Bifurcation analysis of a two-species competitive discrete model of plankton allelopathy
© Wu and Zhang; licensee Springer. 2014
Received: 16 September 2013
Accepted: 7 February 2014
Published: 19 February 2014
This paper studies the dynamical behaviors of a two-species competitive discrete model of plankton allelopathy. The system undergoes a flip bifurcation as we see by using the center manifold theorem and bifurcation theory. Numerical simulations not only illustrate our results, but they also exhibit the complex dynamical behaviors of the system, such as the period-doubling bifurcation in periods 2, 4, 8, and 16, and chaotic sets.
The study of tremendous fluctuations in the abundance of many phytoplankton communities is an important subject in aquatic ecology. These changes of size and density of phytoplankton have been attributed to several factors, such as physical factors, variation of necessary nutrients, or a combination of these by various workers (see cf. [1–5]). Another important observation made by many workers is that the increased population of one species might affect the growth of another species or several other species by the production of allelopathic toxins or stimulators, thus influencing seasonal succession .
where , are the population densities (number of cells per liter) of two competing species; , are the rates of cell proliferation per hour; , are the rate of intra-specific competition of first and second species, respectively; , are the rate of inter specific competition of first and second species respectively and , are environmental carrying capacities (representing number of cells per liter). The units of , , and are per hour per cell and the unit of time is hours.
where and are the rates of toxic inhibition of the first species by the second and vice versa, respectively, and , , , , , , , and are positive constants.
On the other hands, many scholars have paid attention to the discrete population models, since the discrete-time models governed by discrete systems are more appropriate than the continuous ones when the populations have nonoverlapping generations (cf. [9–12]). Moreover, since the discrete-time models can also provide efficient computational models of continuous models for numerical simulations, it is reasonable to study discrete-time models governed by discrete systems.
where is the step size.
The dynamical behaviors of discrete system of plankton allelopathy have been investigated in the mathematics literature (cf. [13–15]). The purpose of this paper is to investigate the bifurcation and chaos of the map (1.3) by using bifurcation theory (cf. [16, 17]) and center manifold theory (cf. [16–18]). Meanwhile, numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors.
This paper is organized as follows. In Section 2, we discuss the existence and stability of the positive fixed points for the system (1.3). In Section 3, we show that there exist some values of parameters such that the system (1.3) undergoes the flip bifurcation. In Section 4, we present numerical simulations which illustrate our results with the theoretical analysis. A brief discussion is given in Section 5.
2 Fixed points and stability analysis
In order to study the stability of the fixed points of the system (1.3), we first give the following lemma, which can easily be proved by the relations between roots and coefficients of a quadratic equation.
and if and only if and ;
and (or and ) if and only if ;
and if and only if and ;
and if and only if and ;
and are complex and and if and only if and .
Let and be two roots of (2.2), which are called eigenvalues of the fixed point . We recall some definitions of topological types for a fixed point . is called a sink if and . A sink is locally asymptotic stable. is called a source if and . A source is locally unstable. is called a saddle if and (or and ). is called non-hyperbolic if either or .
Note that , so there exist and leading to .
Regarding the stability of , we have the following results.
Since , has two unequal real roots and . Furthermore, we obtain the following.
(A1) If , then and . By Lemma 2.1 we have and . Therefore, is a sink.
(A2) If , then and . By Lemma 2.1 we have and . Therefore, is a source.
(A3) If or , then and . By Lemma 2.1 we have and . Therefore, is non-hyperbolic.
(A4) If , then . By Lemma 2.1 we have and (or and ). Therefore, is a saddle.
From the above analysis, we obtain the result that for the fixed point , if , , where , then one of the two eigenvalues of the positive fixed point is −1 and the other is neither 1 nor −1. Therefore, there may be a flip bifurcation of if the parameters vary in the small neighborhood of , .
Remark Since , one can see that 1 is not the eigenvalue of the positive fixed point . Therefore, fold bifurcations, transcritical bifurcations, and pitchfork bifurcations do not occur at the positive fixed point . Similarly, since , , and there does not exist a pair of conjugate complex roots with modulus 1. Hence, the Neimark-Sacker bifurcation does not occur at the positive fixed point .
3 Flip bifurcation
In this section, we choose parameter δ as a bifurcation parameter to study the flip bifurcation of by using the center manifold theorem and the bifurcation theory in [16–18]. For convenience, for a function , we denote by , , and the first order, second order and the third order partial derivative of , respectively.
We first discuss the flip bifurcation of the system (1.3) at when the parameter varies in a small neighborhood of . Similar arguments can be applied to the other case, . Taking the parameters arbitrarily, we consider the system (1.3) at .
where is a small perturbation parameter.
with , and , , respectively.
Theorem 3.1 If , then the map (3.1) undergoes a Flip bifurcation at the fixed point when the parameter varies in the small neighborhood of the origin. Moreover, if (resp., ), then the period-2 points that bifurcate from are stable (resp., unstable).
4 Numerical simulations
In this section, we present the bifurcation diagrams, phase portraits, and maximum Lyapunov exponents for the system (1.3) to confirm the above theoretical analysis and show the new interesting complex dynamical behaviors by using numerical simulations. The bifurcation parameters are considered for the following parameters.
Choosing , , , , , , , , initial value and varying δ in the range .
We see that the system (1.3) has only one positive fixed point, . After calculation, by Theorem (3.1), the flip bifurcation emerges from the fixed point at with and .
We can know that the dynamics of the system (1.2) is trivial with the condition (2.1). In fact, Samanta  has shown that the unique positive equilibrium of the system (1.2) is globally asymptotically stable with the condition (2.1). However, the discrete-time system (1.3) has complex dynamics. In this paper, we show that the unique positive fixed point of the system (1.3) can undergo a flip bifurcation with the condition (2.1). Moreover, numerical simulations display interesting dynamical behaviors for the system (1.3), including period-doubling orbits and chaotic sets.
This work is jointly supported by the Programs of Educational Commission of Anhui Province of China under Grant nos. KJ2011A197 and KJ2013Z186.
- Nozawa K: The effect of peridinium toxin on the algae. Bull. Misaki Mar. Biol. Inst. Kyoto Univ. 1968, 12: 21.Google Scholar
- Harris DO: A model system for the study of algal growth inhibitors. Arch. Protistenkd. 1971, 113: 230.Google Scholar
- Huntsman SA, Barber RT: Modification of phytoplankton growth by excreted compounds in low-density populations. J. Phycol. 1975, 11: 10.Google Scholar
- Mukhopadhyay A, Chattopadhyay J, Tapaswi PK: A delay differential model of plankton allelopathy. Math. Biosci. 1998, 149: 167–189. 10.1016/S0025-5564(98)00005-4MathSciNetView ArticleGoogle Scholar
- Wolfe JM, Rice EL: BRCA1 protein products: functional motifs. Nat. Genet. 1996, 13: 266–267. 10.1038/ng0796-266View ArticleGoogle Scholar
- Rice EL: Allelopathy. Academic Press, New York; 1984.Google Scholar
- Maynard J: Models in Ecology. Cambridge University Press, Cambridge; 1974.Google Scholar
- Chattopadhyay J: Effect of toxic substances on a two-species competitive system. Ecol. Model. 1996, 84: 287–289. 10.1016/0304-3800(94)00134-0View ArticleGoogle Scholar
- Chen FD: Permanence in a discrete Lotka-Volterra competition model with deviating arguments. Nonlinear Anal., Real World Appl. 2008, 9: 2150–2155. 10.1016/j.nonrwa.2007.07.001MathSciNetView ArticleGoogle Scholar
- Chen YM, Zhou Z: Stable periodic solution of a discrete periodic Lotka-Volterra competition system. J. Math. Anal. Appl. 2003, 277: 358–366. 10.1016/S0022-247X(02)00611-XMathSciNetView ArticleGoogle Scholar
- Chen GY, Teng ZD: On the stability in a discrete two-species competition system. J. Appl. Math. Comput. 2012, 38: 25–36. 10.1007/s12190-010-0460-1MathSciNetView ArticleGoogle Scholar
- Kong XZ, Chen LP, Yang WS: Note on the persistent property of a discrete Lotka-Volterra competition system with delays and feedback controls. Adv. Differ. Equ. 2010., 2010: Article ID 249364Google Scholar
- Huo HF, Li WT: Permanence and global stability for nonautonomous discrete model of plankton allelopathy. Appl. Math. Lett. 2004, 17: 1007–1013. 10.1016/j.aml.2004.07.002MathSciNetView ArticleGoogle Scholar
- Qin WJ, Liu ZJ: Asymptotic behaviors of a delay difference system of plankton allelopathy. J. Math. Chem. 2010, 48: 653–675. 10.1007/s10910-010-9698-yMathSciNetView ArticleGoogle Scholar
- Liu ZJ, Chen LS: Positive periodic solution of a general discrete non-autonomous difference system of plankton allelopathy with delays. J. Comput. Appl. Math. 2006, 197: 446–456. 10.1016/j.cam.2005.09.023MathSciNetView ArticleGoogle Scholar
- Gukenheimer J, Holmes P: Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vectro Fields. Springer, New York; 1983.View ArticleGoogle Scholar
- Wiggins S: Introduction to Applied Nonlinear Dynamical System and Chaos. Springer, New York; 2003.Google Scholar
- Carr J: Application of Center Manifold Theory. Springer, New York; 1981.View ArticleGoogle Scholar
- Samanta GP: A two-species competitive system under the influence of toxic substances. Appl. Math. Comput. 2010, 216: 291–299. 10.1016/j.amc.2010.01.061MathSciNetView ArticleGoogle Scholar
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