In this paper, a mathematical model for the interacting dynamics of phytoplankton-zooplankton is proposed. The phytoplankton have the ability to take refuge and release toxins to avoid over predation by zooplankton. The zooplankton are provided some additional food to persist in the system. The phytoplankton are assumed to be affected directly by external toxic substances whereas zooplankton are affected indirectly by feeding on the affected phytoplankton. We incorporate seasonal variations in the model, assuming the level of nutrients, refuge and the rate of toxins released by phytoplankton as functions of time. Our results show that when high toxicity and refuge cause extinction of zooplankton, providing additional food supports the survival of zooplankton population and controls the phytoplankton population. Prey refuge and additional food have stabilizing effects on the system; higher values of the former results in extinction of zooplankton whereas phytoplankton disappear for larger values of the latter. Seasonality in nutrients level and toxins released by phytoplankton generate higher periodic solutions while time-dependent refuge of phytoplankton causes the occurrence of a period-three solution. The possibility of finding additional food for zooplankton may push back the ecosystem to a simple stable state from a complex dynamics.

本文提出了浮游植物-浮游植物相互作用动力学的数学模型。浮游植物具有避难和释放毒素的能力，以避免浮游动物的过度捕食。向浮游动物提供了一些额外的食物，以维持在系统中。假定浮游植物直接受到外部有毒物质的影响，而浮游动物则通过以受影响的浮游植物为食而间接受到影响。我们在模型中纳入季节变化，假设浮游植物释放的养分，避难所和毒素的水平随时间变化。我们的结果表明，当高毒性和避难所导致浮游动物灭绝时，提供更多食物可支持浮游动物种群的生存并控制浮游植物种群。避难所和其他食物对系统具有稳定作用；前者的较高值导致浮游动物的灭绝，而后者的值较大时浮游植物消失。浮游植物释放的营养物水平和毒素的季节性产生更高的周期解，而浮游植物的时间依赖性避难所导致了第三期解的发生。为浮游动物寻找其他食物的可能性可能会使生态系统从复杂的动力学状态退回到简单的稳定状态。浮游植物释放的营养物水平和毒素的季节性产生更高的周期解，而浮游植物的时间依赖性避难所导致了第三期解的发生。为浮游动物寻找其他食物的可能性可能会使生态系统从复杂的动力学状态退回到简单的稳定状态。浮游植物释放的营养物水平和毒素的季节性产生更高的周期解，而浮游植物的时间依赖性避难所导致了第三期解的发生。为浮游动物寻找其他食物的可能性可能会使生态系统从复杂的动力学状态退回到简单的稳定状态。