Spreading speeds and traveling waves for monotone systems of impulsive reaction-diffusion equations: application to tree-grass interactions in fire-prone savannas

J. Banasiak; Y. Dumont; I. V. Yatat Djeumen

Home/Archive/arXiv

arXiv · 1908.09909

Spreading speeds and traveling waves for monotone systems of impulsive reaction-diffusion equations: application to tree-grass interactions in fire-prone savannas

J. Banasiak·Y. Dumont·I. V. Yatat Djeumen

View original source ↗Read source document ↗Export metadata ↓

Abstract

Many systems in life sciences have been modeled by reaction-diffusion equations. However, under some circumstances, these biological systems may experience instantaneous and periodic perturbations (e.g. harvest, birth, release, fire events, etc) such that an appropriate formalism is necessary, using, for instance, impulsive reaction-diffusion equations. While several works tackled the issue of traveling waves for monotone reaction-diffusion equations and the computation of spreading speeds, very little has been done in the case of monotone impulsive reaction-diffusion equations. Based on vector-valued recursion equations theory, we aim to present in this paper results that address two main issues of monotone impulsive reaction-diffusion equations. First, they deal with the existence of traveling waves for monotone systems of impulsive reaction-diffusion equations. Second, they allow the computation of spreading speeds for monotone systems of impulsive reaction-diffusion equations. We apply our methodology to a planar system of impulsive reaction-diffusion equations that models tree-grass interactions in fire-prone savannas. Numerical simulations, including numerical approximations of spreading speeds, are finally provided in order to illustrate our theoretical results and support the discussion.

Explore related subjects

math.DS math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

J. Banasiak, Y. Dumont, I. V. Yatat Djeumen. 2019-08-26. Spreading speeds and traveling waves for monotone systems of impulsive reaction-diffusion equations: application to tree-grass interactions in fire-prone savannas. https://arxiv.org/abs/1908.09909

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Toric Differential Inclusions and a Proof of the Global Attractor Conjecture

The global attractor conjecture says that toric dynamical systems have a globally attracting point (up to linear conservation relations), or equivalently, complex balanced systems have a globally attracting point within each stoichiometric compatibility class. A proof of this conjecture implies that a large class of nonlinear dynamical systems on the positive orthant have very simple and stable dynamics. The conjecture originates from the 1972 breakthrough work by Fritz Horn and Roy Jackson, and was formulated in its current form by Horn in 1974. Toric dynamical systems can be embedded into toric differential inclusions. We show that each bounded positive solution of a toric differential inclusion is contained in an invariant region that prevents it from approaching the boundary of the positive orthant. We use this result to prove the global attractor conjecture. In particular, it follows that all detailed balanced mass action systems and all deficiency zero weakly reversible systems have the global attractor property.

math.DS↗

arXiv2026

Pinched Arnol'd tongues for Families of circle maps

We prove that generically for a family of circle maps \begin{equation*} f_{b, ω} (x) = x + ω+ b\, ϕ(x) \end{equation*} with $ϕ$ a piecewise linear forcing with $k>2$ breakpoints there is no pinching in any of its Arnol'd tongues. This is in contrast to a theorem of Campbell, Galeeva, Tresser, and Uherka who showed that with two break points there are always multiple pinchpoints in its rational tongues. We also prove that the absence of pinching is generic for Lipschitz and $C^r$ ($r>0$) forcing. The family $f_{b, ω}$ is used as a simple model for a periodically forced oscillator. The rational tongue $T_{p/q}$ represents parameter values where the system is mode-locked into a $p/q$-periodic response. The pinching of the tongues to a point represents parameter values where the system's periodic response is unstable to all perturbations in the frequency parameter $ω$. The theorems in this paper show that typically this type of instability does not occur in the families under consideration.

math.DS↗

arXiv2026

Mostly nonuniformly sectional expanding systems

We introduce the notion of \emph{mostly nonuniform sectional expanding} (MNUSE) for singular flows which encompasses the notions of sectional hyperbolicity, asymptotically sectional and multisingular hyperbolicity. We construct examples of a vector field of class $C^r, r \ge 1$, whose flow exhibits a nonuniformly sectional hyperbolic set satisfying MNUSE, which is neither sectional hyperbolic nor asymptotically sectional hyperbolic. We obtain sufficient conditions for the existence of physical/SRB measures for asymptotically sectionally hyperbolic attracting sets with any finite codimension, extending the codimension two case. We provide examples of such attractors, either with non-sectional hyperbolic equilibria, or with sectional hyperbolic equilibria of mixed type, i.e., with a Lorenz-like singularity together with a Rovella-like singularity in a transitive set. These are higher-dimensional versions of contracting Lorenz-like attractors (also known as Rovella-like attractors) to which we apply our criteria to obtain a physical/SRB measure with full ergodic basin. We also adapt the previous examples to obtain higher codimensional (i.e. with central direction of dimension greater than $2$) nonuniformly sectional expanding attractors.

math.DS↗