Discrete Dynamics in Nature and Society

Volume 2008 (2008), Article ID 310425, 22 pages

http://dx.doi.org/10.1155/2008/310425

## Dynamic Behaviors of a General Discrete Nonautonomous System of Plankton Allelopathy with Delays

College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350002, China

Received 6 May 2008; Revised 8 September 2008; Accepted 22 November 2008

Academic Editor: Juan Jose Nieto

Copyright © 2008 Yaoping Chen 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 study the dynamic behaviors of a general discrete nonautonomous system of plankton allelopathy with delays. We first show that under some suitable assumption, the system is permanent. Next, by constructing a suitable Lyapunov functional, we obtain a set of sufficient conditions which guarantee the global attractivity of the two species. After that, by constructing an extinction-type Lyapunov functional, we show that under some suitable assumptions, one species will be driven to extinction. Finally, two examples together with their numerical simulations show the feasibility of the main results.

#### 1. Introduction

The aim of this paper is to investigate the dynamic behaviors of the following general discrete nonautonomous system of plankton allelopathy with delay: together with the initial conditionwhere is a positive integer, represent the densities of population at the th generation, are the intrinsic growth rate of population at the th generation, measure the intraspecific influence of the th generation of population on the density of own population, stand for the interspecific influence of the th generation of population on the density of own population, and stand for the effect of toxic inhibition of population by population at the th generation, and Also, and are all bounded nonnegative sequences defined for denoted by the set of all nonnegative integers, and such thathere, for any bounded sequence define

As was pointed out by Chattopadhyay [1] the effects of toxic substances on ecological communities are an important problem from an environmental point of view. Chattopadhyay [1] and Maynard-Smith [2] proposed the following two species Lotka-Volterra competition system, which describes the changes of size and density of phytoplankton:where and denote the population density of two competing species at time for a common pool of resources. The terms and denote the effect of toxic substances. Here, they made the assumption that each species produces a substance toxic to the other, only when the other is present. Noticing that the production of the toxic substance allelopathic to the competing species will not be instantaneous, but delayed by different discrete time lags required for the maturity of both species, thus, Mukhopadhyay et al. [3] also incorporated the discrete time delay into the above system. Tapaswi and Mukhopadhyay [4] also studied a two-dimensional system that arises in plankton allelopathy involving discrete time delays and environmental fluctuations. They assumed that the environmental parameters are assumed to be perturbed by white noise characterized by a Gaussian distribution with mean zero and unit spectral density. They focus on the dynamic behavior of the stochastic system and the fluctuations in population. For more works on system (1.5), one could refer to [1–3, 5–24] and the references cited therein.

Since the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping generations, and discrete time models can also provide efficient computational models of continuous models for numerical simulations, corresponding to system (1.5), Huo and Li [25] argued that it is necessary to study the following discrete two species competition system:where and are the population sizes of the two competitors at generation and have respectively, shown that each species produces a toxic substance to the other but the other only is present. In [25], sufficient conditions were obtained to guarantee the permanence of the above system, they also investigated the existence and stability property of the positive periodic solution of system (1.6). Recently, Li and Chen [26] further investigated the dynamic behaviors of the system (1.6). For general nonatonomous case, they obtain a set of sufficient conditions which guarantee the extinction of species and the global stability of species when species is eventually extinct. For periodic case, the other set of sufficient conditions, which concerned with the average condition of the coefficients of he system, were obtained to ensure the eventual extinction of species and the global stability of positive periodic solution of species when species is eventually extinct. For more works on discrete population dynamics, one could refer to [7, 10, 25–45].

Liu and Chen [32] argued that for a more realistic model, both seasonality of the changing environment and some of the past states, that is, the effects of time delays, should be taken into account in a model of multiple species growth. They proposed and studied the system (1.1), which is more general than system (1.6). By applying the coincidence degree theory, they obtained a set of sufficient conditions for the existence of at least one positive periodic solution of system (1.1)-(1.2). Zhang and Fang [46] also investigated the periodic solution of the system (1.1), they showed that under some suitable assumption, system (1.1) could admit at least two positive periodic solution. As we can see, the works [32, 46] are all concerned with the positive periodic solution of the system. However, since few things in the nature are really periodic, it is nature to study the general nonautonomous system (1.1), in this case, it is impossible to study the periodic solution of the system, however, such topics as permanence, extinction, and stability become the most important things. In this paper, we will further investigate the dynamics behaviors of the system (1.1). More precisely, by developing the analysis technique of Liu [31] and Muroya [35, 36], we study the permanence, global attractivity and extinction of system (1.1)-(1.2).

The organization of this paper is as follows. We study the persistence property of the system in Section 2 and the stability property in Section 3. Then in Section 4, by constructing a suitable Lyapunov functional, sufficient conditions which ensure the extinction of species of system (1.1)-(1.2) are studied. In Section 5, two examples together with their numeric simulations show the feasibility of main results. For more relevant works, one could refer to [2, 3, 5–9, 12, 13, 27–30, 33, 34, 37–45] and the references cited therein.

#### 2. Permanence

In this section, we study the persistent property of system (1.1)-(1.2).

Lemma 2.1. *For any positive solution of system (1.1)-(1.2), **where *

*Proof. *Let be any positive solution of system
(1.1)-(1.2), in view of the system (1.1) for all
we haveApplying Lemma 2.1 of Yang [44] to (2.3), we can obtainThis completes the proof of
Lemma 2.1.

*Lemma 2.2. Assume that hold, where and are defined in (2.2). Then for any positive
solution of system (1.1)-(1.2), where *

*Proof. *In view of (2.5), we can choose a constant small enough such thatIn view of (2.1), for above
there exists an integer such thatWe consider the following two cases.*Case (i). *We
assume that there exists an integer such that
Note thatSo we can obtainIt follows from (2.8) thatThen we haveLetNote thatthus
and so, for above or

We can claim thatBy way of contradiction, assume
that there exists an integer such that Then Let be the smallest integer such that Then The above argument produces that a contradiction. Thus (2.19) proved.*Case (ii). *We
assume that for all
then exists, denoted by
We can claim thatBy the way of contradiction, assume
thatTaking limit in the first
equation of (1.1) giveswhich is a contradiction sinceThe claim is thus proved.

From (2.20), we
see thatCombining Cases (i) and (ii), we see thatSetting it follows that

So we can easily see thatFrom the second equation of
(1.1), similar to above analysis, we havewhere is defined in (2.6). This completes the proof
of Lemma 2.2.

*It immediately follows from Lemmas 2.1 and 2.2 that the following theorem holds.*

*Theorem 2.3. Assume that (2.5) hold, then
system (1.1)-(1.2) is permanent.*

*3. Global attractivity*

*3. Global attractivity*

*This section devotes to study the stability
property of the positive solution of system (1.1)-(1.2).*

*Theorem 3.1. Assume that there exists a constant such that where, for and are defined in (2.2), then for any two positive
solutions and of system (1.1)-(1.2), *

*Proof. *First, letThen from the first equation of
(1.1), we haveNoticing that by mean-value
theorywhere
ThenSubstituting (3.6) into (3.4)
leads toSo it follows thatAccording to (2.1), for any
constant
there exists an integer such thatSo for all
it follows thatSo for all
it follows from (3.4) thatNext, letand we can obtainNow, we define bySo for all
it follows from (3.6) and (3.9) thatSimilar to above arguments, we
can definewhereThen for all we can obtainwhere lies between and

Now, we define byIt is easy to see that for all and For the arbitrariness of and by (*H _{0}*),
we can choose small enough such that for So for all
it follows from (3.15) and (3.18) thatSo we havewhich impliesIt follows thatThenwhich implies that that is, This completes the proof of Theorem 3.1.

*4. Extinction of species *

*4. Extinction of species*

*This section
devotes to study the extinction of the species *

*Lemma 4.1. For any positive solution of system (1.1)-(1.2), there exists a constant such that *

*Proof. *By (2.1), there exists a constant such thatIn view of (1.1) for all it follows thatSo we haveLetwhere For all it follows from (1.1) and (4.4) thatso we haveLet then we haveNote that for all so similar to the proof of Lemma 2.2 of Chen [27], we haveThen, there is a positive
constant such thatThis completes the proof of
Lemma 4.1.

*Theorem 4.2. Assume that where is defined in (2.2). Let be any positive solution of system
(1.1)-(1.2), then as *

*Corollary 4.3. Assume that for all
the following inequalities
hold, where is defined in (2.2). Let be any positive solution of system
(1.1)-(1.2), then as *

*Proof of Corollary 4.3. *Obviously, if
condition (*H _{1}^{*}*) holds, one could easily see that condition (

*H*) holds, thus, the conclusion of Corollary 4.3 follows from Theorem 4.2. The proof is complete.

_{1}*Proof of Theorem 4.2. *It follows from (*H _{1}*) that we can choose a constant small enough such thatSetFor above
from (2.1), there is an integer such that for Lemma 4.1 also implies that
there exists such thatSetSo for all
it follows from (1.1), (4.13), and (4.14) that

That is, for all So from the definition of it follows thatThe above analysis shows thatThis completes the proof of Theorem 4.2.

*5. Examples*

*5. Examples*

*The following two examples show
the feasibility of our results.*

*Example 5.1. *Consider the following systemOne could easily see thatClearly, conditions (2.5) are satisfied. From Theorem 2.3, it follows
that system (5.1) is permanent. Also, by simple computation, we haveThe above inequality shows that (*H _{0}*) is fulfilled. From Theorem 3.1, it follows
thatFigures 1 and 2 are the
numeric simulations of the solution of system (5.1) with initial condition and

*Example 5.2. *Consider the following system:One could easily see thatThen, for The above inequality shows that (*H _{1}^{*}*) is fulfilled. From Theorem 4.2, it follows
that
Numeric simulation of the dynamic behaviors of system (5.5) with the initial
conditions is presented in Figure 3.

*Remark 5.3. *In the above two examples, we can take as the perturbation terms. Our numeric
simulations show that if the perturbation terms are large enough, then those
terms will greatly influence the dynamic behaviors of the system, and in some cases,
may lead to the extinction of the species.

*Acknowledgments*

*Acknowledgments*

*The authors are grateful to anonymous referees for their excellent suggestions, which greatly improve the presentation of the paper. Also, this work was supported by the Program for New Century Excellent Talents in Fujian Province University (0330-003383).*

*References*

*References*

- J. Chattopadhyay, “Effect of toxic substances on a two-species competitive system,”
*Ecological Modelling*, vol. 84, no. 1–3, pp. 287–289, 1996. View at Publisher · View at Google Scholar - J. Maynard-Smith,
*Models in Ecology*, Cambridge University Press, Cambridge, UK, 1974. View at Zentralblatt MATH - A. Mukhopadhyay, J. Chattopadhyay, and P. K. Tapaswi, “A delay differential equations model of plankton allelopathy,”
*Mathematical Biosciences*, vol. 149, no. 2, pp. 167–189, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. K. Tapaswi and A. Mukhopadhyay, “Effects of environmental fluctuation on plankton allelopathy,”
*Journal of Mathematical Biology*, vol. 39, no. 1, pp. 39–58, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. E. Crone, “Delayed density dependence and the stability of interacting populations and subpopulations,”
*Theoretical Population Biology*, vol. 51, no. 1, pp. 67–76, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - F. Chen, “On a periodic multi-species ecological model,”
*Applied Mathematics and Computation*, vol. 171, no. 1, pp. 492–510, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Chen, “Permanence and global attractivity of a discrete multispecies Lotka-Volterra competition predator-prey systems,”
*Applied Mathematics and Computation*, vol. 182, no. 1, pp. 3–12, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Chen, Z. Li, X. Chen, and J. Laitochová, “Dynamic behaviors of a delay differential equation model of plankton allelopathy,”
*Journal of Computational and Applied Mathematics*, vol. 206, no. 2, pp. 733–754, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Chen and C. Shi, “Global attractivity in an almost periodic multi-species nonlinear ecological model,”
*Applied Mathematics and Computation*, vol. 180, no. 1, pp. 376–392, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Chen, L. Wu, and Z. Li, “Permanence and global attractivity of the discrete Gilpin-Ayala type population model,”
*Computers & Mathematics with Applications*, vol. 53, no. 8, pp. 1214–1227, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. A. Cui and L. S. Chen, “Asymptotic behavior of the solution for a class of time-dependent competitive system,”
*Annals of Differential Equations*, vol. 9, no. 1, pp. 11–17, 1993. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Zhen and Z. Ma, “Periodic solutions for delay differential equations model of plankton allelopathy,”
*Computers & Mathematics with Applications*, vol. 44, no. 3-4, pp. 491–500, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Y. Song and L. S. Chen, “Periodic solution of a delay differential equation of plankton allelopathy,”
*Acta Mathematica Scientia. Series A*, vol. 23, no. 1, pp. 8–13, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X.-Z. Meng, L.-S. Chen, and Q.-X. Li, “The dynamics of an impulsive delay predator-prey model with variable coefficients,”
*Applied Mathematics and Computation*, vol. 198, no. 1, pp. 361–374, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. Fergola, M. Cerasuolo, A. Pollio, G. Pinto, and M. DellaGreca, “Allelopathy and competition between Chlorella vulgaris and Pseudokirchneriella subcapitata: experiments and mathematical model,”
*Ecological Modelling*, vol. 208, no. 2–4, pp. 205–214, 2007. View at Publisher · View at Google Scholar - R. R. Sarkar, B. Mukhopadhyay, R. Bhattacharyya, and S. Banerjee, “Time lags can control algal bloom in two harmful phytoplankton-zooplankton system,”
*Applied Mathematics and Computation*, vol. 186, no. 1, pp. 445–459, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Mulderij, E. H. van Nes, and E. van Donk, “Macrophyte-phytoplankton interactions: the relative importance of allelopathy versus other factors,”
*Ecological Modelling*, vol. 204, no. 1-2, pp. 85–92, 2007. View at Publisher · View at Google Scholar - J. Jia, M. Wang, and M. Li, “Periodic solutions for impulsive delay differential equations in the control model of plankton allelopathy,”
*Chaos, Solitons and Fractals*, vol. 32, no. 3, pp. 962–968, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - A. Wan and J. Wei, “Bifurcation analysis in an approachable haematopoietic stem cells model,”
*Journal of Mathematical Analysis and Applications*, vol. 345, no. 1, pp. 276–285, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - S. Gao, L. Chen, J. J. Nieto, and A. Torres, “Analysis of a delayed epidemic model with pulse vaccination and saturation incidence,”
*Vaccine*, vol. 24, no. 35-36, pp. 6037–6045, 2006. View at Publisher · View at Google Scholar - H. Zhang, L. Chen, and J. J. Nieto, “A delayed epidemic model with stage-structure and pulses for pest management strategy,”
*Nonlinear Analysis: Real World Applications*, vol. 9, no. 4, pp. 1714–1726, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - R. Yafia, “Hopf bifurcation in a delayed model for tumor-immune system competition with negative immune response,”
*Discrete Dynamics in Nature and Society*, vol. 2006, Article ID 95296, 9 pages, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Bodnar and U. Foryś, “A model of immune system with time-dependent immune reactivity,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 70, no. 2, pp. 1049–1058, 2009. View at Publisher · View at Google Scholar - Z. Liu, J. Wu, Y. Chen, and M. Haque, “Impulsive perturbations in a periodic delay differential equation model of plankton allelopathy,”
*Nonlinear Analysis: Real World Applications*. In press. View at Publisher · View at Google Scholar - H.-F. Huo and W.-T. Li, “Permanence and global stability for nonautonomous discrete model of plankton allelopathy,”
*Applied Mathematics Letters*, vol. 17, no. 9, pp. 1007–1013, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Li and F. Chen, “Extinction in two dimensional discrete Lotka-Volterra competitive system with the effect of toxic substances,”
*Dynamics of Continuous, Discrete & Impulsive Systems. Series B*, vol. 15, no. 2, pp. 165–178, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Chen, “Permanence for the discrete mutualism model with time delays,”
*Mathematical and Computer Modelling*, vol. 47, no. 3-4, pp. 431–435, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Chen, “Permanence and global stability for nonlinear discrete model,”
*Advances in Complex Systems*, vol. 9, no. 1-2, pp. 31–40, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Chen and Z. Zhou, “Stable periodic solution of a discrete periodic Lotka-Volterra competition system,”
*Journal of Mathematical Analysis and Applications*, vol. 277, no. 1, pp. 358–366, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Fan and K. Wang, “Periodic solutions of a discrete time nonautonomous ratio-dependent predator-prey system,”
*Mathematical and Computer Modelling*, vol. 35, no. 9-10, pp. 951–961, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Q. Liu,
*Studies on continuous and discrete population dynamics system with time-delays*, Ph. D. thesis, Academia Sinica, Taipei, Taiwan, 2002. - Z. Liu and L. Chen, “Positive periodic solution of a general discrete nonautonomous difference system of plankton allelopathy with delays,”
*Journal of Computational and Applied Mathematics*, vol. 197, no. 2, pp. 446–456, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Liu and L. Chen, “Periodic solution of a two-species competitive system with toxicant and birth pulse,”
*Chaos, Solitons and Fractals*, vol. 32, no. 5, pp. 1703–1712, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Lu and W. Wang, “Permanence and global attractivity for Lotka-Volterra difference systems,”
*Journal of Mathematical Biology*, vol. 39, no. 3, pp. 269–282, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Muroya, “Persistence and global stability in discrete models of pure-delay nonautonomous Lotka-Volterra type,”
*Journal of Mathematical Analysis and Applications*, vol. 293, no. 2, pp. 446–461, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Muroya, “Persistence and global stability in discrete models of Lotka-Volterra type,”
*Journal of Mathematical Analysis and Applications*, vol. 330, no. 1, pp. 24–33, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Saito, W. Ma, and T. Hara, “A necessary and sufficient condition for permanence of a Lotka-Volterra discrete system with delays,”
*Journal of Mathematical Analysis and Applications*, vol. 256, no. 1, pp. 162–174, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. Turchin, “Chaos and stability in rodent population dynamics: evidence from nonlinear time-series analysis,”
*Oikos*, vol. 68, no. 1, pp. 167–172, 1993. View at Publisher · View at Google Scholar - P. Turchin and A. D. Taylor, “Complex dynamics in ecological time series,”
*Ecology*, vol. 73, no. 1, pp. 289–305, 1992. View at Publisher · View at Google Scholar - S. Tang and Y. Xiao, “Permanence in Kolmogorov-type systems of delay difference equations,”
*Journal of Difference Equations and Applications*, vol. 7, no. 2, pp. 167–181, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. Wendi and L. Zhengyi, “Global stability of discrete models of Lotka-Volterra type,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 35, no. 8, pp. 1019–1030, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. Wendi, G. Mulone, F. Salemi, and V. Salone, “Global stability of discrete population models with time delays and fluctuating environment,”
*Journal of Mathematical Analysis and Applications*, vol. 264, no. 1, pp. 147–167, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Xu, M. A. J. Chaplain, and F. A. Davidson, “Periodic solutions of a discrete nonautonomous Lotka-Volterra predator-prey model with time delays,”
*Discrete and Continuous Dynamical Systems. Series B*, vol. 4, no. 3, pp. 823–831, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Yang, “Uniform persistence and periodic solutions for a discrete predator-prey system with delays,”
*Journal of Mathematical Analysis and Applications*, vol. 316, no. 1, pp. 161–177, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Y. Zhang, Z. C. Wang, Y. Chen, and J. Wu, “Periodic solutions of a single species discrete population model with periodic harvest/stock,”
*Computers & Mathematics with Applications*, vol. 39, no. 1-2, pp. 77–90, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Zhang and H. Fang, “Multiple periodic solutions for a discrete time model of plankton allelopathy,”
*Advances in Difference Equations*, vol. 2006, Article ID 90479, 14 pages, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet

*
*