You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'The transient dynamics of the Verhulst model perturbed by arbitrary non-Gaussian white noise is investigated. Based on the infinitely divisible distribution of the Lévy process we study the nonlinear relaxation of the population density for three cases of white non-Gaussian noise: (i) shot noise, (ii) noise with a probability density of increments expressed in terms of Gamma function, and (iii) Cauchy stable noise. We obtain exact results for the probability distribution of the population density in all cases, and for Cauchy stable noise the exact expression of the nonlinear relaxation time is derived. Moreover starting from an initial delta function distribution, we find a transition induced by the multiplicative Lévy noise, from a trimodal probability distribution to a bimodal probability distribution in asymptotics. Finally we find a nonmonotonic behavior of the nonlinear relaxation time as a function of the Cauchy stable noise intensity.'
author:
- 
- 
title: Verhulst model with Lévy white noise excitation
---

Introduction
============

The nonlinear stochastic systems with noise excitation have attracted extensive attention and the concept of noise-induced transitions has got a wide variety of applications in physics, chemistry, and biology [@Hor84]. Noise-induced transitions are conventionally defined in terms of changes in the number of extrema in the probability distribution of a system variable and may depend both quantitatively and qualitatively on the character of the noise, i.e. on the properties of stochastic process which describes the noise excitation. The Verhulst model, which is a cornerstone of empirical and theoretical ecology, is one of the classic examples of self-organization in many natural and artificial systems [@Eig79]. This model, also known as the logistic model, is relevant to a wide range of situations including population dynamics [@Hor84; @Mor82; @Ciu93; @Mat00], self-replication of macromolecules [@Eig71], spread of viral epidemics [@Ace06], cancer cell population [@Bao03], biological and biochemical systems [@Der90; @Ciu96], population of photons in a single mode laser [@McN74; @Oga83], autocatalytic chemical reactions [@Sch72; @Cha76; @Gar77; @Bou82; @Leu87], freezing of supercooled liquids [@Das83], social sciences [@Her72; @Mon78], etc.

In considering how the population density $x\left(t\right)$ may change with time $t$, Verhulst proposed the following equation $$\frac{dx}{dt}=rx\left( 1-\frac{x}{\Omega }\right) .
\label{F-1}$$ where there is the Malthus term with the rate constant $r$ and a saturation term with the $\Omega$ factor, which is the upper limit for the population growth due to the availability of the resources.

Really the parameters $r$ and $\Omega$ are not constant. In fact the parameter $r$ changes randomly due to season fluctuations, and the parameter $\Omega$ fluctuates due to the environmental interaction which causes the random availability of resources. As a consequence we have the following stochastic Verhulst equation $$\frac{dx}{dt}=r\left( t\right) x\left[ 1-\frac{x}{\Omega
\left(t\right) }\right] .
\label{F-2}$$ In the context of macromolecular self-replication, the model equation (\[F-2\]), with constant $\Omega$ and a white Gaussian noise in $r\left( t\right)$, was numerically studied in Ref. [@Leu88] and the critical slowing down, i.e. a divergence of the relaxation time at some noise intensity, was found. Later Jackson and co-authors [@Jac89] investigated the same model, by analog experiment and digital simulations. They analyzed specifically in detail the nonlinear relaxation time defined as [@Bin73] $$T=\frac{\int\nolimits_{0}^{\infty}\left[\left\langle x\left(
t\right) \right\rangle -\left\langle x\left(\infty\right)
\right\rangle \right] dt}{x\left(0\right) - \left\langle x\left(
\infty\right) \right\rangle}
\label{F-3}$$ and did not observe the critical slowing down. They explained this discrepancy by the incorrect approximate truncation of the asymptotic power series for $T$ used in Ref. [@Leu88]. The stability conditions were derived in Ref. [@Gol03]. Similar investigations for colored Gaussian noise $r(t)$ were performed in Ref. [@Man90], where a monotonic dependence of the relaxation time and the correlation time on the noise intensity was found. The generalization of Eq. (\[F-2\]), to study a Bernoulli-Malthus-Verhulst model driven by a multiplicative colored noise, was analyzed recently in Ref. [@Cal07].\
The evolution of the mean value in the case of Eq. (\[F-2\]) with constant $r$ and white Gaussian noise excitation $\beta\left( t\right) =r/\Omega \left( t\right)$ was considered in Refs. [@Ciu93; @Suz82; @Suz82a; @Bre82; @Mak85; @Mor86]. In Refs. [@Mak85; @Mor86] the authors, using perturbation technique, obtained the exact expansion in power series on noise intensity of all the moments and found the long-time decay of $t^{-1/2}$. In Ref. [@Ciu93] the authors derived the long-time behavior of all the moments of the population density by means of an exact asymptotic expansion of the time averaged process generating function, and found the same asymptotic behavior of $t^{-1/2}$ at the critical point. This very slow relaxation of the moments near the critical point is the phenomenon of critical slowing down.\
In the present paper, using the previously obtained results for a generalized Langevin equation with a Lévy noise source [@Dub05; @Dub08], we investigate the transient dynamics of the stochastic Verhulst model with a fluctuating growth rate and a constant value for the saturation population density $\Omega$, that is $\Omega=1$. The exact results for the mean value of the population density and its nonstationary probability distribution for different types of white non-Gaussian excitation $r\left(
t\right)$ are obtained. We find the interesting noise-induced transitions for the probability distribution of the population density and the relaxation dynamics of its mean value for Cauchy stable noise. Finally we obtain a nonmonotonic behavior of the nonlinear relaxation time as a function of the Cauchy noise intensity.

Stochastic Verhulst equation with non-Gaussian fluctuations of growth rate
==========================================================================

Let us consider Eq. (\[F-2\]) with a constant saturation value $\Omega=1$, namely $$\frac{dx}{dt}=r\left(  t\right)  x\left(  1-x\right)  .
\label{F-4}$$ After changing variable $y=\ln[x/(1-x)]$, we obtain $$y\left(t\right) = y\left(0\right) +\int\nolimits_{0}^{t}r\left(
\tau\right) d\tau \,$$ and the exact solution of Eq. (\[F-4\]) is $$x\left(t\right) = \left(1+\frac{1-x_{0}}{x_{0}}\exp\left\{ -\int
\nolimits_{0}^{t}r\left(\tau\right) d\tau\right\} \right)^{-1},
\label{F-5}$$ where $x_{0}=x\left(0\right)$. Now by substituting in Eq. (\[F-5\]) the following expression for the random rate $r(t)$ $$r\left( t\right) =r+\xi\left( t\right),
\label{F-6}$$ where $r>0$ and $\xi\left( t\right)$ is an arbitrary white non-Gaussian noise with zero mean, we can rewrite the solution (\[F-5\]) as $$x\left(t\right) =
\left(1+\frac{1-x_{0}}{x_{0}}e^{-rt-L\left(t\right)}\right)^{-1}.
\label{F-7}$$ Here $L\left(t\right)$ denotes the so-called Lévy random process with $L\left(0\right)=0$, and $\xi\left(t\right)=\dot{L}\left(t\right) $. As it was shown in Refs. [@Dub05; @Dub08; @Fel71], Lévy processes having stationary and statistically independent increments on non-overlapping time intervals belongs to the class of stochastic processes with infinitely divisible distributions. As a consequence, the characteristic function of $L\left( t\right)$ can be represented in the following form (see Eq. (6) in  [@Dub05]) $$\left\langle e^{iuL\left(  t\right)  }\right\rangle =\exp\left\{
t\int\nolimits_{-\infty}^{+\infty}\frac{e^{iuz}-1-iu\sin
z}{z^{2}}\rho\left( z\right)  dz\right\}  ,
\label{F-8}$$ where $\rho\left(z\right)$ is some non-negative kernel function. The case $\rho\left(z\right)=2D\delta\left(  z\right)$ corresponds to a white Gaussian noise excitation $\xi\left(t\right)$, while for a symmetric Lévy stable noise $\xi\left( t\right)$ with index $\alpha$ we have a power-law kernel $\rho\left( z\right)=Q\left\vert
z\right\vert ^{1-\alpha}$, with $0<\alpha<2$.

In the model under consideration the stationary probability distribution has: (i) a singularity at the stable point $x = 1$ for white Gaussian noise, and (ii) two singularities at both stable points $x=0$ and $x=1$ for Lévy noise. To analyze the time behavior of the probability distribution in the transient dynamics it is better not to use the Kolmogorov equation for the probability density $P\left(x,t\right)$, but rather the exact solution (\[F-7\]). Using the standard theorem of the probability theory regarding a nonlinear transformation of a random variable, we find from Eq. (\[F-7\]) $$P\left(x,t\right) =\frac{1}{x\left(1-x\right)}P_{L}\left(\ln
\left[\frac{\left(1-x_{0}\right)x}{x_{0}\left(1-x\right)}\right]
-rt,t\right),
\label{F-9}$$ where $P_{L}\left(z,t\right)$ is the probability density corresponding to the characteristic function (\[F-8\]). For a white Gaussian noise $\xi\left(t\right)$, this distribution reads $$P_{L}\left(  z,t\right)  =\frac{1}{2\sqrt{\pi Dt}}\exp\left\{
-\frac{z^{2} }{4Dt}\right\}  .
\label{F-10}$$ The time evolution of the probability distribution $P\left(x,t\right) $ for $D=0.5$, $r=1$, and $x_{0}=0.2$ is plotted in Fig. \[fig-1\].

As it is easily seen, the maximum of the unimodal distribution with initial position at $x=0.2$ shifts with time towards the stable point at $x=1$. At the same time, as it follows from Eqs. (\[F-9\]) and (\[F-10\]), for all $t>0$ we have $$\lim_{x\rightarrow 0^+}P\left(x,t\right) = \lim_{x\rightarrow
1^-}P\left(x,t\right) = 0.
\label{F-11}$$

The same picture is observed for another kernel function $\rho\left(z\right) = Kz/\left(2\sinh z\right)$ $\left(K>0\right)$, corresponding to a Lévy process $\eta\left(t\right)$ with finite moments and the following probability density of increments $$P_{L}\left(  z,t\right)
=\frac{2^{Kt-1}}{\pi^{2}\Gamma\left(Kt\right)} \Gamma\left(
\frac{Kt}{2}+\frac{iz}{\pi}\right) \Gamma\left(\frac{Kt}
{2}-\frac{iz}{\pi}\right)  ,
\label{F-12}$$ where $\Gamma\left(x\right)$ is the Gamma function. The corresponding time evolution of the probability distribution $P\left( x,t\right)$ for $K=0.43$, $r=1$, and $x_{0}=0.2$ is shown in Fig. \[fig-2\].

-0.4cm

A different situation we have for a Cauchy stable noise $\xi\left(t\right)$ with constant kernel $\rho\left(z\right) = Q$ $\left( \alpha=1\right)$. After evaluation of the integral in Eq. (\[F-8\]), the probability density of the Lévy process increments takes the form of the well-known Cauchy distribution [@Fel71] $$P_{L}\left(z,t\right) = \frac{D_{1}t}{\pi\left[z^{2}+
\left(D_{1}t\right)^{2}\right]},
\label{F-13}$$ where $D_{1}=\pi Q$ is the noise intensity parameter. In such a case from Eqs. (\[F-9\]) and (\[F-13\]) for all $t>0$ we find $$\lim_{x\rightarrow 0^+}P\left(  x,t\right) =\lim_{x\rightarrow
1^-}P\left( x,t\right)  = \infty.
\label{F-14}$$ As a result, from an initial delta function we immediately obtain a trimodal distribution for $t>0$ and then after some transition time $t_c$ a bimodal one with two singularities at the stable points $x=0$ and $x=1$ (see Figs. \[fig-3\], \[fig-4\] and \[fig-5\]). We should note that the transition from trimodal to bimodal distribution is a general feature of the model in the presence of a Cauchy stable noise, and it is not limited to some range of parameters. In fact, from Eq. (\[F-14\]) and a delta function initial distribution inside the interval (0,1), this transition always takes place.

In the following Figs. \[fig-4\] and \[fig-5\] we show the time evolution of the probability distribution of the population density for two other values of the noise intensity, namely $D_1 = 1$ and $D_1 = 5$. As the noise intensity increases the probability distribution shows two singularities near $x = 0$ and $x = 1$ with different amplitude.

This transition in the shape of the probability distribution of the population density is due to both the multiplicative noise and the Lévy noise source. Using Eqs. (\[F-9\]) and (\[F-13\]) and equating to zero the derivative of $P(x,t)$ with respect to $x$, we obtain the following condition for the extrema in the range $0<x<1$, and particularly for a minimum in the same interval $$\frac{z(x,t)}{z(x,t)^2 + (D_1 t)^2} = x - \frac{1}{2} \; ,
\label{F-15}$$ with $$z(x,t) = \ln
\left[\frac{\left(1-x_{0}\right)x}{x_{0}\left(1-x\right)}\right] -rt
\,.
\label{F-16}$$ This condition can be solved graphically by finding the intersection between the functions $y_1 = z(x,t)/(z(x,t)^2 + (D_1 t)^2)$ and $y_2
= x - 1/2$. This is done in the following Figs. \[fig-6\], \[fig-7\], \[fig-8\], where the function $y_1$ is plotted for three different values of time and noise intensity. In each figure the black blue curve (color on line) corresponds to the critical value of time $t_c$ for which we have a noise induced transition of the probability distribution of the population density from trimodal to bimodal, that is from two minima and one maximum to one minimum inside the interval $0 < x < 1$. The appearance of one minimum in the probability distribution is the signature of this transition.

The three values of the critical time $t_c$ corresponding to the three values of the Lévy noise intensity investigated are: $D_1
= 0.5,~t_c = 2.493;~D_1 = 1,~t_c = 1.635;~D_1 = 5,~t_c = 0.247$. One rough evaluation of the critical time $t_c$ is obtained by putting equal to $1$ the scale parameter of the Cauchy distribution of Eq. (\[F-13\]), that is $t_c \sim 1/D_{1}$. The critical time $t_c$ is the time at which the maximum and one minimum of the probability distribution (see Figs. \[fig-3\], \[fig-4\], and \[fig-5\]) coalesce in one inflection point and in this point $x$ the function $y_2 = x - 1/2$ becomes tangent at the function $y_1$ (see Figs. \[fig-6\], \[fig-7\], and \[fig-8\]). It is interesting to note that the critical time $t_c$ decreases with the noise intensity $D_1$. This is because by increasing the noise intensity, more quickly the population density reaches the two points near the boundaries $x = 0$ and $x = 1$.

Nonlinear relaxation time of the mean population density
========================================================

It must be emphasized that to find the time evolution of the mean population density one can use two different approaches. The first one was proposed in Ref. [@Jac89]. According to the exact solution (\[F-7\]) of the Verhulst equation (\[F-4\]), we can rewrite this expression in the following form $$x\left(t\right) = f\left(e^{-rt-L\left(t\right)}\right),
\label{F-17}$$ where $$f\left(q\right)=\left(1+\frac{1-x_{0}}{x_{0}}q(t) \right)^{-1}.
\label{F-18}$$ Then, by expanding the smooth function (\[F-18\]) in a standard Taylor power series in $q$ around the point $q=0$ we have $$f\left(q\right) = \sum\limits_{n=0}^{\infty}\frac{f^{\left(
n\right)} \left(0\right)}{n!}q^{n}.
\label{F-19}$$ After substitution of Eq. (\[F-19\]) in Eq. (\[F-17\]) and averaging we obtain $$\left\langle x\left(t\right) \right\rangle =
\sum\limits_{n=0}^{\infty} \frac{f^{\left(n\right)} \left(
0\right)e^{-nrt}}{n!}\left\langle e^{-nL\left(t\right)}\right\rangle
\label{F-20}$$ or, in accordance with Eq. (\[F-8\]), $$\begin{aligned}
\left\langle x\left(  t\right)  \right\rangle
&=&\sum\limits_{n=0}^{\infty} \frac{f^{\left(  n\right)  }\left(
0\right) e^{-nrt}}{n!}\label{F-21} \\
&\times&\exp\left\{ t\int_{-\infty}^{+\infty}\frac{e^{-nz}-1+n\sin
z}{z^{2}}\,\rho\left( z\right) dz\right\} .\nonumber\end{aligned}$$ For white Gaussian noise $\xi\left(t\right)$ with kernel $\rho\left(z\right) = 2D \delta(z)$ we obtain from Eq. (\[F-21\]) the following asymptotic series $$\left\langle x\left(t\right) \right\rangle
=\sum\limits_{n=0}^{\infty}
\frac{f^{\left(n\right)}\left(0\right)}{n!}e^{Dtn^{2}-nrt}.
\label{F-22}$$ By considering a finite number of terms in this expansion leads to a wrong conclusion about the critical slowing down phenomenon in such a system, as found in Ref. [@Leu88]. The exact result is obtained, of course, by summing all the terms in Eq. (\[F-22\]). Moreover, for most of the kernels $\rho(z)$ the integral in Eq. (\[F-21\]) diverges. Thus, this approach is inappropriate for our purposes, and it is better to use the direct average in Eq. (\[F-7\]). Therefore, using this second approach we have $$\left\langle x\left(  t\right)  \right\rangle
=\int\nolimits_{-\infty }^{+\infty}\left(
1+\frac{1-x_{0}}{x_{0}}e^{-rt-z}\right)  ^{-1}P_{L}\left(
z,t\right)dz.
\label{F-23}$$

Let us consider now different models of white non-Gaussian noise $\xi\left(t\right)$. We start with the white shot noise $$\xi\left(t\right) = \sum\limits_{i}a_{i}\delta\left(t-t_{i}\right)
\label{F-24}$$ having the symmetric dichotomous distribution of the pulse amplitude $P(a) = \left[\delta\left(a-a_{0}\right) + \delta
\left(a+a_{0}\right)\right] /2$, mean frequency $\nu$ of pulse train, and kernel $\rho\left(z\right) = \nu z^{2} P\left(z\right)$. From Eq. (\[F-8\]) we have $$\left\langle e^{iuL\left(t\right)}\right\rangle = e^{-\nu
t\left(1-\cos a_{0}u\right)}.
\label{F-25}$$ By making the reverse Fourier transform in Eq. (\[F-25\]) we find the probability distribution of the corresponding Lévy process $$P_{L}\left( z,t\right) =e^{-\nu
t}\sum_{n=-\infty}^{+\infty}I_n\left( \nu t\right) \delta \left(
z-na_0\right) ,
\label{F-26}$$ where $I_n\left( x\right) $ is the $n$-order modified Bessel function of the first kind. The relaxation of the mean population density $x(t)$ is shown in Fig. \[fig-9\]. According to the Eq. (\[F-23\]) and (\[F-26\]) the stationary value of the population density in such a case is $\left\langle x\right\rangle
_{st}=1$, but the relaxation time (\[F-3\]) increases with increasing the mean frequency of pulses.

For white non-Gaussian noise with the kernel $\rho(z) =
Kz/\left(2\sinh z\right)$ we observe a similar transient dynamics, which is shown in Fig. \[fig-10\]. We have the same stationary value $\left\langle x\right\rangle_{st}$, and the relaxation time $T$ increases with increasing the parameter $K$, which is proportional to the noise intensity.

Finally, in the case of white Cauchy noise $\xi\left(t\right)$ we obtain interesting exact analytical results. First of all, substituting Eq. (\[F-13\]) in Eq. (\[F-23\]) and changing the variable $z=D_{1}ty$ under the integral, we obtain $$\left\langle x\left(t\right) \right\rangle = \frac{1}{\pi}\int
\nolimits_{-\infty}^{+\infty}\left[
1+\frac{1-x_{0}}{x_{0}}e^{-t\left( r+D_{1}y\right)}\right]
^{-1}\frac{dy}{1+y^{2}}.
\label{F-27}$$ For the stationary mean value $\left\langle x\right\rangle _{st}$ we find from Eq. (\[F-27\]) $$\left\langle x\right\rangle
_{st}=\lim_{t\rightarrow\infty}\left\langle x\left(  t\right)
\right\rangle =\frac{1}{\pi}\int\nolimits_{-\infty
}^{+\infty}\frac{1\left(  r+D_{1}y\right)  dy}{1+y^{2}},
\label{F-28}$$ where $1\left(x\right)$ is the step function. After evaluation of the integral in Eq. (\[F-28\]) we obtain finally $$\left\langle x\right\rangle
_{st}=\frac{1}{2}+\frac{1}{\pi}\arctan\frac {r}{D_{1}}.
\label{F-29}$$ As it is seen from Fig. \[fig-11\] and Eq. (\[F-29\]), for small noise intensity $D_{1}$, with respect to the value of the rate parameter $r = 1$, the stationary mean value of the population density is approximately $1$, as for the other white non-Gaussian noise excitations considered. But for large values of $D_{1}$, this asymptotic value, which is independent from the initial value of population density $x_{0}$, tends to $0.5$.

It is interesting also to analyze, for this case of white Cauchy noise, the dependence of the relaxation time $T$ from the noise intensity $D_{1}$. Substituting Eq. (\[F-27\]) in Eq. (\[F-3\]) and changing the order of integration, for initial condition $x_0 =
0.5$, we are able to calculate analytically the double integral in $t$ and in $y$ obtaining the final result $$T=\frac{\pi\ln2}{r\left(  1+D_{1}^{2}/r^{2}\right) arccot(D_{1}/r)}.
\label{F-30}$$ We find a non-monotonic behavior of the relaxation time $T$ versus the noise intensity $D_{1}$ with a maximum at the noise intensity $D_1 = 0.43$, as shown in Fig. \[fig-12\].

This nonmonotonic behavior is also visible for another initial position $x_0 = 0.2$, as shown in Fig. \[fig-11\]. Here the relaxation time to reach the stationary value of population density $x_{st}$ increases from very low noise intensity ($D_1 = 0.1$) to moderate low intensity ($D_1 = 0.5$), while decreases for higher noise intensities ($D_1 = 1.5$). This is also due to the dependence of $x_{st}$ from the noise intensity $D_1$ (see Eq. (\[F-30\])). We note that this non-monotonic behavior of the relaxation time $T$ is related to the peculiarities of the transient dynamics of the mean population density and it will be object of further investigations.

Conclusions
===========

The transient dynamics of the Verhulst model, perturbed by arbitrary non-Gaussian white noise, is investigated. This well-known equation is an appropriate ecological and biological model to describe closed-population dynamics, self-replication of  macromolecules under constraint, cancer growth, spread of viral epidemics, etc... By using the properties of the infinitely divisible distribution of the generalized Wiener process, we analyzed the effect of different non-Gaussian white sources on the nonlinear relaxation of the mean population density and on the time evolution of the probability distribution of the population density. We obtain exact results for the nonstationary probability distribution in all cases investigated and for the Cauchy stable noise we derive the exact analytical expression of the nonlinear relaxation time. Due to the presence of a Lévy multiplicative noise, the probability distribution of the population density exhibits a transition from a trimodal to a bimodal distribution in asymptotics. This transition, characterized by the appearance of a minimum, happens at a critical time $t_c$, which can be roughly evaluated as $t_c \sim 1/D_1$ (where $D_1$ is the noise intensity) and exactly evaluated from the condition (\[F-15\]). Finally a nonmonotonic behavior of the nonlinear relaxation time of the population density as a function of the Cauchy noise intensity was found.

Acknowledgements {#acknowledgements .unnumbered}
================

We acknowledge support by MIUR, CNISM-INFM, and Russian Foundation for Basic Research (project 08-02-01259).

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