A Reaction Diffusion Model to Describe the Toxin Effect on the Fish-Plankton Population (original) (raw)
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Mathematical analysis of toxin-phytoplankton-fish model with self-diffusion and cross-diffusion
BIOMATH, 2019
In this paper we propose a nonlinear reaction-diffusion system describing the interaction between toxin-producing phytoplankton and fish population. We analyze the effect of self- and cross-diffusion on the dynamics of the system. The existence, uniqueness and uniform boundedness of solutions are established in the positive octant. The system is analyzed for various interesting dynamical behaviors which include boundedness, persistence, local stability, global stability around each equilibria based on some conditions on self- and cross-diffusion coefficients. The analytical findings are verified by numerical simulation.
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.
Mathematical analysis of plankton population dynamics
Malaysian Journal of Fundamental and Applied Sciences
Harmful algal blooms (HABs) event that causes enormous economic loss and health effect raises concerns among environmentalists. In this paper, a mathematical model of interaction between nutrient, toxin-producing phytoplankton (TPP), non-toxic phytoplankton (NTP), zooplankton, and toxic chemicals is proposed to study on how the process of these HABs occurred. The model of interaction is represented by Ordinary Differential Equations (ODEs) and stability analysis of the model is conducted. Several conditions for the system to be stable around trivial and interior equilibrium point are obtained. From the analysis, it is observed that under nutrient limitation, the amounts of toxic chemicals secreted out by the TPP are increased. As a result, NTP population and zooplankton population are affected by the situation. If this situation is prolonged, this will result in the extinction of both populations. Overall, this study shows that TPP release more toxic chemicals when the nutrient is l...
Nutrient-phytoplankton-zooplankton models with a toxin
Mathematical and Computer Modelling, 2006
Models of nutrient-plankton interaction with a toxic substance that inhibits either the growth rate of phytoplankton, zooplankton or both populations are proposed and studied. For simplicity, it is assumed that both nutrient and the toxin have the same constant input and washout rate. The effects of toxin upon the existence, magnitude, and stability of the steady states are examined. Numerical simulations demonstrate that the system can have multiple attractors when phytoplankton's nutrient uptake rate is inhibited by the toxin.
Role of toxin producing phytoplankton for coexistence of planktonic ecosystem
2005
Effect of toxin producing plankton and its control is an intriguing problem in marine plankton ecology. In this paper we have proposed a three component model consisting of a non-toxic phytoplankton (NTP), toxin producing phytoplankton (TPP) and zooplankton (Z), where the growth of zooplankton species reduce due to toxic chemicals released by phytoplankton species. The nature of the dynamics of the system is observed around the positive equilibrium. Conditions for co-existence or extinction of populations are derived. It is observed that the three components persist if the predation rate of zooplankton population on toxic phytoplankton is bounded in certain regions. It is observed that when toxin is liberated by the toxin producing phytoplankton then stability zone increases. To validate the analytical results, numerical experiments and field collected sample observations are also presented. Our analytical as well as numerical study reveals the essential mathematical features regard...
American Journal of Computational and Applied Mathematics, 2012
In this paper, a system consisting of two competing harmfu l phytoplankton and a zooplankton with Holling type-II functional response and discrete time lag is considered. A stable coexistence of all the species has been observed for the system without delay and the Hopf-b ifurcation phenomenon is observed for the interior equilibriu m po int. The Hopf-bifurcating solution is obtained for the critical values of parameters like predation rates and half saturation constants. Further, using the normal form theory, we have determined the direction and the stability of the Hopf-bifurcation solution. The introduction of time delay in the system also shows the Hopf-bifurcat ion as the delay parameter passes through a critical value. Finally, the numerical simu lation is carried out to support the theoretical results.
The role of space in marine plankton ecosystem in presence of toxin producing phytoplankton
2015
In this paper, we investigate the complex dynamics of a nutrient phytoplankton zooplankton reaction-diffusion system with Holling type-II functional response. 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. Moreover we have found out a condition for diffusive instability of a locally stable equilibrium. Also it is shown that the system undergoes Hopf bifurcation when the maximal zooplankton conversion rate crosses a certain critical value. Computer simulations have been carried out to illustrate various analytical results.
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.
Journal of Theoretical Biology, 2002
Termination of planktonic blooms is of great importance to human health, ecosystem, environment, tourism and fisheries. Toxic substances released by plankton play an important role in this context. The effect of toxin-producing plankton (TPP) on zooplankton is observed from the field-collected samples and mathematical modelling. Information from both the studies led us to suggest that TPP may terminate the planktonic blooms by decreasing the grazing pressure of zooplankton and thus acts as a biological control.
Aggregation of toxin-producing phytoplankton acts as a defence mechanism – a model-based study
Mathematical and Computer Modelling of Dynamical Systems, 2013
We propose a simple model of toxin producing phytoplanktonzooplankton interactions in which the former is assumed to be able to detect the presence of zooplankton and to counteract it by forming patches and by releasing some toxic chemicals in the surrounding water. We observe that the formation of patch by the toxin producing phytoplankton decreases the grazing pressure of zooplankton resulting in stronger coupling between the interacting species determined by the fraction of the phytoplankton population that aggregates to form patches. Finally the results were validated by comparing them with an alternative spatial model.