Table 1.
Parameter values for both dimensional and dimensionless models.
Fig 1.
Bifurcation diagram for the temporal model.
(a) Three-parametric bifurcation diagram for the non-trivial equilibrium point for the temporal model, and (b) the possible equilibrium points corresponding to the water density (u) for the bifurcation parameters α and γ with a fixed value μ = 0.8. The blue and magenta colour surfaces (curves) represent the transcritical and Hopf bifurcations, respectively. The grey and cyan colour surfaces represent the stable and unstable equilibrium points. The fixed parameter values for both figures are β = 0.45, h = 1, and η = 0.05.
Fig 2.
Patterns for the plants in the absence of downhill water flow for various combinations of diffusion rates for water and herbivores.
The fixed parameter values are: α = 2.8125, β = 0.45, γ = 0.1, h = 1, μ = 0.8, and η = 0.05.
Fig 3.
Stationary and non-stationary solutions for the plant.
Parameter values: α = 2.8125, β = 0.45, γ = 0.1, h = 1, μ = 0.8, η = 0.05, ay = 0, du = 500, dw = 100; (a) ax = 0, (b) ax = 100, (c) ax = 150, and (d) ax = 182.5.
Fig 4.
Stationary and non-stationary solutions for the plant for different groundwater flow values and directions.
Parameter values: α = 2.8125, β = 0.45, γ = 0.1, h = 1, μ = 0.8, η = 0.05, du = 10, dw = 20; (a) ax = ay = 0, (b) ax = 10, ay = 0, (c) ax = 0, ay = 10, (d) ax = 10, ay = 10, (e) ax = 30, ay = 10, and (f) ax = 10, ay = 30.
Fig 5.
Non-homogeneous non-stationary solutions for the plant.
Parameter values: α = 2.8125, β = 0.45, γ = 0.11, h = 1, μ = 0.8, η = 0.05, ay = 0, du = 2.5, dw = 1.5; (a) ax = 0, (b) the spatial average of (a) in time, and (c) ax = 182.5.
Fig 6.
Non-homogeneous non-stationary solutions corresponding to the second equation of (3).
Parameter values: α = 2.8125, β = 0.45, h = 1, μ = 0.8, η = 0.05, ax = 0, ay = 0, du = 500, dw = 100; (a) γ = 0.1 and (b) γ = 0.11 with (a) as the initial conditions.