=2,\theta =2,q=0.3,ET=0.5,r=2.5,\sigma =1$ and $(H_0,P_0)=(0.5,0.1)$.

Figure 13. Attractors' switch-like behavior of system (5(5) $\dot{Z}(t)=\{\begin{array}{ll}{F}_{S_1}(Z),& Z\in S_1,\\ F_{S_2}(Z),& Z\in S_2,\end{array}$(5) ) with $q_t=q+\eta u$ which random perturbation every 90 generations. Parameters are: $a=0.8,\theta =5,q=0.5,ET=0.4,r=1,\eta =0.3$ and $(H_0,P_0)=(0.5,0.4)$.

Figure 14. Switching frequency (S-F) and switching time (S-T) of system (5(5) $\dot{Z}(t)=\{\begin{array}{ll}{F}_{S_1}(Z),& Z\in S_1,\\ F_{S_2}(Z),& Z\in S_2,\end{array}$(5) ). Parameters are $a=2,\theta =2,q=0.01,ET=0.35,r=2.1$. The initial densities from top to bottom are $(0.3,0.4),(0.2,0.6)$ and $(0.7,0.6)$.