Shock-fronted travelling waves in a reaction-diffusion model with nonlinear forward-backward-forward diffusion

Reaction-diffusion equations (RDEs) are often derived as continuum limits of lattice-based discrete models. Recently, a discrete model which allows the rates of movement, proliferation and death to depend upon whether the agents are isolated has been proposed, and this approach gives various RDEs where the diffusion term is convex and can become negative (Johnston et al., Sci. Rep. 7, 2017), i.e. forward-backward-forward diffusion. Numerical simulations suggest these RDEs support shock-fronted travelling waves when the reaction term includes an Allee effect. In this work we formalise these preliminary numerical observations by analysing the shock-fronted travelling waves through embedding the RDE into a larger class of higher order partial differential equations (PDEs). Subsequently, we use geometric singular perturbation theory to study this larger class of equations and prove the existence of these shock-fronted travelling waves. Most notable, we show that different embeddings yield shock-fronted travelling waves with different properties.Comment: 41 pages, 11 figure

A complex pattern of chemokine receptor expression is seen in osteosarcoma

<p>Abstract</p> <p>Background</p> <p>Osteosarcoma is the most frequent bone tumor in childhood and adolescence. Patients with primary metastatic disease have a poor prognosis. It is therefore important to better characterize the biology of this tumor to define new prognostic markers or therapeutic targets for tailored therapy. Chemokines and their receptors have been shown to be involved in the development and progression of malignant tumors. They are thought to be active participants in the biology of osteosarcoma. The function of specific chemokines and their receptors is strongly associated with the biological context and microenvironment of their expression. In this report we characterized the expression of a series of chemokine receptors in the complex environment that defines osteosarcoma.</p> <p>Methods</p> <p>The overall level of chemokine receptor mRNA expression was determined using TaqMan RT-PCR of microdissected archival patient biopsy samples. Expression was then verified at the protein level by immunohistochemistry using a series of receptor specific antibody reagents to elucidate the cellular association of expression.</p> <p>Results</p> <p>Expression at the RNA level was found for most of the tested receptors. CCR1 expression was found on infiltrating mononuclear and polynuclear giant cells in the tumor. Cells associated with the lining of intratumoral vessels were shown to express CCR4. Infiltrating mononuclear cells and tumor cells both showed expression of the receptor CCR5, while CCR7 was predominantly expressed by the mononuclear infiltrate. CCR10 was only very rarely detected in few scattered infiltrating cells.</p> <p>Conclusion</p> <p>Our data elucidate for the first time the cellular context of chemokine receptor expression in osteosarcoma. This is an important issue for better understanding potential chemokine/chemokine receptor function in the complex biologic processes that underlie the development and progression of osteosarcoma. Our data support the suggested involvement of chemokines and their receptors in diverse aspects of the biology of osteosarcoma, but also contradict aspects of previous reports describing the expression of these receptors in this tumor.</p

On the Determinants of Currency Crises: The Role of Model Uncertainty

We tackle explicitly the issue of model uncertainty in the framework of binary variable models of currency crises. Using Bayesian model averaging techniques, we assess the robustness of the explanatory variables proposed in the recent literature for both static and dynamic models. Our results indicate that the variables belonging to the set of macroeconomic fundamentals proposed by the literature are very fragile determinants of the occurrence of currency crises. The results improve if the crisis index identifies a crisis period (defined as the period up to a year before a crisis) instead of a crisis occurrence. In this setting, the extent of real exchange rate misalignment and financial market indicators appear as robust determinants of crisis periods

Canards in R3

AbstractWe give a geometric analysis of canard solutions in three-dimensional singularly perturbed systems with a folded two-dimensional critical manifold. By analysing the reduced flow we obtain singular canard solutions passing through a singularity on the fold-curve. We classify these singularities, called canard points, as folded saddles, folded nodes, and folded saddle-nodes. We prove the existence of canard solutions in the case of the folded saddle. We show the existence of canards in the folded node case provided a generic non-resonance condition is satisfied and in a subcase of the folded saddle-node. The proof is based on the blow-up method

Existence of travelling wave solutions for a model of tumor invasion

The existence of travelling wave solutions to a haptotaxis dominated model is analysed. A version of this model has been derived in Perumpanani et al. (1999) to describe tumour invasion, where diffusion is neglected as it is assumed to play only a small role in the cell migration. By instead allowing diffusion to be small, we reformulate the model as a singular perturbation problem, which can then be analysed using geometric singular perturbation theory. We prove the existence of three types of physically realistic travelling wave solutions in the case of small diffusion. These solutions reduce to the no diffusion solutions in the singular limit as diffusion as is taken to zero. A fourth travelling wave solution is also shown to exist, but that is physically unrealistic as it has a component with negative cell population. The numerical stability, in particular the wavespeed of the travelling wave solutions is also discussed