Effect of Spatial Average on the Spatiotemporal Pattern Formation of Reaction-Diffusion Systems

Abstract

Some quantities in reaction-diffusion models from cellular biology or ecology depend on the spatial average of density functions instead of local density functions. We show that such nonlocal spatial average can induce instability of constant steady state, which is different from classical Turing instability. For a general scalar equation with spatial average, the occurrence of the steady state bifurcation is rigorously proved, and the formula to determine the bifurcation direction and the stability of the bifurcating steady state is given. For the two-species model, spatially non-homogeneous time-periodic orbits could arise due to spatially non-homogeneous Hopf bifurcation from the constant equilibrium. Examples from a nonlocal cooperative Lotka-Volterra model and a nonlocal Rosenzweig-MacArthur predator-prey model are used to demonstrate the bifurcation of spatially non-homogeneous patterns.

Appendix

1.1 The proof of Theorem 2.6

Proof

Firstly, we integrate both sides of Eq. (2.17) on $\Omega$ and divide by $|\Omega |$ which is the spatial domain size, then we obtain a ODE system of ${\bar{u}}$:

$$\begin{aligned} {\bar{u}}_t=a-(b+c){\bar{u}}-({\tilde{d}}+e){\bar{u}}^2. \end{aligned}$$

(A.1)

Then, the equilibrium of Eq. (A.1) also satisfies (2.17) which admits a unique positive root $u=u_*$. In addition, for Eq. (A.1), the unique equilibrium $u=u_*$ is globally stable, and all the solutions of (A.1) will converge to $u=u_*$ as $t\rightarrow +\infty$. Note that ${\bar{u}}$ is a function of t. Then, we rewrite Eq. (2.17) as:

$$\begin{aligned} u_t=d\Delta u+A(t)-B(t)u, \end{aligned}$$

(A.2)

where $A(t)=a-b{\bar{u}}-{\tilde{d}}{\bar{u}}^2$ and $B(t)=c+e{\bar{u}}$. Denote ${\tilde{A}}=\lim \nolimits _{t\rightarrow +\infty }A(t)>0,~{\tilde{B}}=\lim \nolimits _{t\rightarrow +\infty }B(t)>0$, then for any $0<\epsilon \ll 1$, there exists $T>0$ such that for arbitrary $t>T$, we have

$$\begin{aligned} {\left\{ \begin{array}{ll} u_t\le d\Delta u+({\tilde{A}}+\epsilon )-({\tilde{B}}-\epsilon )u,\\ u_t\ge d\Delta u+({\tilde{A}}-\epsilon )-({\tilde{B}}+\epsilon )u. \end{array}\right. } \end{aligned}$$

Therefore, we can use the $u_1\ge u(x,t)$ as the upper solution with $u_1$ is the solution of the following equation:

$$\begin{aligned} {\left\{ \begin{array}{ll} u_{1}^{\prime }=({\tilde{A}}+\epsilon )-({\tilde{B}}-\epsilon )u_1,~t>T,\\ u_{1}(0)=\max \limits _{x\in \Omega } u(x,T), \end{array}\right. } \end{aligned}$$

(A.3)

and the lower solution $u_2\le u(x,t)$ satisfying

$$\begin{aligned} {\left\{ \begin{array}{ll} u_{2}^{\prime }=({\tilde{A}}-\epsilon )-({\tilde{B}}+\epsilon )u_2,~t>T,\\ u_{2}(0)=\max \limits _{x\in \Omega } u(x,T). \end{array}\right. } \end{aligned}$$

(A.4)

Moreover, by the theory of ODE, we know the asymptotic behavior of Eqs. (A.3) and (A.4):

$$\begin{aligned} \lim _{t\rightarrow +\infty }u_1(t)=\frac{{\tilde{A}}+\epsilon }{{\tilde{B}}-\epsilon },~~\lim _{t\rightarrow +\infty }u_2(t)=\frac{{\tilde{A}}-\epsilon }{{\tilde{B}}+\epsilon }. \end{aligned}$$

By the arbitrariness of $\epsilon$, we obtain that $\lim \nolimits _{t\rightarrow +\infty }u(x,t)={\tilde{A}}/{\tilde{B}}=u_*$. We complete the proof. $\square$