Dynamic Behaviors of a General Discrete Nonautonomous System of Plankton Allelopathy with Delays
Yaoping Chen,1Fengde Chen,1and Zhong Li1
Academic Editor: Juan Jose Nieto
Received06 May 2008
Revised08 Sept 2008
Accepted22 Nov 2008
Published26 Feb 2009
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 [1β3, 5β24] 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, 25β45].
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, 5β9, 12, 13, 27β30, 33, 34, 37β45] 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
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 (H0),
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
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 (H1*) holds, one could easily see that condition (H1) holds, thus, the conclusion of Corollary 4.3
follows from Theorem 4.2. The proof is complete.
Proof of Theorem 4.2. It follows from (H1) 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
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 (H0) 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 (H1*) 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
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
J. Chattopadhyay, βEffect of toxic substances on a two-species competitive system,β Ecological Modelling, vol. 84, no. 1β3, pp. 287β289, 1996.
A. Mukhopadhyay, J. Chattopadhyay, and P. K. Tapaswi, βA delay differential equations model of plankton allelopathy,β Mathematical Biosciences, vol. 149, no. 2, pp. 167β189, 1998.
P. K. Tapaswi and A. Mukhopadhyay, βEffects of environmental fluctuation on plankton allelopathy,β Journal of Mathematical Biology, vol. 39, no. 1, pp. 39β58, 1999.
E. E. Crone, βDelayed density dependence and the stability of interacting populations and subpopulations,β Theoretical Population Biology, vol. 51, no. 1, pp. 67β76, 1997.
F. Chen, βPermanence and global attractivity of a discrete multispecies Lotka-Volterra competition predator-prey systems,β Applied Mathematics and Computation, vol. 182, no. 1, pp. 3β12, 2006.
F. Chen, Z. Li, X. Chen, and J. LaitochovΓ‘, βDynamic behaviors of a delay differential equation model of plankton allelopathy,β Journal of Computational and Applied Mathematics, vol. 206, no. 2, pp. 733β754, 2007.
F. Chen and C. Shi, βGlobal attractivity in an almost periodic multi-species nonlinear ecological model,β Applied Mathematics and Computation, vol. 180, no. 1, pp. 376β392, 2006.
F. Chen, L. Wu, and Z. Li, βPermanence and global attractivity of the discrete Gilpin-Ayala type population model,β Computers & Mathematics with Applications, vol. 53, no. 8, pp. 1214β1227, 2007.
J. A. Cui and L. S. Chen, βAsymptotic behavior of the solution for a class of time-dependent competitive system,β Annals of Differential Equations, vol. 9, no. 1, pp. 11β17, 1993.
J. Zhen and Z. Ma, βPeriodic solutions for delay differential equations model of plankton allelopathy,β Computers & Mathematics with Applications, vol. 44, no. 3-4, pp. 491β500, 2002.
X. Y. Song and L. S. Chen, βPeriodic solution of a delay differential equation of plankton allelopathy,β Acta Mathematica Scientia. Series A, vol. 23, no. 1, pp. 8β13, 2003.
X.-Z. Meng, L.-S. Chen, and Q.-X. Li, βThe dynamics of an impulsive delay predator-prey model with variable coefficients,β Applied Mathematics and Computation, vol. 198, no. 1, pp. 361β374, 2008.
P. Fergola, M. Cerasuolo, A. Pollio, G. Pinto, and M. DellaGreca, βAllelopathy and competition between Chlorella vulgaris and Pseudokirchneriella subcapitata: experiments and mathematical model,β Ecological Modelling, vol. 208, no. 2β4, pp. 205β214, 2007.
R. R. Sarkar, B. Mukhopadhyay, R. Bhattacharyya, and S. Banerjee, βTime lags can control algal bloom in two harmful phytoplankton-zooplankton system,β Applied Mathematics and Computation, vol. 186, no. 1, pp. 445β459, 2007.
G. Mulderij, E. H. van Nes, and E. van Donk, βMacrophyte-phytoplankton interactions: the relative importance of allelopathy versus other factors,β Ecological Modelling, vol. 204, no. 1-2, pp. 85β92, 2007.
J. Jia, M. Wang, and M. Li, βPeriodic solutions for impulsive delay differential equations in the control model of plankton allelopathy,β Chaos, Solitons and Fractals, vol. 32, no. 3, pp. 962β968, 2007.
A. Wan and J. Wei, βBifurcation analysis in an approachable haematopoietic stem cells model,β Journal of Mathematical Analysis and Applications, vol. 345, no. 1, pp. 276β285, 2008.
S. Gao, L. Chen, J. J. Nieto, and A. Torres, βAnalysis of a delayed epidemic model with pulse vaccination and saturation incidence,β Vaccine, vol. 24, no. 35-36, pp. 6037β6045, 2006.
H. Zhang, L. Chen, and J. J. Nieto, βA delayed epidemic model with stage-structure and pulses for pest management strategy,β Nonlinear Analysis: Real World Applications, vol. 9, no. 4, pp. 1714β1726, 2008.
R. Yafia, βHopf bifurcation in a delayed model for tumor-immune system competition with negative immune response,β Discrete Dynamics in Nature and Society, vol. 2006, Article ID 95296, 9 pages, 2006.
M. Bodnar and U. ForyΕ, βA model of immune system with time-dependent immune reactivity,β Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 2, pp. 1049β1058, 2009.
Z. Liu, J. Wu, Y. Chen, and M. Haque, βImpulsive perturbations in a periodic delay differential equation model of plankton allelopathy,β Nonlinear Analysis: Real World Applications. In press.
H.-F. Huo and W.-T. Li, βPermanence and global stability for nonautonomous discrete model of plankton allelopathy,β Applied Mathematics Letters, vol. 17, no. 9, pp. 1007β1013, 2004.
Z. Li and F. Chen, βExtinction in two dimensional discrete Lotka-Volterra competitive system with the effect of toxic substances,β Dynamics of Continuous, Discrete & Impulsive Systems. Series B, vol. 15, no. 2, pp. 165β178, 2008.
F. Chen, βPermanence for the discrete mutualism model with time delays,β Mathematical and Computer Modelling, vol. 47, no. 3-4, pp. 431β435, 2008.
Y. Chen and Z. Zhou, βStable periodic solution of a discrete periodic Lotka-Volterra competition system,β Journal of Mathematical Analysis and Applications, vol. 277, no. 1, pp. 358β366, 2003.
M. Fan and K. Wang, βPeriodic solutions of a discrete time nonautonomous ratio-dependent predator-prey system,β Mathematical and Computer Modelling, vol. 35, no. 9-10, pp. 951β961, 2002.
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, βPositive periodic solution of a general discrete nonautonomous difference system of plankton allelopathy with delays,β Journal of Computational and Applied Mathematics, vol. 197, no. 2, pp. 446β456, 2006.
Z. Liu and L. Chen, βPeriodic solution of a two-species competitive system with toxicant and birth pulse,β Chaos, Solitons and Fractals, vol. 32, no. 5, pp. 1703β1712, 2007.
Z. Lu and W. Wang, βPermanence and global attractivity for Lotka-Volterra difference systems,β Journal of Mathematical Biology, vol. 39, no. 3, pp. 269β282, 1999.
Y. Muroya, βPersistence and global stability in discrete models of pure-delay nonautonomous Lotka-Volterra type,β Journal of Mathematical Analysis and Applications, vol. 293, no. 2, pp. 446β461, 2004.
Y. Muroya, βPersistence and global stability in discrete models of Lotka-Volterra type,β Journal of Mathematical Analysis and Applications, vol. 330, no. 1, pp. 24β33, 2007.
Y. Saito, W. Ma, and T. Hara, βA necessary and sufficient condition for permanence of a Lotka-Volterra discrete system with delays,β Journal of Mathematical Analysis and Applications, vol. 256, no. 1, pp. 162β174, 2001.
P. Turchin, βChaos and stability in rodent population dynamics: evidence from nonlinear time-series analysis,β Oikos, vol. 68, no. 1, pp. 167β172, 1993.
S. Tang and Y. Xiao, βPermanence in Kolmogorov-type systems of delay difference equations,β Journal of Difference Equations and Applications, vol. 7, no. 2, pp. 167β181, 2001.
W. Wendi and L. Zhengyi, βGlobal stability of discrete models of Lotka-Volterra type,β Nonlinear Analysis: Theory, Methods & Applications, vol. 35, no. 8, pp. 1019β1030, 1999.
W. Wendi, G. Mulone, F. Salemi, and V. Salone, βGlobal stability of discrete population models with time delays and fluctuating environment,β Journal of Mathematical Analysis and Applications, vol. 264, no. 1, pp. 147β167, 2001.
R. Xu, M. A. J. Chaplain, and F. A. Davidson, βPeriodic solutions of a discrete nonautonomous Lotka-Volterra predator-prey model with time delays,β Discrete and Continuous Dynamical Systems. Series B, vol. 4, no. 3, pp. 823β831, 2004.
X. Yang, βUniform persistence and periodic solutions for a discrete predator-prey system with delays,β Journal of Mathematical Analysis and Applications, vol. 316, no. 1, pp. 161β177, 2006.
R. Y. Zhang, Z. C. Wang, Y. Chen, and J. Wu, βPeriodic solutions of a single species discrete population model with periodic harvest/stock,β Computers & Mathematics with Applications, vol. 39, no. 1-2, pp. 77β90, 2000.
J. Zhang and H. Fang, βMultiple periodic solutions for a discrete time model of plankton allelopathy,β Advances in Difference Equations, vol. 2006, Article ID 90479, 14 pages, 2006.