The threshold contact process: A continuum limit

Summary In the threshold contact process on thed-dimensional integer lattice with ranger, healthy sites become infected at rate if they have at least one infectedr-neighbour, and recover at rate 1. We show that the critical value _c (r) is asymptotic tor ^–d _c asr, where _c is the critical value of the birth rate for a continuum threshold contact process which may be described in terms of an oriented continuous percolation model driven by a Poisson process of rate ind+1 dimensions. We have bounds of 0.7320 < _c < 3 ford=1.     

Evolution in predator–prey systems

We study the adaptive dynamics of predator–prey systems modeled by a dynamical system in which the traits of predators and prey are allowed to evolve by small mutations. When only the prey are allowed to evolve, and the size of the mutational change tends to 0, the system does not exhibit long term prey coexistence and the trait of the resident prey type converges to the solution of an ODE. When only the predators are allowed to evolve, coexistence of predators occurs. In this case, depending on the parameters being varied, we see that (i) the number of coexisting predators remains tight and the differences in traits from a reference species converge in distribution to a limit, or (ii) the number of coexisting predators tends to infinity, and we calculate the asymptotic rate at which the traits of the least and most “fit” predators in the population increase. This last result is obtained by comparison with a branching random walk killed to the left of a linear boundary and a finite branching–selection particle system.     

A posteriori error estimates for linear equations

Summary This paper describes upper and lowerp-norm error bounds for approximate solutions of the linear system of equationsAx=b. These bounds imply that the error is proportional to the quantity wherer is the residual andq is the conjugate index top. The constant of proportionality is larger than 1 and lies in a specified range. Similar results are obtained for approximations toA ^–1 and solutions of nonsingular linear equations on general spaces.Research was partially supported by NSF Grant DMS8901477     

Partial functional differential equation with an integral condition and applications to population dynamics

In this work we consider a semilinear functional partial differential equation with an integral condition. We apply the method of semidiscretization in time, to establish the existence and uniqueness of solutions. We also study the continuation of the solution to the maximal interval of existence. Finally we give examples to demonstrate the applications of our results.     

On the mixing property and the ergodic principle for nonhomogeneous Markov chains

We study different types of limit behavior of infinite dimension discrete time nonhomogeneous Markov chains. We show that the geometric structure of the set of those Markov chains which have asymptotically stationary density depends on the considered topologies. We generalize and correct some results from Ganikhodjaev et al. (2006) [3].     

Asynchronous exponential growth for an age dependent population equation with delayed birth process

We show that the solutions of a population equation with age structure and delayed birth process have asynchronous exponential growth (AEG). We use operator matrices and Hille-Yosida operators as well as Perron-Frobenius techniques.     

Probabilistic interpretation for a system of quasilinear parabolic partial differential equation combined with algebra equations

In this paper, we study a kind of system of second order quasilinear parabolic partial differential equation combined with algebra equations. Introducing a family of coupled forward–backward stochastic differential equations, and by virtue of some delicate analysis techniques, we give a probabilistic interpretation for it in the viscosity sense.     

Fundamental solutions, transition densities and the integration of Lie symmetries

In this paper we present some new applications of Lie symmetry analysis to problems in stochastic calculus. The major focus is on using Lie symmetries of parabolic PDEs to obtain fundamental solutions and transition densities. The method we use relies upon the fact that Lie symmetries can be integrated with respect to the group parameter. We obtain new results which show that for PDEs with nontrivial Lie symmetry algebras, the Lie symmetries naturally yield Fourier and Laplace transforms of fundamental solutions, and we derive explicit formulas for such transforms in terms of the coefficients of the PDE.     

Lyapunov exponents and asymptotic dynamics in random Kolmogorov models

The current paper is devoted to the investigation of asymptotic dynamics in random Kolmogorov models. Applying the theory of principal Lyapunov exponents and the principal spectrum developed in the authors previous papers together with the concept of part metric it provides conditions for the existence of a globally attracting positive random equilibrium, the existence of a globally attracting uniformly positive random equilibrium, and the extinction in random Kolmogorov models. These results are an important complement to the existing ones.     

Generation of interfaces for Lotka-Volterra competition-diffusion system with large interaction rates

We discuss the generation and motion of interfaces for Lotka-Volterra competition-diffusion system with large interaction. An asymptotic analysis of solutions shows that the two competing species are segregated and an interface appears on the common boundary of their habitats. The motion of the interface is governed by a free boundary problem. In this paper we establish a mathematical theory for the formation of interfaces (at the initial stage) by using an upper and lower solutions method. In addition, combining our results and a known result for the motion of interfaces (after the initial stage), we obtain some information on the generation and motion of interfaces for given almost any smooth initial data.     

Cugliandolo-Kurchan equations for dynamics of Spin-Glasses

We study the Langevin dynamics for the family of spherical p-spin disordered mean-field models of statistical physics. We prove that in the limit of system size N approaching infinity, the empirical state correlation and integrated response functions for these N-dimensional coupled diffusions converge almost surely and uniformly in time, to the non-random unique strong solution of a pair of explicit non-linear integro-differential equations intensively studied by Cugliandolo and Kurchan. Research partially supported by NSF grants #DMS-0406042, #DMS-FRG-0244323     

Well-posedness for a model of prion proliferation dynamics

The model considered consists of an ordinary differential equation coupled with an integro-partial differential equation and describes the interaction between non-infectious and infectious prion proteins. We provide sufficient conditions for uniqueness of monomer-preserving weak solutions. In addition, we also prove existence of weak solutions under rather general assumptions on the involved degradation rates.     

Diffusion-limited aggregation on a tree

Summary. We study the following growth model on a regular d-ary tree. Points at distance n adjacent to the existing subtree are added with probabilities proportional to α ⁻ⁿ , where α < 1 is a positive real parameter. The heights of these clusters are shown to increase linearly with their total size; this complements known results that show the height increases only logarithmically when α≧ 1. Results are obtained using stochastic monotonicity and regeneration results which may be of independent interest. Our motivation comes from two other ways in which the model may be viewed: as a problem in first-passage percolation, and as a version of diffusion-limited aggregation (DLA), adjusted so that “fingering” occurs. Received: 16 August 1994 / In revised form: 18 March 1996     

A diffusive predator-prey model with a protection zone

In this paper we study the effects of a protection zone Ω₀ for the prey on a diffusive predator-prey model with Holling type II response and no-flux boundary condition. We show the existence of a critical patch size described by the principal eigenvalue of the Laplacian operator over Ω₀ with homogeneous Dirichlet boundary conditions. If the protection zone is over the critical patch size, i.e., if is less than the prey growth rate, then the dynamics of the model is fundamentally changed from the usual predator-prey dynamics; in such a case, the prey population persists regardless of the growth rate of its predator, and if the predator is strong, then the two populations stabilize at a unique coexistence state. If the protection zone is below the critical patch size, then the dynamics of the model is qualitatively similar to the case without protection zone, but the chances of survival of the prey species increase with the size of the protection zone, as generally expected. Our mathematical approach is based on bifurcation theory, topological degree theory, the comparison principles for elliptic and parabolic equations, and various elliptic estimates.     

A class of quasilinear stochastic partial differential equations of McKean-Vlasov type with mass conservation

Summary A system ofN particles inR ^d with mean field interaction and diffusion is considered. Assuming adiabatic elimination of the momenta the positions satisfy a stochastic ordinary differential equation driven by Brownian sheets (microscopic equation), where all coefficients depend on the position of the particles and on the empirical mass distribution process. This empirical mass distribution process satisfies a quasilinear stochastic partial differential equation (SPDE). This SPDE (mezoscopic equation) is solved for general measure valued initial conditions by extending the empirical mass distribution process from point measure valued initial conditions with total mass conservation. Starting with measures with densities inL ₂(R ^d ,dr), wheredr is the Lebesgue measure, the solution will have densities inL ₂(R ^d ,dr) and strong uniqueness (in the Itô sense) is obtained. Finally, it is indicated how to obtain (macroscopic) partial differential equations as limits of the so constructed SPDE's.This research was supported by NSF grant DMS92-11438 and ONR grant N00014-91J-1386     

Large deviations for Langevin spin glass dynamics

Summary We study the asymptotic behaviour of asymmetrical spin glass dynamics in a Sherrington-Kirkpatrick model as proposed by Sompolinsky-Zippelius. We prove that the annealed law of the empirical measure on path space of these dynamics satisfy a large deviation principle in the high temperature regime. We study the rate function of this large deviation principle and prove that it achieves its minimum value at a unique probability measureQ which is not markovian. We deduce that the quenched law of the empirical measure converges to _Q . Extending then the preceeding results to replicated dynamics, we investigate the quenched behavior of a single spin. We get quenched convergence toQ in the case of a symmetric initial law and even potential for the free spin.     

Large deviation principles for the stochastic quasi-geostrophic equations

In this paper we establish the large deviation principle for the stochastic quasi-geostrophic equation with small multiplicative noise in the subcritical case. The proof is mainly based on the weak convergence approach. Some analogous results are also obtained for the small time asymptotics of the stochastic quasi-geostrophic equation.     

Global bifurcation of positive equilibria in nonlinear population models

Existence of nontrivial nonnegative equilibrium solutions for age-structured population models with nonlinear diffusion is investigated. Introducing a parameter measuring the intensity of the fertility, global bifurcation is shown of a branch of positive equilibrium solutions emanating from the trivial equilibrium. Moreover, for the parameter-independent model we establish existence of positive equilibria by means of a fixed point theorem for conical shells.