About this Journal Submit a Manuscript Table of Contents
Advances in Mathematical Physics

Volume 2014 (2014), Article ID 597012, 6 pages

http://dx.doi.org/10.1155/2014/597012
Research Article

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

1Departamento de Física, Facultad de Ciencias, Universidad de Tarapacá, 1000000 Arica, Chile

2Escuela Universitaria de Ingeniería Eléctrica-Electrónica, Universidad de Tarapacá, 1000000 Arica, Chile

3Instituto de Alta Investigación IAI, Universidad de Tarapacá, 1000000 Arica, Chile

Received 8 January 2014; Revised 23 April 2014; Accepted 5 May 2014; Published 4 June 2014

Academic Editor: Klaus Kirsten

Copyright © 2014 Héctor Calisto et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.

1. Introduction

During the past decades, the phenomena induced by noise in nonlinear systems have received considerable attention [16]. In the ecological context we can mention resource management, new techniques for pollution and waste control, and a better understanding of biophysical processes in living organisms (see, e.g., [7, 8]). In this paper, we study the stochastic effects on the environment in a successful colonizing population. In absence of fluctuations of the environment, it is assumed that the population grows according to the generalized Malthus-Verhulst model where the dot represents the time derivative. The factor has been made explicit so as to include in (1) the Gompertz model [9] by taking the limit .

In recent decades, several generalizations of the Malthus-Verhulst model have been applied, for example, to laser physics [10, 11], and have been widely used in literature (see [12, 13] for a large list of references). In the above references, the reader can find many applications to the processes of different species growth, such as animals that inhabit the rivers and seas of our planet, human populations, components of the central nervous system in living organisms, transmission of diseases by different types of viruses, growth of malignant tumors caused by abnormal and uncontrolled cell division, interaction of vortices in a turbulent fluid, coupled chemical reactions that occur in our upper atmosphere, interactions between galaxies, competition between different political parties, companies business, and talks between different countries. Furthermore, the generalized Malthus-Verhulst model, among other possible models, has the applications for other areas of knowledge, such as social science [14, 15], autocatalytic chemical reactions [3, 16], biological and biochemical processes, population of photons in a single mode laser [2], freezing of supercooled liquids [17], grain growth in polycrystalline materials [18], and cell growth in foam [19].

A fluctuating environment can affect the dynamics of (1) in different ways as follows.

(i) Changes in Net Growth Rate. Stochastic elements in the growth rate can be introduced in the form where represents noise, , and is a parameter that controls the noise intensity. In [9, 20], the case where is Gaussian white noise is studied. In [21], the authors studied the case where is given by the Ornstein-Uhlenbeck process [22]. In [23], several cases of white non-Gaussian noise are examined. In [24], the problem of asymmetric Poissonian dichotomic noise is solved and an exact expression for the probability density was found. Finally in [25], an extension to non-Poissonian dichotomic noise has been addressed.

(ii) Changes in the Upper Limit to Growth. In [9], in order to incorporate randomness in the carrying capacity , the following modification has been considered: with the deterministic value of the carrying capacity, a constant parameter measuring the intensity of the noise, and Gaussian white noise.

The authors obtained the probability density function at steady state. In [26], the authors considered only stochastic discrete populations growth models. The two aspects mentioned in the previous paragraph are considered and a large list of references is provided.

In this paper, we will study the effects of random fluctuations in the environment on a successful population. By this, we mean a population with a magnitude of the critical size order of the system. Thus, the number of individuals and the changes produced by time dependent fluctuations can be treated as a continuous variable. In [9], the authors modeled environment fluctuations by white Gaussian noise which has zero correlation time. In this case, the process is Markovian and a Fokker-Planck equation for the probability density can be obtained. However, this noise cannot always replace real noise, which has a finite correlation time, perhaps small, but not zero. In this case, the hypothesis of Gaussian white noise could be inadequate to describe the stochastic process.

Willing to take into account the color of the noise, we should choose a mathematically tractable colored noise [6]. Among many possible noises, two have drawn a lot of attention in the literature. The former is the Ornstein-Uhlenbeck process. Stochastic processes driven by this kind of noise have been studied, for example, in [5, 27] and, with respect to the white noise assumption, different features have been found. The latter noise is the two-step Markov process or dichotomous noise. This noise is not Gaussian but Markovian and its influence in the stochastic processes has been studied in [5, 28]. Experimental evidences of the dichotomic noise have been studied in literature [5, 2931]. Interesting results, some of them quite similar to the former case, have been obtained in the stationary state. Few dynamical properties are known in this case [28, 32]. This type of noise has found a wide application in the construction of models [5]. Furthermore, by appropriate processes of limit, it can be demonstrated that the asymmetric dichotomic noise converges to Gaussian white noise, and it also converges to the white shot noise [33, 34].

This paper will focus mainly on the analytical evaluation of the average number of individuals , exploiting the properties of the correlation function of the stochastic variable without using the probability density . The analytical formula for will be supported by numerical simulations showing an excellent agreement with the analytical calculations.

2. Mean Value of

As in all stochastic processes, the evaluation of the average of the stochastic variable under study is important. In this section, we will evaluate the mean value of without evaluating the probability density but with using the exponential properties of the correlation function of the stochastic variable . Note that the equation for the probability density can be written utilizing the Shapiro-Loginov derivative [35]. Instead, to evaluate the mean value of , we will follow a different approach based on the properties of the correlation function of . In spite of the fact that, usually, the calculations performed using the correlation function are a hard task, sound analytical results can be found (see, e.g., [36]). In this section, we will show a technique that in principle can be applied to other stochastic equations. We may rewrite (1) as where for sake of compactness we set and we considered the symmetric case . Performing the substitutions we obtain The above equation can be solved via standard methods and we obtain for the variable the following result: where is related to the initial condition of the function through the relation To make handier the above expression, let us define the following symbols: and, taking the average with respect to the realizations, we may write [37] with . We focus our attention on a symmetric dichotomic noise with an exponential correlation function (Poissonian process); that is, Exploiting the factorizing of the correlation functions [37, 38], after long but straightforward algebra, we obtain where and . To obtain an analytical expression of the multiple integral, we use the Laplace transform and we obtain The quantity is so reduced to a convolution integral where For , we may write (14) as where is the Laplace transform of evaluated in and is the hypergeometric function (for more details, see, e.g., [39]). Expressing the result in terms of , (9), we have or, going back to the time , From (8), we deduce that, for , ; that is, the more we start near to the point , the more extended on time is the transient region (Figures 1, 2, and 3). For a deeper discussion about the dynamic near the point , see [40]. Considering the limit for , we may write the asymptotic expression as Some interesting conclusions can be extrapolated from the above expression taking the limits and . The limit gives the following approximated expression: As expected for a vanishing intensity of the noise, the number of individuals takes the maximum population size allowed, namely, . More, (20) shows that in general the average value of the population is always smaller than the unperturbed value of the carrying capacity even if half of the stochastic realizations exceed this value, that is, . The deduction holds true also for an arbitrary value . Indeed, in this range of the noise intensity, the hypergeometric function is decreasing and for we have . For , we have the following approximate expression: By a direct inspection of (21), we note that there is a relationship among the parameters , given by such that the function argument vanishes, producing a divergent value of the numerical coefficient in (21). Such a divergence is only apparent and it disappears considering further terms in the expansion of (19). Nevertheless, if we consider the derivative of with respect to we have that This means that when the fluctuations are very large, that is, , for , the quantity vanishes with an infinite slope, that is to say, vanishes faster than the case (see Figures 4 and 5).

597012.fig.001
Figure 1: The plot of the mean value of versus time in arbitrary units. The values of the parameters are , , , , , and . The analytical value of is given by (18). The agreement between the analytical expression (dashed line) and the numerical simulation (continuous line) is remarkable.
597012.fig.002
Figure 2: The plot of the mean value of versus time in arbitrary units. The values of the parameters are , , , , , and . The analytical value of is given by (18). The agreement between the analytical expression (dashed line) and the numerical simulation (continuous line) is remarkable.
597012.fig.003
Figure 3: The plot of the mean value of versus time in arbitrary units. The values of the parameters are , , , , , and . The analytical value of is given by (18). The agreement between the analytical expression (dashed line) and the numerical simulation (continuous line) is remarkable.
597012.fig.004
Figure 4: The plot of as function of the noise intensity . The values of the parameters are , , and . In this case, and, for , the number of individuals decreases with an infinite slope.
597012.fig.005
Figure 5: The plot of as function of the noise intensity . The values of the parameters are , , and . In this case, and, for , the number of individuals decreases with a finite slope.

3. Numerical Check

In order to validate the analytical results obtained in Section 2, we solved numerically the stochastic differential equation (3). The numerical simulation is done by creating an ensemble of trajectories following the prescription of (3), where the random variable fluctuates between the two values . We simulated a symmetric process so the probability to take the value or is the same. The initial value of , selected by tossing a fair coin, lasts for a time . At the end of this time interval, we toss again the coin and the random variable may or may not change sign according to the result of tossing the coin procedure. The random variable keeps the new sign for the whole time . This procedure is iterated at the succeeding times .

For a dichotomous process with an exponential correlation function, the waiting time distribution density is an exponential function. The time durations are drawn from a waiting time distribution . The numerical simulation is used to check the theoretical prediction of (19). As showed in Figures 1, 2, and 3, the theoretical and numerical values are in very good agreement.

4. Concluding Remarks

In this paper, we studied a generalized logistic equation (Malthus-Verhulst model) with a fluctuating carrying capacity. We found an analytical expression for the average of the number of individuals that allowed us to make few nontrivial remarks. We showed that, even in presence of a fluctuating carrying capacity, the average tends, asymptotically, to a constant quantity, (19). The asymptotic value is always smaller than the unperturbed value of the carrying capacity even if half of the stochastic realizations exceed this value. Finally, we found that, in case of very large fluctuation, in particular of the unperturbed carrying capacity size order, the number of individuals, for , decreases with an infinite slope, while, for , the number of individuals decreases with a finite slope. Our theoretical study has been supported by numerical simulations showing an excellent agreement with the analytical results.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

Mauro Bologna acknowledges financial support from FONDECYT Project no. 1120344.

References

  1. R. L. Stratonovich, Topics in the Theory of Random Noise, Vol. I and Vol. II, Gordon and Breach Science, New York, NY, USA, 1963. View at MathSciNet
  2. H. Haken, Synergetic: An Introduction, Springer, Berlin, Germany, 2nd edition, 1978.
  3. C. W. Gardiner, Handbook of Stochastic Methods, Springer, Berlin, Germany, 1983. View at Publisher · View at Google Scholar · View at MathSciNet
  4. H. Risken, The Fokker-Planck Equation: Methods of Solutions and Applications, Springer, Berlin, Germany, 1984. View at Publisher · View at Google Scholar · View at MathSciNet
  5. W. Horsthemke and R. Lefever, Noise-Induced Transitions, Springer, Berlin, Germany, 1984. View at MathSciNet
  6. N. G. van Kampen, Stochastic Processes in Physics and Chemistry, North-Holland, Amsterdam, The Netherlands, 1992.
  7. S. Ciuchi, F. De Pasquale, and B. Spagnolo, “Self-regulation mechanism of an ecosystem in a non-Gaussian fluctuation regime,” Physical Review E, vol. 54, no. 1, pp. 706–716, 1996. View at Scopus
  8. B. Spagnolo and A. La Barbera, “Role of the noise on the transient dynamics of an ecosystem of interacting species,” Physica A, vol. 315, no. 1-2, pp. 114–124, 2002. View at Publisher · View at Google Scholar · View at Scopus
  9. N. S. Goel and N. Richter-Dyn, Stochastic Models in Biology, The Blackburn Press, Caldwell, NJ, USA, 2003.
  10. W. E. Lamb Jr., “Theory of an optical maser,” Physical Review, vol. 134, no. 6A, pp. A1429–A1450, 1964. View at Publisher · View at Google Scholar · View at Scopus
  11. B. Pariser and T. C. Marshall, “Time development of a laser signal,” Applied Physics Letters, vol. 6, no. 12, pp. 232–234, 1965. View at Publisher · View at Google Scholar · View at Scopus
  12. R. Zygadło, “Relaxation and stationary properties of a nonlinear system driven by white shot noise: an exactly solvable model,” Physical Review E, vol. 47, no. 6, pp. 4067–4075, 1993. View at Publisher · View at Google Scholar · View at Scopus
  13. B. J. West and P. Grigolini, Complex Webs: Anticipating the Improbable, Cambridge University Press, Cambridge, UK, 2011. View at MathSciNet
  14. R. Herman and E. W. Montroll, “A manner of characterizing the development of countries,” Proceedings of the National Academy of Sciences of the United States of America, vol. 69, pp. 3019–3023, 1972. View at Publisher · View at Google Scholar
  15. E. W. Montroll, “Social dynamics and the quantifying of social forces,” Proceedings of the National Academy of Sciences of the United States of America, vol. 75, no. 10, pp. 4633–4637, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. V. Bouché, “A new instability phenomenon in the Malthus-Verhulst model,” Journal of Physics A. Mathematical and General, vol. 15, no. 6, pp. 1841–1848, 1982. View at Publisher · View at Google Scholar · View at MathSciNet
  17. A. K. Das, “A stochastic approach to the freezing of supercooled liquids,” Canadian Journal of Physics, vol. 61, pp. 1046–1049, 1983. View at Publisher · View at Google Scholar
  18. C. S. Pande, “On a stochastic theory of grain growth,” Acta Metallurgica, vol. 35, no. 11, pp. 2671–2678, 1987. View at Scopus
  19. C. H. Wörner, A. Olguín, M. Ortiz, O. Herrera, J. C. Flores, and H. Calisto, “Final cell distribution in a Zener pinned structure,” Acta Materialia, vol. 51, no. 20, pp. 6263–6267, 2003. View at Publisher · View at Google Scholar · View at Scopus
  20. N. S. Goel, S. C. Maitra, and E. W. Montroll, “On the Volterra and other nonlinear models of interacting populations,” Reviews of Modern Physics, vol. 43, pp. 231–276, 1971. View at Publisher · View at Google Scholar · View at MathSciNet
  21. H. Calisto and M. Bologna, “Exact probability distribution for the Bernoulli-Malthus-Verhulst model driven by a multiplicative colored noise,” Physical Review E, vol. 75, no. 5, Article ID 050103, 2007. View at Publisher · View at Google Scholar · View at Scopus
  22. G. E. Uhlenbeck and L. S. Ornstein, “On the theory of the Brownian motion,” Physical Review, vol. 36, no. 5, pp. 823–841, 1930. View at Publisher · View at Google Scholar · View at Scopus
  23. A. A. Dubkov and B. Spagnolo, “Verhulst model with Lévy white noise excitation,” The European Physical Journal B, vol. 65, no. 3, pp. 361–367, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. G. Aquino, M. Bologna, and H. Calisto, “An exact analytical solution for generalized growth models driven by a Markovian dichotomic noise,” Europhysics Letters, vol. 89, no. 5, Article ID 50012, 2010. View at Publisher · View at Google Scholar · View at Scopus
  25. M. Bologna and H. Calisto, “Effects on generalized growth models driven by a non-Poissonian dichotomic noise,” European Physical Journal B, vol. 83, no. 3, pp. 409–414, 2011. View at Publisher · View at Google Scholar · View at Scopus
  26. J. H. Matis and T. R. Kiffe, Stochastic Population Models, Springer, New York, NY, USA, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  27. P. Hanggi and P. Jung, “Colored noise in dynamical systems,” Advances in Chemical Physics, vol. 89, pp. 239–326, 1995.
  28. I. Bena, “Dichotomous Markov noise: exact results for out-of-equilibrium systems,” International Journal of Modern Physics B, vol. 20, no. 20, pp. 2825–2888, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  29. P. Allegrini, M. Bologna, L. Fronzoni, P. Grigolini, and L. Silvestri, “Experimental quenching of harmonic stimuli: universality of linear response theory,” Physical Review Letters, vol. 103, no. 3, Article ID 030602, 2009. View at Publisher · View at Google Scholar · View at Scopus
  30. I. Broussell, I. L'Heureux, and E. Fortin, “Experimental evidence for dichotomous noise-induced states in a bistable interference filter,” Physics Letters A, vol. 225, no. 1–3, pp. 85–91, 1997. View at Scopus
  31. M. Wu and C. D. Andereck, “Effects of external noise on the Fredericksz transition in a nematic liquid crystal,” Physical Review Letters, vol. 65, no. 5, pp. 591–594, 1990. View at Publisher · View at Google Scholar · View at Scopus
  32. N. G. van Kampen, “Stochastic differential equations,” Physics Reports, vol. 24, no. 3, pp. 171–228, 1976. View at MathSciNet
  33. C. Van den Broeck, “On the relation between white shot noise, Gaussian white noise, and the dichotomic Markov process,” Journal of Statistical Physics, vol. 31, no. 3, pp. 467–483, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. A. Fuliński, “Non-Markovian noise,” Physical Review E, vol. 50, no. 4, pp. 2668–2681, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. V. E. Shapiro and V. M. Loginov, “‘Formulae of differentiation’ and their use for solving stochastic equations,” Physica A, vol. 91, no. 3-4, pp. 563–574, 1978. View at Publisher · View at Google Scholar · View at MathSciNet
  36. M. Bologna, K. J. Chandía, and B. Tellini, “Stochastic resonance in a non-Poissonian dichotomous process: a new analytical approach,” Journal of Statistical Mechanics: Theory and Experiment, no. 7, Article ID P07006, 2013. View at MathSciNet
  37. P. Allegrini, P. Grigolini, and B. J. West, “Dynamical approach to Lévy processes,” Physical Review E, vol. 54, no. 5, pp. 4760–4767, 1996. View at Scopus
  38. M. Bologna and P. Grigolini, “Lévy diffusion: the density versus the trajectory approach,” Journal of Statistical Mechanics: Theory and Experiment, vol. 2009, no. 3, Article ID P03005, 2009. View at Publisher · View at Google Scholar · View at Scopus
  39. M. Bologna, “Asymptotic solution for first and second order linear Volterra integro-differential equations with convolution kernels,” Journal of Physics A. Mathematical and Theoretical, vol. 43, no. 37, Article ID 375203, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  40. S. Ciuchi, F. De Pasquale, and B. Spagnolo, “Nonlinear relaxation in the presence of an absorbing barrier,” Physical Review E, vol. 47, no. 6, pp. 3915–3926, 1993. View at Publisher · View at Google Scholar · View at Scopus