Dynamic analysis of stochastic delay mutualistic system of leaf-cutter ants with stage structure and their fungus garden

Ruimin ZhangSchool of Science, Jimei University, Xiamen, People's Republic of ChinaView further author information

Xiaohui LiuSchool of Science, Jimei University, Xiamen, People's Republic of ChinaView further author information

Chunjin WeiSchool of Science, Jimei University, Xiamen, People's Republic of ChinaCorrespondence[email protected] [email protected] [email protected]
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Table 1. The meaning of some parameters.

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Figure 1. The evolution of $(x_{1} (t), x_{2} (t), y (t))$ for model (Equation4(4) ${\begin{aligned} d x_{1} (t) & = (r_{1} x_{2} (t) y (t) - d_{1} x_{1} (t) - c x_{1} (t)) d t + σ_{1} x_{1} (t) d B_{1} (t), \\ d x_{2} (t) & = (c x_{1} (t) - d_{2} x_{2} (t)) d t + σ_{2} x_{2} (t) d B_{2} (t), \\ d y (t) & = [\frac{r_{2} a x_{2} (t - τ) y (t - τ)}{m + m_{1} y (t - τ) + m_{2} x_{2} (t - τ)} - d_{3} y (t) - r_{3} x_{2} (t) y (t)] d t \\ + σ_{3} y (t) d B_{3} (t), \end{aligned}$ (4) ) and its corresponding deterministic model (Equation3(3) ${\begin{aligned} \frac{d x_{1} (t)}{d t} & = r_{1} x_{2} (t) y (t) - d_{1} x_{1} (t) - c x_{1} (t), \\ \frac{d x_{2} (t)}{d t} & = c x_{1} (t) - d_{2} x_{2} (t), \\ \frac{d y (t)}{d t} & = \frac{r_{2} a x_{2} (t - τ) y (t - τ)}{m + m_{1} y (t - τ) + m_{2} x_{2} (t - τ)} - d_{3} y (t) - r_{3} x_{2} (t) y (t), \end{aligned}$ (3) ) with initial value $(x_{1} (0), x_{2} (0), y (0)) = (2, 1, 3)$ : (a) time series diagram of $x_{1} (t), x_{2} (t), y (t)$ in case 1: $σ_{1} = 0.05, σ_{2} = 0.05, σ_{3} = 0.01$ and (b) phase diagram of $x_{1} (t), x_{2} (t), y (t)$ in case 1: $σ_{1} = 0.04, σ_{2} = 0.03, σ_{3} = 0.02$ .

Figure 1. The evolution of (x1(t),x2(t),y(t)) for model (Equation4(4) {dx1(t)=(r1x2(t)y(t)−d1x1(t)−cx1(t))dt+σ1x1(t)dB1(t),dx2(t)=(cx1(t)−d2x2(t))dt+σ2x2(t)dB2(t),dy(t)=[r2ax2(t−τ)y(t−τ)m+m1y(t−τ)+m2x2(t−τ)−d3y(t)−r3x2(t)y(t)]dt+σ3y(t)dB3(t),(4) ) and its corresponding deterministic model (Equation3(3) {dx1(t)dt=r1x2(t)y(t)−d1x1(t)−cx1(t),dx2(t)dt=cx1(t)−d2x2(t),dy(t)dt=r2ax2(t−τ)y(t−τ)m+m1y(t−τ)+m2x2(t−τ)−d3y(t)−r3x2(t)y(t),(3) ) with initial value (x1(0),x2(0),y(0))=(2,1,3): (a) time series diagram of x1(t),x2(t),y(t) in case 1: σ1=0.05,σ2=0.05,σ3=0.01 and (b) phase diagram of x1(t),x2(t),y(t) in case 1: σ1=0.04,σ2=0.03,σ3=0.02.

Figure 2. The evolution of $(x_{1} (t), x_{2} (t), y (t))$ for model (Equation4(4) ${\begin{aligned} d x_{1} (t) & = (r_{1} x_{2} (t) y (t) - d_{1} x_{1} (t) - c x_{1} (t)) d t + σ_{1} x_{1} (t) d B_{1} (t), \\ d x_{2} (t) & = (c x_{1} (t) - d_{2} x_{2} (t)) d t + σ_{2} x_{2} (t) d B_{2} (t), \\ d y (t) & = [\frac{r_{2} a x_{2} (t - τ) y (t - τ)}{m + m_{1} y (t - τ) + m_{2} x_{2} (t - τ)} - d_{3} y (t) - r_{3} x_{2} (t) y (t)] d t \\ + σ_{3} y (t) d B_{3} (t), \end{aligned}$ (4) ) and its corresponding deterministic model (Equation3(3) ${\begin{aligned} \frac{d x_{1} (t)}{d t} & = r_{1} x_{2} (t) y (t) - d_{1} x_{1} (t) - c x_{1} (t), \\ \frac{d x_{2} (t)}{d t} & = c x_{1} (t) - d_{2} x_{2} (t), \\ \frac{d y (t)}{d t} & = \frac{r_{2} a x_{2} (t - τ) y (t - τ)}{m + m_{1} y (t - τ) + m_{2} x_{2} (t - τ)} - d_{3} y (t) - r_{3} x_{2} (t) y (t), \end{aligned}$ (3) ) with initial value $(x_{1} (0), x_{2} (0), y (0)) = (2, 1, 3)$ : (a) time series diagram of $x_{1} (t), x_{2} (t), y (t)$ in case 2: $σ_{1} = 0.4, σ_{2} = 0.05, σ_{3} = 0.01$ ; (b) phase diagram of $x_{1} (t), x_{2} (t), y (t)$ in case 2: $σ_{1} = 0.4, σ_{2} = 0.05, σ_{3} = 0.01$ ; (c) time series diagram of $x_{1} (t), x_{2} (t), y (t)$ in case 3: $σ_{1} = 1.4, σ_{2} = 0.05, σ_{3} = 0.01$ ; (d) phase diagram of $x_{1} (t), x_{2} (t), y (t)$ in case 3: $σ_{1} = 1.4, σ_{2} = 0.05, σ_{3} = 0.01$ .

Figure 2. The evolution of (x1(t),x2(t),y(t)) for model (Equation4(4) {dx1(t)=(r1x2(t)y(t)−d1x1(t)−cx1(t))dt+σ1x1(t)dB1(t),dx2(t)=(cx1(t)−d2x2(t))dt+σ2x2(t)dB2(t),dy(t)=[r2ax2(t−τ)y(t−τ)m+m1y(t−τ)+m2x2(t−τ)−d3y(t)−r3x2(t)y(t)]dt+σ3y(t)dB3(t),(4) ) and its corresponding deterministic model (Equation3(3) {dx1(t)dt=r1x2(t)y(t)−d1x1(t)−cx1(t),dx2(t)dt=cx1(t)−d2x2(t),dy(t)dt=r2ax2(t−τ)y(t−τ)m+m1y(t−τ)+m2x2(t−τ)−d3y(t)−r3x2(t)y(t),(3) ) with initial value (x1(0),x2(0),y(0))=(2,1,3): (a) time series diagram of x1(t),x2(t),y(t) in case 2:σ1=0.4,σ2=0.05,σ3=0.01; (b) phase diagram of x1(t),x2(t),y(t) in case 2:σ1=0.4,σ2=0.05,σ3=0.01; (c) time series diagram of x1(t),x2(t),y(t) in case 3:σ1=1.4,σ2=0.05,σ3=0.01; (d) phase diagram of x1(t),x2(t),y(t) in case 3:σ1=1.4,σ2=0.05,σ3=0.01.

Figure 3. The evolution of $(x_{1} (t), x_{2} (t), y (t))$ for model (Equation4(4) ${\begin{aligned} d x_{1} (t) & = (r_{1} x_{2} (t) y (t) - d_{1} x_{1} (t) - c x_{1} (t)) d t + σ_{1} x_{1} (t) d B_{1} (t), \\ d x_{2} (t) & = (c x_{1} (t) - d_{2} x_{2} (t)) d t + σ_{2} x_{2} (t) d B_{2} (t), \\ d y (t) & = [\frac{r_{2} a x_{2} (t - τ) y (t - τ)}{m + m_{1} y (t - τ) + m_{2} x_{2} (t - τ)} - d_{3} y (t) - r_{3} x_{2} (t) y (t)] d t \\ + σ_{3} y (t) d B_{3} (t), \end{aligned}$ (4) ) and its corresponding deterministic model (Equation3(3) ${\begin{aligned} \frac{d x_{1} (t)}{d t} & = r_{1} x_{2} (t) y (t) - d_{1} x_{1} (t) - c x_{1} (t), \\ \frac{d x_{2} (t)}{d t} & = c x_{1} (t) - d_{2} x_{2} (t), \\ \frac{d y (t)}{d t} & = \frac{r_{2} a x_{2} (t - τ) y (t - τ)}{m + m_{1} y (t - τ) + m_{2} x_{2} (t - τ)} - d_{3} y (t) - r_{3} x_{2} (t) y (t), \end{aligned}$ (3) ) with initial value $(x_{1} (0), x_{2} (0), y (0)) = (2, 1, 3)$ : (a) time series diagram of $x_{1} (t), x_{2} (t), y (t)$ in case 4: $σ_{1} = 0.05, σ_{2} = 0.2, σ_{3} = 0.01$ ; (b) phase diagram of $x_{1} (t), x_{2} (t), y (t)$ in case 4: $σ_{1} = 0.05, σ_{2} = 0.2, σ_{3} = 0.01$ ; (c) time series diagram of $x_{1} (t), x_{2} (t), y (t)$ in case 5: $σ_{1} = 0.05, σ_{2} = 0.5, σ_{3} = 0.01$ and (d) phase diagram of $x_{1} (t), x_{2} (t), y (t)$ in case 5: $σ_{1} = 0.05, σ_{2} = 0.5, σ_{3} = 0.01$ .

Figure 3. The evolution of (x1(t),x2(t),y(t)) for model (Equation4(4) {dx1(t)=(r1x2(t)y(t)−d1x1(t)−cx1(t))dt+σ1x1(t)dB1(t),dx2(t)=(cx1(t)−d2x2(t))dt+σ2x2(t)dB2(t),dy(t)=[r2ax2(t−τ)y(t−τ)m+m1y(t−τ)+m2x2(t−τ)−d3y(t)−r3x2(t)y(t)]dt+σ3y(t)dB3(t),(4) ) and its corresponding deterministic model (Equation3(3) {dx1(t)dt=r1x2(t)y(t)−d1x1(t)−cx1(t),dx2(t)dt=cx1(t)−d2x2(t),dy(t)dt=r2ax2(t−τ)y(t−τ)m+m1y(t−τ)+m2x2(t−τ)−d3y(t)−r3x2(t)y(t),(3) ) with initial value (x1(0),x2(0),y(0))=(2,1,3): (a) time series diagram of x1(t),x2(t),y(t) in case 4: σ1=0.05,σ2=0.2,σ3=0.01; (b) phase diagram of x1(t),x2(t),y(t) in case 4:σ1=0.05,σ2=0.2,σ3=0.01; (c) time series diagram of x1(t),x2(t),y(t) in case 5: σ1=0.05,σ2=0.5,σ3=0.01 and (d) phase diagram of x1(t),x2(t),y(t) in case 5: σ1=0.05,σ2=0.5,σ3=0.01.

Figure 4. The evolution of $(x_{1} (t), x_{2} (t), y (t))$ for model (Equation4(4) ${\begin{aligned} d x_{1} (t) & = (r_{1} x_{2} (t) y (t) - d_{1} x_{1} (t) - c x_{1} (t)) d t + σ_{1} x_{1} (t) d B_{1} (t), \\ d x_{2} (t) & = (c x_{1} (t) - d_{2} x_{2} (t)) d t + σ_{2} x_{2} (t) d B_{2} (t), \\ d y (t) & = [\frac{r_{2} a x_{2} (t - τ) y (t - τ)}{m + m_{1} y (t - τ) + m_{2} x_{2} (t - τ)} - d_{3} y (t) - r_{3} x_{2} (t) y (t)] d t \\ + σ_{3} y (t) d B_{3} (t), \end{aligned}$ (4) ) and its corresponding deterministic model (Equation3(3) ${\begin{aligned} \frac{d x_{1} (t)}{d t} & = r_{1} x_{2} (t) y (t) - d_{1} x_{1} (t) - c x_{1} (t), \\ \frac{d x_{2} (t)}{d t} & = c x_{1} (t) - d_{2} x_{2} (t), \\ \frac{d y (t)}{d t} & = \frac{r_{2} a x_{2} (t - τ) y (t - τ)}{m + m_{1} y (t - τ) + m_{2} x_{2} (t - τ)} - d_{3} y (t) - r_{3} x_{2} (t) y (t), \end{aligned}$ (3) ) with initial value $(x_{1} (0), x_{2} (0), y (0)) = (2, 1, 3)$ : (a) time series diagram of $x_{1} (t), x_{2} (t), y (t)$ in case 6: $σ_{1} = 0.05, σ_{2} = 0.05, σ_{3} = 0.4$ ; (b) phase diagram of $x_{1} (t), x_{2} (t), y (t)$ in case 6: $σ_{1} = 0.05, σ_{2} = 0.05, σ_{3} = 0.4$ ; (c) time series diagram of $x_{1} (t), x_{2} (t), y (t)$ in case 7: $σ_{1} = 0.05, σ_{2} = 0.05, σ_{3} = 0.9$ ; (d) phase diagram of $x_{1} (t), x_{2} (t), y (t)$ in case 7: $σ_{1} = 0.05, σ_{2} = 0.05, σ_{3} = 0.9$ .

Dynamic analysis of stochastic delay mutualistic system of leaf-cutter ants with stage structure and their fungus garden

Table 1. The meaning of some parameters.

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Dynamic analysis of stochastic delay mutualistic system of leaf-cutter ants with stage structure and their fungus garden

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Table 1. The meaning of some parameters.

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