Interactions of time delay and spatial diffusion induce the periodic oscillation of the vegetation system

Abstract

Empirical data exhibit a common phenomenon that vegetation biomass fluctuates periodically over time in ecosystem, but the corresponding internal driving mechanism is still unclear. Simultaneously, considering that the conversion of soil water absorbed by roots of the vegetation into vegetation biomass needs a period time, we thus introduce the conversion time into Klausmeier model, then a spatiotemporal vegetation model with time delay is established. Through theoretical analysis, we not only give the occurence conditions of stability switches for system without and with diffusion at the vegetation-existence equilibrium, but also derive the existence conditions of saddle-node-Hopf bifurcation of non-spatial system and Hopf bifurcation of spatial system at the coincidence equilibrium. Our results reveal that the conversion delay induces the interaction between the vegetation and soil water in the form of periodic oscillation when conversion delay increases to the critical value. By comparing the results of system without and with diffusion, we find that the critical value decreases with the increases of spatial diffusion factors, which is more conducive to emergence of periodic oscillation phenomenon, while spatial diffusion factors have no effects on the amplitude of periodic oscillation. These results provide a theoretical basis for understanding the spatiotemporal evolution behaviors of vegetation system.

Bifurcation analysis of an extended Klausmeier–Gray–Scott model with infiltration delay

“…Thus we do not give $$\tau _{10}<\tau _{20}$$ when $$\omega _{1}>\omega _{2}$$, which does not conform to the conventional result. Thus, such result shows that system () gives rise to theoretical illusion of stability switches at the coincidence steady state $$E_{*}$$ 23,24 …”

“…Theorems 3 and 4 indicate that the density of the vegetation presents a periodic oscillation pattern with time, which is consistent with periodic fluctuation of the vegetation observed in nature. 24 We further verify the conditions under which Hopf bifurcation occurs through Figure 3, and find that the decrease of 𝑘 causes value of expression 𝑏 01 (𝑘) to gradually vary from greater than 0 to less than 0. The reduction of 𝐴 and the increase of 𝑀 can both promote 𝑏 01 (𝑘) > 0 in Figure 3A,B, which in turn shows that the range of 𝑘 satisfying 𝑏 01 (𝑘) < 0 is shortened.…”

“…Thus, such result shows that system (4) gives rise to theoretical illusion of stability switches at the coincidence steady state 𝐸 * . 23,24 Case 2: When 𝑆 1 (𝜔 𝑖 ) < 0, 𝜏 𝑖0 = 2𝜋−arccos(𝐶 1 (𝜔 𝑖 )) 𝜔 𝑖…”

Rich dynamics of a vegetation–water system with the hydrotropism effect

“…The research showed that the pattern trend changes gradually in the spatially expanded ecosystem [16]. Moreover, other scholars have also established relevant systems to study vegetation patterns [17][18][19][20][21][22][23][24][25].…”

Temporal analysis of desertification vulnerability in Northeast Brazil using Google Earth Engine

“…Evaluating changes in land use and occupation in a given region is one of the factors used to determine the desertification degree of an area, while the temporal analysis is of great importance to define the speed of conversion of vegetation in arid environments (Cao, Li, Fu, & Wu, 2020; Li, Sun, & Jin, 2022; Wen, 2020). Degradation indices are applied together with land‐use history to monitor the degradation patterns of these areas (Tomasella et al., 2018).…”