A mathematical model of facultative mutualism with populations interacting in a food chain

Complex behaviour in four species food-web model

“…Proceeding along the lines of Kumar and Freedman [19], it is concluded that no γ i lies in (X). Thus, (X) lies in the positive octant.…”

“…Observe that for w 2 ∈ (3.5, 5.5) in data set (19), the singularity E 2 = (0.206186, 1.02296, 0, 0) is an attracting focus (n 1 < 0) in the xy-plane and has an unstable manifold along the z-axis (n 3 > 0). The singularity E 4 = (0.57193, 0, 0, 1.6522) is attracting focus (n 1 < 0) in the xz-plane having an unstable manifold (n 2 > 0) in y-direction for w 2 ∈ (4.105, 5.5).…”

“…Now, due to the first condition of Theorem 5.5, the three-dimensional subsystem X η (x, y, 0, z) is uniformly persistent. The attractor for w 2 = 5.0 for data set (19) in xyz-phase space is shown in Figure 5.…”

“…Identical attractor Figure 4. Maximal Liapunov exponent versus w 2 for data set (19). is obtained in the xwz-phase space.…”

“…It should be noted that the data in Equation (19) is symmetric (in parameters) about the middle predators (see data sets (17) and (18)). The data set (19) is now made asymmetric with slightly different death rates w 5 and w 13 as follows: For the data set (20), the periodic orbit in the xwz-phase space and time series of predator y are shown in Figure 6. The coexistence of three species (x, w, z) and the extinction of predator y with time are evident.…”