Abstract
We consider an ensemble of active particles, i.e., of agents endowed by internal variables u(t). Namely, we assume that the nonlinear dynamics of u is perturbed by realistic bounded symmetric stochastic perturbations acting nonlinearly or linearly. In the absence of birth, death and interactions of the agents (BDIA) the system evolution is ruled by a multidimensional Hypo-Elliptical Fokker–Plank Equation (HEFPE). In presence of nonlocal BDIA, the resulting family of models is thus a Partial Integro-differential Equation with hypo-elliptical terms. In the numerical simulations we focus on a simple case where the unperturbed dynamics of the agents is of logistic type and the bounded perturbations are of the Doering–Cai–Lin noise or the Arctan bounded noise. We then find the evolution and the steady state of the HEFPE. The steady state density is, in some cases, multimodal due to noise-induced transitions. Then we assume the steady state density as the initial condition for the full system evolution. Namely we modeled the vital dynamics of the agents as logistic nonlocal, as it depends on the whole size of the population. Our simulations suggest that both the steady states density and the total population size strongly depends on the type of bounded noise. Phenomena as transitions to bimodality and to asymmetry also occur.
Highlights
Two of the more active fields of application of statistical physics to biology are theoretical population dynamics, mathematical epidemiology, sociophysics [3,4,5] and mathematical oncology [6,7]
At the best of our knowledge, this work has some novelties of potential interest in statistical physics: it is the first kinetic model where the impact of bounded stochastic processes is included and it is investigated its interplay with logistic non-local birth–death dynamics
N, i.e., numerically exploring the steady state behavior of the system in absence of birth and death, i.e., assessing the steady state of the hypoelliptic Fokker–Planck equation; Assessing the steady state of the full model, assuming in the phase where birth and deaths occur that the system in absence of vital dynamics was at its equilibrium, i.e., at the steady state of the above mentioned Fokker–Planck equation
Summary
Two of the more active fields of application of statistical physics to biology are theoretical population dynamics, mathematical epidemiology (including behavioral aspects [1,2]), sociophysics [3,4,5] and mathematical oncology [6,7]. White noise perturbations cannot be applied to parameters on which a system depends nonlinearly, and often even Gaussian colored perturbation cannot These and other critical issues imply that bounded stochastic processes ought to be used in most case in biophysics: an increasingly important approach [36]. The above mentioned interplays can be modeled by assuming that the dynamics is affected by bounded stochastic perturbations This is our key assumption here, which will lead us to define a family of partial integro-differential models that extends hypo-elliptic nonlinear Fokker–Planck equation. At the best of our knowledge, this work has some novelties of potential interest in statistical physics: it is the first kinetic model where the impact of bounded stochastic processes is included (resulting in hypo-ellictic integro-differential equations) and it is investigated its interplay with logistic non-local birth–death dynamics. Noise induced transition to bimodality and asymmetry are observed
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