Figures & data: Modelling releases of sterile mosquitoes with different strategies

2,795

Views

CrossRef citations to date

Altmetric

Figures & data

Figure 1. With parameters $a = 20$ and $k = 0.3$ as given in Example 3.2, four different values $b = 1.3, 2.5, 5, 7$ , are used. With the same initial value $w = 1$ , chaotic behaviour is exhibited in the upper left figure, and then stable 4-cycle, 2-cycle, and 1-cycle are presented in the upper right figure, the lower left and right figures, respectively.

Figure 2. This is a schematic diagram to show the existence of positive fixed points. The figures on the left and right correspond to $k \geq 1$ and $k < 1$ , respectively. The intersection between the curve of $((1 + w)^{2} / w) + b$ and the curve of $a (1 + w) e^{- k w}$ gives a positive fixed point of Equation (Equation19(19) $w_{n + 1} = \frac{a w_{n}}{1 + w_{n} + B (w_{n})} w_{n} e^{- k w_{n}} = \frac{a w_{n} (1 + w_{n})}{(1 + w_{n})^{2} + b w_{n}} w_{n} e^{- k w_{n}} .$ (19) ). Keep the curve of $a (1 + w) e^{- k w}$ fixed. As parameter b increases gradually from zero to exceeding $b_{c}$ , the curve of $((1 + w)^{2} / w) + b$ moves up gradually. Accordingly, there exist two, one, or no intersections, and hence Equation (Equation19(19) $w_{n + 1} = \frac{a w_{n}}{1 + w_{n} + B (w_{n})} w_{n} e^{- k w_{n}} = \frac{a w_{n} (1 + w_{n})}{(1 + w_{n})^{2} + b w_{n}} w_{n} e^{- k w_{n}} .$ (19) ) has two, one, or no positive fixed points.

Figure 2. This is a schematic diagram to show the existence of positive fixed points. The figures on the left and right correspond to k≥1 and k<1, respectively. The intersection between the curve of ((1+w)2/w)+b and the curve of a(1+w)e−kw gives a positive fixed point of Equation (Equation19(19) wn+1=awn1+wn+B(wn)wne−kwn=awn(1+wn)(1+wn)2+bwnwne−kwn.(19) ). Keep the curve of a(1+w)e−kw fixed. As parameter b increases gradually from zero to exceeding bc, the curve of ((1+w)2/w)+b moves up gradually. Accordingly, there exist two, one, or no intersections, and hence Equation (Equation19(19) wn+1=awn1+wn+B(wn)wne−kwn=awn(1+wn)(1+wn)2+bwnwne−kwn.(19) ) has two, one, or no positive fixed points.

Figure 3. The model equation for the upper figure (a) is (Equation8(8) $w_{n + 1} = \frac{a w_{n}}{w_{n} + b} w_{n} e^{- k w_{n}} .$ (8) ) with the constant releases. The lower two figures (b) on the left and (c) on the right are based on equation (Equation14(14) $w_{n + 1} = \frac{a w_{n}}{1 + (1 + b) w_{n}} w_{n} e^{- k w_{n}} .$ (14) ) where the number of releases is proportional to the wild mosquito population size, and Equation (Equation19(19) $w_{n + 1} = \frac{a w_{n}}{1 + w_{n} + B (w_{n})} w_{n} e^{- k w_{n}} = \frac{a w_{n} (1 + w_{n})}{(1 + w_{n})^{2} + b w_{n}} w_{n} e^{- k w_{n}} .$ (19) ) where the number of releases is proportional to the wild mosquito population size plus the saturation, respectively. We fix the same parameters $a = 20$ and $k = 0.3$ , and use b as the bifurcation parameter. For $b_{s} < b < b_{c}$ , there exists a unique positive fixed point. As b decreases, backward period-doubling bifurcations occur for all of the model equations. Notice, in all of the figures, $b_{s}$ corresponds to the b point where there exists two fixed points on the left but only one on the right. The threshold value $b_{c}$ corresponds to the b value where there is a positive fixed on the left and no positive fixed point on the right.

Figure 3. The model equation for the upper figure (a) is (Equation8(8) wn+1=awnwn+bwne−kwn.(8) ) with the constant releases. The lower two figures (b) on the left and (c) on the right are based on equation (Equation14(14) wn+1=awn1+(1+b)wnwne−kwn.(14) ) where the number of releases is proportional to the wild mosquito population size, and Equation (Equation19(19) wn+1=awn1+wn+B(wn)wne−kwn=awn(1+wn)(1+wn)2+bwnwne−kwn.(19) ) where the number of releases is proportional to the wild mosquito population size plus the saturation, respectively. We fix the same parameters a=20 and k=0.3, and use b as the bifurcation parameter. For bs<b<bc, there exists a unique positive fixed point. As b decreases, backward period-doubling bifurcations occur for all of the model equations. Notice, in all of the figures, bs corresponds to the b point where there exists two fixed points on the left but only one on the right. The threshold value bc corresponds to the b value where there is a positive fixed on the left and no positive fixed point on the right.

Table 1. Summary table for the threshold release values.

Display Table

Modelling releases of sterile mosquitoes with different strategies

Table 1. Summary table for the threshold release values.

Information for

Open access

Opportunities

Help and information

Your download is now in progress and you may close this window

Login or register to access this feature

Modelling releases of sterile mosquitoes with different strategies

Figures & data

Table 1. Summary table for the threshold release values.

Related research

To cite this article:

Download citation

Information for

Open access

Opportunities

Help and information

Keep up to date