The existence and uniqueness are proved for the global positive solution to the system of stochastic differential equations describing a two-species mutualism model disturbed by the white noise, the centered and non-centered Poisson noises. We obtain sufficient conditions for stochastic ultimate boundedness, stochastic permanence, nonpersistence in the mean, strong persistence in the mean and extinction of the solution to the considered system. Keywords Stochastic mutualism model, global solution, stochastic ultimate boundedness, stochastic permanence, extinction, nonpersistence in the mean, strong persistence in the mean 2010 MSC 92D25, 60H10, 60H30
Deep learning applications require global optimization of non-convex objective functions, which have multiple local minima. The same problem is often found in physical simulations and may be resolved by the methods of Langevin dynamics with Simulated Annealing, which is a well-established approach for minimization of many-particle potentials. This analogy provides useful insights for non-convex stochastic optimization in machine learning. Here we find that integration of the discretized Langevin equation gives a coordinate updating rule equivalent to the famous Momentum optimization algorithm. As a main result, we show that a gradual decrease of the momentum coefficient from the initial value close to unity until zero is equivalent to application of Simulated Annealing or slow cooling, in physical terms. Making use of this novel approach, we propose CoolMomentum—a new stochastic optimization method. Applying Coolmomentum to optimization of Resnet-20 on Cifar-10 dataset and Efficientnet-B0 on Imagenet, we demonstrate that it is able to achieve high accuracies.
The existence and uniqueness of a global positive solution is proven for the system of stochastic differential equations describing a nonautonomous stochastic predator-prey model with a modified version of the Leslie-Gower term and Holling-type II functional response disturbed by white noise, centered and noncentered Poisson noises. Sufficient conditions are obtained for stochastic ultimate boundedness, stochastic permanence, nonpersistence in the mean, weak persistence in the mean and extinction of a solution to the considered system.
This paper presents the explicit form of positive global solution to stochastically perturbed nonautonomous logistic equation
where w(t) is the standard one-dimensional Wiener process,ν(t, A) = ν(t, A) − tΠ(A), ν(t, A) is the Poisson measure, which is independent on w(t), E[ν(t, A)] = tΠ(A), Π(A) is a finite measure on the Borel sets in R. If coefficients a(t), b(t), α(t) and γ(t, z) are continuous on t, T -periodic on t functions, a(t) > 0, b(t) > 0 andthen there exists unique, positive T -periodic solution to equation for E[1/N (t)].
We study the non-autonomous stochastic predator-prey model with a modified version of Leslie-Gower term and Holling-type II functional response driven by the system of stochastic differential equations with white noise, centered and non-centered Poisson noises. The sufficient conditions of strong persistence in the mean of the solution to the considered system are obtained.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.