Fig. 4: Spatio-temporal plots by changing μ.

The diffusion-induced instability region around the interior equilibrium point Q_*, satisfying the analytical conditions (12a) and (12b), is shown in the κ-μ parameter space (a) for the system (6). This region highlights the emergence of instability driven by diffusion mechanisms near the interior fixed point Q_*. To investigate the corresponding spatio-temporal dynamics, a two-dimensional spatial domain of size 100 × 100 is considered with spatial steps Δx = Δy = 0.33, where the dynamics are governed by the spatial abundance of the undefended prey u. Fixing the inverse of the carrying capacity, controlling the strength of density-dependent limitations, κ = 0.15, five increasing values of the mortality rate due to the intra-species conflict rate among predators, i.e., the strength of density-dependent predator competition μ = 0.6, 0.72, 0.8, 0.89, and 0.95 are selected, each marked by black dots in (a). These values correspond to a range of distinct spatio-temporal patterns observed in the system: b a spot pattern at μ = 0.6, c a mixture of spots and stripes at μ = 0.72, d a stripe pattern at μ = 0.8, e a mixture of stripes and holes at μ = 0.89, and f a hole pattern at μ = 0.95. All pattern snapshots are taken at 5 × 10⁵ time iterations with Δt = 0.025.