taming the diffusion approximation through a controlling factor wkb method the diffusion approximation da is widely used in the analysis of stochastic population dynamics from population genetics to ecology and evolution the da is an uncontrolled approximation that assumes the smoothness of the calculated quantity over the relevant state space and fails when this property is not satisfied this failure becomes severe in situations where the direction of selection switches sign here we employ the wkb wentzel kramers brillouin large deviations method which requires only the logarithm of a given quantity to be smooth over its state space combining the wkb scheme with asymptotic matching techniques we show how to derive the diffusion approximation in a controlled manner and how to produce better approximations applicable for much wider regimes of parameters we also introduce a scalable independent of population size wkb based numerical technique the method is applied to a central problem in population genetics and evolution finding the chance of ultimate fixation in a zero sum two types competition

on the existence of solutions for strongly nonlinear differential equations the objectives of this paper are twofold firstly to prove the existence of an approximate solution in the mean for some nonlinear differential equations we also investigate the behavior of the class of solutions which may be associated with the differential equation secondly we aim to implement the homotopy perturbation method hpm to find analytic solutions for strongly nonlinear differential equations

survival behavior in the cyclic lotka volterra model with a randomly switching reaction rate we study the influence of a randomly switching reproduction predation rate on the survival behavior of the nonspatial cyclic lotka volterra model also known as the zero sum rock paper scissors game used to metaphorically describe the cyclic competition between three species in large and finite populations demographic fluctuations internal noise drive two species to extinction in a finite time while the species with the smallest reproduction predation rate is the most likely to be the surviving one law of the weakest here we model environmental external noise by assuming that the reproduction predation rate of the strongest species the fastest to reproduce and predate in a given static environment randomly switches between two values corresponding to more and less favorable external conditions we study the joint effect of environmental and demographic noise on the species survival probabilities and on the mean extinction time in particular we investigate whether the survival probabilities follow the law of the weakest and analyze their dependence on the external noise intensity and switching rate remarkably when on average there is a finite number of switches prior to extinction the survival probability of the predator of the species whose reaction rate switches typically varies nonmonotonically with the external noise intensity with optimal survival about a critical noise strength we also outline the relationship with the case where all reaction rates switch on markedly different time scales

semianalytical solution of the random product problem of matrices and discrete time random evolution we consider the product of a large number of matrices chosen randomly with some correlation as a or b we introduce an exact functional equation whose solution gives the limiting distribution and the lyapunov exponent for the product of large number of matrices thus we suggest new semianalytical method to solve the random product problem of correlated matrices the method can be easily generalized to high order matrices as we deal with a larger systems of functional equations we have provided a known numerical algorithm for the solution of the matrix product problem we have also applied our method for a solution of discrete time random evolution our functional equation method provides exact transition point

existence and uniqueness results for some nonlinear boundary value problems u a s u b s0 dirichlet 2 u9 a s u9 b s 0 neumann 3 u a s u b and u9 a s u9 b periodic 4 w x here f r a r is an odd increasing homeomorphism k a b a r and w x 2 py2 f a b r a r equation 1 with f s s s s where p 1 and k t s n y 1 rt arises in the study of radial solutions for the p laplacian equation on an annular domain in n dimensions py2 n div u u q f x u u s 0 a x b x g r 5 w x and has been studied extensively see for instance 1 4 6 in this paper we shall establish existence results for 1 with each of the above boundary conditions 2 3 and 4 under various growth conditions on f in particular our results give the existence of radial solutions

theory of time averaged neutral dynamics with environmental stochasticity competition is the main driver of population dynamics which shapes the genetic composition of populations and the assembly of ecological communities neutral models assume that all the individuals are equivalent and that the dynamics is governed by demographic shot noise with a steady state species abundance distribution sad that reflects a mutation extinction equilibrium recently many empirical and theoretical studies emphasized the importance of environmental variations that affect coherently the relative fitness of entire populations here we consider two generic time averaged neutral models in both the relative fitness of each species fluctuates independently in time but its mean is zero the first model a describes a system with local competition and linear fitness dependence of the birth death rates while in the second model b the competition is global and the fitness dependence is nonlinear due to this nonlinearity model b admits a noise induced stabilization mechanism that facilitates the invasion of new mutants a self consistent mean field approach is used to reduce the multispecies problem to two species dynamics and the large n asymptotics of the emerging set of fokker planck equations is presented and solved our analytic expressions are shown to fit the sads obtained from extensive monte carlo simulations and from numerical solutions of the corresponding master equations

solvability of strongly nonlinear boundary value problems for second order differential inclusions abstract in this paper we study a problem for a second order differential inclusion with dirichlet neumann and mixed boundary conditions the equation is driven by a nonlinear not necessarily homogeneous differential operator satisfying certain conditions and containing as a particular case the p laplacian operator we prove the existence of solutions both for the case in which the multivalued nonlinearity has convex values and for the case in which it has not convex values the presence of a maximal monotone operator in the equation make the results applicable to gradient systems with non smooth time invariant convex potential and differential variational inequalities

phase diagram for logistic systems under bounded stochasticity extinction is the ultimate absorbing state of any stochastic birth death process hence the time to extinction is an important characteristic of any natural population here we consider logistic and logisticlike systems under the combined effect of demographic and bounded environmental stochasticity three phases are identified an inactive phase where the mean time to extinction t increases logarithmically with the initial population size an active phase where t grows exponentially with the carrying capacity n and a temporal griffiths phase with a power law relationship between t and n the system supports an exponential phase only when the noise is bounded in which case the continuum diffusion approximation breaks down within the griffiths phase this breakdown is associated with a crossover between qualitatively different survival statistics and decline modes to study the power law phase we present a new wkb scheme which is applicable both in the diffusive and in the nondiffusive regime

periodic problems for strongly nonlinear second order differential inclusions in this paper we study periodic problems for second order differential inclusions in rn with a maximal monotone term the nonlinear differential operator is not necessarily homogeneous does not obey any growth condition and incorporates as a special case the one dimensional p laplacian using techniques from multivalued analysis and the theory of operators of monotone type we prove the existence of solutions for both the convex and nonconvex problems when the maximal monotone term a is defined everywhere and when it is not defined everywhere case of variational inequalities

giant fluctuations in logistic growth of two species competing for limited resources we analyze the fluctuation of the number of individuals when two competing species beginning with a few initial individuals are submitted to a logistic growth we show that when the total number of individuals reaches the carrying capacity the number of each species is subject to giant fluctuations variance sim mean 2 if the two species have similar growth rate we show that the deterministic logistic equation can be used only when the growth rates are significantly different otherwise such growth has to be investigated by stochastic processes tools these results generalize to a wide class of growth law

evolution of a fluctuating population in a randomly switching environment environment plays a fundamental role in the competition for resources and hence in the evolution of populations here we study a well mixed finite population consisting of two strains competing for the limited resources provided by an environment that randomly switches between states of abundance and scarcity assuming that one strain grows slightly faster than the other we consider two scenarios one of pure resource competition and one in which one strain provides a public good and investigate how environmental randomness external noise coupled to demographic internal noise determines the population s fixation properties and size distribution by analytical means and simulations we show that these coupled sources of noise can significantly enhance the fixation probability of the slower growing species we also show that the population size distribution can be unimodal bimodal or multimodal and undergoes noise induced transitions between these regimes when the rate of switching matches the population s growth rate

population dynamics in a changing environment random versus periodic switching environmental changes greatly influence the evolution of populations here we study the dynamics of a population of two strains one growing slightly faster than the other competing for resources in a time varying binary environment modeled by a carrying capacity switching either randomly or periodically between states of abundance and scarcity the population dynamics is characterized by demographic noise birth and death events coupled to a varying environment we elucidate the similarities and differences of the evolution subject to a stochastically and periodically varying environment importantly the population size distribution is generally found to be broader under intermediate and fast random switching than under periodic variations which results in markedly different asymptotic behaviors between the fixation probability of random and periodic switching we also determine the detailed conditions under which the fixation probability of the slow strain is maximal

on a pseudo fuccik spectrum for strongly nonlinear second order odes and an existence result abstract we study here the existence of solutions to the nonlinear dirichlet problem p u f t u q t a e for t a b u a u b 0 where is an increasing odd homeomorphism of r f a b r r satisfies the caratheodory assumptions and q l 1 a b r with a view towards defining a pseudo fucik spectrum pfs we study using a time mapping approach the eigenvalue like problem u a u b u 0 u a 0 u b where a 0 b 0 we show here that this pfs which we denote by l consists of a set of curves in r 2 resembling the classical fucik spectrum i e when s s our main existence result which deals with nonresonance with respect to the pfs can be roughly stated as follows if for s 0 sufficiently large and for almost every t a b the pair f t s s f t s s lies in a compact rectangle contained in an open component of r 2 l which intersects the diagonal then problem p has at least a solution

intrinsic noise in systems with switching environments we study individual based dynamics in finite populations subject to randomly switching environmental conditions these are inspired by models in which genes transition between on and off states regulating underlying protein dynamics similarly switches between environmental states are relevant in bacterial populations and in models of epidemic spread existing piecewise deterministic markov process approaches focus on the deterministic limit of the population dynamics while retaining the randomness of the switching here we go beyond this approximation and explicitly include effects of intrinsic stochasticity at the level of the linear noise approximation specifically we derive the stationary distributions of a number of model systems in good agreement with simulations this improves existing approaches which are limited to the regimes of fast and slow switching

a log linear model for the birnbaum saunders distribution the birnbaum saunders distribution was derived to model times to failure for metals subject to fatigue in this article we formulate and develop a log linear model for the birnbaum saunders distribution the model may be used for accelerated life testing or to compare the median lives of several populations methods of analyzing data for this log linear model are discussed with maximum likelihood and least squares methods being compared it is found that for commonly occurring conditions the notorious intractability of the birnbaum saunders distribution is not a serious problem because least squares and normal theory procedures provide a reasonable alternative