Invasion pinning in a periodically fragmented habitat

Abstract

Biological invasions can cause great damage to existing ecosystems around the world. Most landscapes in which such invasions occur are heterogeneous. To evaluate possible management options, we need to understand the interplay between local growth conditions and individual movement behaviour. In this paper, we present a geometric approach to studying pinning or blocking of a bistable travelling wave, using ideas from the theory of symmetric dynamical systems. These ideas are exploited to make quantitative predictions about how spatial heterogeneities in dispersal and/or reproduction rates contribute to halting biological invasion fronts in reaction–diffusion models with an Allee effect. Our theoretical predictions are confirmed using numerical simulations, and their ecological implications are discussed.

Appendix: Proof of Theorem 2.5

We rewrite (17) as

(29)

The existence of the two-dimensional center manifold $\mathcal {S}_{\varepsilon }$ follows from Sandstede et al. (1997) and LeBlanc and Roy (2013). To get the dynamics of (29) restricted to $\mathcal {S}_{\varepsilon }$, we follow (Sandstede et al. 1997) and use the following parametrization of a local neighborhood of the relative equilibrium at $\varepsilon =0$:

$$\begin{aligned} \begin{pmatrix} u \\ c \end{pmatrix} = \mathcal {T}_{a} \left[ \begin{pmatrix} u^* \\ 0 \end{pmatrix} + c \begin{pmatrix} 0 \\ 1 \end{pmatrix} + \varepsilon \begin{pmatrix} w \\ 0 \end{pmatrix} \right] , \end{aligned}$$

(30)

where w is a bounded uniformly continuous function with bounded and uniformly continuous derivative.

Following closely LeBlanc and Roy (2013), we substitute parameterization (30) into system (29), and obtain

$$\begin{aligned} {\dot{a}}(t) \mathcal {T}_{a(t)} \left( \begin{pmatrix} \xi u^* \\ 0 \end{pmatrix} + O(\varepsilon ) \right)&= \mathcal {T}_{a(t)} \left( \begin{pmatrix} \mathcal {A} u^* \\ 0 \end{pmatrix} + \begin{pmatrix} \varepsilon \mathcal {A} w \\ 0 \end{pmatrix} \right) \\&\quad + \mathcal {T}_{a(t)} \begin{pmatrix} \mathcal {F}(u^*+\varepsilon w,c) \\ 0 \end{pmatrix} \\&\quad + \varepsilon \begin{pmatrix} \mathcal {G}(\mathcal {T}_{a} (\xi u^* + \varepsilon \xi w), \mathcal {T}_{a} (u^* + \varepsilon w), x, \varepsilon ) \\ 0 \end{pmatrix} . \end{aligned}$$

Applying $\mathcal {T}_{-a(t)}$ to both sides of the equation above gives

$$\begin{aligned} {\dot{a}}(t) \left( \begin{pmatrix} \xi u^* \\ 0 \end{pmatrix} + O(\varepsilon ) \right)&= \left( \begin{pmatrix} \mathcal {A} u^* \\ 0 \end{pmatrix} + \begin{pmatrix} \varepsilon \mathcal {A} w \\ 0 \end{pmatrix} \right) + \begin{pmatrix} \mathcal {F}(u^*+\varepsilon w,c) \\ 0 \end{pmatrix} \\&\quad + \varepsilon \begin{pmatrix} \mathcal {T}_{-a(t)} \mathcal {G}(\mathcal {T}_{a} (\xi u^* + \varepsilon \xi w), \mathcal {T}_{a} (u^* + \varepsilon w), x, \varepsilon ) \\ 0 \end{pmatrix}. \end{aligned}$$

Since $u^*$ is the wave profile for the unperturbed system know that

$$\begin{aligned} \begin{pmatrix} \mathcal {A} u^* \\ 0 \end{pmatrix} = - \begin{pmatrix} \mathcal {F}(u^*,0) \\ 0 \end{pmatrix}. \end{aligned}$$

From the definition of the linear operator in (13), we have

$$\begin{aligned} L \begin{pmatrix} 0 \\ c \end{pmatrix} = \begin{pmatrix} \mathcal {F}_c(u^*,0)c \\ 0 \end{pmatrix},&L \begin{pmatrix} \varepsilon w \\ 0 \end{pmatrix} = \begin{pmatrix} \varepsilon \mathcal {A} w + \varepsilon \mathcal {F}_u (u^*,0) w \\ 0 \end{pmatrix}, \end{aligned}$$

and so we write the previous expression as

$$\begin{aligned} {\dot{a}}(t) \left( \begin{pmatrix} \xi u^* \\ 0 \end{pmatrix} + O(\varepsilon ) \right)&= L \begin{pmatrix} 0 \\ c \end{pmatrix} + \varepsilon L \begin{pmatrix} w \\ 0 \end{pmatrix} + \begin{pmatrix} \mathcal {F}(u^*+\varepsilon w,c) \\ 0 \end{pmatrix} \\&\quad - \begin{pmatrix} \mathcal {F}(u^*,0) \\ 0 \end{pmatrix} - \begin{pmatrix} \mathcal {F}_c(u^*,0)c \\ 0 \end{pmatrix} - \varepsilon \begin{pmatrix} \mathcal {F}_u(u^*,0)w \\ 0 \end{pmatrix} \\&\quad + \varepsilon \begin{pmatrix} \mathcal {T}_{-a(t)} \mathcal {G}(\mathcal {T}_{a(t)} (\xi u^* + \varepsilon \xi w), \mathcal {T}_{a(t)} (u^* + \varepsilon w), x, \varepsilon ) \\ 0 \end{pmatrix} . \end{aligned}$$

Hypothesis 2.2, along with the fact that w is a uniformly continuous function and bounded with respect to $\xi$, tells us that as $(\varepsilon ,c) \rightarrow (0,0)$ the difference

$$\begin{aligned} \mathcal {T}_{-a} \mathcal {G}( \mathcal {T}_{a} (\xi u^* + \varepsilon \xi w), \mathcal {T}_{a} (u^* + \varepsilon w), x, \varepsilon ) - \mathcal {T}_{-a} \mathcal {G}( \mathcal {T}_{a} (\xi u^*), \mathcal {T}_{a}u^*, x, 0) \end{aligned}$$

(31)

tends to zero. Thus we introduce the function

$$\begin{aligned} q(a,c,\varepsilon )&= \frac{1}{\varepsilon } (\mathcal {F}(u^* + \varepsilon w,c) - \mathcal {F}(u^*,c)) - \mathcal {F}_u(u^*,c)w \\&\quad + \mathcal {F}_u(u^*,c)w - \mathcal {F}_u(u^*,0)w \\&\quad + \mathcal {T}_{-a} \mathcal {G}( \mathcal {T}_{a} (\xi u^* + \varepsilon \xi w), \mathcal {T}_{a} (u^* + \varepsilon w), x, \varepsilon ) \\&\quad - \mathcal {T}_{-a} \mathcal {G}( \mathcal {T}_{a} (\xi u^*), \mathcal {T}_{a}u^*, x, 0) \end{aligned}$$

where $q(a,c,\varepsilon ) \rightarrow 0$ as $(\varepsilon ,c) \rightarrow (0,0)$. Adding and subtracting $(\varepsilon q, 0 )^T$ from the equation gives us

$$\begin{aligned} {\dot{a}}(t) \left( \begin{pmatrix} \xi u^* \\ 0 \end{pmatrix} + O(\varepsilon ) \right)&= L \begin{pmatrix} 0 \\ c \end{pmatrix} + \varepsilon L \begin{pmatrix} w \\ 0 \end{pmatrix} + \varepsilon \begin{pmatrix} q \\ 0 \end{pmatrix} \\&\quad + \varepsilon \begin{pmatrix} \mathcal {T}_{-a} \mathcal {G}( \mathcal {T}_{a} (\xi u^*), \mathcal {T}_{a}u^*, x, 0) \\ 0 \end{pmatrix} . \end{aligned}$$

Applying the projection P to the above equation and using the fact that $L(0,c)^T = c(\xi u^*,0)^T$ and $P(\xi u^*,0) = (\xi u^*,0) = \psi _1$ gives us

$$\begin{aligned} {\dot{a}} ( \psi _1 + O(\varepsilon )) = c \psi _1 + \varepsilon P \begin{pmatrix} \mathcal {T}_{-a} \mathcal {G}( \xi \mathcal {T}_{a} u^*, \mathcal {T}_{a}u^*, x, 0) \\ 0 \end{pmatrix} + \varepsilon P \begin{pmatrix} q \\ 0 \end{pmatrix}. \end{aligned}$$

Projecting onto $\psi _1$ gives

$$\begin{aligned} {\dot{a}} ( \langle \psi _1, \psi _1 \rangle (1 + O(\varepsilon ) ))&= c \langle \psi _1, \psi _1 \rangle + \varepsilon \left\langle \begin{pmatrix} \mathcal {T}_{-a(t)} \mathcal {G}( \xi \mathcal {T}_{a} u^*, \mathcal {T}_{a}u^*, x, 0) \\ 0 \end{pmatrix}, \psi _1 \right\rangle \\&\quad + \varepsilon \left\langle \begin{pmatrix} q \\ 0 \end{pmatrix}, \psi _1 \right\rangle . \end{aligned}$$

Since $\varepsilon$ and c are assumed to be small we can neglect the higher order terms that appear as a result of dividing by $1+O(\varepsilon )$. Letting

$$\begin{aligned} r(a)= & {} - \frac{\left\langle \begin{pmatrix} \mathcal {T}_{-a} \mathcal {G}( \xi \mathcal {T}_{a} u^*, \mathcal {T}_{a}u^*, x, 0) \\ 0 \end{pmatrix}, \psi _1 \right\rangle }{ \langle \psi _1, \psi _1 \rangle } , \nonumber \\ {\hat{q}}= & {} \frac{\left\langle \begin{pmatrix} q \\ 0 \end{pmatrix}, \psi _1 \right\rangle }{ \langle \psi _1, \psi _1 \rangle }, \end{aligned}$$

(32)

we get the system of differential equations

$$\begin{aligned} a_t&= c - \varepsilon r(a) + \varepsilon {\hat{q}} (a,c,\varepsilon ), \\ c_t&= 0, \end{aligned}$$

where ${\hat{q}} (a,c,\varepsilon ) \rightarrow 0$ uniformly as $(\varepsilon ,c) \rightarrow (0,0)$.

Now, an application of the implicit function theorem tells us that for $(\varepsilon , c)$ close enough to (0, 0) there exists a curve of equilibria such that $a_t = 0$. This curve is the graph of the function $c = \varepsilon r(a) + \varepsilon \sigma (a,\varepsilon )$ where $\sigma$ is some bounded function with the property $\sigma (a,0) = 0$ (LeBlanc and Roy 2013; Roy 2013). Thus, it follows that if

$$\begin{aligned} \inf _{a \in \mathbb {R}} \{ r(a) + \sigma (a, \varepsilon ) \}< \frac{c}{\varepsilon } < \sup _{a \in \mathbb {R}} \{ r(a) + \sigma (a, \varepsilon ) \} \end{aligned}$$

(33)

then at some point between $a \in \mathbb {R}$ we expect $a_t = 0$. This means that at some point in time the travelling front will reach an equilibrium point at which the wave is stationary. In other words, propagation of the wave is blocked. We remark that for $(\varepsilon , c)$ close enough to (0, 0) the term $\sigma (a, \varepsilon )$ becomes negligible. Hence, close to (0, 0), the parameter region where we expect propagation failure is approximated by $\inf _{a \in \mathbb {R}} \{r(a)\}$ and $\sup _{a \in \mathbb {R}} \{r(a)\}$. $\square$