Travelling waves in a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypical trait

We consider a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypical trait. To sustain the possibility of invasion in the case where an underlying principal eigenvalue is negative, we investigate the existence of travelling wave solutions. We identify a minimal speed $c^*>0$, and prove the existence of waves when $c\geq c^*$ and the non existence when $0\leq

The Evolution of Reaction-diffusion Controllers for Minimally Cognitive Agents

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An elementary model for an advancing autoignition front in laminar reactive co-flow jets injected into supercritical water

In this paper we formulate and analyze an elementary model for the propagation of advancing autoignition fronts in reactive co-flow fuel/oxidizer jets injected into an aqueous environment at high pressure. This work is motivated by the experimental studies of autoignition of hydrothermal flames performed at the high pressure laboratory of NASA Glenn Research Center. Guided by experimental observations, we use several simplifying assumptions that allow the derivation of a simple, still experimentally feasible, mathematical model for the propagation of advancing ignition fronts. The model consists of a single diffusion-absorption-advection equation posed in an infinite cylindrical domain with a non-linear condition on the boundary of the cylinder and describes the temperature distribution within the jet. This model manifests an interplay of thermal diffusion, advection and volumetric heat loss within a fuel jet which are balanced by the weak chemical reaction on the jet's boundary. We analyze the model by means of asymptotic and numerical techniques and discuss feasible regimes of propagation of advancing ignition fronts. In particular, we show that in the most interesting parametric regime when the advancing ignition front is on the verge of extinction this model reduces to a one dimensional reaction-diffusion equation with bistable non-linearity. We hope that the present study will be helpful for the interpretation of existing experimental data and guiding of future experiments.Comment: 17 pages, 9 figure

Wave propagation and earth satellite radio emission studies

Radio propagation studies of the ionosphere using satellite radio beacons are described. The ionosphere is known as a dispersive, inhomogeneous, irregular and sometimes even nonlinear medium. After traversing through the ionosphere the radio signal bears signatures of these characteristics. A study of these signatures will be helpful in two areas: (1) It will assist in learning the behavior of the medium, in this case the ionosphere. (2) It will provide information of the kind of signal characteristics and statistics to be expected for communication and navigational satellite systems that use the similar geometry

Effect of selection on ancestry: an exactly soluble case and its phenomenological generalization

We consider a family of models describing the evolution under selection of a population whose dynamics can be related to the propagation of noisy traveling waves. For one particular model, that we shall call the exponential model, the properties of the traveling wave front can be calculated exactly, as well as the statistics of the genealogy of the population. One striking result is that, for this particular model, the genealogical trees have the same statistics as the trees of replicas in the Parisi mean-field theory of spin glasses. We also find that in the exponential model, the coalescence times along these trees grow like the logarithm of the population size. A phenomenological picture of the propagation of wave fronts that we introduced in a previous work, as well as our numerical data, suggest that these statistics remain valid for a larger class of models, while the coalescence times grow like the cube of the logarithm of the population size.Comment: 26 page

Tumor growth instability and the onset of invasion

Motivated by experimental observations, we develop a mathematical model of chemotactically directed tumor growth. We present an analytical study of the model as well as a numerical one. The mathematical analysis shows that: (i) tumor cell proliferation by itself cannot generate the invasive branching behaviour observed experimentally, (ii) heterotype chemotaxis provides an instability mechanism that leads to the onset of tumor invasion and (iii) homotype chemotaxis does not provide such an instability mechanism but enhances the mean speed of the tumor surface. The numerical results not only support the assumptions needed to perform the mathematical analysis but they also provide evidence of (i), (ii) and (iii). Finally, both the analytical study and the numerical work agree with the experimental phenomena.Comment: 12 pages, 8 figures, revtex

Propagation in a non local reaction diffusion equation with spatial and genetic trait structure

We study existence and uniqueness of traveling fronts, and asymptotic speed of propagation for a non local reaction diffusion equation with spatial and genetic trait structure

Pulsating fronts for bistable on average reaction-diffusion equations in a time periodic environment

This paper is devoted to reaction-diffusion equations with bistable nonlinearities depending periodically on time. These equations admit two linearly stable states. However, the reaction terms may not be bistable at every time. These may well be a periodic combination of standard bistable and monostable nonlinearities. We are interested in a particular class of solutions, namely pulsating fronts. We prove the existence of such solutions in the case of small time periods of the nonlinearity and in the case of small perturbations of a nonlinearity for which we know there exist pulsating fronts. We also study uniqueness, monotonicity and stability of pulsating fronts