Pattern formation of a biomass–water reaction–diffusion model

Abstract

In this paper, we are concerned with a biomass–water reaction–diffusion model subject to the homogeneous Neumann boundary condition. We derive several sufficient conditions on the existence and non-existence of non-constant stationary solutions with respect to large or small diffusion rate, which give the criteria for the possibility of Turing patterns in this system. Our results confirm the numerical findings of Manor and Shnerb, (2006) and also complement the theoretical results of Wang et al., (2017) for the corresponding ODE model.

Introduction

In order to explore the formation of Turing pattern in the field of vegetation, Manor and Shnerb proposed and studied a biomass–water reaction–diffusion model [1]. Denote the water density by $u$ and the biomass density by $v$. Then the system considered in [1] in the dimensionless form reads as

$$\left\{\begin{array}{cc}\frac{\partial u}{\partial t}-d_1\Delta u=a-\lambda uv-u,\hfill & t>0,x\in \Omega ,\hfill \\ \frac{\partial v}{\partial t}-d_2\Delta v=uv-\left({\mu }_0+\frac{{\mu }_1}{v+1}\right)v,\hfill & t>0,x\in \Omega ,\hfill \\ \frac{\partial u}{\partial \nu }=\frac{\partial v}{\partial \nu }=0,\hfill & t>0,x\in \partial \Omega ,\hfill \\ u(x,0)=u_0\left(x\right)\ge 0,v(x,0)=v_0\left(x\right)\ge 0,\hfill & x\in \Omega ,\hfill \end{array}\right.$$

where $d_1,d_2,a,\lambda ,{\mu }_0,{\mu }_1>0$ are constants, $\nu$ is the unit outward normal on $\partial \Omega$ with $\Omega$ a smooth bounded domain of $R^N(N\ge 1)$. In (1.1), the first equation is used to describe the water (or soil moisture) dynamics, with the deposition $a$, inorganic losses $-u$ due to the effect of percolation and evaporation, and the consumption of water is represented as the biomass $\lambda uv$ and $d_1$ is the diffusion rate of water (or soil moisture). The second equation is used to describe the dynamics of the biomass: it decays at the rate ${\mu }_0+{\mu }_1/(v+1)$ and grows upon water intake $\left(uv\right)$, and $d_2$ is its diffusion rate with $d_2$ being much smaller than $d_1$. One may refer to [1], [2] for the detailed interpretation of the background of (1.1).

In [1], Manor and Shnerb performed numerical analysis by taking $d_2$ to be small and observed the appearance of Turing pattern; in particular, their simulations show that the resulting patterns are disordered and robust. In [2], Wang, Shi and Zhang studied the corresponding ordinary differential equation (ODE) model and proved that the model possesses a very rich dynamics including multiple stable equilibria, backward bifurcation of positive equilibria, supercritical or subcritical Hopf bifurcations, bubble loop of limit cycles, homoclinic bifurcation and Bogdanov–Takens bifurcation. In this paper, our main aim is to establish various sufficient conditions on the existence and non-existence of non-constant stationary solutions associated with (1.1) with respect to large or small diffusion rate $d_1$ or $d_2$, which therefore give the criteria for the possibility of Turing patterns in (1.1). Our results confirm and complement the numerical findings of [1] and those of [2]; see Section 5 for further discussions. For related studies on (1.1), one may further refer to [3], [4], [5], [6], [7] and the references therein.

The stationary solution problem associated with (1.1) satisfies the following elliptic systems:

$$\left\{\begin{array}{cc}-d_1\Delta u=a-\lambda uv-u,\hfill & x\in \Omega ,\hfill \\ -d_2\Delta v=uv-\left({\mu }_0+\frac{{\mu }_1}{v+1}\right)v,\hfill & x\in \Omega ,\hfill \\ \frac{\partial u}{\partial \nu }=\frac{\partial v}{\partial \nu }=0,\hfill & x\in \partial \Omega .\hfill \end{array}\right.$$

It is easy to check that if ${({\mu }_0\lambda +{\mu }_0+{\mu }_1\lambda -a)}^2>4{\mu }_0\lambda ({\mu }_0+{\mu }_1-a)$, (1.2) has two constant solutions $({\hat{u}}_1,{\hat{v}}_1)$ and $({\hat{u}}_2,{\hat{v}}_2)$ with

$${\hat{v}}_1=\frac{a-({\mu }_0+{\mu }_1)\lambda -{\mu }_0+\sqrt{{({\mu }_0\lambda +{\mu }_0+{\mu }_1\lambda -a)}^2-4{\mu }_0\lambda ({\mu }_0+{\mu }_1-a)}}{2{\mu }_0\lambda },{\hat{u}}_1=\frac{a}{\lambda {\hat{v}}_1+1};$$

$${\hat{v}}_2=\frac{a-({\mu }_0+{\mu }_1)\lambda -{\mu }_0-\sqrt{{({\mu }_0\lambda +{\mu }_0+{\mu }_1\lambda -a)}^2-4{\mu }_0\lambda ({\mu }_0+{\mu }_1-a)}}{2{\mu }_0\lambda },{\hat{u}}_2=\frac{a}{\lambda {\hat{v}}_2+1}.$$

Note that ${\hat{v}}_2<0$ if $a>{\mu }_0+{\mu }_1$ and so $({\hat{u}}_1,{\hat{v}}_1)$ is the unique positive constant solution, and if $a<{\mu }_0+{\mu }_1$, then (1.2) has two positive constant solutions $({\hat{u}}_1,{\hat{v}}_1)$ and $({\hat{u}}_2,{\hat{v}}_2)$.

Throughout the paper, unless otherwise stated, all solutions to be considered for system (1.2) will be classical positive solutions. The remainder of the paper is organized as follows. In Section 2, we will obtain a priori positive upper and lower bounds of solutions of (1.2). In Section 3, we obtain some results on the non-existence of non-constant solutions while Section 4 concerns the existence of non-constant solutions. Section 5 ends the paper with a brief discussion.