The Nonlinear Dynamics In The Growth Of Unicellular Aquatic Algae

The value of unicellular aquatic algae is significant, but the harm induced by unicellular aquatic algae blooms is enormous. It is very valuable to study the growth mechanism of unicellular aquatic algae. Based on theories of cell growth dynamics, pattern dynamics and impulsive control dynamics, the present paper studied the nonlinear dynamics in the growth of unicellular aquatic algae using dynamic modelling, dynamic analysis and numerical experiment, the main results are as follows:1. The effect of zooplankton on the growth of unicellular aquatic algae was studied, and a unicellular aquatic algae-zooplankton dynamic model with time delay was proposed. The results showed that delay in the growth of zooplankton results in the unicellular aquatic algae bloom. It was found from analysis that there is timescale difference between the growth of unicellular aquatic algae and the growth of zooplankton, and the delay enlarge the timescale difference, which give time for the reproduction of unicellular aquatic algae, then the blooms occurred. Researches suggested that zooplankton density is much lower before unicellular aquatic algae blooms, which is another important factor inducing unicellular aquatic algae blooms. The model predicted four processes of unicellular aquatic algae blooms, including initial accumulation, rapid increase, sustained bloom and rapid decline, and revealed a kind of dynamic mechanism on unicellular aquatic algae blooms.2. The effect of delay on the growth of unicellular aquatic algae was studied, and a nutrient-unicellular aquatic algae dynamic model with delay was proposed. The results showed that the delay in the growth of unicellular aquatic algae causes the uniform stationary solution to lose its stability, which induce the dynamic oscillation in the change of cells and the emergence of patchiness patterns in the spatial distribution of unicellular aquatic algae. Addionally, when the concentration of nutrient changed in certain range, the numerical experiment indicated that the change of cells of unicellular aquatic algae is positive correlated with the input rate of nutrient, and it is negative correlated with the loss rate of nutrient, which means that the increase of nutrient promotes reproduction of unicellular aquatic algae.3. The effects of delay and flow on the growth of unicellular aquatic algae was studied, and a nutrient-unicellular aquatic algae dynamics model with delay and flow was given. The research indicated that delay and flow influence temporal-spatio dynamics of growth of unicellular aquatic algae. The dynamics analysis revealed that delay determines the nature of chinge in cells of unicellular aquatic algae, and the flow dominates the evolutionary process in the spatial distribution of unicellular aquatic algae. Especially, the dynamic oscillation in the change of cells was likely to occur when flow exist. The numerical experiment shown that the band patterns emerge in the spatial distribution of unicellular aquatic algae along the direction of flow, the simulation is in good agreement with the field observation.4. The effects of the sinking effect of unicellular aquatic algae and species diffusion on the growth of unicellular aquatic algae were studied, a nutrient-unicellular aquatic algae-zooplankton dynamic model with sinking effect was given. The research showed that the spatial distribution of unicellular aquatic algae is heterogeneous due to the diffusion of zooplankton and the change of cells is temporal-spatio oscillation when sinking effect does not exist. In contrast, the results indicated that the sinking speed can influence the nature of change of cells and enhance the probability of heterogeneous spatial distribution of unicellular aquatic algae. The research revealed that multi-patterns emerge in the spatial distribution of unicellular aquatic algae including spot pattern, strip pattern and patchiness pattern when the sinking speed is much lower. However, the spatial distribution of unicellular aquatic algae is band pattern when the sinking speed is much faster, because the sinking effect directly influences the spatial dynamics of the growth of unicellular aquatic algae. Moreover, the research suggested that the increase of nutrient contribute to the static change of cells under the constant sinking speed.5. The control strategy on unicellular aquatic algae blooms was investigated, a unicellular aquatic algae-herbivory dynamic model with impulsive control depending on state was given. Some novel methods were developed. For a kind of impulsive control system depending on state, some problems have been solved, such as uniqueness of order-1 periodic solution, and so on. And the effect of artificial disturbance on the growth dynamics of unicellular aquatic algae was dissected. The research suggested that artificial disturbance enhances the complexity of change of cells and even causes chaos to emerge, which increase difficulty in the dynamic prediction of cells of unicellular aquatic algae. On this basis, the control strategy on the unicellular aquatic algae blooms was studied, and a control strategy, which can prevent and govern unicellular aquatic algae blooms, was given, and the strategy was of high efficiency and durability. In addition, the research indicated that excessively released amount of herbivory is not good at a time in the optimal control.