Dynamics of a stochastic modified Leslie–Gower predator–prey system with hunting cooperation

Figures & data

Figure 1. Solutions of system (4(4) $\{\begin{array}{l}{\displaystyle \mathrm{d}x\left(t\right)=x\left(t\right)[a-\mathrm{bx}(t)-\frac{(\lambda +\mathrm{\alpha y}(t\left)\right)y\left(t\right)}{1+h(\lambda +\mathrm{\alpha y}(t\left)\right)x\left(t\right)}]\mathrm{d}t+{\sigma }_{1}x\left(t\right)\mathrm{d}{B}_1\left(t\right),}\\ {\displaystyle \mathrm{d}y\left(t\right)=y\left(t\right)(r-\frac{\mathrm{fy}\left(t\right)}{m+x\left(t\right)})\mathrm{d}t+{\sigma }_{2}y\left(t\right)\mathrm{d}{B}_2\left(t\right).}\end{array}$(4) ) with $a=0.8,b=0.3,\lambda =0.5,\alpha =0.25,h=1,r=0.5,f=0.8,m=1.$ Green lines: ${\sigma }_1={\sigma }_2=0$. Red lines: ${\sigma }_1={\sigma }_2=0.15$. Blue lines: ${\sigma }_1={\sigma }_2=0.3$.

Figure 2. Solutions of system (4(4) $\{\begin{array}{l}{\displaystyle \mathrm{d}x\left(t\right)=x\left(t\right)[a-\mathrm{bx}(t)-\frac{(\lambda +\mathrm{\alpha y}(t\left)\right)y\left(t\right)}{1+h(\lambda +\mathrm{\alpha y}(t\left)\right)x\left(t\right)}]\mathrm{d}t+{\sigma }_{1}x\left(t\right)\mathrm{d}{B}_1\left(t\right),}\\ {\displaystyle \mathrm{d}y\left(t\right)=y\left(t\right)(r-\frac{\mathrm{fy}\left(t\right)}{m+x\left(t\right)})\mathrm{d}t+{\sigma }_{2}y\left(t\right)\mathrm{d}{B}_2\left(t\right).}\end{array}$(4) ) with $a=1,b=0.6,\lambda =0.8,\alpha =0.25,h=1,r=1.5,f=1,m=\mathrm{0.2.}$ Green lines: ${\sigma }_1={\sigma }_2=0$. Red lines: ${\sigma }_1={\sigma }_2=0.2$. Blue lines: ${\sigma }_1={\sigma }_2=0.5$.

Figure 3. Solutions of system (4(4) $\{\begin{array}{l}{\displaystyle \mathrm{d}x\left(t\right)=x\left(t\right)[a-\mathrm{bx}(t)-\frac{(\lambda +\mathrm{\alpha y}(t\left)\right)y\left(t\right)}{1+h(\lambda +\mathrm{\alpha y}(t\left)\right)x\left(t\right)}]\mathrm{d}t+{\sigma }_{1}x\left(t\right)\mathrm{d}{B}_1\left(t\right),}\\ {\displaystyle \mathrm{d}y\left(t\right)=y\left(t\right)(r-\frac{\mathrm{fy}\left(t\right)}{m+x\left(t\right)})\mathrm{d}t+{\sigma }_{2}y\left(t\right)\mathrm{d}{B}_2\left(t\right).}\end{array}$(4) ) with $a=1,b=0.6,\lambda =0.8,\alpha =0.25,h=1,r=1.5,f=1,m=\mathrm{0.2.}$ Red lines: ${\sigma }_1=0.2,{\sigma }_2=1.8$. Blue lines: ${\sigma }_1={\sigma }_2=0$.

Figure 4. Solutions of system (4(4) $\{\begin{array}{l}{\displaystyle \mathrm{d}x\left(t\right)=x\left(t\right)[a-\mathrm{bx}(t)-\frac{(\lambda +\mathrm{\alpha y}(t\left)\right)y\left(t\right)}{1+h(\lambda +\mathrm{\alpha y}(t\left)\right)x\left(t\right)}]\mathrm{d}t+{\sigma }_{1}x\left(t\right)\mathrm{d}{B}_1\left(t\right),}\\ {\displaystyle \mathrm{d}y\left(t\right)=y\left(t\right)(r-\frac{\mathrm{fy}\left(t\right)}{m+x\left(t\right)})\mathrm{d}t+{\sigma }_{2}y\left(t\right)\mathrm{d}{B}_2\left(t\right).}\end{array}$(4) ) with $a=1,b=0.6,\lambda =0.8,\alpha =0.25,h=1,r=1.5,f=1,m=\mathrm{0.2.}$ Red lines: ${\sigma }_1=1.5,{\sigma }_2=0.2$. Blue lines: ${\sigma }_1={\sigma }_2=0$.

Figure 5. Solutions of system (4(4) $\{\begin{array}{l}{\displaystyle \mathrm{d}x\left(t\right)=x\left(t\right)[a-\mathrm{bx}(t)-\frac{(\lambda +\mathrm{\alpha y}(t\left)\right)y\left(t\right)}{1+h(\lambda +\mathrm{\alpha y}(t\left)\right)x\left(t\right)}]\mathrm{d}t+{\sigma }_{1}x\left(t\right)\mathrm{d}{B}_1\left(t\right),}\\ {\displaystyle \mathrm{d}y\left(t\right)=y\left(t\right)(r-\frac{\mathrm{fy}\left(t\right)}{m+x\left(t\right)})\mathrm{d}t+{\sigma }_{2}y\left(t\right)\mathrm{d}{B}_2\left(t\right).}\end{array}$(4) ) with $a=1,b=0.6,\lambda =0.8,\alpha =0.25,h=1,r=1.5,f=1,m=\mathrm{0.2.}$ Red lines: ${\sigma }_1=1.5,{\sigma }_2=1.8$. Blue lines: ${\sigma }_1={\sigma }_2=0$.