Travelling-wave analysis of a model of tumour invasion with degenerate, cross-dependent diffusion.

In this paper, we carry out a travelling-wave analysis of a model of tumour invasion with degenerate, cross-dependent diffusion. We consider two types of invasive fronts of tumour tissue into extracellular matrix (ECM), which represents healthy tissue. These types differ according to whether the density of ECM far ahead of the wave front is maximal or not. In the former case, we use a shooting argument to prove that there exists a unique travelling-wave solution for any positive propagation speed. In the latter case, we further develop this argument to prove that there exists a unique travelling-wave solution for any propagation speed greater than or equal to a strictly positive minimal wave speed. Using a combination of analytical and numerical results, we conjecture that the minimal wave speed depends monotonically on the degradation rate of ECM by tumour cells and the ECM density far ahead of the front.

Turing-Hopf patterns on growing domains: The torus and the sphere.

This paper deals with the study of spatial and spatio-temporal patterns in the reaction-diffusion FitzHugh-Nagumo model on growing curved domains. This is carried out on two exemplar cases: a torus and a sphere. We compute bifurcation boundaries for the homogeneous steady state when the homogeneous system is monostable. We exhibit Turing and Turing-Hopf bifurcations, as well as additional patterning outside of these bifurcation regimes due to the multistability of patterned states. We consider static and growing domains, where the growth is slow, isotropic, and exponential in time, allowing for a simple analytical calculation of these bifurcations in terms of model parameters. Numerical simulations allow us to discuss the role played by the growth and the curvature of the domains on the pattern selection on the torus and the sphere. We demonstrate parameter regimes where the linear theory can successfully predict the kind of pattern (homogeneous and heterogeneous oscillations and stationary spatial patterns) but not their detailed nonlinear structure. We also find parameter regimes where the linear theory fails, such as Hopf regimes which give rise to spatial patterning (depending on geometric details), where we suspect that multistability plays a key role in the departure from homogeneity. Finally we also demonstrate effects due to the evolution of nonuniform patterns under growth, suggesting important roles for growth in reaction-diffusion systems beyond modifying instability regimes.

Reactive-Diffusive-Advective Traveling Waves in a Family of Degenerate Nonlinear Equations.

This paper deals with the analysis of existence of traveling wave solutions (TWS) for a diffusion-degenerate (at D(0) = 0) and advection-degenerate (at h'(0) = 0) reaction-diffusion-advection (RDA) equation. Diffusion is a strictly increasing function and the reaction term generalizes the kinetic part of the Fisher-KPP equation. We consider different forms of the convection term h(u): (1) h'(u) is constant k, (2) h'(u) = ku with k > 0, and (3) it is a quite general form which guarantees the degeneracy in the advective term. In Case 1, we prove that the task can be reduced to that for the corresponding equation, where k = 0, and then previous results reported from the authors can be extended. For the other two cases, we use both analytical and numerical tools. The analysis we carried out is based on the restatement of searching TWS for the full RDA equation into a two-dimensional dynamical problem. This consists of searching for the conditions on the parameter values for which there exist heteroclinic trajectories of the ordinary differential equations (ODE) system in the traveling wave coordinates. Throughout the paper we obtain the dynamics by using tools coming from qualitative theory of ODE.

A Turing-Hopf Bifurcation Scenario for Pattern Formation on Growing Domains.

In this paper, we study the emergence of different patterns that are formed on both static and growing domains and their bifurcation structure. One of these is the so-called Turing-Hopf morphogenetic mechanism. The reactive part we consider is of FitzHugh-Nagumo type. The analysis was carried out on a flat square by considering both fixed and growing domain. In both scenarios, sufficient conditions on the parameter values are given for the formation of specific space-time structures or patterns. A series of numerical solutions of the corresponding initial and boundary value problems are obtained, and a comparison between the resulting patterns on the fixed domain and those arising when the domain grows is established. We emphasize the role of growth of the domain in the selection of patterns. The paper ends by listing some open problems in this area.

Role reversal in a predator-prey interaction.

Predator-prey relationships are one of the most studied interactions in population ecology. However, little attention has been paid to the possibility of role exchange between species, despite firm field evidence of such phenomena in nature. In this paper, we build a mathematical model capable of reproducing the main phenomenological features of role reversal in a classical system and present results for both the temporal and spatio-temporal cases. We show that, depending on the choice of parameters, our role-reversal dynamical system exhibits excitable-like behaviour, generating waves of species' concentrations that propagate through space. Our findings fill a long-standing gap in modelling ecological interactions and can be applicable to better understanding ecological niche shifts and planning of sustainable ecosystems.

Searching for spatial patterns in a pollinator-plant-herbivore mathematical model.

This paper deals with the spatio-temporal dynamics of a pollinator-plant-herbivore mathematical model. The full model consists of three nonlinear reaction-diffusion-advection equations defined on a rectangular region. In view of analyzing the full model, we firstly consider the temporal dynamics of three homogeneous cases. The first one is a model for a mutualistic interaction (pollinator-plant), later on a sort of predator-prey (plant-herbivore) interaction model is studied. In both cases, the interaction term is described by a Holling response of type II. Finally, by considering that the plant population is the unique feeding source for the herbivores, a mathematical model for the three interacting populations is considered. By incorporating a constant diffusion term into the equations for the pollinators and herbivores, we numerically study the spatiotemporal dynamics of the first two mentioned models. For the full model, a constant diffusion and advection terms are included in the equation for the pollinators. For the resulting model, we sketch the proof of the existence, positiveness, and boundedness of solution for an initial and boundary values problem. In order to see the separated effect of the diffusion and advection terms on the final population distributions, a set of numerical simulations are included. We used homogeneous Dirichlet and Neumann boundary conditions.