Existence, uniqueness and stability analysis of allelopathic stimulatory phytoplankton model

The term plankton refers to the freely floating and weakly swimming organisms within aquatic environment. The plant species commonly known as phytoplankton are unicellular and microscopic in size. Phytoplankton have a significant utility in marine life and they play a vital role at the base of the marine food chain. They also control the global carbon cycle which has a significant impact on the climate regulation. The regular change and abrupt fluctuation of phytoplankton density within aquatic environment are controlled by several factors, e.g., variation of available nutrients, environmental forcing due to seasonal change in environment and many others (for details, see [1]). A remarkable feature associated with many phytoplankton populations is the occurrence of bloom formation. A drastic increase in phytoplankton population, up by several orders of magnitude which is shortly followed by a sudden collapse whereby phytoplankton population returns to its original low-level. This kind of rapid growth followed by sudden decay in phytoplankton population is known as ‘phytoplankton bloom’. There has been a global increase in harmful plankton blooms in the last two decades and considerable scientific attention has been paid towards harmful algal blooms in recent years [2–5]. An important observation made by several researchers is that the change in population density of one phytoplankton species has the ability to affect the growth of several other species by producing toxic substances, and this is a responsible factor for bloom formation of various phytoplankton species. The term “allelopathy” was first introduced by Molisch (1937), and then defined extensively for plankton communities by Rice [6]. According to Rice [6], allelopathy is the effect of one plant species on the growth of another by releasing a chemical compound into the surrounding environment. These types of chemical compounds released by various plants are known as “allelochemicals” [7]. Allelochemicals released by one species of phytoplankton may have both positive and negative effects on the growth of another species. For example, the green alga,