Two-dimensional traveling waves arising from planar front interaction in a three-species competition-diffusion system

Abstract

From the ecological point of view of competitor-mediated coexistence, we consider a three-species competition-diffusion system which models one exotic competing species W invading the native system of two strongly competing species U and V. Even if W is weaker than the native species, it is found that both competitive exclusion and competitor-mediated coexistence may occur, depending on the growth rate of W. Firstly, we show that there are two different planarly stable traveling waves involving the species (U, V) and (U, V, W), respectively. Studying the interaction of these waves in one dimension offers insight about whether or not coexistence occurs in two-dimensional domains. However, when planar fronts collide at a certain angle, phenomena which are not immediately reducible to the one-dimensional case can be observed, such as wedge-shaped patterns. In particular, the interaction between a radial and a planar front produces different types of moving patterns which seem to tend to truly two-dimensional traveling waves, such as wedge-, zipper- and biwedge-shaped traveling waves. All these waves arise from the interaction of stable planar fronts and their velocities can be computed by only knowing their asymptotic behaviour. We also show how such waves exist only if the angle between the planar fronts is under a certain critical value.

Appendices

Appendix A: Proof of the velocity formula for asymptotically planar traveling waves

We will now give a more formal proof for the relation $\mathbf {c} - \mathbf {c}_p \perp \mathbf {c}_p$, where $\mathbf {c}$ is the velocity of a two-dimensional traveling wave whose shape in one asymptotic direction tends to that of a planar front with velocity $\mathbf {c}_p$. This relation was used in Sect. 3.2 to derive the formulas (11)–(12) for the velocity of asymptotically planar traveling waves, of which wedge-shaped traveling waves are an example.

Theorem

Consider the following equation in the variables $\mathbf {u}:{\mathbb {R}}^{n}\rightarrow {\mathbb {R}}^{m}$ and $\mathbf {c}\in {\mathbb {R}}^{n}$

$$\begin{aligned} \mathbf {D}\varDelta \mathbf {u} + \nabla \mathbf {u}\ \mathbf {c} + \mathbf {f}(\mathbf {u}) = \mathbf {0} \quad \text {in }{\mathbb {R}}^{n}, \end{aligned}$$

(15)

where $\mathbf {D}\in {\mathbb {R}}^{m\times m}$ is a positive diagonal matrix, $\nabla \mathbf {u}:{\mathbb {R}}^{n}\rightarrow {\mathbb {R}}^{m\times n}$ is the Jacobian of $\mathbf {u}$ and $\mathbf {f}:{\mathbb {R}}^{m}\rightarrow {\mathbb {R}}^{m}$.

Suppose that:

• 1. $\left( \mathbf {u},\mathbf {c}\right)$ and $\left( \mathbf {u}_{p},\mathbf {c}_{p}\right)$ are solutions of (15) such that $\mathbf {c}_{p}\ne 0$ and $\mathbf {u},\mathbf {u}_{p}\in \mathscr {C}^2\left( {\mathbb {R}}^{n},{\mathbb {R}}^{m}\right)$;

• 2. $\mathbf {f}:{\mathbb {R}}^m\rightarrow {\mathbb {R}}^m$ is Lipschitz continuous on the set $\mathbf {u}\left( {\mathbb {R}}^n\right) \cup \mathbf {u}_{p}\left( {\mathbb {R}}^n\right)$;

• 3. for every $\varepsilon >0$ there exists $\varOmega _{\varepsilon }\subset {\mathbb {R}}^{n}$ bounded such that

  $$\begin{aligned} \left\| \left( \mathbf {u}-\mathbf {u}_{p}\right) |_{\varOmega _{\varepsilon }}\right\| _{\mathscr {C}^2\left( \varOmega _{\varepsilon },{\mathbb {R}}^{m}\right) }<\varepsilon . \end{aligned}$$

  (16)

• 4. there exists a constant $\delta >0$ such that for every $\varepsilon >0$ there exists $\mathbf {\alpha }_\varepsilon \in {\mathbb {R}}^m$, $\mathbf {n}_\varepsilon \in {\mathbb {R}}^n$ and $x_\varepsilon \in \varOmega _\varepsilon$ satisfying

  $$\begin{aligned} \nabla \mathbf {u}_p(x_\varepsilon )= & {} \mathbf {\alpha }_\varepsilon \mathbf {n}_\varepsilon ^{t}, \end{aligned}$$

  (17)
  $$\begin{aligned} \left| \mathbf {n}_\varepsilon \right|= & {} 1, \end{aligned}$$

  (18)
  $$\begin{aligned} \left| \mathbf {\alpha }_\varepsilon \right|> & {} \delta , \end{aligned}$$

  (19)

  where $\left| \cdot \right|$ is the Euclidean norm.

Then, there exists a sequence $\varepsilon _k \rightarrow 0$ and $\mathbf {n}\in {\mathbb {R}}^n$ such that $\mathbf {n}_{\varepsilon _k}\rightarrow \mathbf {n}$ and $\left| \mathbf {n}\right| = 1$. Moreover, we have that $\mathbf {c} - \mathbf {c}_{p} \perp \mathbf {n}$.

Proof

By (18) we have that $\left\{ \mathbf {n}_\varepsilon \right\} _{\varepsilon >0}$ is contained in the compact boundary of the unit ball of ${\mathbb {R}}^n$. Thus existence of a limit vector $\mathbf {n}\in {\mathbb {R}}^n$ such that $\left| \mathbf {n}\right| = 1$ is immediately established.

Subtracting the equations obtained from (15) by substitution with $\left( \mathbf {u},\mathbf {c}\right)$ and $\left( \mathbf {u}_{p},\mathbf {c}_{p}\right)$ respectively, we get

$$\begin{aligned} \mathbf {D}\left( \varDelta \mathbf {u} - \varDelta \mathbf {u}_{p}\right) + \nabla \mathbf {u}\mathbf {c} - \nabla \mathbf {u}_{p}\mathbf {c}_{p} + \mathbf {f}(\mathbf {u})-\mathbf {f}(\mathbf {u}_{p}) = \mathbf {0} \quad \text {in }{\mathbb {R}}^n. \end{aligned}$$

If we add $\nabla \mathbf {u}_p\mathbf {c}$ to both sides, we obtain after rearrangement

$$\begin{aligned} \nabla \mathbf {u}_p ( \mathbf {c} - \mathbf {c}_{p} ) = \mathbf {D}(\varDelta \mathbf {u}_p - \varDelta \mathbf {u}) + (\nabla \mathbf {u}_p - \nabla \mathbf {u})\mathbf {c} + \mathbf {f}(\mathbf {u}_p)-\mathbf {f}(\mathbf {u}) \quad \text {in }{\mathbb {R}}^n. \end{aligned}$$

Now fix $\varepsilon >0$ and consider only $x\in \varOmega _{\varepsilon }$. We take the Euclidean norm on both sides. By Lipschitz continuity of $\mathbf {f}$ and (16), we get

$$\begin{aligned} \left| \nabla \mathbf {u}_p \left( \mathbf {c} - \mathbf {c}_{p} \right) \right| < K \varepsilon \quad \text {in }\varOmega _{\varepsilon } \end{aligned}$$

(20)

for $K = {{\mathrm{tr}}}\mathbf {D} + K' \left| \mathbf {c}\right| + L$, where ${{\mathrm{tr}}}$ is the matrix trace operator, $K'$ is due to the conversion from the operatorial norm of $\nabla \mathbf {u}_p - \nabla \mathbf {u}$ to its maximum norm and L is the Lipschitz constant for $\mathbf {f}$.

Evaluating (20) at $x_\varepsilon$ and using (17) we get,

$$\begin{aligned} \left| \mathbf {\alpha }_\varepsilon \mathbf {n}_\varepsilon ^{t} \left( \mathbf {c} - \mathbf {c}_{p} \right) \right| < K \varepsilon . \end{aligned}$$

By applying the lower bound on $\left| \mathbf {\alpha }_\varepsilon \right|$ given in (19), we obtain

$$\begin{aligned} K \varepsilon > \left| \mathbf {\alpha }_\varepsilon \right| \left| \mathbf {n}_\varepsilon ^{t} \left( \mathbf {c} - \mathbf {c}_{p} \right) \right| > \delta \left| \mathbf {n}_\varepsilon ^{t} \left( \mathbf {c} - \mathbf {c}_{p} \right) \right| . \end{aligned}$$

Since this inequality holds for every $\varepsilon >0$, by considering it on the sequence $\left\{ \varepsilon _k \right\} _k$ and taking the limit we obtain $\mathbf {n}^{t} \left( \mathbf {c} - \mathbf {c}_{p} \right) = 0$, which is equivalent to saying that $\mathbf {c} - \mathbf {c}_{p} \perp \mathbf {n}$. $\square$

Corollary

In the setting of the above Theorem, consider the following additional hypothesis: $\mathbf {u}_p$ is a planar front and $\mathbf {c}_p = \left| \mathbf {c}_p \right| \mathbf {n}$, where $\mathbf {n}$ is the unit vector normal to the front. Then,

$$\begin{aligned} \mathbf {c} - \mathbf {c}_{p} \perp \mathbf {c}_{p}. \end{aligned}$$

(21)

Proof

Note that we may denote the normal to the front with the symbol $\mathbf {n}$ without any abuse of notation. Indeed, we have that $\mathbf {n}_\varepsilon = \mathbf {n}$ for all $\varepsilon >0$ due to the front being planar. By the above Theorem and by our choice of $\mathbf {c}_p$ we get

$$\begin{aligned} \mathbf {n}^{t} \left( \mathbf {c} - \mathbf {c}_{p} \right) = \frac{\mathbf {c}_p^t}{\left| \mathbf {c}_p \right| } \left( \mathbf {c} - \mathbf {c}_{p} \right) = 0, \end{aligned}$$

from which (21) immediately follows. $\square$

Remark

The lower bound on the norm of $\nabla \mathbf {u}_{p}(x_\varepsilon )$ is essential. Otherwise, we could take any two traveling wave solutions sharing one asymptotic equilibrium state and the above Theorem would apply. For example, it could be applied to two different orientations of the same planar front, which is clearly impossible.

Appendix B: Numerical solution of the traveling wave equation in two dimensions

As stated at the beginning of Sect. 3.3, we need to find a numerical method appropriate to the computation of traveling wave solutions. We are interested both in solving the stationary problem given by the traveling equation (14) and the time-evolution problem given by (1). The main difficulty lies in the fact that the domain in which these problems are posed is the whole space (in our case ${\mathbb {R}}^2$), whereas numerical methods are in general limited to bounded domains. However, in the case of traveling wave solutions the asymptotic behavior at spatial infinity is given. This means that a bounded domain $\varOmega$ is sufficient if it is large enough to cover the area where the solution differs substantially from the asymptotic behaviour, which is where the non-trivial and interesting features of the wave lie. The faster the solution converges to the asymptotic state for $\left| x \right| \rightarrow \infty$, the smaller the domain $\varOmega$ can be taken. Apart from the choice of $\varOmega$, we must determine which boundary conditions the numerical solution should satisfy on $\partial \varOmega$. These conditions should be chosen such that also the true traveling wave solution satisfies them, at least approximately.

As a simple and well-known example, consider the traveling wave equation (14) in one spatial dimension. In order to specify the asymptotic behaviour, it is sufficient to set the two limit values of the solution for $x\rightarrow \pm \infty$, as it is done for example in the problem (5)–(6). In many cases, by choosing $\varOmega =(-L,L)$ with $L>0$ large enough, we can capture all the most relevant information about the one-dimensional traveling wave, since such information is often concentrated around the wave front. Outside $\varOmega$, the solution is approximately equal to the asymptotic equilibrium states. This means that the exact traveling wave not only approximately satisfies a couple of Dirichlet boundary conditions on $\partial \varOmega =\{-L,L\}$, but it is also approximately compatible with Neumann zero-flux boundary conditions, since its derivatives must tend to 0 for $x\rightarrow \pm \infty$.

Now consider the two-dimensional case. The asymptotic behaviour consists in two or more fronts which propagate independently. While in ${\mathbb {R}}$ the possible choices for the bounded domain $\varOmega$ are essentially intervals, in ${\mathbb {R}}^2$ a great variety of shapes is possible in theory. As for the boundary conditions, Dirichlet type conditions can be used for any large enough choice of $\varOmega$, since the asymptotic behaviour is known. A possible drawback is that the asymptotic state must be computed beforehand. In our case, since the traveling wave is asymptotically planar, this computation can be reduced to a one-dimensional traveling wave equation for each front. While such simplification helps in term of computational time, using different discretizations for the one- and two-dimensional problems may introduce an additional source of error. Moreover, when studying the interaction of different waves, the value at the boundary changes with time, making Dirichlet conditions inapplicable.

In order to derive boundary conditions other than Dirichlet, the asymptotic behaviour of the gradient of the traveling wave should be studied. Far from all the fronts the solution tends to one of the equilibrium states and thus its gradient tends to $\mathbf {0}$. On the other hand, for a point $x\in {\mathbb {R}}^2$ in the vicinity of a front the solution asymptotically satisfies

$$\begin{aligned} \frac{\partial \mathbf {u}}{\partial \nu _f(x)}(x) = \mathbf {0}, \end{aligned}$$

(22)

where $\nu _f(x)$ is the unit vector perpendicular to the asymptotic direction of propagation of the front nearest to x. Note that points for which the gradient is approximately $\mathbf {0}$ satisfy also condition (22). Then, we may take (22) as the boundary condition for the whole $\partial \varOmega$. This is a case of an oblique boundary condition (see for example [16]). This choice reduces to the Neumann zero-flux boundary conditions in the case $\nu _f(x)=\nu (x)$ for every $x\in \partial \varOmega$, where $\nu (x)$ is the unit vector normal to $\partial \varOmega$ at x. This is also true if we let the shape of $\partial \varOmega$ violate $\nu _f=\nu$ far from all the fronts, since the gradient is approximately $\mathbf {0}$ there: this fact allows $\partial \varOmega$ to be a closed curve. In practice, this means that, if we want to use zero-flux boundary conditions, $\varOmega$ must be a polygon whose edges are normal to the front they intersect.

Thus, in the case of wedge-shaped traveling waves and rectangular domains as those used in Figs. 3, 4, 5 and 6, zero-flux boundary conditions do not yield the correct result: all fronts are forced to gradually become perpendicular to the boundary, which results in the sides of the wedge becoming parallel and eventually colliding, leading to the disappearance of the wedge feature. While a rectangular domain may be employed by implementing the oblique boundary conditions in the numerical method, we preferred to use the simpler zero-flux conditions by adopting a polygonal domain whose edges are perpendicular to the fronts. For example, in our computations for wedge-shaped traveling waves we use a pentagonal domain with the two topmost edges normal to the wedge sides, the base orthogonal to the wedge bisector and the remaining two edges parallel to the fronts. More complicated domains are needed for zipper- and biwedge-shaped traveling waves, as can be seen for example in Fig. 28. In order to deal with non-rectangular domains, we employ a continuous piecewise-linear finite element discretization using the software FreeFem++ [10]. Adaptive mesh refinement is also adopted in order to reduce the size of the discretized system.

Other well-known difficulties concerning the traveling equation (14) are that $\mathbf {c}$ is an unknown to be solved for, which leads to a non-square linear system in the discretized problem, and the solution is no longer unique, since the traveling wave equation is invariant with respect to translation. These two problems can be solved at the same time by specifying additional constraints. In particular, we enforce integral phase conditions [3], which are also used by the popular continuation package AUTO [6] in the context of numerical computation of orbits of ODEs [5] (in one dimension traveling waves are essentially homoclinic orbits of the ODE associated to the traveling wave equation). Since in our case the position of two planar fronts uniquely determines the position of the full traveling wave, we can apply two one-dimensional phase conditions to two edges of $\partial \varOmega$ instead of one two-dimensional phase condition to the whole $\varOmega$.

We are also interested in solving the time-evolution problem (1), especially in order to obtain an approximate solution to bootstrap the Newton solver for the associated two-dimensional traveling wave equation. Here the difficulty is that traveling waves leave the computational domain $\varOmega$ in a finite time. As usual in such cases, we employ a moving reference frame, which is equivalent to consider the time-evolution problem associated to the traveling wave equation (14). Since the velocity of the traveling wave is not known a priori, we have to adapt the frame velocity to the current velocity of the wave in order to keep it centered inside $\varOmega$. To this purpose, we periodically track the points where two of the planar fronts intersect the boundary and use their velocities to compute the velocity of the whole wave by applying formulas (11)–(12). This expedient is adequate for our needs, though we remark that more elegant, but more complicated, methods may be employed. For example, the frame velocity can be considered as a variable evolving with time and by imposing the phase condition at all times a partial differential algebraic equation (PDAE) can be obtained. Using the method of lines this PDAE can be reduced to a differential algebraic equation (DAE) and solved numerically. This approach, which is called freezing the traveling wave, has been proposed independently in [2] and [25]. An introduction to the theory and numerical methods for DAEs can be found in [24] instead.