Abstract
Plankton blooms and its control is an intriguing problem in ecology. To investigate the oscillatory nature of blooms, a two-dimensional model for plankton species is considered where one species is toxic phytoplankton and other is zooplankton. The delays required for the maturation time of zooplankton, the time for phytoplankton digestion and the time for phytoplankton cells to mature and release toxic substances are incorporated and the delayed model is analyzed for stability and bifurcation phenomena. It proves that periodic plankton blooms can occur when the delay (the sum of the above three delays) changes and crosses some threshold values. The phenomena described by this mechanism can be controlled through the toxin release rates of phytoplankton. Then, a delay feedback controller with the coefficient depending on delay is introduced to system. It concludes that the onset of the bifurcation can be delayed as negative feedback gain (the decay rate) decreases (increases). Some numerical examples are given to verify the effectiveness of the delay feedback control method and the existence of crossing curve. These results show that the oscillatory nature of blooms can be controlled by human behaviors.
Highlights
With the development of society and economy, the marine environment pollution, such as harmful algal bloom and overfishing, has seriously damaged the ecological balance of phytoplankton-zooplankton system which is an important part of the marine ecosystem
We study a minimal model for phytoplankton-zooplankton interaction relationship and Hopf bifurcation control problem
From our analytical study we have observed that enhancing the rate of toxin production it can increase the region of stability of the system and as a result can control the periodicity of planktonic blooms
Summary
With the development of society and economy, the marine environment pollution, such as harmful algal bloom and overfishing, has seriously damaged the ecological balance of phytoplankton-zooplankton system which is an important part of the marine ecosystem. We improve system (1.1) to describe TPP and zooplankton interactions and combine the following functional forms: g(x, δ) = r1(1 − x/δ), p(x) = mf (x) and q(y/x) = r2(1 − y/γx), that is, Leslie-Gower type system with generalized functional response and two delays is investigated:. The significance of the control is that by artificially increasing or decreasing the number of predators of the σ age, we can modify the bifurcation characteristics of nonlinear system to obtain some specific dynamical behaviors. Note that the strength of feedback control is in the form of ke−dσ, and the function decreases exponentially with delay σ This means that the feedback effects of past states diminish over time.
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