Malthus-Verhulst Model Research Articles

Carrying capacity in growing networks

.In this work, a growing network model that can generate a random network with a finite degree in infinite time is studied. The dynamics are governed by a rule where the degree increases under a scheme similar to the Malthus–Verhulst model in the context of population growth. The degree distribution is analysed in both stationary and time-dependent regimes through some exact results and simulations, and a scaling behaviour is found in asymptotically large time. For finite times, the time-dependent degree distribution displays an accumulation of hubs as a result of competition between attractive and repulsive terms in linking probability.

Effects of a Fluctuating Carrying Capacity on the Generalized Malthus-Verhulst Model

We consider a generalized Malthus-Verhulst model with a fluctuating carrying capacity and we study its effects on population growth. The carrying capacity fluctuations are described by a Poissonian process with an exponential correlation function. We will find an analytical expression for the average of a number of individuals and show that even in presence of a fluctuating carrying capacity the average tends asymptotically to a constant quantity.

Influence of strong noise on the decline and propagation of population in the delayed Malthus–Verhulst model

The effects of strong noise on the decline and propagation processes of a population in the Malthus–Verhulst model with time delay are investigated by a stochastic simulation. Time delays in two different processes are concurrent in ecosystems. The simulation results indicate that: The stability of the population is enhanced by the decreasing multiplicative noise intensity and the increasing delay time. The replacement of old individuals with young ones is accelerated by an increasing multiplicative noise intensity, an increasing additive noise intensity and a decreasing delay time. An increasing multiplicative noise intensity will drive the population of species to fluctuate more largely.

Noise-induced decline and propagation of population in the delayed Malthus–Verhulst model

The effects of multiplicative and additive colored noises on decline and propagation processes of population in the delayed stochastic Malthus–Verhulst model are investigated by numerically computing and stochastically simulating. From the biological point of view, our results indicate that: the increasing correlation time of multiplicative noise strengthens the stability of population, and the correlation time of additive noise does not affect it. The increasing correlation time of multiplicative noise slows down the replacement of old individuals with young ones, while the increasing correlation time of additive noise with short delay time and the optimal correlation time of additive noise with long delay time quicken it.

The effects of time delay on the decline and propagation processes of population in the Malthus-Verhulst model with cross-correlated noises

The effects of time delay on the decline and propagation processes of population in the Malthus-Verhulst model with cross-correlated noises are investigated separately. Through numerically computing and stochastically simulating, we find that: (i) inclusion of time delay in the decline process, increasing the delay time τ weakens the stability of population with short delay and strengthens it with long delay. The stability of population reduces monotonically as the cross-correlated intensity λ increasing. The population of a species goes to extinction with increasing τ and increasing λ; (ii) inclusion of time delay in the propagation process, the increasing τ strengthens the stability of population and the increasing λ weakens it. The increasing τ slows down the growth process of a species while the increasing λ speeds it up. That is, the increasing delay time does not affect roughly the stability of population with short delay but strengthens it with long delay, and the population of species is restricted in the lower level by the larger delay time. The stability of population is weakened and the replacement of old individuals with young ones is accelerated by the increasing cross-correlation intensity between two noises.

Exact probability distribution for the Bernoulli-Malthus-Verhulst model driven by a multiplicative colored noise

We report an exact result for the calculation of the probability distribution of the Bernoulli-Malthus-Verhulst model driven by a multiplicative colored noise. We study the conditions under which the probability distribution of the Malthus-Verhulst model can exhibit a transition from a unimodal to a bimodal distribution depending on the value of a critical parameter. Also we show that the mean value of x(t) in the latter model always approaches asymptotically the value 1.

Maximum-entropy approach with higher moments for solving Fokker–Planck equation

The maximum-entropy method with higher number of moments is used to solve the Fokker–Planck equation. An adopted Newton method is used to iterate the maximum entropy set of equations. The method is used to calculate the probability density function of the Fokker–Planck equation. The calculations are carried out for three examples. (1) The bistable systems of double well potential that is used in many problems related to the fluctuation and relaxation processes in far from equilibrium systems. (2) The Malthus–Verhulst model, which is used to study the evolution of the number of individuals of an ecological species and the evolution of the intensity of the laser light. (3) The Black–Scholes equation used in financial market option pricing. Although the maximum-entropy approach has several advantages, it is not convergent at large times and so cannot be used to calculate the steady state solution.

Exact solutions of the Fokker-Planck equation for the Malthus-Verhulst model

A class of particular solutions of the Fokker-Planck equation associated with the Malthus-Verhulst model is obtained. These time-dependent solutions are exact and allow us to study the evolution of both the distribution function and the moments. A careful analysis is carried out for the two simplest cases, showing the different possible types of relaxation.

Spatial correlations in the decay from an unstable state in the Malthus-Verhulst model

We study the transient behaviour during the decay of an initial unstable state of a reaction diffusion model describing the inhomogeneous fluctuations in the Malthus-Verhulst problem. The enhancement of fluctuations during the transient leads to a new phenomenon,i.e. the growth of the correlation length above its steady-state value. The analysis of the transient behaviour is performed in a particular region of the control parameter space, where a systematic approximation scheme can be introduced.

A new instability phenomenon in the Malthus-Verhulst model

The population dynamics model known as the Malthus-Verhulst model is shown to exhibit, for particular values of the rate parameters, a new regime characterised by a relaxation time which is exponentially small in inverse of the competition rate between individuals of the species. This result is obtained by means of a representation of the time evolution of the system in terms of a stochastic process which describes the Brownian motion of a particle under the action of a unidimensional periodic potential. The long-time behaviour of the first moment of the population is calculated by a suitable extrapolation of a perturbative expansion around the Wiener process result.

Transient behaviour near an instability point: Vectorial stochastic representation of the Malthus-Verhulst model

We study the fluctuations in the Malthus-Verhulst model of population dynamics in the vicinity of the instability point. We introduce a representation in terms of an-dimensional simple stochastic process, wheren is an integer related to the rate parameters of the model. We are thus able to obtain the transient behaviour for the decay from an unstable state by means of the quasi-deterministic approximation. Forn=1 and 2 we recover the well-known case of a double-well potential and the single-mode laser, respectively. We show that it is also possible to perform an analytic continuation of the results for any noninteger positiven.