Propagation Phenomena for A Reaction–Advection–Diffusion Competition Model in A Periodic Habitat

Abstract

This paper is devoted to the study of propagation phenomena for a Lotka–Volterra reaction–advection–diffusion competition model in a periodic habitat. We first investigate the global attractivity of a semi-trivial steady state for the periodic initial value problem. Then we establish the existence of the rightward spreading speed and its coincidence with the minimal wave speed for spatially periodic rightward traveling waves. We also obtain a set of sufficient conditions for the rightward spreading speed to be linearly determinate. Finally, we apply the obtained results to a prototypical reaction–diffusion model.

Appendix

In this section, we extend the abstract results in [6] and [21] on spreading speeds and traveling waves to the case of a periodic habitat.

Let ${\mathcal {C}}$ be the set of all bounded and continuous functions from $\mathbb {R}$ to $\mathbb {R}^m$ with $m\ge 1$ and ${\mathcal {C}}_+=\{\phi \in {\mathcal {C}}:\phi (x)\ge 0,\ \forall x\in \mathbb {R}\}$. Clearly, any vector in $\mathbb {R}^m$ can be regarded as a function in ${\mathcal {C}}$. For $u=(u_1,\ldots ,u_m), w=(w_m,\ldots ,w_m)\in {\mathcal {C}}$, we write $u\ge w (u\gg w)$ provided $u_j(x)\ge w_j(x)(u_j(x)>w_j(x)), \forall 1\le j\le m,\, x\in \mathbb {R}$, and $u>w$ provided $u\ge w$ but $u\ne w$. Assume that $\beta$ is a strongly positive $L$-periodic continuous function from $\mathbb {R}$ to $\mathbb {R}^m$. Set

$$\begin{aligned} {\mathcal {C}}_{\beta }=\{u\in {\mathcal {C}}:\, 0\le u(x)\le \beta (x),\ \forall x\in \mathbb {R}\},\ {\mathcal {C}}^{per}_{\beta }=\{u\in \mathcal {C_\beta }:\, u(x)=u(x+L),\ \forall x\in \mathbb {R}\}. \end{aligned}$$

Let $X=C([0,L],\mathbb {R}^m)$ equipped with the maximum norm $|\cdot |_X, \, X_+=C([0,L],\mathbb {R}_+^m), \,$

$$\begin{aligned} X_{\beta }=\{u\in X:\ 0\le u(x)\le {\beta }(x),\ \forall x\in [0,L]\},\ \text {and} \ \overline{X}_{\beta }=\{u\in X_{\beta }: u(0)=u(L)\}. \end{aligned}$$

Let $BC(\mathbb {R}, X)$ be the set of all continuous and bounded functions from $\mathbb {R}$ to $X$. Then we define

$$\begin{aligned}&{\mathcal {X}}=\{v\in BC(\mathbb {R},X):v(s)(L)=v(s+L)(0),\forall s\in \mathbb {R}\}, \\&{\mathcal {X}}_+=\{v\in {\mathcal {X}}:v(s)\in X_+,\forall s\in \mathbb {R}\}, \end{aligned}$$

and

$$\begin{aligned} {\mathcal {X}}_{\beta }=\{v\in BC(\mathbb {R},X_{\beta }):v(s)(L)=v(s+L)(0),\forall s\in \mathbb {R}\}. \end{aligned}$$

Let

$$\begin{aligned} {\mathcal {K}}_{\beta }:=\{v\in BC(L\mathbb {Z},X_{\beta }):v(i)(L)=v(i+L)(0),\forall i\in L\mathbb {Z}\}. \end{aligned}$$

Clearly, any element in $\overline{X}_{\beta }$ can be regarded as a constant function in ${\mathcal {X}}_{\beta }$, that is, any element in ${\mathcal {C}}^{per}_\beta$ corresponds to a constant function in ${\mathcal {X}}_{\beta }$. We equip ${\mathcal {C}}$ and ${\mathcal {X}}$ with the compact open topology, that is, $u_n\rightarrow u$ in ${\mathcal {C}}$ or ${\mathcal {X}}$ means that the sequence of $u_n(s)$ converges to $u(s)$ in $\mathbb {R}^m$ or $X$ uniformly for $s$ in any compact set. We equip ${\mathcal {C}}$ and ${\mathcal {X}}$ with the norm $\Vert \cdot \Vert _{\mathcal {C}}$ and $\Vert \cdot \Vert _{\mathcal {X}}$, respectively, which are defined by

$$\begin{aligned} \Vert u\Vert _{{\mathcal {C}}}=\sum \limits _{k=1}^{\infty }\frac{\max _{|x|\le k}|u(x)|}{2^k},\ \forall u\in {\mathcal {C}}, \end{aligned}$$

where $|\cdot |$ denotes the usual norm in $\mathbb {R}^m$, and

$$\begin{aligned} \Vert u\Vert _{{\mathcal {X}}}=\sum \limits _{k=1}^{\infty }\frac{\max _{|x|\le k}|u(x)|_X}{2^k},\ \forall u\in {\mathcal {X}}. \end{aligned}$$

Define a translation operator ${\mathcal {T}}_a$ by ${\mathcal {T}}_a[u](x)=u(x-a)$ for any given $a\in L\mathbb {Z}$. Let $Q$ be an operator on ${\mathcal {C}}_{\beta }$, where $\beta \in \mathcal {C}$ is strongly positive and $L$-periodic. In order to use the theory developed in [6] and [21], we need the following assumptions on $Q$:

(A1):

$Q$ is $L$-periodic, that is, ${\mathcal {T}}_a[Q[u]] =Q[{\mathcal {T}}_a[u]],\quad \forall u\in {\mathcal {C}}_{\beta },\, a\in L\mathbb {Z}$.

(A2):

$Q:\, {\mathcal {C}}_{\beta } \rightarrow {\mathcal {C}}_{\beta }$ is continuous with respect to the compact open topology.

(A3):

$Q:\, {\mathcal {C}}_{\beta } \rightarrow {\mathcal {C}}_{\beta }$ is monotone (order preserving) in the sense that $Q[u] \ge Q[w]$ whenever $u \ge w$.

(A4):

$Q$ admits two $L$-periodic fixed points $0$ and $\beta$ in ${\mathcal {C}}_+$, and for any $z\in {\mathcal {C}}^{per}_{\beta }$ with $0\ll z\le \beta$, there holds $\lim \limits _{n\rightarrow \infty }Q^n[z](x)=\beta (x)$ uniformly for $x\in \mathbb {R}$.

(A5):

$Q[{\mathcal {C}}_{\beta }]$ is precompact in ${\mathcal {C}}_{\beta }$ with respect to the compact open topology.

Define a homeomorphsim $F:{\mathcal {C}}\rightarrow {\mathcal {K}}$ by

$$\begin{aligned} F[\phi ](i)(\theta )=\phi (i+\theta ),\ i\in L\mathbb {Z},\ \theta \in [0,L], \end{aligned}$$

and an operator $P:{\mathcal {K}}_{\beta }\rightarrow {\mathcal {K}}_{\beta }$ by

$$\begin{aligned} P=F\circ Q\circ F^{-1}. \end{aligned}$$

(5.10)

Next, we define $\tilde{P}: {\mathcal {X}}\rightarrow {\mathcal {X}}$ by

$$\begin{aligned} \tilde{P}[v](s):=P[v(\cdot +s)](0),\quad \forall v\in {\mathcal {X}},\ s\in \mathbb {R}. \end{aligned}$$

(5.11)

We further claim that

$$\begin{aligned} \tilde{P}[v](s)(\theta )=Q[v_s](\theta ),\quad \forall v\in {\mathcal {X}},\ s\in \mathbb {R},\ \theta \in [0,L], \end{aligned}$$

(5.12)

where $v_s\in {\mathcal {C}}$ is defined by

$$\begin{aligned} v_s(x)=v(s+n_x)(\theta _x),\quad \forall x=n_x+\theta _x\in \mathbb {R},\ n_x=L\left[ \frac{x}{L}\right] ,\ \theta _x\in [0,L). \end{aligned}$$

Indeed, since

$$\begin{aligned} F[\phi ](i)(\theta )=\phi (i+\theta ),\quad F^{-1}[\psi ](x)=\psi (n_x)(\theta _x), \end{aligned}$$

it then follows that

$$\begin{aligned} \tilde{P}[v](s)&= P[v(\cdot +s)](0)=FQF^{-1}[v(\cdot +s)](0)\\&= F[Q[v(n_\cdot +s)(\theta _\cdot )]](0)=F[Q(v_s)](0), \end{aligned}$$

and hence,

$$\begin{aligned} \tilde{P}[v](s)(\theta )=F[Q(v_s)](0)(\theta )=Q[v_s](\theta ). \end{aligned}$$

Let $r\in Int(X_+)$ with $r(0)=r(L)$. In order to apply the results in [6] to $\tilde{P}$, we need to verify that $\tilde{P}$ satisfies the following assumptions:

(B1):

${\mathcal {T}}_a[\tilde{P}[u]] =\tilde{P}[{\mathcal {T}}_a[u]],\quad \forall u\in {\mathcal {X}}_{r},\, a\in \mathbb {R}$.

(B2):

$\tilde{P}:\, {\mathcal {X}}_{r} \rightarrow {\mathcal {X}}_{r}$ is continuous with respect to the compact open topology.

(B3):

$\tilde{P}:\, {\mathcal {X}}_{r} \rightarrow {\mathcal {X}}_{r}$ is monotone (order preserving) in the sense that $\tilde{P}[u] \ge \tilde{P}[w]$ whenever $u \ge w$.

(B4):

$\tilde{P}$ admits two fixed points $0$ and $r$ in $\overline{X}_{r}$, and for any $z\in \overline{X}_{r}$ with $0\ll z\le r$, there holds $\lim \limits _{n\rightarrow \infty }{\tilde{P}}^n[z]=r$.

(B5):

There exists $k\in [0,1)$ such that for any ${\mathcal {U}}\subset {\mathcal {X}}_r, \, \alpha (\tilde{P}[{\mathcal {U}}](0))\le k\alpha ({\mathcal {U}}(0))$, where $\alpha$ denotes the Kuratowski measure of noncompactness in ${\mathcal {X}}_r$.

Proposition 5.1

Let $\beta \in \mathcal {C}$ is strongly positive and $L$-periodic. Assume that $Q:{\mathcal {C}}_{\beta } \rightarrow {\mathcal {C}}_{\beta }$ satisfies assumptions (A1)–(A5). Then $\tilde{P}$: ${\mathcal {X}}_{ \beta }\rightarrow {\mathcal {X}}_{\beta }$ satisfies assumptions (B1)–(B5).

Proof

For any $c\in \mathbb {R}$, let $u(\cdot )=v(\cdot +c), \forall v\in {\mathcal {X}}$. Then

$$\begin{aligned} T_{-c}\tilde{P}[v](s)&= \tilde{P}[v](s+c)\\&= Q[v_{s+c}]=Q[u_{s}]=\tilde{P}[u(\cdot )](s)\\&= \tilde{P}[T_{-c}v](s), \quad \forall v\in {\mathcal {X}},\ s\in \mathbb {R}, \end{aligned}$$

and hence, (B1) holds. (B2) can be verified by similar arguments to those in [20, Lemma 2.1], and (B3) directly follows from (A3). Clearly, $0$ is the fixed point of $\tilde{P}$ since $Q(0)=0$. To verify (B4), we need to show that $\beta |_{[0,L]}$ is the fixed point of $\tilde{P}$. Note that $\beta (x)$ is a constant function in ${\mathcal {X}}$ with $x\in [0,L]$ we have

$$\begin{aligned} \beta _s(\cdot )=\beta (s+n_\cdot )(\theta _\cdot )=\beta (\theta _\cdot ),\quad \forall s \in \mathbb {R}. \end{aligned}$$

Therefore, $\beta _s=\beta$ in ${\mathcal {C}}, \forall s\in \mathbb {R}$. Moreover,

$$\begin{aligned} \tilde{P}[\beta ](s)(\theta )=Q[ \beta _s](\theta )=Q[\beta ](\theta )=\beta (\theta ),\quad \forall \theta \in [0,L]. \end{aligned}$$

This implies that $\tilde{P}[\beta ]=\beta$ in ${\mathcal {X}}$. Thus, $(B4)$ follows from $(A4)$. Now we prove (B5) holds. For any given ${\mathcal {U}}\subset {\mathcal {X}}_{\beta }$, it is easy to see that $\tilde{P}({\mathcal {U}})(0)$ is uniformly bounded. By (A5), it follows for any $\varepsilon >0$, there exists $\delta >0$ such that

$$\begin{aligned} |Q(v)(x_1)-Q(v)(x_2)|<\varepsilon , \quad \forall v\in {\mathcal {C}}_\beta \end{aligned}$$

provided that $x_1,x_2\in [0,L]$ with $|x_1-x_2|<\delta$. So for any $v\in {\mathcal {U}}$,

$$\begin{aligned} |\tilde{P}(v)(0)(\theta _1)-\tilde{P}(v)(0)(\theta _2)|=|Q(v_0)(\theta _1)-Q(v_0)(\theta _2)|<\varepsilon \end{aligned}$$

provided that $\theta _1,\theta _2\in [0,L]$ with $|\theta _1-\theta _2|<\delta$. This implies that $\tilde{P}({\mathcal {U}})(0)$ is equicontinuous. By Arzelà–Ascoli theorem, it follows that $\tilde{P}({\mathcal {U}})(0)$ is precompact in ${\mathcal {X}}_\beta$, and hence, $\alpha (\tilde{P}({\mathcal {U}})(0))=0$. This proves (B5) with $k=0$. $\square$

Let $\omega \in \overline{X}_\beta$ with $0\ll \omega \ll \beta$. Choose $\phi \in {\mathcal {X}}_\beta$ such that the following properties hold:

(C1):

$\phi (s)$ is nonincreasing in $s$;

(C2):

$\phi (s)\equiv 0$ for all $s\ge 0$;

(C3):

$\phi (-\infty )=\omega$.

Let $c$ be a given real number. According to [31], we define an operator $R_{c}$ by

$$\begin{aligned} R_c[a](s):=\max \{\phi (s),T_{-c}\tilde{P}[a](s)\}, \end{aligned}$$

and a sequence of functions $a_n(c;s)$ by the recursion:

$$\begin{aligned} a_0(c;s)=\phi (s),\quad a_{n+1}(c;s)=R_{c}[a_n(c;\cdot )](s). \end{aligned}$$

As a consequence of similar arguments to those in [6, Lemmas 3.1–3.3], we have the following result.

Lemma 5.4

The following statements are valid:

• (1) For each $s\in \mathbb {R}, \, a_n(c,s)$ converges to $a(c;s)$ in $X$, where $a(c;s)$ is nonincreasing in both $c$ and $s$, and $a(c;\cdot )\in {\mathcal {X}}_{\beta }$.

• (2) $a(c,-\infty )= \beta$ and $a(c,+\infty )$ is a fixed point of $\tilde{P}$.

Following [6, 33], we define two numbers

$$\begin{aligned} c^*_+=\sup \{c:a(c,+\infty )=\beta \},\quad \overline{c}_+=\sup \{c:a(c,+\infty )>0\}. \end{aligned}$$

(5.13)

Clearly, $c^*_+\le \overline{c}_+$ due to the monotonicity of $a(c;\cdot )$ with respect to $c$. For each $t\ge 0$, let $P_t$ and $\tilde{P}_t$ be defined as in (5.10) and (5.12) with $Q=Q_t$, respectively. By [6, Remark3.2], we have the following result.

Theorem 5.4

Let $\{Q_t\}_{t\ge 0}$ be a continuous-time semifow on ${\mathcal {C}}_\beta$ with $Q_t[0]=0,Q_t[\beta ]=\beta$ for all $t\ge 0$ and $\{\tilde{P}_t\}_{t\ge 0}$ be defined as in (5.12) for each $t\ge 0$, and $c^*_+$ and $\overline{c}_+$ be denoted by (5.13) with $\tilde{P}=\tilde{P}_1$. Suppose that $Q_t$ satisfies (A1)–(A5) for each $t>0$. Then the following statements are valid:

• (i) If $\phi \in {\mathcal {C}}_{\beta }, \, 0\le \phi \le \omega \ll \beta$ for some $\omega \in {\mathcal {C}}^{per}_{\beta }$, and $\phi (x)=0, \forall x\ge H$, for some $H\in \mathbb {R}$, then $\lim _{t\rightarrow \infty ,x\ge ct}Q_t(\phi )=0$ for any $c>\overline{c}_+$.

• (ii) If $\phi \in {\mathcal {C}}_{\beta }$ and $\phi (x)\ge \sigma , \, \forall x\le K$, for some $\sigma \gg 0$ and $K\in \mathbb {R}$, then $\lim _{t\rightarrow \infty ,x\le ct}(Q_t(\phi )(x)-\beta (x))=0$ for any $c<c^*_+$.

Proof

Since $\{Q_t\}_{t\ge 0}$ is a continuous-time semifow on ${\mathcal {C}}_\beta$ with $Q_t(0)=0$ and $Q_t(\beta )=\beta$ for all $t\ge 0$, it follows that $\{\tilde{P}_t\}_{t\ge 0}$ is a continuous-time semiflow on ${\mathcal {X}}_{\beta }$ with $\tilde{P}_t(0)=0$ and $\tilde{P}_t(\beta )=\beta$ for all $t\ge 0$. By Proposition 5.1, $\tilde{P}_t$ satisfies (B1)–(B5). For any $\phi \in {\mathcal {C}}_{\beta }, \, 0\le \phi \le \omega \ll \beta$ with $\omega \in {\mathcal {C}}^{per}_{\beta }$, let

$$\begin{aligned} u(s)(\theta )=[\phi (n_s+L+\theta )-\phi (n_s+\theta )]\theta _s+\phi (n_s+\theta ). \end{aligned}$$

for $s\in \mathbb {R}, \, s=n_s+\theta _s, \, n_s=L\left[ \displaystyle \frac{s}{L}\right] ,\ \theta _s\in [0,L), \, \theta \in [0,L]$. Then $u\in {\mathcal {X}}_{{\beta }}$, and $0\le u\le \omega \ll \beta$.

To prove statement $(i)$, we suppose that there exists some $H\in \mathbb {R}$ such that $\phi (x)=0, \, x\ge H$ and $\phi (x)\not \equiv 0$ (otherwise, it is trivial). Thus, $u(s)=0, \, s\ge H+L$. By [6, Remark 3.2], it follows that $\lim _{t\rightarrow \infty ,s\ge ct}\tilde{P}_t(u)(s)=0$ in $X$ for any $c>\overline{c}_+$. On the other hand, we have

$$\begin{aligned} \tilde{P}_t[u](n_x)(\theta _x)&= Q_t[u_{n_x}](\theta _x)=Q_t[u(n_x+n_\cdot )(\theta _\cdot )](\theta _x),\\&= Q_t[\phi (n_x+\cdot )](\theta _x)=Q_t[\phi (\cdot )](x),\quad x\in \mathbb {R}, \end{aligned}$$

and for $s\in L\mathbb {Z}$, $\lim _{t\rightarrow \infty ,s\ge ct}\tilde{P}_t(u)(s)=0$ in $X$ holds true for any $c>\overline{c}_+$. Choose a $c'\in (\overline{c}_+,c)$, we obtain

$$\begin{aligned} {\left| Q_t[\phi ](x)\right| }\le \left| \tilde{P}_{t}[u](n_x)\right| _{X},\quad \forall x\ge ct, \ t\ge \frac{L}{c-c^{\prime }}, \end{aligned}$$

(5.14)

and $n_x\ge ct-L\ge c't$. Letting $t\rightarrow \infty$ in (5.14), we have $\lim _{t\rightarrow \infty ,x\ge ct}Q_t(\phi )=0$ for any $c>\overline{c}_+$.

By similar arguments to the above, we can show that statement (ii) is also valid. $\square$

In view of the above theorem, we may regard $\overline{c}_+$ and $c^*_+$, respectively, as the fastest and slowest rightward spreading speeds for $\{Q_t\}_{t\ge 0}$ on ${\mathcal {C}}_\beta$. If $\overline{c}_+=c^*_+$, then we say that this system admits a single rightward spreading speed.

Next, we address the existence and non-existence of traveling waves in a periodic habitat for the continuous-time semiflow $\{Q_t\}_{t\ge 0}$. Given a continuous-time semiflow $\{Q_t\}_{t\ge 0}$ on ${\mathcal {C}}_\beta$, we say that $V(x-ct,x)$ is an $L$-periodic rightward traveling wave of $\{Q_t\}_{t\ge 0}$ if $V(\cdot +a,\cdot )\in {\mathcal {C}}_\beta$, $\forall a\in \mathbb {R}, \, Q_t[U](x)=V(x-ct,x), \, \forall t\ge 0$, and $V(\xi ,x)$ is an $L$-periodic function in $x$ for any fixed $\xi \in \mathbb {R}$, where $U(x):=V(x,x)$. Moreover, we say that $V(\xi ,x)$ connects $\beta$ to $0$ if $\lim _{\xi \rightarrow -\infty }|V(\xi ,x)-\beta (x)|=0$ and $\lim _{\xi \rightarrow +\infty }|V(\xi ,x)|=0$ uniformly for $x\in \mathbb {R}$.

Since we have only shown the weak compactness (B5) for $\tilde{P}_t$, we cannot directly apply [6, Theorem 4.1] to $\{\tilde{P}_t\}_{t\ge 0}$ on ${\mathcal {X}}_{\beta }$. However, $\{P_t\}_{t\ge 0}$ on ${\mathcal {K}}_{\beta }$ has the compactness because any element in ${\mathcal {K}}_{\beta }$ is defined on the discrete domain. Following the proof of Case 1 in [21, Theorem 4.2] and the argument in [6, Theorem 3.1], we obtain the existence and non-existence of traveling waves for the discrete-time dynamical system $\{P_1^n\}$ on ${\mathcal {K}}_{\beta }$. Thus, the existence and non-existence of traveling waves for the continuous-time dynamical system $\{P_t\}_{t\ge 0}$ on ${\mathcal {K}}_{\beta }$ follows from the arguments in [21, Theorem 4.4]. By similar arguments to those in [21, Theorem 5.3], we can extend [6, Theorem 4.1] to the case of a periodic habitat so that the following result holds true.

Theorem 5.5

Let $\{Q_t\}_{t\ge 0}$ be a continuous-time semifow on ${\mathcal {C}}_\beta$ with $Q_t[0]=0,Q_t[\beta ]=\beta$ for all $t\ge 0, \, \{\tilde{P}_t\}_{t\ge 0}$ be defined as in (5.12), and $c^*_+$ and $\overline{c}_+$ be denoted by (5.13) with $\tilde{P}=\tilde{P}_1$. Suppose that $Q_t$ satisfies (A1)–(A5) for each $t>0$. Then the following statements are valid:

• (1) For any $c\ge c^*_+$, there is an $L$-periodic traveling wave $W(x-ct,x)$ connecting $\beta$ to some equilibrium $\beta _1\in \mathcal {C}^{per}_\beta \backslash \{\beta \}$ with $W(\xi ,x )$ be continuous and nonincreasing in $\xi \in \mathbb {R}$.

• (2) If, in addition, $0$ is an isolated equilibrium of $\{Q_t\}_{t\ge 0}$ in ${\mathcal {C}}^{per}_\beta$, then for any $c\ge \overline{c}_+$, either of the following holds true:

  • (i) there exists an $L$-periodic traveling wave $W(x-ct,x)$ connecting $\beta$ to $0$ with $W(\xi ,x )$ be continuous and nonincreasing in $\xi \in \mathbb {R}$.

  • (ii) $\{Q_t\}_{t\ge 0}$ has two ordered equilibria $\alpha _1$,$\alpha _2 \in {\mathcal {C}}^{per}_\beta \backslash \{0,\beta \}$ such that there exist an $L$-periodic traveling wave $W_1(x-ct,x)$ connecting $\alpha _1$ and $0$ and an $L$-periodic traveling wave $W_2(x-ct,x)$ connecting $\beta$ and $\alpha _2$ with $W_i(\xi ,x ), i=1,2$ be continuous and nonincreasing in $\xi \in \mathbb {R}$.

• (3) For any $c< c^*_+$, there is no $L$-periodic traveling wave connecting $\beta$, and for any $c<\overline{c}_+$, there is no $L$-periodic traveling wave connecting $\beta$ to $0$.