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The growth of vegetation is undeniably subject to random fluctuations arising from environmental variability and internal effects due to periodic forcing. To address these issues, we investigated a spatial version of a vegetation model including seasonal rainfall, noise, and diffusion. By numerical simulations, we found that noise can induce the pattern transition from stationary pattern to other patterns. More specifically, when noise intensity is small, patch invasion is induced. As noise intensity further increases, chaotic patterns emerge. For the system with noise and seasonal rainfall, it exhibits frequency-locking phenomena. Patterns transition may be a warning signal for the onset of desertification and thus the obtained results may provide some measures to protect vegetation, such as reducing random factors or changing irrigation on vegetation.

Understanding the effect of external variability on vegetation systems is a problem of great interest. An obvious and most important source of external variability is seasonality. The consequences of the cyclic variation of seasonality have been well investigated [

In the past years, the influences of noise and periodic forcing have been well studied in ecosystems [

Regular patterns of vegetation have been observed in many arid and semiarid regions of the world. The formation of regular vegetation bands on hillsides of semiarid catchments is often attributed to a low scale process of water redistribution by runoff [

The main purpose of this paper is to investigate the effects of noise and seasonal rainfall on the vegetation patterns. In particular, we want to check whether pattern transition or frequency locking emerges. The rest of our paper is arranged as follows. In Section

Von Hardenberg et al. proposed a partial differential equation model, with equations for the biomass density

where

When combined with noise and seasonal rainfall, the original spatially extended model is written as the following system:

The seasonal rainfall is assumed to be sinusoidal with amplitude

In (

Before proceeding to the spatially explicit case, the first step is to have a look at the properties of the local dynamics. The local system of systems (

We assume that the local system has a stable equilibrium

We make the following substitute:

The initial conditions are assumed as

By solving the linear equations (

Straightforward manipulation of (

The condition for a spatial mode

In this section, extensive testing was performed through numerical integration to describe systems (

In order to well show the effects of noise and seasonal rainfall, we pay attention to the spatial pattern of systems (

In Figure

Snapshots of contour pictures of the vegetation at

For the systems (

In recent years, noise-sustained and noise-induced spatial pattern formations have been discussed in ecological systems [

Snapshots of contour pictures of the vegetation at

For the case

Snapshots of contour pictures of the vegetation at

In order to see the effects of noise intensity and temporal correlation on pattern dynamics, we give regions of pattern structures with respect to the two parameters in Figures

Regions of pattern structures of systems (

Regions of pattern structures of systems (

It is well known that an external periodic force applied to a nonlinear pendulum can cause the pendulum to become entrained at a frequency which is rationally related to the applied frequency, a phenomenon known as frequency locking. It is useful to reveal the complexity of the ecosystem. It can be found that systems (

It is checked by numerical simulations that when systems (

The 1 : 1 frequency-locking oscillation with the values of the parameters

The 2 : 1 frequency-locking oscillation with the values of the parameters

where

Temporal correlation

Phase diagram in

In this paper, we investigated a vegetation model combined with seasonal rainfall, noise, and spatial diffusion. By performing a series of numerical simulations, we found that there was emergence of pattern transition from stationary pattern to patch invasion. What is more, chaotic pattern will appear if noise intensity is large. And for the system with both noise and periodic forcing, it exhibits frequency-locking phenomena. The results showed that noise and seasonal rainfall play an important role in vegetation patterns.

Climate fluctuation is also considered as a source of vegetation spatial pattern, which means that all the parameters in systems (

The mechanisms inducing the change of structure or dynamics of vegetation populations are among the most challenging research areas in ecology [

The authors declare that there is no conflict of interests regarding the publication of this paper.