Revisiting the stability of spatially heterogeneous predator-prey systems under eutrophication

We employ partial integro-differential equations to model trophic interaction in a spatially extended heterogeneous environment. Compared to classical reaction-diffusion models, this framework allows us to more realistically describe the situation where movement of individuals occurs on a faster time scale than the demographic (population) time scale, and we cannot determine population growth based on local density. However, most of the results reported so far for such systems have only been verified numerically and for a particular choice of model functions, which obviously casts doubts about these findings. In this paper, we analyse a class of integro-differential predator-prey models with a highly mobile predator in a heterogeneous environment, and we reveal the main factors stabilizing such systems. In particular, we explore an ecologically relevant case of interactions in a highly eutrophic environment, where the prey carrying capacity can be formally set to 'infinity'. We investigate two main scenarios: (i) the spatial gradient of the growth rate is due to abiotic factors only, and (ii) the local growth rate depends on the global density distribution across the environment (e.g. due to non-local self-shading). For an arbitrary spatial gradient of the prey growth rate, we analytically investigate the possibility of the predator-prey equilibrium in such systems and we explore the conditions of stability of this equilibrium. In particular, we demonstrate that for a Holling type I (linear) functional response, the predator can stabilize the system at low prey density even for an 'unlimited' carrying capacity. We conclude that the interplay between spatial heterogeneity in the prey growth and fast displacement of the predator across the habitat works as an efficient stabilizing mechanism.Comment: 2 figures; appendices available on request. To appear in the Bulletin of Mathematical Biolog

Singular Kernel Problems in Materials with Memory

In recent years the interest on devising and study new materials is growing since they are widely used in different applications which go from rheology to bio-materials or aerospace applications. In this framework, there is also a growing interest in understanding the behaviour of materials with memory, here considered. The name of the model aims to emphasize that the behaviour of such materials crucially depends on time not only through the present time but also through the past history. Under the analytical point of view, this corresponds to model problems represented by integro-differential equations which exhibit a kernel non local in time. This is the case of rigid thermodynamics with memory as well as of isothermal viscoelasticity; in the two different models the kernel represents, in turn, the heat flux relaxation function and the relaxation modulus. In dealing with classical materials with memory these kernels are regular function of both the present time as well as the past history. Aiming to study new materials integro-differential problems admitting singular kernels are compared. Specifically, on one side the temperature evolution in a rigid heat conductor with memory characterized by a heat flux relaxation function singular at the origin, and, on the other, the displacement evolution within a viscoelastic model characterized by a relaxation modulus which is unbounded at the origin, are considered. One dimensional problems are examined; indeed, even if the results are valid also in three dimensional general cases, here the attention is focussed on pointing out analogies between the two different materials with memory under investigation. Notably, the method adopted has a wider interest since it can be applied in the cases of other evolution problems which are modeled by analogue integro-differential equations. An initial boundary value problem with homogeneneous Neumann boundary conditions is studied.In recent years the interest on devising and study new materials is growing since they are widely used in different applications which go from rheology to bio-materials or aerospace applications. In this framework, there is also a growing interest in understanding the behaviour of materials with memory, here considered. The name of the model aims to emphasize that the behaviour of such materials crucially depends on time not only through the present time but also through the past history. Under the analytical point of view, this corresponds to model problems represented by integro-differential equations which exhibit a kernel non local in time. This is the case of rigid thermodynamics with memory as well as of isothermal viscoelasticity; in the two different models the kernel represents, in turn, the heat flux relaxation function and the relaxation modulus. In dealing with classical materials with memory these kernels are regular function of both the present time as wel

A simple mathematical model of gradual Darwinian evolution: Emergence of a Gaussian trait distribution in adaptation along a fitness gradient

We consider a simple mathematical model of gradual Darwinian evolution in continuous time and continuous trait space, due to intraspecific competition for common resource in an asexually reproducing population in constant environment, while far from evolutionary stable equilibrium. The model admits exact analytical solution. In particular, Gaussian distribution of the trait emerges from generic initial conditions.Comment: 21 pages, 2 figures, as accepted to J Math Biol 2013/03/1

Asymptotics of Reaction-Diffusion Fronts with One Static and One Diffusing Reactant

The long-time behavior of a reaction-diffusion front between one static (e.g. porous solid) reactant A and one initially separated diffusing reactant B is analyzed for the mean-field reaction-rate density R(\rho_A,\rho_B) = k\rho_A^m\rho_B^n. A uniformly valid asymptotic approximation is constructed from matched self-similar solutions in a reaction front (of width w \sim t^\alpha where R \sim t^\beta enters the dominant balance) and a diffusion layer (of width W \sim t^{1/2} where R is negligible). The limiting solution exists if and only if m, n \geq 1, in which case the scaling exponents are uniquely given by \alpha = (m-1)/2(m+1) and \beta = m/(m+1). In the diffusion layer, the common ad hoc approximation of neglecting reactions is given mathematical justification, and the exact transient decay of the reaction rate is derived. The physical effects of higher-order kinetics (m, n > 1), such as the broadening of the reaction front and the slowing of transients, are also discussed.Comment: final version, new title & combustion reference

Power spectrum and diffusion of the Amari neural field

We study the power spectrum of a space-time dependent neural field which describes the average membrane potential of neurons in a single layer. This neural field is modelled by a dissipative integro-differential equation, the so-called Amari equation. By considering a small perturbation with respect to a stationary and uniform configuration of the neural field we derive a linearized equation which is solved for a generic external stimulus by using the Fourier transform into wavevector-freqency domain, finding an analytical formula for the power spectrum of the neural field. In addition, after proving that for large wavelengths the linearized Amari equation is equivalent to a diffusion equation which admits space-time dependent analytical solutions, we take into account the nonlinearity of the Amari equation. We find that for large wavelengths a weak nonlinearity in the Amari equation gives rise to a reaction-diffusion equation which can be formally derived from a neural action functional by introducing a dual neural field. For some initial conditions, we discuss analytical solutions of this reaction-diffusion equation.Comment: 8 pages, 2 figures, improved version with inclusion of reaction-diffusion equation and dual neural field. To be published in the open access journal Symmetr