Abstract
We propose here a new model to describe biological invasions in the plane when a strong diffusion takes place on a line. We establish the main properties of the system, and also derive the asymptotic speed of spreading in the direction of the line. For low diffusion, the line has no effect, whereas, past a threshold, the line enhances global diffusion in the plane and the propagation is directed by diffusion on the line. It is shown here that the global asymptotic speed of spreading in the plane, in the direction of the line, grows as the square root of the diffusion on the line. The model is much relevant to account for the effects of fast diffusion lines such as roads on spreading of invasive species.
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Notes
A subsolution (resp. supersolution) is a couple satisfying the system (in the classical sense) with the \(=\) signs replaced by \(\le \) (resp. \(\ge \)) signs, which is also continuous up to time \(0\).
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Acknowledgments
This study was supported by the French “gence Nationale de la Recherche” through the project PREFERED (ANR 08-BLAN-0313). H.B. was also supported by an NSF FRG grant DMS-1065979. L.R. was partially supported by the Fondazione CaRiPaRo Project “Nonlinear Partial Differential Equations: models, analysis, and control-theoretic problems”.
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Dedicated in friendship to Odo Diekmann
Appendices
Appendix A: Existence result for the Cauchy problem
Proof of the existence part of Proposition 3.1. We prove the result for an initial datum \((u_0,v_0)\) which is locally Hölder continuous, together with its derivatives up to order 2, and satisfies the compatibility condition
The regularity of the initial datum is therefore inherited by the solution of the Cauchy problem for all time \(t\ge 0\). The case of a merely continuous initial datum can then be handled by a standard regularization technique (see, e.g., Ladyzhenskaya et al. 1968).
We will obtain a solution to (3.1)–(3.2) as the limit of a subsequence of solutions \(((u_n,v_n))_n\) of the following problems:
starting from \(v_0\).
Step 1 Solvability of (9.1), (9.2).
We say that a function \(w(z,t)\) has admissible growth in \(z\) if it satisfies \(|w(z,t)|\le \beta e^{\sigma |z|^2}\), for some \(\sigma ,\beta >0\). It is well known that the linear Cauchy problem is uniquely solvable in the class of functions with admissible growth in the space variable. If \(v_{n-1}\) is a continuous function with admissible growth, then problem (9.1) admits a unique classical solution \(u_n\) with admissible growth. In order to solve (9.2), notice that it can be reduced to a homogeneous system by replacing \(v_n\) with \(v_n-v_0-\mu (u_n-u_0)\). It then follows from the standard parabolic theory that it admits a unique classical solution with admissible growth. Let \(((u_n,v_n))_n\) denote the family of solutions constructed in this way, starting from \(v_0\).
Step 2 \(L^\infty \) estimates.
We show, with a recursive argument, that
The property trivially holds for \(n=0\). Assume that it holds for some value \(n-1\). Since \(0\) and \(\frac{1}{\mu }H\) are respectively a sub and a supersolution of (9.1), the comparison principle yields \(0\le u_n\le \frac{1}{\mu }H\). It then follows that \(H\) is a supersolution of (9.2), whence \(0\le v_n\le H\).
Step 3 \(W^{2,1}_p\) estimates.
By step 1 we know that \(0\le v_{n-1}\le H\). Thus, applying the local boundary estimates to (9.1) we infer that, for any given \(\rho ,T>0\) and \(1<p<\infty \),
where \(B_\rho \) denotes the \(N\)-dimensional ball of radius \(\rho \) and centre \(0\) and \(C\) is a constant only depending on \(N\), \(D\), \(q\), \(\mu \), \(\rho \), \(T\), \(p\) and \(\Vert u_0\Vert _{W^2_p(B_{\rho +2})}\) (and not on \(n\)). Set \(Q_\rho :=B_\rho \times (0,\rho )\). Since \(0\le v_n\le H\) too, the estimates yield
with \(C^{\prime }\) only depending on \(N\), \(d\), \(r\), \(\mu \), \(\rho \), \(T\), \(p\), \(\Vert f\Vert _\infty \), \(\Vert u_0\Vert _{W^2_p(B_{\rho +1})}\) and \(\Vert v_0\Vert _{W^2_p(Q_{\rho +1})}\). This shows that the \((u_n)_n\) and \((v_n)_n\) are uniformly bounded in \(W^{2,1}_p(B_\rho \times (0,T))\) and \(W^{2,1}_p(Q_\rho \times (0,T))\) respectively.
Step 4 Existence of a solution. Now that we know that \((u_n)_n\) and \((v_n)_n\) are uniformly bounded in compact sets with respect to the \(W^{2,1}_p\) norm, taking \(p>N+1\) and using the Morrey inequality, we infer that this is also true with respect to the \(C^{\alpha }\) norm, for some \(0<\alpha <1\). Then, by the Schauder estimates, the time derivative and the space derivatives up to order \(2\) are uniformly Hölder continuous in compact sets too. As a consequence, \(((u_n,v_n))_n\) converges (up to subsequences) in \(C^{2,1}_{loc}\) to some \((u,v)\). Passing to the limit as \(n\rightarrow \infty \) in (9.1), (9.2) we eventually find that \((u,v)\) satisfies (3.1)–(3.2). Form step 1 we know that \(u\) and \(v\) are bounded and nonnegative.\(\square \)
Appendix B: The equation \(h^L(c,\beta )=0\)
In this section we describe in detail how, for \(\xi >0\) small enough, Eq. (6.4) admits two solutions close to \(\tau _+\) and \(\tau _-\) respectively. We recall that \(\tau _\pm =\pm i\sqrt{(e/a)\xi }+O(\xi )\) are the roots of the trinomial \(g(\tau ):=a\tau ^2+d\xi \tau +e\xi \). Let us focus on \(\tau _+\), the other case being analogous.
Let \(B\) be the ball of radius \(A\xi \), centred at \(\tau _+\); \(A\) large and to be adjusted. For \(\tau \in \partial B\), we have
for \(\xi \) small and \(h>0\) independent of \(A\). On the other hand we have \(|\varphi (\tau ,\xi )|\le C\xi ^{3/2}+O(\xi ^2)\), where now \(O(\xi ^2)\) depends on \(A\). We can therefore choose \(A\) large enough and then \(\xi \) small enough in such a way that \(|g|>|\varphi |\) on \(\partial B\). Since \(g\) and \(\varphi \) are holomorphic, by Rouché’s theorem the Eq. (6.4) has the same number of solutions in \(B\) as \(g=0\), that is \(1\). Notice that such a solution has positive imaginary part proportional to \(\sqrt{\xi }\) and real part of order \(\xi \).
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Berestycki, H., Roquejoffre, JM. & Rossi, L. The influence of a line with fast diffusion on Fisher-KPP propagation. J. Math. Biol. 66, 743–766 (2013). https://doi.org/10.1007/s00285-012-0604-z
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DOI: https://doi.org/10.1007/s00285-012-0604-z