Threshold dynamics of a periodic stoichiometric model

Yequan Liang; Xiuxiang Liu; Xiao Yu

doi:10.3934/dcdsb.2023065

Article Contents

2023, Volume 28, Issue 11: 5542-5558. Doi: 10.3934/dcdsb.2023065

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Threshold dynamics of a periodic stoichiometric model

School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China

^*Corresponding author: Xiao Yu
^*Corresponding author: Xiao Yu

Received: December 2022

Revised: March 2023

Early access: April 3, 2023

Published online: April 3, 2023

This work was supported by the NSF of China (12001205, 12171214) and Guangdong Basic and Applied Basic Research Foundation (2019A1515110179).

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Abstract

In this paper, we investigate a general time-periodic stoichiometric ODE model, which describes the algal growth. The model system has singularities induced by the zero nitrogen concentration. We first observe that there exists a threshold value $ \lambda_0 $, which is exactly the principal eigenvalue of a nonlinear eigenvalue problem associated with a homogeneous of degree one system, and further show the global dynamics of the model system in terms of $ \lambda_0 $. In particular, we obtain the uniqueness of the positive periodic solution when $ \lambda_0>0 $. Finally, we carry out simulations to illustrate the analytic results.

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Mathematics Subject Classification: Primary: 92D25, 34D23, 34C60; Secondary: 37C65.

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Figure 1. Global dynamics of system (3)

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Table 1. The parameters and their values of the system (1)

Parameter	Description	Value	Unit
$ \mathit{K} $	Light-dependent carrying capacity of algae	0–2	(mg C)/l
$ T $	Total $ N $ in the system	0–0.5	(mg N)/l
$ \mu $	Maximum growth rate of algae	1.2	$ \text{day}^{-1} $
$ q $	Minnimum N:C ratio of algae	0.0064	(mg N)/(mg C)
$ d $	N loss/recycling rate of algae	0.05	$ \text{day}^{-1} $
$ c $	Maximum N uptake rate of algae	3.2	(mg N)/(mg C)/day
$ a $	N-dependent half saturation constant of algae	0.128	(mg N)/l
^* Parameter values are referenced from [6,10].

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Table 2. Parameter values for simulation

Parameter	Value$ {_1} $	Value$ {_2} $
$ \mathit{K} $	$ 1.0+0.5\sin(2\pi t/365) $	$ 1.2+0.6\sin(2\pi t/365) $
$ T $	$ 0.2 $	$ 0.5 $
$ \mu $	$ 0.07+0.05\sin(2\pi t/365) $	$ 0.8+0.5\sin(2\pi t/365) $
$ q $	$ 0.05+0.014\sin(2\pi t/365) $	$ 0.05+0.014\sin(2\pi t/365) $
$ d $	$ 0.03+0.02\cos(2\pi t/365) $	$ 0.03+0.02\cos(2\pi t/365) $
$ c $	$ 0.003+0.002\sin(2\pi t/365) $	$ 0.003+0.002\sin(2\pi t/365) $
$ a $	$ 0.08+0.048\sin(2\pi t/365) $	$ 0.08+0.048\sin(2\pi t/365) $

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Table 3. Initial values

Set	$ I_1 $	$ I_2 $	$ I_3 $	$ I_4 $	$ I_5 $
$ L_1 $	$ (0.1,0.02) $	$ (0.3,0.05) $	$ (0.8,0.1) $	$ (1.2,0.12) $	$ (1.5,0.08) $
$ L_2 $	$ (0.1,0.02) $	$ (0.3,0.05) $	$ (0.8,0.3) $	$ (1.2,0.35) $	$ (1.5,0.3) $

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