The coexistence of fast and slow diffusion processes in the life cycle of Aedes aegypti mosquitoes

A new model that describes the life cycle of mosquitoes of the species Aedes aegypti, main carriers of vector-borne diseases, is proposed. The novelty is to include in the model the coexistence of two independent diffusion processes, one fast which obeys the constitutive Fick’s law, the other slow which satisfies the Cattaneo evolution equation. The analysis of the corresponding ODE model shows the overall stability of the Mosquitoes-Free Equilibrium (MFE), together with the local stability of the other equilibrium point admitted by the system. Traveling wave type solutions have been investigated, providing an estimate of the minimal speed for which there are monotone waves that connect the homogeneous equilibria allowed by the system. A special section is dedicated to the analysis of the hyperbolic model obtained neglecting the fast diffusive contribution. This particular case is suitable to describe the biological process as it overcomes the paradox of infinite speed propagation, typical of parabolic systems. Several numerical simulations compare the existing models in the literature with those presented in this discussion, showing that although the generalized model is parabolic, the associated wave velocity admits upper bound represented by the speed of the waves linked to the classic parabolic model present in the published literature, so the presence of a slow flux together with a fast one slows down the speed with which a population spreads.

Centre-of-mass and minimal speed limits of the great hammerhead

The great hammerhead is denser than water, and hence relies on hydrodynamic lift to compensate for its lack of buoyancy, and on hydrodynamic moment to compensate for a possible misalignment between centres of mass and buoyancy. Because hydrodynamic forces scale with the swimming speed squared, whereas buoyancy and gravity are independent of it, there is a critical speed below which the shark cannot generate enough lift to counteract gravity, and there are anterior and posterior centre-of-mass limits beyond which the shark cannot generate enough pitching moment to counteract the buoyancy–gravity couple. The speed and centre-of-mass limits were found from numerous wind-tunnel experiments on a scaled model of the shark. In particular, it was shown that the margin between the anterior and posterior centre-of-mass limits is a few tenths of the product between the length of the shark and the ratio between its weight in and out of water; a diminutive 1% body length. The paper presents the wind-tunnel experiments, and discusses the roles that the cephalofoil and the pectoral and caudal fins play in longitudinal balance of a shark.

Spreading in kinetic reaction–transport equations in higher velocity dimensions

In this paper, we extend and complement previous works about propagation in kinetic reaction–transport equations. The model we study describes particles moving according to a velocity-jump process, and proliferating according to a reaction term of monostable type. We focus on the case of bounded velocities, having dimension higher than one. We extend previous results obtained by the first author with Calvez and Nadin in dimension one. We study the large time/large-scale hyperbolic limit via an Hamilton–Jacobi framework together with the half-relaxed limits method. We deduce spreading results and the existence of travelling wave solutions. A crucial difference with the mono-dimensional case is the resolution of the spectral problem at the edge of the front, that yields potential singular velocity distributions. As a consequence, the minimal speed of propagation may not be determined by a first-order condition.

Traveling wave solutions for a diffusive three-species intraguild predation model

The aim of this work is to investigate the existence and non-existence of traveling wave solutions for a diffusive three-species intraguild predation model which means that one predator can eat its potential resource competitors. The method of upper–lower solution is implemented to show the existence of traveling wave solutions. In order to simplify the construction of an admissible pair of upper–lower solution, the scheme of strictly contracting rectangle is applied. Finally, the minimal speed [Formula: see text] of traveling wave solutions of the model is characterized. If the wave speed is greater than [Formula: see text], we show the existence of traveling wave solutions connecting trivial and positive equilibria by combining the upper and lower solutions with the contracting rectangle. On the other hand, if the wave speed is less than [Formula: see text], the non-existence of such solutions is also established. Furthermore, to illustrate our theoretical results, some numerical simulations are performed and biological meanings are interpreted.

FKPP equation with impulses on unbounded domain

This paper deals with the problem of travelling wave solutions in a scalar impulsive FKPP-like equation. It is a first step of a more general study that aims to address existence of travelling wave solutions for systems of impulsive reaction-diffusion equations that model ecological systems dynamics such as fire-prone savannas. Using results on scalar recursion equations, we show existence of populated vs. extinction travelling waves invasion and compute an explicit expression of their spreading speed (characterized as the minimal speed of such travelling waves). In particular, we find that the spreading speed explicitly depends on the time between two successive impulses. In addition, we carry out a comparison with the case of time-continuous events. We also show that depending on the time between two successive impulses, the spreading speed with pulse events could be lower, equal or greater than the spreading speed in the case of time-continuous events. Finally, we apply our results to a model of fire-prone grasslands and show that pulse fires event may slow down the grassland vs. bare soil invasion speed.

Monotone traveling waves for delayed neural field equations

We study the existence of traveling wave solutions and spreading properties for single-layer delayed neural field equations. We focus on the case where the kinetic dynamics are of monostable type and characterize the invasion speeds as a function of the asymptotic decay of the connectivity kernel. More precisely, we show that for exponentially bounded kernels the minimal speed of traveling waves exists and coincides with the spreading speed, which further can be explicitly characterized under a KPP type condition. We also investigate the case of algebraically decaying kernels where we prove the non-existence of traveling wave solutions and show the level sets of the solutions eventually locate in-between two exponential functions of time. The uniqueness of traveling waves modulo translation is also obtained.

Asymptotic analysis of a monostable equation in periodic media

We consider a multidimensional monostable reaction-diffusion equation whose nonlinearity involves periodic heterogeneity. This serves as a model of invasion for a population facing spatial heterogeneities. As a rescaling parameter tends to zero, we prove the convergence to a limit interface, whose motion is governed by the minimal speed (in each direction) of the underlying pulsating fronts. This dependance of the speed on the (moving) normal direction is in contrast with the homogeneous case and makes the analysis quite involved. Key ingredients are the recent improvement \cite{A-Gil} %[4]of the well-known spreading properties \cite{Wein02}, %[32], \cite{Ber-Ham-02}, %[9],and the solution of a Hamilton-Jacobi equation.