How do productivity gradient and diffusion shape patterns in a plant–herbivore grazing system?

Natural systems show heterogeneous patchy distributions of vegetation over large landscapes. Reaction–diffusion systems can demonstrate such heterogeneity of species distributions. Here, we analyse a reaction–diffusion model of plant–herbivore interactions in two-dimensional space to illustrate non-homogeneous distributions of plants and herbivores. The non-spatial system shows bottom-up control, where herbivore density is low under low and high primary productivity but increased at intermediate productivity. In addition, the non-spatial system provides bistability between a dense vegetation state devoid of herbivores and a coexisting state of plants and herbivores. In the spatiotemporal model, we give analytical conditions of occurring diffusion-driven (Turing) instability, where a novel point in our model is the relative dispersal of herbivores, which represents the movement of herbivores from a higher to a lower vegetation state in addition to the self-diffusion of both species. It is shown that heterogeneity in the population distribution does not occur if the relative dispersal of herbivores is low, but it appears in the opposite case. Due to bistability in the underlying non-spatial system, the spatiotemporal model produces initial value-dependent patterns. The two initial values make different patterns despite having the same primary productivity and relative dispersal rate. As productivity increases with a given relative herbivore dispersal, pattern transition occurs from a blend of stripes and spots of low vegetation state to a predominantly low-density vegetation state with smaller patches of densely vegetated states with one initial value. On the contrary, a discernible change in vegetation patterns from cold spots in the dense vegetation to hot stripes in the primarily low-vegetated state is noticed under the other initial population value. Furthermore, the population distributions of plants and herbivores in the entire domain after a long period are heterogeneous for both initial values, provided the relative herbivore dispersal is substantial. We estimated mean population densities to observe species fitness in the whole domain under variable productivity. When productivity is high, the mean population density of plants may go up or down, depending on the herbivore’s relative dispersal rate. In contrast to the bottom-up control dynamics of the non-spatial system, the system exhibits a top-down control under high relative dispersal, where the herbivore regulates vegetation growth under high productivity. On the other hand, herbivores are extinct under high productivity if the relative dispersal is low.