Role of herbivory in shaping the dryland vegetation ecosystem: Linking spiral vegetation patterns and nonlinear, nonlocal grazing

Self-organized vegetation patterns are an amazing aspect of dryland ecosystems; in addition to being visually appealing, patterns control how these water-deprived systems react to escalating environmental stress. Although there is a wide variety of vegetation patterns, little is known about the mechanisms behind spiral patterns. The well-known models that explain other vegetation patterns such stripes, rings, and fairy circles cannot account for these spirals. Here we have adopted a modeling approach in which the interplay between herbivore grazing and vegetation is found to be the reason why spirals form. To comprehend the nonlinear dependence of grazing on the availability vegetation, we have introduced a grazing term that gets saturated when forage is abundant. To account for the impact of the spatial nonhomogeneity in vegetation layout, it is thought that grazing is dependent on mean vegetation density rather than density at a single site. Results show how the system dynamics is changed fundamentally depending on the different types of grazing response. Incorporation of nonlocality into the herbivore grazing leads to spiral-shaped vegetation patterns only in natural grazing scenarios; however, no patterning is seen in human controlled herbivory. Overall, our research points to the nonlocal, nonlinear grazing behavior of herbivores as one of the major driving forces for the development of spiral patterns.

Figure 1

Phase portrait of system (3) with Type I grazing at a particular grid position [we considered the (0,0) spatial point in computational domain $\mathrm{\Omega }=[-40,40]\times [-40,40](\approx 1m^2)], D=3.2$. Each trajectory (cyan line) starts at a point (black square) near HSS, (a) $E_0(0,0)$, (b) $E_1(0.9075,0)$, (c) $E_2(11.01925,0)$, (d) $E_3(0.8,0.99439)$, and ends at the point marked by a red asterisk. The time-series evaluations are shown in the inset figures, the blue (dotted) and red lines represent vegetation biomass and grazing field respectively.

Figure 2

(a) Phase portrait of system (3) with Type III grazing (with $D=3.2$) at a particular grid position. The trajectory (cyan line) starts at a point (black square) near the HSS $E_3$ and ends exactly at $E_3(0.666667,0.773942)$, marked by a red asterisk. (b) Corresponding time-series evaluations; the blue (dotted) and red lines represent vegetation biomass and grazing field respectively.

Figure 3

Snapshots of vegetation biomass $n$ taken at different instances as the numerical simulation of model (4) with Type III grazing goes on (time is increasing from left to right, row-wise: $t=50,100,150,200,250,300,350,400,450,500$). The horizontal and vertical directions of each plot are the $x_1$ and $x_2$ axes respectively; here the computational domain is $\mathrm{\Omega }=[-40,40]\times [-40,40](\approx 1m^2)$ and $D=3.2$. A random and inhomogeneous perturbation [in the range (0,0.05)] around HSS $E_1(0.9075,0)$, is taken as initial condition for the simulation.

Figure 4

Same setting as Fig. 3. Here, a random and inhomogeneous perturbation [in the range (0,0.05)] around HSS $E_3(0.666667,0.773942)$ is taken as initial condition for the simulation.

Figure 5

Snapshots of vegetation biomass $n$ taken at different instances ($t=50,150,250,350,450$) as the numerical simulation of model (4) with Type III grazing goes on. Here the computational domain is $\mathrm{\Omega }=[-160,160]\times [-160,160](\approx 16m^2)$ and $D=3.2$. A random and inhomogeneous perturbation [in the range (0,0.05)] around HSS $E_3(0.666667,0.773942)$ is taken as initial condition for the simulation.

Figure 6

Phase portrait and time series (inset figures) of system (4) having Type III grazing (with $D=3.2$) at a particular grid position, corresponding to the simulations with initial conditions that are random perturbations around the HSS: (a) $E_0(0,0)$, (b) $E_1(0.09075,0)$, (c) $E_2(11.01925,0)$, (d) $E_3(0.666667,0.773942)$. Here (b) and (d) correspond to Figs. 3 and 4 respectively.

Figure 7

Phase portrait and time series (inset figures) of system (4) having Type III grazing (with $D=4.8$) at a particular grid position, corresponding to the simulations with initial conditions that are random perturbation around the HSS: (a) $E_0(0,0)$, (b) $E_1(0.09075,0)$, (c) $E_2(11.01925,0)$, (d) $E_3(0.840168,0.704451)$

Figure 8

Same setting as Fig. 3. Here, a random and inhomogeneous perturbation [in the range (0,0.05)] around HSS $E_1(0.09075,0)$ is taken as initial condition for the simulation and $D=4.8$.

Figure 9

Same setting as Fig. 3. Here, a random and inhomogeneous perturbation [in the range (0,0.05)] around HSS $E_3(0.840168,0.704451)$ is taken as initial condition for the simulation and $D=4.8$.

Figure 10

Same setting as Fig. 5. Here, a random and inhomogeneous perturbation around HSS $E_3(0.840168,0.704451)$ is taken as initial condition for the simulation and $D=4.8$.

Figure 11

Poincaré maximum return maps for simulations having the same setting as (a) Fig. 6 [i.e., model (4) with Type III and $D=3.2]$ and (b) Fig. 7 [i.e., model (4) with Type III and $D=4.8]$.