Dynamic behaviors of a delay differential equation model of plankton allelopathy

Department of Mathematics , Palacký University of Olomouc, Olmütz, Olomoucký, Czech Republic
Journal of Computational and Applied Mathematics (Impact Factor: 1.08). 09/2007; 206(2):733-754. DOI: 10.1016/

ABSTRACT In this paper, we consider a modified delay differential equation model of the growth of n-species of plankton having competitive and allelopathic effects on each other. We first obtain the sufficient conditions which guarantee the permanence of the system. As a corollary, for periodic case, we obtain a set of delay-dependent condition which ensures the existence of at least one positive periodic solution of the system. After that, by means of a suitable Lyapunov functional, sufficient conditions are derived for the global attractivity of the system. For the two-dimensional case, under some suitable assumptions, we prove that one of the components will be driven to extinction while the other will stabilize at a certain solution of a logistic equation. Examples show the feasibility of the main results.

  • Source
    • "Now we will state a lemma which will be required later [31]. "
    [Show abstract] [Hide abstract]
    ABSTRACT: In this paper, we investigate a non-autonomous competitive phytoplankton model with periodic coefficients in deterministic and stochastic environment, respectively. We prove the existence of at least one positive periodic solution together with it’s global asymptotic stability. The existence of periodic solution has been obtained by using the continuation theorem of coincidence degree theory proposed by Gaines and Mawhin. We formulate the corresponding stochastic model by perturbing the growth rate parameters by white noise terms. We prove that all the higher order moments of the solution to the stochastic system is uniformly bounded which ensure that the solution of the stochastic system is stochastically bounded. We provide easily verifiable sufficient conditions for non-persistence in mean, extinction and stochastic permanence of the stochastic system. Sufficient condition for permanence shows that if the noise intensity is very low then the solution of the stochastic system persists in the periodic coexistence domain of the deterministic system. We perform exhaustive numerical simulations to validate our analytical findings.
    Applied Mathematics and Computation 07/2014; 238:300–318. DOI:10.1016/j.amc.2014.04.009 · 1.60 Impact Factor
  • Source
    • "Now we establish persistence of the delayed model, using the positivity of the dependent variables. We need to recall the following two lemmas, whose proofs can be found in [4] and [15] respectively. "
    [Show abstract] [Hide abstract]
    ABSTRACT: Persistence and global stability of the coexistence equilibrium of a recently published model in biocontrol of crops are here shown both in the absence and the presence of delays, introduced to simulate the handling time of the prey. In the latter case, the system can behave in two different ways, in dependence of whether a suitably defined key parameter exceeds a certain threshold. Namely, below the threshold the delay is shown not to be able to influence the stability of the coexistence equilibrium; above it, existence of a Hopf bifurcation is analytically proven. Further, in this range, numerical simulations reveal a route to chaotic behavior as function of the size of the delay. Some operative conclusions for agroecosystem management are drawn, although they ultimately depend on each particular situation.
    Mathematics and Computers in Simulation 01/2013; 87:30–44. DOI:10.1016/j.matcom.2013.02.001 · 0.86 Impact Factor
  • Source
    • "It has been observed that some algae produce auxins which stimulate the growth of the other algae, for more details we refer to [8] [13]. The production of the allelopathic substance will not be instantaneous, but delayed by some discrete time lag required for maturity of the species. "
    [Show abstract] [Hide abstract]
    ABSTRACT: In this paper we consider the two species competitive delay plankton allelopathy stimulatory model system. We show the existence and uniqueness of the solution of the deterministic model. Moreover, we study the persistence of the model and the stability properties of its equilibrium points. We illustrate the theoretical results by some numerical simulations.
    Journal of Mathematical Analysis and Applications 07/2010; 367(1-367):249-259. DOI:10.1016/j.jmaa.2010.01.024 · 1.12 Impact Factor
Show more


Available from