Dynamics induced by delay in a nutrient–phytoplankton model with diffusion

Stability switches and chaos induced by delay in a reaction-diffusion nutrient-plankton model

In this paper, we investigate a reaction-diffusion model incorporating dynamic variables for nutrient, phytoplankton, and zooplankton. Moreover, we account for the impact of time delay in the growth of phytoplankton following nutrient uptake. Our theoretical analysis reveals that the time delay can trigger the emergence of persistent oscillations in the model via a Hopf bifurcation. We also analytically track the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions. Our simulation results demonstrate stability switches occurring for the positive equilibrium with an increasing time lag. Furthermore, the model exhibits homogeneous periodic-2 and 3 solutions, as well as chaotic behaviour. These findings highlight that the presence of time delay in the phytoplankton growth can bring forth dynamical complexity to the nutrient-plankton system of aquatic habitats.

Investigation of a nutrient-plankton model with stochastic fluctuation and impulsive control

In this paper, we investigate a stochastic nutrient-plankton model with impulsive control of the nutrient concentration and zooplankton population. Analytically, we find that the population size is nonnegative for a sufficiently long time. We derive some sufficient conditions for the existence of stable periodic oscillations, which indicate that the plankton populations will behave periodically. The numerical results show that the plankton system experiences a transition from extinction to the coexistence of species due to the emergence of impulsive control. Additionally, we observe that the nutrient pulse has a stronger relationship with phytoplankton growth than the zooplankton pulse. Although the frequency of impulsive control and appropriate environmental fluctuations can promote the coexistence of plankton populations, an excessive intensity of noise can result in the collapse of the entire ecosystem. Our findings may provide some insights into the relationships among nutrients, phytoplankton and zooplankton in a stochastic environment.

Dynamics of a predator-prey model with strong Allee effect and nonconstant mortality rate

In this paper, dynamics analysis for a predator-prey model with strong Allee effect and nonconstant mortality rate are taken into account. We systematically studied the existence and stability of the equilibria, and detailedly analyzed various bifurcations, including transcritical, saddle-node, Hopf and Bogdanov-Takens bifurcation. In addition, the theoretical results are verified by numerical simulations. The results indicate that when the mortality is large, the nonconstant death rate can be approximated to a constant value. However, it cannot be considered constant under small mortality rate conditions. Unlike the extinction of species for the constant mortality, the nonconstant mortality may result in the coexistence of prey and predator for the predator-prey model with Allee effect.

Dynamics induced by environmental stochasticity in a phytoplankton-zooplankton system with toxic phytoplankton

Environmental stochasticity and toxin-producing phytoplankton (TPP) are the key factors that affect the aquatic ecosystems. To investigate the effects of environmental stochasticity and TPP on the dynamics of plankton populations, a stochastic phytoplankton-zooplankton system with two TPP is studied theoretically and numerically in this paper. Theoretically, we first prove that the system possesses a unique and global positive solution with positive initial values, and then derive some sufficient conditions guaranteeing the extinction and persistence in the mean of the system. Significantly, it is shown that the system has a stationary distribution when toxin liberation rate reaches some a critical value. Additionally, numerical analysis shows that the white noise can affect the survival of plankton populations directly. Furthermore, it has been observed that the increasing one toxin liberation rate can increase the survival chance of phytoplankton and reduce the biomass of zooplankton, but the combined effects of two liberation rates on the changes in plankton populations are stronger than that of controlling any one of the two TPP.

Bifurcation and patterns induced by flow in a prey-predator system with Beddington-DeAngelis functional response

In this paper, a prey-predator system described by a couple of advection-reaction-diffusion equations is studied theoretically and numerically, where the migrations of both prey and predator are considered and depicted by the unidirectional flow (advection term). To investigate the effect of population migration, especially the relative migration between prey and predator, on the population dynamics and spatial distribution of population, we systematically study the bifurcation and pattern dynamics of a prey-predator system. Theoretically, we derive the conditions for instability induced by flow, where neither Turing instability nor Hopf instability occurs. Most importantly, linear analysis indicates the instability induced by flow depends only on the relative flow velocity. Specifically, when the relative flow velocity is zero, the instability induced by flow does not occur. Moreover, the diffusion-driven patterns at the same flow velocity may not be stationary because of the contribution of flow. Numerical bifurcation analyses are consistent with the analytical results and show that the patterns induced by flow may be traveling waves with different wavelengths, amplitudes, and speeds, which are illustrated by numerical simulations.

Fractional order oxygen-plankton system under climate change

Global climate change affects marine species including phytoplankton, which constitute the base of the marine food web, by changing the primary productivity. Global warming affects the ocean surface temperature, in turn leading to a change in the oxygen production of phytoplankton. In this work, the fractional oxygen-phytoplankton-zooplankton mathematical model is considered by the Caputo fractional operator. The production rate of photosynthesis is determined by a temperature function. The model is, therefore, based on the idea that the rate of photosynthesis changes due to the impact of global warming, while phytoplankton oxygen production increases and decreases. We analyze the model with the Caputo fractional derivative differently from the classical case of the model and we compare the results with the integer order derivative when α tends to 1. Existence and uniqueness properties of the oxygen-plankton model have been proved by means of a local Lipschitz condition. It was shown that the species are more sustainable than its corresponding classical case in the Caputo model. Our results show that the effect of global warming on the oxygen production rate has been observed to be quite severe, resulting in oxygen depletion and plankton extinction.

Analysis of the interaction among rice, weeds, inorganic fertilizer, and a herbivore in a composite farming paddy ecosystem

As one of the Globally Important Agricultural Heritage Systems (GIAHS), rice field composite farming is an ecological measure in rice production, which can reduce the amount of chemical fertilizers, pesticides and herbicides. This research studies the interaction among rice, weed, inorganic fertilizer and herbivore in a composite farming paddy ecosystem. We develop a differential equation model to analyze the relations and interactions among those components. Results show the existence of an equilibrium for paddy and weed extinction, one or two equilibria for rice extinction, an equilibrium for weed extinction, and an equilibrium for rice and weed coexistence. Based on the obtained stability conditions of these equilibria, measures are proposed to avoid the existence or the stability of equilibria for rice extinction. Other measures are proposed to lead to a stable equilibrium for weed extinction, which is the most desirable result in rice production. Conditions for maximizing the yield of rice are also obtained by taking the relative mortality of rice as variable. In addition, we discover the existence of Hopf bifurcation phenomenon in the system, and develop the critical value of Hopf bifurcation by taking the artificial fertilizer rate as the bifurcation parameter. Our findings provide effective guidance and insights for rice production in a composite farming paddy ecosystem.

Analysis of the interaction among weed, inorganic fertilizer and herbivore in paddy ecosystem in fallow season

Paddy growth is influenced by the amount of inorganic fertilizer in paddy ecosystem in fallow season. To discover the interaction among weed, inorganic fertilizer and herbivore in the system, we put forward a differential equation model and investigate its properties. Results show that the system has a weed and herbivore extinct equilibrium and a herbivore extinct equilibrium. The two equilibria are proven to be unstable using the center manifold method. Under certain conditions, the system also has a positive equilibrium point. We give the stable region and the unstable region of the positive equilibrium point, which are determined by some parameters. We find that the system has the Hopf bifurcation phenomenon, and give the critical value of Hopf bifurcation by taking a system parameter as the bifurcation parameter. By comparing the equilibrium states between a paddy ecosystem with herbivore and one without herbivore, we find that the content of inorganic fertilizer can be improved by putting herbivore into a paddy field. An example is given to illustrate the feasibility of the results. Numerical simulation shows that Hopf bifurcation phenomena exist in the system.