International Journal of Stochastic Analysis

Volume 2011 (2011), Article ID 489386, 7 pages

http://dx.doi.org/10.1155/2011/489386

## A Stochastic Two Species Competition Model: Nonequilibrium Fluctuation and Stability

Department of Mathematics, Bengal Engineering and Science University, Shibpur Howrah 711103, India

Received 2 December 2010; Accepted 7 February 2011

Academic Editor: Josefa Linares-Perez

Copyright © 2011 G. P. Samanta. 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

The object of this paper is to study the stability behaviours of the deterministic and stochastic versions of a two-species symmetric competition model. The logistic parameters of the competitive species are perturbed by colored noises or Ornstein-Uhlenbeck processes due to random environment. The Fokker-Planck equation has been used to obtain probability density functions. Here, we have also discussed the relationship between stability behaviours of this model in a deterministic environment and the corresponding model in a stochastic environment.

#### 1. Introduction

In recent years, scientists have become increasingly aware of the fact that most natural phenomena do not follow strictly deterministic laws but rather oscillate randomly about some average behaviour. This is especially true in the ecological models where environmental influences should be taken as stochastic. Many models in theoretical ecology take the Volterra-Lotka model of interacting species as a starting point. In an improved analysis, the influence of the environment has to be taken into account. This is often done by arbitrarily augmenting the deterministic equations with stochastic terms or taking the environmental parameters as time dependent and rapidly varying. In both cases, the corresponding stochastic properties have to be postulated [1]. In 1995, Renshaw mentioned that the most natural phenomena do not follow strictly deterministic laws but rather oscillate randomly about some average so that the deterministic equilibrium is not an absolutely fixed state; instead, it is a “fuzzy” value around which the biological system fluctuates. In fact, randomness or stochasticity plays a vital role in the structure and function of biological systems. The environmental factors are time dependent, randomly varying and should be taken as stochastic. In ecology, we have two types of stochasticity, namely, the demographic stochasticity and the environmental stochasticity [2, 3]. Both types of stochasticity play a significant part in the realistic dynamical modelling of ecosystems. Several systematic procedures have been developed to obtain the stationary probability density function for the response of general single-degree-of-freedom nonlinear oscillators and nonlinear dynamical systems under parametric and external Gaussian white noise excitations [4–7].

In this paper, we have studied the relationship between stability behaviours of a two-species symmetric competition model in which environmental parameters are prescribed constant (deterministic environment) and the corresponding model in which these parameters have an element of random fluctuation (stochastic environment), where it is assumed that the fluctuations in the environment manifest themselves as random fluctuations in the logistic parameters about some mean value, as they usually do in nature [8, 9].

#### 2. Two-Species Symmetric Competition: Stochastic Differential Equations

Two or more species-populations compete for the same limited food source or in some way the growth of each is inhibited by members both of its own and of the other species. For example, competition may be for territory which is directly related to food resources. Some interesting phenomena have been observed from the study of practical competition models [10, 11]. Here, we will discuss a competition model which demonstrates a fairly general principle which is observed to hold in nature. Assume that the two-species symmetric competition model in a deterministic environment satisfies the following deterministic differential equations:
where , denote the number or density of individuals of two competing species. Thus, it is assumed that the *per capita* growth rate of each population at any instant is a linear function of the sizes of the two competing populations at that instant. Each population would grow logistically if it were alone with a constant environmental parameter and a parameter which measures the (symmetric) competition between the two species (i.e., it measures the degree to which the presence of one species affects the growth of the other species), we assume that .

Let us rewrite the system of differential equations (2.1) in the following form:

The deterministic equilibrium populations , are given by The interaction matrix in the neighbourhood of this equilibrium is The two eigenvalues of this interaction matrix are , , and the deterministic stability criterion is satisfied since .

To take into account the random environment, we extend model (2.2) to the form of the following stochastic differential equations: where the fluctuations are colored noises or Ornstein-Uhlenbeck processes which are more realistic noises than white noises. These are extremely useful to model rapidly fluctuating phenomena, because it can be seen by studying their spectra that thermal noises in electrical resistance and climate fluctuations, disregarding the periodicities of astronomical origin and so forth, are colored noises to a very good approximation. These examples support the usefulness of the colored noise idealization in applications to natural systems. The mathematical expectations and correlation functions of the processes are given by where , are, respectively, the intensity and the correlation time of the colored noise and represents the average over the ensemble of the stochastic process. The are the solutions of the stochastic differential equations [12] where are standard white noises characterized by where denotes the Dirac delta function.

Let ; therefore, . Hence, Put ; therefore, , where . Hence,

Therefore, From (2.7) and (2.11), we have Put, , , and .

Therefore, where,

#### 3. Stationary Probability Density

The reduced Fokker-Planck equation is given by [13] where are the state variables (representing the possible values) of , is the stationary probability density, the drift and diffusion coefficients are where After some simplifications using the results of Cai and Lin [13], we have where

By integrating over all values of , we obtain Similarly, we have

#### 4. Concluding Remarks

In this paper, we have studied the relationship between stability behaviours of a two-species symmetric competition model for interacting species, in which environmental parameters are prescribed constants (deterministic environment), and the corresponding model in which these parameters have an element of random fluctuation (stochastic environment). In the stochastic environment, the environmental parameter has been perturbed by colored noises or Ornstein-Uhlenbeck processes, which are more realistic noises than white noises.

Now, in the deterministic environment, the maximum of the eigenvalues of the interaction matrix is . Therefore, the stability determining quantity is , and the deterministic stability criterion is satisfied, since (as ). In the stochastic environment, whose random fluctuation has intensity , the stability provided by the population interaction dynamics is again characterized by . It is no longer enough that , for if , population fluctuations are relatively small; in this case, the probability cloud persists for an appreciable time, and the environment is effectively deterministic. For , but not much less, populations are likely to undergo significant fluctuations, even though they persist for long times. Finally, if , population exhibit large fluctuations in this case, the interaction dynamics provides an ever-weaker stabilizing influence to offset the randomizing , which rapidly lead to extinction. These results are in good agreement with those of Samanta [14], Samanta and Maiti [15], and Maiti and Samanta [16].

The correlation time of the colored noise is . The limiting process for of the colored noise is the Gaussian white noise, and in this case we obtain the well-known results of May [8]. Now, for , we see that the population fluctuations are very much small, and in this case, the environment is effectively deterministic.

#### Acknowledgment

The author is very grateful to the anonymous referees and the editor (Dr. Josefa Linares-Perez) for their careful reading, valuable comments, and helpful suggestions, which have helped him to improve the presentation of this paper significantly.

#### References

- J. Roerdink and A. Weyland, “A generalized Fokker-Planck equation in the case of the Volterra model,”
*Bulletin of Mathematical Biology*, vol. 43, no. 1, pp. 69–79, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. M. Nisbet and W. S. C. Gurney,
*Modelling Fluctuating Populations*, Wiley-Interscience, New York, NY, USA, 1982. - C. W. Gardiner,
*Handbook of Stochastic Methods*, vol. 13 of*Springer Series in Synergetics*, Springer, Berlin, Germany, 1983. - R. Wang and Z. Zhang, “Exact stationary solutions of the Fokker-Planck equation for nonlinear oscillators under stochastic parametric and external excitations,”
*Nonlinearity*, vol. 13, no. 3, pp. 907–920, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Zhang, R. Wang, and K. Yasuda, “On joint stationary probability density function of nonlinear dynamic systems,”
*Acta Mechanica*, vol. 130, no. 1-2, pp. 29–39, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Wang, K. Yasuda, and Z. Zhang, “A generalized analysis technique of the stationary FPK equation in nonlinear systems under Gaussian white noise excitations,”
*International Journal of Engineering Science*, vol. 38, no. 12, pp. 1315–1330, 2000. View at Publisher · View at Google Scholar - R. Wang, Y.-B. Duan, and Z. Zhang, “Resonance analysis of the finite-damping nonlinear vibration system under random disturbances,”
*European Journal of Mechanics. A. Solids*, vol. 21, no. 6, pp. 1083–1088, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. M. May, “Stability in randomly fluctuating versus deterministic environments,”
*The American Naturalist*, vol. 107, pp. 621–650, 1973. View at Publisher · View at Google Scholar - G. P. Samanta, “A two-species competitive system under the influence of toxic substances,”
*Applied Mathematics and Computation*, vol. 216, no. 1, pp. 291–299, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. B. Hsu, S. P. Hubbell, and P. A. Waltman, “A contribution to the theory of competing predators,”
*Ecological Monographs*, vol. 48, pp. 337–349, 1979. View at Google Scholar - J. D. Murray,
*Mathematical Biology*, vol. 19 of*Biomathematics*, Springer, New York, NY, USA, 2nd edition, 1993. View at Publisher · View at Google Scholar - G. E. Uhlenbeck and I. S. Ornstein, “On the theory of Brownian motion,” in
*Selected Papers on Noise and Stochastic Process*, N. Wax, Ed., Dover, New York, NY, USA, 1954. View at Google Scholar - G. Q. Cai and Y. K. Lin, “On exact stationary solutions of equivalent nonlinear stochastic systems,”
*International Journal of Non-Linear Mechanics*, vol. 23, no. 4, pp. 315–325, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. P. Samanta, “Influence of environmental noises on the Gomatam model of interacting species,”
*Ecological Modelling*, vol. 91, no. 1–3, pp. 283–291, 1996. View at Publisher · View at Google Scholar - G. P. Samanta and Alakes Maiti, “Stochastic Gomatam model of interacting species: non-equilibrium fluctuation and stability,”
*Systems Analysis Modelling Simulation*, vol. 43, no. 6, pp. 683–692, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. Maiti and G. P. Samanta, “Deterministic and stochastic analysis of a prey-dependent predator-prey system,”
*International Journal of Mathematical Education in Science and Technology*, vol. 36, no. 1, pp. 65–83, 2005. View at Publisher · View at Google Scholar · View at MathSciNet