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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.


Related in: MedlinePlus

a, b The asymptotical stability of the coexistence equilibrium  with . c–e Coexistence equilibrium  loses its stability when . Stable periodic solution arising from Hopf bifurcation at . Here
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Fig2: a, b The asymptotical stability of the coexistence equilibrium with . c–e Coexistence equilibrium loses its stability when . Stable periodic solution arising from Hopf bifurcation at . Here

Mentions: First, we consider the special case of system (1), that is, and . In order to verify the results of Theorem 7, we consider as bifurcation parameter and for case (a) taken parameters in Table 1. It is easy to compute that Our numerical simulations show that for all , interior equilibrium is stable. Figure 1 shows the simulation result for the system (1) with . For case (b) of Theorem 7, we take parameters as , , and other parameters the same as that in Table 1. A direct computation gives , and holds. After calculations we find the minimum value of the delay parameter ‘’ for system (1) for which the stability behaviour changes and the first critical values are given by , such that is locally stable for and is unstable for . From our analytical findings we have seen that is locally asymptotically stable for . Figure 2 shows the simulation result for system (1) with . Interior equilibrium point looses its stability as passes through its critical value and a Hopf bifurcation occurs. A periodic solution is depicted in Fig. 2d, e.Fig. 1


Mathematical analysis of a nutrient-plankton system with delay.

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

a, b The asymptotical stability of the coexistence equilibrium  with . c–e Coexistence equilibrium  loses its stability when . Stable periodic solution arising from Hopf bifurcation at . Here
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4940349&req=5

Fig2: a, b The asymptotical stability of the coexistence equilibrium with . c–e Coexistence equilibrium loses its stability when . Stable periodic solution arising from Hopf bifurcation at . Here
Mentions: First, we consider the special case of system (1), that is, and . In order to verify the results of Theorem 7, we consider as bifurcation parameter and for case (a) taken parameters in Table 1. It is easy to compute that Our numerical simulations show that for all , interior equilibrium is stable. Figure 1 shows the simulation result for the system (1) with . For case (b) of Theorem 7, we take parameters as , , and other parameters the same as that in Table 1. A direct computation gives , and holds. After calculations we find the minimum value of the delay parameter ‘’ for system (1) for which the stability behaviour changes and the first critical values are given by , such that is locally stable for and is unstable for . From our analytical findings we have seen that is locally asymptotically stable for . Figure 2 shows the simulation result for system (1) with . Interior equilibrium point looses its stability as passes through its critical value and a Hopf bifurcation occurs. A periodic solution is depicted in Fig. 2d, e.Fig. 1

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