Verhulst-type kinetics driven by white shot noise: Exact solution by direct averaging

Probability characteristics of nonlinear dynamical systems driven by δ-pulse noise

For a nonlinear dynamical system described by the first-order differential equation with Poisson white noise having exponentially distributed amplitudes of δ pulses, some exact results for the stationary probability density function are derived from the Kolmogorov-Feller equation using the inverse differential operator. Specifically, we examine the "effect of normalization" of non-Gaussian noise by a linear system and the steady-state probability density function of particle velocity in the medium with Coulomb friction. Next, the general formulas for the probability distribution of the system perturbed by a non-Poisson δ-pulse train are derived using an analysis of system trajectories between stimuli. As an example, overdamped particle motion in the bistable quadratic-cubic potential under the action of the periodic δ-pulse train is analyzed in detail. The probability density function and the mean value of the particle position together with average characteristics of the first switching time from one stable state to another are found in the framework of the fast relaxation approximation.

Transient and stationary characteristics of the Malthus-Verhulst-Bernoulli model with non-Gaussian fluctuating parameters

Two generalized Verhulst equations with non-Gaussian fluctuations of the reproduction rate and the volume of resources are under analytical investigation. For the first model, using the central limit theorem, we find the asymptotic behavior of the probability distribution of population density for an arbitrary non-Gaussian colored noise with nonzero power spectral density at zero frequency. Specifically, we confirm this result in the case of Markovian dichotomous noise and examine the evolution of mean population density. For fluctuating resources with one-sided stable distribution the transient dynamics of probability density function and statistical characteristics in the steady state are obtained. As shown, the scenario of the population's evolution depends on the parameter of nonlinearity in the original stochastic equation.

Data collapse, scaling functions, and analytical solutions of generalized growth models

We consider a nontrivial one-species population dynamics model with finite and infinite carrying capacities. Time-dependent intrinsic and extrinsic growth rates are considered in these models. Through the model per capita growth rate we obtain a heuristic general procedure to generate scaling functions to collapse data into a simple linear behavior even if an extrinsic growth rate is included. With this data collapse, all the models studied become independent from the parameters and initial condition. Analytical solutions are found when time-dependent coefficients are considered. These solutions allow us to perceive nontrivial transitions between species extinction and survival and to calculate the transition's critical exponents. Considering an extrinsic growth rate as a cancer treatment, we show that the relevant quantity depends not only on the intensity of the treatment, but also on when the cancerous cell growth is maximum.

Power-law distribution as a result of asynchronous random switching between Malthus and Verhulst kinetics

It is shown analytically that the flashing annihilation term of a Verhulst kinetic leads to the power-law distribution in the stationary state. For the frequency of switching slower than twice the free growth rate this provides the quasideterministic source of a Lévy noise at the macroscopic level.

Numerical method for solving stochastic differential equations with Poissonian white shot noise

We propose a numerical integration scheme to solve stochastic differential equations driven by Poissonian white shot noise. Our formula, which is based on an integral equation, which is equivalent to the stochastic differential equation, utilizes a discrete time approximation with fixed integration time step. We show that our integration formula approaches the Euler formula if the Poissonian noise approaches the Gaussian white noise. The accuracy and efficiency of the proposed algorithm are examined by studying the dynamics of an overdamped particle driven by Poissonian white shot noise in a spatially periodic potential. We find that the accuracy of the proposed algorithm only weakly depends on the parameters characterizing the Poissonian white shot noise; this holds true even if the limit of Gaussian white noise is approached.

Martingale integrals over Poissonian processes and the Ito-type equations with white shot noise

The construction of the Ito-type stochastic integrals and differential equations for compound Poisson processes is provided. The general martingale and nonanticipating properties of the ordinary (Gaussian) Ito theory are conserved. These properties appear particularly important if the stochastic description has to be proposed according to game theory or the linear relaxation (or the exponential growth) requirements. In contrast to the ordinary Ito theory the (uncorrelated) parametric fluctuation of a definite sign can be still modeled by asymmetric white shot noise, so the general scope of applications is not restricted by the positivity requirements. The possible use of the developed formalism in econophysics is addressed.