Abstract

Algal blooms are increasing in coastal waters worldwide. The study on the features of algal pollution in water bodies and the ways to eliminate them is of vital importance. Preventing, treating, and monitoring algal blooms can be an unanticipated cost for a water system. To tame algal bloom in a lake, the government provides funds through budget allocation. In this paper, we propose a mathematical model to investigate the effect of budget allocation on the control of algal bloom in a lake. We assume that the growth of budget follows logistic law and also increases in proportion to the algal density in the lake. A part of the budget is utilized for the control of inflow of nutrients, while the remaining is used in the removal of algae from the lake. Our results show that algal bloom can be mitigated from the lake by reducing the inflow rate of nutrients to a very low value, which can be achieved for very high efficacy of budget allocation for the control of nutrients inflow from outside sources. Also, increasing the efficacy of budget allocation for the removal of algae helps to control the algal bloom. Further, more budget should be used on the control of nutrient’s inflow than on the removal of algae, as the presence of nutrients in high concentration will immediately proliferate the growth of algae. Moreover, the combined effects of controlling the inflow of nutrients and removing algae at high rates will result in nutrients and algae-free aquatic environment. Further, we modify the model by considering a discrete time delay involved in the increment of budget due to increased density of algae in the lake. We observe that chaotic oscillations may arise via equilibrium destabilization on increasing the values of time delay. We apply basic tools of nonlinear dynamics such as Poincaré section and maximum Lyapunov exponent to confirm the chaotic behavior of the system.

中文翻译：

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藻类繁殖的动力学：预算分配和时间延迟的影响

摘要

全球沿海水域中的藻华不断增加。研究水体中藻类污染的特征及其消除方法至关重要。预防，治疗和监测藻华的发生可能是水系统不可预期的成本。为了驯服湖中的藻花，政府通过预算分配提供资金。在本文中，我们提出了一个数学模型来研究预算分配对湖泊藻华控制的影响。我们假设预算的增长遵循逻辑规律，并且也与湖中的藻类密度成比例地增长。预算的一部分用于控制养分的流入，而其余部分则用于从湖中去除藻类。我们的结果表明，可以通过将营养物质的流入量降低到非常低的值来缓解湖泊中的藻华，这可以通过非常高的预算分配效率来控制外部营养物质的流入。同样，增加预算拨款去除藻类的功效有助于控制藻类繁殖。此外，用于控制养分流入的预算应比用于去除藻类的预算更多，因为高浓度养分的存在会立即促进藻类的生长。此外，控制养分流入和高效率去除藻类的综合效果将导致养分和无藻的水生环境。进一步，我们通过考虑由于湖中藻类密度增加而导致预算增加的离散时间延迟来修改模型。我们观察到，通过增加时间延迟值的平衡不稳定可能会引起混沌振荡。我们应用非线性动力学的基本工具（例如庞加莱截面和最大Lyapunov指数）来确认系统的混沌行为。