Figure 1

The dynamic behavior of system (1.2) with initial values of \(x(0)=1\), \(y(0)=1\), \(z(0)=1\), (a) time series of \(x(t)\), \(y(t)\), \(z(t)\), which are simulated 100 cycles in the interval \([0,200]\). (b) phase portrait of \(x(t)\), \(y(t)\), \(z(t)\), the solution goes gradually into the stable state from the initial point \([1,1,1]\). (c) dynamic behavior of system (1.2) with the mutual interference of the predator \(m=0.8\) and the prey refuge \(\theta=0.7\), (d) phase portrait of \(x(t)\), \(y(t)\), \(z(t)\), other parameters are the same as in Fig. 1(a)