A wetland ecosystem is studied theoretically and numerically to reveal the rules of dynamics which can be quite accurate to better describe the observed spatial regularity of tussock vegetation. Mathematical theoretical works mainly investigate the stability of constant steady states, the existence of nonconstant steady states, and bifurcation, which can deduce a standard parameter control relation and in return can provide a theoretical basis for the numerical simulation. Numerical analysis indicates that the theoretical works are correct and the wetland ecosystem can show rich dynamical behaviors not only regular spatial patterns. Our results further deepen and expand the study of dynamics in the wetland ecosystem. In addition, it is successful to display tussock formation in the wetland ecosystem may have important consequences for aquatic community structure, especially for species interactions and biodiversity. All these results are expected to be useful in the study of the dynamic complexity of wetland ecosystems.
Ecosystems consist of organisms, the environment, and the interactions with each other [
Tussock formation is a global phenomenon that enhances microtopography and increases biodiversity by adding structure to ecological communities [
van de Koppel and Crain [
Model (
The rest of the paper is organized as follows. In Section
Obviously,
If
From the first equation of system (
If
From the proof of Theorem
Then
From (
In this section, we will discuss the local and global stability of constant steady states. In the remaining part of this paper, we always assume that
Recall that
For system ( the trivial constant steady state if
The linearization of (
For each If If
Now, we give the result of global stability of
Assume that
If
Inequalities (
Since
Therefore, from the above analysis, it can be concluded that the constant steady state
In this section, we discuss the existence and nonexistence of nonconstant steady states of (
Now, we first cite two important lemmas. For notational convenience, we denote
Suppose that Assume that Assume that
Let
For any positive solution
Let
For any positive constant
Applying Lemma
Note that
Let
In this subsection, we show the nonexistence of nonconstant positive solutions of (
Let
Assume that
Multiplying the first equation in (
From the Poincaré inequality, we can obtain
Since
In this subsection, we discuss the existence of nonconstant positive steady-state solutions of (
For simplicity, we denote
It is known that
To calculate the index of
Suppose
From Lemma
If
Since
In the following, we will prove that, for any
Fixing
Our earlier analysis has shown that (
From Theorems
Set
In this section, we study the existence of periodic solutions of system ( There exists
And for the unique pair of complex eigenvalues near the imaginary axis
For convenience, (
We will identify the Hopf bifurcation values
(a) When
Assume that
In the following, we look for spatially nonhomogeneous Hopf bifurcation for
(b) When
Since
Obviously,
Clearly,
Assume that the bifurcation periodic orbits from the bifurcation periodic orbits from
The stability and bifurcation direction of the spatially homogeneous periodic orbits bifurcating from
In this section, we will study Turing bifurcation of the positive constant steady state
The roots of (
It is obvious that the stability conditions for
If
In this section, we present some numerical simulations to verify the feasibility of the theoretical results obtained in previous sections. In these numerical simulations, we consider one-dimensional spatial domain
Stable positive constant steady state
Illustration of the existence of spatially homogeneous periodic solutions of system (
Illustration of the existence of spatially nonhomogeneous periodic solutions of system (
Illustration of the existence of nonconstant positive steady-state solutions of system (
In Figure
Bifurcation diagram of system (
In this paper, we have investigated a diffusive plant-wrack model with homogeneous Neumann boundary conditions theoretically and numerically. Mathematical analysis is used to study dissipation, persistence properties, and local and global stability of constant steady states. To prove global stability, a new method based on the upper and lower solution method is applied. Furthermore, the conditions of existence and nonexistence of nonconstant positive steady-state solutions of the reaction-diffusion system have been derived. Moreover, it has been found that system (
The results presented here indicate that system (
By comparing Figures
Using the model proposed by van de Koppel and Crain [
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China (Grant no. 31170338), by the Key Program of Zhejiang Provincial Natural Science Foundation of China (Grant no. LZ12C03001), and by the Zhejiang Provincial Natural Science Foundation (nos. LY14C030006 and LY13A010010).