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Mathematical analysis of a nutrient-plankton system with delay.

Rehim M, Zhang Z, Muhammadhaji A - Springerplus (2016)

Bottom Line: In order to account for the time needed by the phytoplankton to mature after which they can release toxins, a discrete time delay is incorporated into the system.Next, the existence of the Hopf bifurcation as time delay crosses a threshold value is established and, meanwhile, the phenomenon of stability switches is found under certain conditions.Numerical simulations are presented to illustrate the analytical results.

View Article: PubMed Central - PubMed

Affiliation: College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046 Xinjiang China.

ABSTRACT
A mathematical model describing the interaction of nutrient-plankton is investigated in this paper. In order to account for the time needed by the phytoplankton to mature after which they can release toxins, a discrete time delay is incorporated into the system. Moreover, it is also taken into account discrete time delays which indicates the partially recycled nutrient decomposed by bacteria after the death of biomass. In the first part of our analysis the sufficient conditions ensuring local and global asymptotic stability of the model are obtained. Next, the existence of the Hopf bifurcation as time delay crosses a threshold value is established and, meanwhile, the phenomenon of stability switches is found under certain conditions. Numerical simulations are presented to illustrate the analytical results.

No MeSH data available.


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a, b The coexistence equilibrium  becomes stable for  with . c Coexistence equilibrium  loses its stability at  with . Stable periodic solution arising from Hopf bifurcation at . d, e Stable limit cycle is observed at  with . Here
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Fig4: a, b The coexistence equilibrium becomes stable for with . c Coexistence equilibrium loses its stability at with . Stable periodic solution arising from Hopf bifurcation at . d, e Stable limit cycle is observed at with . Here

Mentions: Next, We present some numerical results on the case of system (1) that and . Take , , , , and other parameters the same as that in Table 1. With the help of this parameter set we obtain the interior equilibrium as . Let us fix and gradually increase the value of . After some calculations one can find the minimum value of the delay parameter “” for the model system (1) for which the stability behaviour changes and the first critical values are given by , such that is stable for and unstable for . Figure 3 shows the simulation result for the model system (1) with . Interior equilibrium point looses its stability as passes through its critical value and a Hopf bifurcation occurs, a stable Hopf-bifurcating periodic solution is depicted in Fig. 3d, e. The equilibrium point remains locally asymptotically stable whenever the delay parameter lies in the range (6.759, 10.6156). again switches from stability to instability as passes through and an unstable solution for the model system (1) is shown in Fig. 4. The numerical simulations we have done here illustrate the stable periodic solution arising from Hopf bifurcation at and , respectively, and the switching of stability that occurs as the magnitude of the delay parameter increases gradually. For the above set of parameter values, when fixing and varying the value of or fixing and varying , the dynamical behavior of the system (1) explored by numerical simulation are the same as above, so we omit it here.Fig. 3


Mathematical analysis of a nutrient-plankton system with delay.

Rehim M, Zhang Z, Muhammadhaji A - Springerplus (2016)

a, b The coexistence equilibrium  becomes stable for  with . c Coexistence equilibrium  loses its stability at  with . Stable periodic solution arising from Hopf bifurcation at . d, e Stable limit cycle is observed at  with . Here
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

License
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getmorefigures.php?uid=PMC4940349&req=5

Fig4: a, b The coexistence equilibrium becomes stable for with . c Coexistence equilibrium loses its stability at with . Stable periodic solution arising from Hopf bifurcation at . d, e Stable limit cycle is observed at with . Here
Mentions: Next, We present some numerical results on the case of system (1) that and . Take , , , , and other parameters the same as that in Table 1. With the help of this parameter set we obtain the interior equilibrium as . Let us fix and gradually increase the value of . After some calculations one can find the minimum value of the delay parameter “” for the model system (1) for which the stability behaviour changes and the first critical values are given by , such that is stable for and unstable for . Figure 3 shows the simulation result for the model system (1) with . Interior equilibrium point looses its stability as passes through its critical value and a Hopf bifurcation occurs, a stable Hopf-bifurcating periodic solution is depicted in Fig. 3d, e. The equilibrium point remains locally asymptotically stable whenever the delay parameter lies in the range (6.759, 10.6156). again switches from stability to instability as passes through and an unstable solution for the model system (1) is shown in Fig. 4. The numerical simulations we have done here illustrate the stable periodic solution arising from Hopf bifurcation at and , respectively, and the switching of stability that occurs as the magnitude of the delay parameter increases gradually. For the above set of parameter values, when fixing and varying the value of or fixing and varying , the dynamical behavior of the system (1) explored by numerical simulation are the same as above, so we omit it here.Fig. 3

Bottom Line: In order to account for the time needed by the phytoplankton to mature after which they can release toxins, a discrete time delay is incorporated into the system.Next, the existence of the Hopf bifurcation as time delay crosses a threshold value is established and, meanwhile, the phenomenon of stability switches is found under certain conditions.Numerical simulations are presented to illustrate the analytical results.

View Article: PubMed Central - PubMed

Affiliation: College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046 Xinjiang China.

ABSTRACT
A mathematical model describing the interaction of nutrient-plankton is investigated in this paper. In order to account for the time needed by the phytoplankton to mature after which they can release toxins, a discrete time delay is incorporated into the system. Moreover, it is also taken into account discrete time delays which indicates the partially recycled nutrient decomposed by bacteria after the death of biomass. In the first part of our analysis the sufficient conditions ensuring local and global asymptotic stability of the model are obtained. Next, the existence of the Hopf bifurcation as time delay crosses a threshold value is established and, meanwhile, the phenomenon of stability switches is found under certain conditions. Numerical simulations are presented to illustrate the analytical results.

No MeSH data available.


Related in: MedlinePlus