Pattern formation of a spatial vegetation system with root hydrotropism
Introduction
Vegetation is the foundation of ecosystem. As the main component of ecosystem, it plays an important role in climate regulation, windbreak and sand fixation, conserving water and soil. At the same time, photosynthesis allows vegetation to absorb carbon dioxide while also releasing oxygen, which is critical for the carbon cycle [1], [2], [3], [4]. For consumers, vegetation is not only the source of food, but also their habitat. At present, the change of vegetation has had an important impact on human society in many aspects. Vegetation degradation is becoming more and more serious due to climate change and unreasonable human activities, such as overgrazing and deforestation [5], [6], [7], [8]. While water is an important material for vegetation growth, it is especially urgent and important to understand the ecological relationship between water and vegetation biomass.
Semi-arid vegetation pattern is a non-uniform macrostructure with some regularity that is used to examine the spatial distribution of vegetation in arid and semi-arid environments [9], [10], [11], [12], [13]. Simultaneously, changes in vegetation pattern structure serve as an early warning system for land degradation [14], [15], [16], [17]. Li and Tao et al. studied the process of aeolian dunes in grassland by numerical simulation. The findings revealed that desertification in the research area was influenced by evolutionary time, human grazing area, and grazing intensity within a grassland unit area. In addition, the relationship between desertification intensity and human grazing area as well as grazing intensity within a grassland unit area is shown [18]. Hardenberg and others put forward a new vegetation model, which can show many vegetation patterns observed in limited water resources. Through this model, the article predicts the transition from bare land with few precipitation to concentrated land with abundant precipitation and lush vegetation. They also predicted the rainfall range where different vegetation patterns coexist. According to the prediction of the model, this paper gives a new explanation of desertification [19].
Due to the climatic characteristics of semi-arid areas, the rainfall is uneven in time and space, which is characterized by less rainfall and concentrated in certain period. Many scholars have done a lot of work in the study of vegetation pattern in semi-arid areas [20], [21], [22], [23], [24], [25]. As early as 1999, based on the realistic ecological assumptions, Klausmeier first proposed a reaction-diffusion equation for vegetation and water. This equation set can be used to describe the pattern structure of vegetation in nature. The nonlinear process has a significant influence on the formation of vegetation patterns in semi-arid environments, according to the study and modeling of the equations [26]. The importance of vegetation to ecological environment is self-evident, and on the other hand noise has a certain influence on the structural changes of vegetation patterns. Sun et al. gave numerical evidence from point vegetation pattern to strip vegetation pattern caused by noise. The findings reveal that noise has a significant impact on the establishment of vegetation patterns in semi-arid environments [27], [28]. Mathematical models have been utilized by many researchers to examine vegetation patterns in dry and semi-arid regions. Yizhaq et al. developed a mathematical model to investigate the impact of soil-water diffusivity on vegetation pattern, and the results suggested that heterogeneity could not be overlooked when researching desertification induced by climate change or human influences [29].
After rainfall, some of the rainwater will infiltrate through the ground surface and then be absorbed by the roots of the vegetation. In fact, the roots of plants absorb water not only from their own location but also from nearby or even the whole study area, but this process takes some time. Britton first applied nonlocal time delay to reaction-diffusion equation in 1999, which can describe nonlocal interaction between individuals [30]. Based on practical significance, in order to more accurately describe the process of water uptake by vegetation, we introduce a mathematical model with non-local time delay to describe this actual situation.
It is well known that water resources are scarce in arid and semi-arid regions, and that required for the growth of vegetation itself mainly come from rainfall. After rainfall, part of water resources will be lost due to evaporation, while another part will infiltrate under the surface and become groundwater, which will then be absorbed by vegetation. Due to a shortage of water resources in semi-arid places, vegetation exhibits hydrotropism, which implies that due to a lack of water resources in its own location, the root system of vegetation will expand to a location with greater water resources. The hydrotropism of vegetation roots can not be described by the classical reaction-diffusion equation. In 1953, Patlak first derived a mathematical expression that can reflect the qualitative movement phenomenon based on the research background of different probabilities of directional movement and other directions [31]. Based on the phenomenon that vegetation roots have hydrotropism, we constructed a vegetation-water model with hydrotropism of vegetation roots.
The full text is presented as follows. The vegetation-water model with nonlocal time delay and root hydrotropism is introduced in the second section. The global weak solution of the system and the detailed analysis of the conditions of the vegetation pattern are showed in the third section. The numerical simulation of different root hydrotropism intensities were carried out in the fourth section. And the last section gives some conclusions.
Section snippets
A mathematical model derivation
Annual precipitation is lower in semi-arid locations, and rainfall is concentrated and unevenly dispersed in time and place. Thus, water, the source of plant life, is mainly obtained through rainfall. After rainfall, some water cannot penetrate underground due to surface crust in semi-arid regions, and become surface water, which then evaporates and disappears. The rest will infiltrate through the surface to the ground and become groundwater. The water resources that can be absorbed by
Vegetation pattern formation
The formation of vegetation pattern we studied is the instability caused by diffusion and root hydrotropism in the uniform steady state of vegetation [40], [41]. In biological terms, this is equivalent to assuming that vegetation is evenly distributed in arid and semi-arid regions at the early stage. However, there is no prior difference in moisture concentration at the onset of rainfall. Therefore, the random diffusion in the absence of water resources and due to vegetative propagation or long
Numerical results
For the dynamic behavior of spatial vegetation pattern in semi-arid area, we can not use analytical method to study it. Therefore, according to the conditions of vegetation pattern formation obtained from the previous mathematical analysis, it is necessary to find the appropriate parameters. The boundary condition of our research system is Neumann boundary condition, that is, the spatial area of the semi-arid area will not have any contact with the environment outside the study region. The
Conclusions
This article mainly discusses the spatio-temporal dynamics of vegetation-water model with hydrotropism of plant roots and nonlocal delay, and analyzes the effect of different root hydrotropism intensities on the spatial distribution of plant in our study region. The results show that changes in the hydrotropism intensity of plant root can induce changes in the structure of vegetation pattern in our study region. With the increase of root hydrotropism intensity of vegetation roots, the
Acknowledgments
The project is funded by National Key Research and Development Program of China (grant no. 2018YFE0109600), National Natural Science Foundation of China under Grant no. 42075029, Program for the Outstanding Innovative Teams (OIT ) of Higher Learning Institutions of Shanxi, and China Postdoctoral Science Foundation (Grant nos. 2017M621110 and 2019T120199).
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