Mathematical analysis of plankton population dynamics (original) (raw)
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Biosystems, 2007
The termination of harmful algal blooms (HABs) and coexistence of phytoplankton-zooplankton populations are of great importance to human health, ecosystem, environment, tourism and fisheries. In this paper we propose a three-component model consisting of dissolved limiting nutrients (N) supplied at constant rate and partially recycled after the death of plankton by bacterial decomposition, phytoplankton (P) and zooplankton (Z), where the growth of zooplankton species reduce due to toxic chemicals released by phytoplankton species. Our analysis leads to different thresholds which are expressible in terms of model parameters and determine the existence and stability of various states of the system. We observe that phytoplankton-zooplankton persist if the maximal zooplankton ingestion rate exceeds a lower threshold value. It is shown that the coexistence equilibrium loses its stability when the dilution rate of the nutrient concentration passes through a critical value and Hopf bifurcation occurs that induces oscillations of the population. Our results indicate that the occurrence of bloom increases when the nutrient concentration is very high, and in that case toxin produced by the phytoplankton plays a very crucial role towards the termination of the planktonic bloom.
Mathematical Modelling of Harmful Algal Blooms on West Coast of Sabah
Mathematics
Algal bloom is a condition in which there is a massive growth of algae in a certain region and it is said to be harmful when the bloom causes damage effects. Due to the tremendous impact of harmful algal bloom (HAB) on some aspects, this research proposes the mathematical modelling of an HAB model to describe the process of HAB together with population dynamics. This research considers the delay terms in the modelling since the liberation of toxic chemicals by toxin-producing phytoplankton (TPP) is not an instantaneous process in which the species need to achieve their maturity. A model of fish interaction is also being studied to show the effect of HAB on fish species. Time delay is incorporated for the mortality of fish due to the consumption of toxic zooplankton. Stability analysis is conducted and numerical simulations are applied to obtain the analytical results which highlight the critical values for the delay parameters. The existence of Hopf bifurcation is established when t...
A delay differential equation model on harmful algal blooms in the presence of toxic substances
Mathematical Medicine and Biology, 2002
The periodic nature of blooms is the main characteristic in marine plankton ecology. Release of toxic substances by phytoplankton species or toxic phytoplankton reduce the growth of zooplankton by decreasing grazing pressure and have an important role in planktonic blooms. A simple mathematical model of phytoplankton-zooplankton systems with such characteristics is proposed and analysed. As the process of liberation of toxic substances by phytoplankton species is still not clear, we try to describe a suitable mechanism to explain the cyclic nature of bloom dynamics by using different forms of toxin liberation process. To substantiate our analytical findings numerical simulations are performed and these adequately resemble the results obtained in our field study.
The dynamics of nutrient, toxic phytoplankton, nontoxic phytoplankton and zooplankton model
International Journal of Applied Mathematical Research, 2016
The objective of this paper is to study the dynamical behavior of an aquatic food web system. A mathematical model that includes nutrients, phytoplankton and zooplankton is proposed and analyzed. It is assumed that, the phytoplankton divided into two compartments namely toxic phytoplankton which produces a toxic substance as a defensive strategy against predation by zooplankton, and a nontoxic phytoplankton. All the feeding processes in this food web are formulating according to the Lotka-Volterra functional response. This model is represented mathematically by the set of nonlinear differential equations. The existence, uniqueness and boundedness of the solution of this model are investigated. The local and global stability conditions of all possible equilibrium points are established. The occurrence of local bifurcation and Hopf bifurcation are investigated. Finally, numerical simulation is used to study the global dynamics of this model.
Ecological Modelling, 2008
a b s t r a c t In this paper we have proposed and analyzed a simple mathematical model consisting of four variables, viz., nutrient concentration, toxin producing phytoplankton (TPP), nontoxic phytoplankton (NTP), and toxin concentration. Limitation in the concentration of the extracellular nutrient has been incorporated as an environmental stress condition for the plankton population, and the liberation of toxic chemicals has been described by a monotonic function of extracellular nutrient. The model is analyzed and simulated to reproduce the experimental findings of Graneli and Johansson [Graneli, E., Johansson, N., 2003. Increase in the production of allelopathic Prymnesium parvum cells grown under N-or P-deficient conditions. Harmful Algae 2, 135-145]. The robustness of the numerical experiments are tested by a formal parameter sensitivity analysis. As the first theoretical model consistent with the experiment of Graneli and Johansson (2003), our results demonstrate that, when nutrientdeficient conditions are favorable for the TPP population to release toxic chemicals, the TPP species control the bloom of other phytoplankton species which are non-toxic. Consistent with the observations made by Graneli and Johansson (2003), our model overcomes the limitation of not incorporating the effect of nutrient-limited toxic production in several other models developed on plankton dynamics.
Biological control of harmful algal blooms: A modelling study
Journal of Marine Systems, 2006
A multispecies dynamic simulation model (ERSEM) was used to examine the influence of allelopathic and trophic interactions causing feeding avoidance by predators, on the formation of harmful algal blooms, under environmental scenarios typical of a Mediterranean harbour (Barcelona). The biological state variables of the model included four functional groups of phytoplankton (diatoms, toxic and non-toxic flagellates and picophytoplankton), heterotrophic flagellates, micro-and mesozooplankton and bacteria. The physical-chemical forcing (irradiance, temperature and major nutrient concentrations) was based on an actual series of measurements taken along a year cycle in the Barcelona harbour. In order to evaluate potential effects of advection, some runs were repeated after introducing a biomass loss term. Numerical simulations showed that allelopathic effects of a toxic alga on a non-toxic but otherwise similar competitor did not have appreciable influence on the dynamics of the system. However, induction of avoidance of the toxic alga by predators, which resulted on increased predation pressure on other algal groups had a significant effect on the development of algal and predator populations. The presence of advection overrided the effect of these interactions and only allowed organisms with sufficiently high potential growth rates to thrive.
Applied Mathematics and Computation, 2011
We consider a plankton-nutrient interaction model consisting of phytoplankton, zooplankton and dissolved limiting nutrient with general nutrient uptake functions and instantaneous nutrient recycling. In this model, it is assumed that phytoplankton releases toxic chemical for self defense against their predators. The model system is studied analytically and the threshold values for the existence and stability of various steady states are worked out. It is observed that if the maximal zooplankton conversion rate crosses a certain critical value, the system enters into Hopf bifurcation. Finally it is observed that to control the planktonic bloom and to maintain stability around the coexistence equilibrium we have to control the nutrient input rate specially caused by artificial eutrophication. In case if it is not possible to control the nutrient input rate, one could use toxic phytoplankton to prevent the recurrence bloom.
Structural Stability Ananlysis of an Algal Bloom Mathematical Model in Trophic Interaction
International Journal of Nonlinear Analysis: Real World Applications, 2010
The paper deals with the dynamical behavior of plankton population ecosystem, mainly found in Sunderban mangrove area. The ecosystem is represented by a set of two dimensional non-linear differential equations involving zooplankton-phytoplankton population. Plankton populations undergo dramatic changes in marine ecology. We propose a description of plankton communities as excitable systems which resemble the behavior of excitable media. The delay parameter dependency of the various ‘excitable’ phenomena, trigger mechanism, threshold, and slow recovery, is clear, and permits ready investigation of the influence of properties of the physical environment, including variations in nutrient fluxes, temperature or population levels. We have analyzed the stability and bifurcation of the model system with and without delay. We have shown the existence and uniqueness of limit cycles in the rapid growth of the plankton population. We also studied the model system into a stochastic one, by incorporating random fluctuations of the environment. And we study the stochastic stability of the dynamical system in mean square sense around the interior equilibrium.
2008
Harmful algal blooms (HABs) have several adverse effects and have drawn considerable scientific attention in recent years. In the present paper, first, we considered the field observations carried out in the North-West coast of Bay of Bengal, India, and observed that single harmful phytoplankton-zooplankton system shows bloom phenomenon and the average biomass levels of the species in this case are higher compare to the situation when two harmful phytoplanktons are present. The presence of two harmful phytoplanktons in the system reduces the high abundance of the population showing considerable decrease in HABs. Next, motivated from our experiment we proposed a deterministic model consisting of two harmful phytoplanktons and zooplankton with an additional factor that the harmful-phytoplankton reduce the growth of zooplankton by toxic substances. To observe the effect of environmental variability in the deterministic model we modified the system into a stochastic model by applying stochastic perturbation in the form of white noise. The dynamics of both deterministic and stochastic model were studied and threshold values of the inhibition rate as well as noise intensity were calculated to obtain stable coexistence of the species. We further observed that intensity of white noise and competition between the species play important roles to reduce the bloom situation. The physical interpretation of our mathematical analysis and numerical simulations suggest the termination of planktonic blooms is possible due to competition between two or more phytoplankton as well as through low intensity of external noise. Our analytical findings support our field observations.
A simple mathematical model for seasonal planktonic blooms
Mathematical Methods in The Applied Sciences, 2009
We show how the inclusion of the defense strategy by different species can alter the prediction of simple models. One of the defense strategy by the phytoplankton population against their grazer is the release of toxic chemicals. In turn the zooplankton population reduces there predation rate over toxin producing phytoplankton (TPP) to protect themselves from those toxic chemicals. Thus, when the level of toxicity is high, the grazing pressure is low and when the level of toxicity is low or when the toxin is absent, the grazing pressure is high. Here we have considered a TPP–zooplankton system where the rate of toxin liberation and the predation rate vary with zooplankton abundance. We observe that our proposed model has the potential to show different dynamical behaviour that are similar to that seen in real-world situations. Further, we consider three different functional forms for the distribution of the toxins and compare them using latin hypercube sampling technique and found that the functional forms seem to have no effect in determining the final outcome of the system. Copyright © 2009 John Wiley & Sons, Ltd.