scholarly journals Bloom Formation and Turing Patterns in an Infochemical Mediated Multitrophic Plankton Model

2020 ◽  
Vol 30 (10) ◽  
pp. 2030028
Author(s):  
Tahani A. S. Al-Karkhi ◽  
Rudy Kusdiantara ◽  
Hadi Susanto ◽  
Edward A. Codling
Keyword(s):  
Phytoplankton Bloom ◽  
Time Integration ◽  
Reaction Diffusion ◽  
Grazing Pressure ◽  
System Stability ◽  
Diffusion Term ◽  
Predator Prey ◽  
Bloom Formation ◽  
Parameter Values ◽  
Plankton Model

A two-species predator–prey plankton model is studied, where the grazing pressure of microzooplankton on phytoplankton is controlled through external infochemical mediated predation. The system stability is analyzed in order to explain the conditions for phytoplankton bloom formation and to explore system bifurcations. The interplay between the level of infochemical-mediated external predation and the phytoplankton carrying capacity is considered over a range of parameter values and the resultant system dynamics is illustrated. The model is extended to include a spatial diffusion term leading to a reaction–diffusion system that is investigated by determining the Turing space of the model. Thereafter, the bifurcation analysis of specific time-independent patterns is explored. Through time integration, the system is also shown to exhibit the potential for temporally varying spatial patterns.

Diffusion-driven instability in oscillating environments

1995 ◽  
Vol 6 (4) ◽  
pp. 355-372 ◽  
Author(s):  
Jonathan A. Sherratt
Keyword(s):  
Diffusion Coefficients ◽  
Numerical Solutions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Model Parameters ◽  
Seasonal Fluctuations ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Ecological Applications ◽  
Parameter Values

Diffusion-driven instability in systems of reaction-diffusion equations is a commonly used model for pattern formation in both embryology and ecology. In ecological applications, model parameters tend to oscillate in time, because of either daily or seasonal fluctuations in the environment. I investigate the effects of such fluctuations on diffusion-driven instability by considering analytically the possibility of Turing bifurcations when the parameter values (diffusion coefficients and kinetic parameters) oscillate in time between two sets of constant values, with a period that is either very short or very long compared to the time scale of the growth and predation kinetics. I show that oscillations in the kinetics can have quite different effects from oscillations in the dispersal terms. I also discuss the comparison between the solution forms predicted by linear theory and the numerical solutions of a simple nonlinear predator-prey model.

scholarly journals An investigation of grazing behaviors that result in winter phytoplankton biomass accumulation

2020 ◽  
Author(s):  
Mara Freilich ◽  
Alexandre Mignot ◽  
Glenn Flierl ◽  
Raffaele Ferrari
Keyword(s):  
North Atlantic ◽  
Phytoplankton Biomass ◽  
Phytoplankton Bloom ◽  
Biomass Accumulation ◽  
Grazing Pressure ◽  
Mathematical Framework ◽  
Phytoplankton Concentration ◽  
The North ◽  
Bloom Formation ◽  
The North Atlantic

Abstract. Recent observations have shown that phytoplankton biomass increases in the North Atlantic during winter, even when the mixed layer is deepening and light is limited. Current theories suggest that this is due to a release from grazing pressure. Here we demonstrate that the often-used grazing models that are linear at low phytoplankton concentration do not allow for a wintertime increase in phytoplankton biomass. However, certain mathematical formulations of grazing that are quadratic (or more generally non-linear) in phytoplankton concentration at low concentrations can reproduce the fall to spring transition in phytoplankton, including wintertime biomass accumulation. We illustrate this point with a minimal model for the annual cycle of North Atlantic phytoplankton designed to simulate phytoplankton concentration as observed by BioGeoChemical-Argo (BGC-Argo) floats in the North Atlantic. This analysis provides a mathematical framework for assessing hypotheses of phytoplankton bloom formation.

scholarly journals Inhomogeneous spatial patterns in diffusive predator-prey system with spatial memory and predator-taxis

2021 ◽  
Author(s):  
Yehu Lv
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Spatial Memory ◽  
Reaction Diffusion ◽  
Diffusion Term ◽  
Predator Prey ◽  
Spatially Inhomogeneous ◽  
Prey System ◽  
Mode 2 ◽  
The Stability

Abstract In this paper, we consider a diffusive predator-prey system with spatial memory and predator-taxis. Since in this system, the memory delay appears in the diffusion term, and the diffusion term is nonlinear, the classical normal form of Hopf bifurcation in the reaction-diffusion system with delay can't be applied to this system. Thus, in this paper, we first derive an algorithm for calculating the normal form of Hopf bifurcation in this system. Then in order to illustrate the effectiveness of our newly developed algorithm, we consider the diffusive Holling-Tanner model with spatial memory and predator-taxis. The stability and Hopf bifurcation analysis of this model are investigated, and the direction and stability of Hopf bifurcation periodic solution are also researched by using our newly developed algorithm for calculating the normal form of Hopf bifurcation. At last, we carry out some numerical simulations, two stable spatially inhomogeneous periodic solutions corresponding to the mode-1 and mode-2 Hopf bifurcations are found, which verifies our theoretical analysis results.

scholarly journals Bifurcation, Chaos, and Pattern Formation for the Discrete Predator-Prey Reaction-Diffusion Model

2019 ◽  
Vol 2019 ◽  
pp. 1-9
Author(s):  
Lili Meng ◽  
Yutao Han ◽  
Zhiyi Lu ◽  
Guang Zhang
Keyword(s):  
Reaction Diffusion ◽  
Turing Instability ◽  
Instability Region ◽  
Flip Bifurcation ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Reaction Diffusion Model ◽  
Parameter Values ◽  
Bifurcation Chaos

In this paper, a discrete predator-prey system with the periodic boundary conditions will be considered. First, we get the conditions for producing Turing instability of the discrete predator-prey system according to the linear stability analysis. Then, we show that the discrete model has the flip bifurcation and Turing bifurcation under the critical parameter values. Finally, a series of numerical simulations are carried out in the Turing instability region of the discrete predator-prey model; some new Turing patterns such as striped, bar, and horizontal bar are observed.

scholarly journals Oscillations and chaos behind predator–prey invasion: mathematical artifact or ecological reality?

1997 ◽  
Vol 352 (1349) ◽  
pp. 21-38 ◽  
Author(s):  
Jonathan A. Sherratt ◽  
Barry T. Eagan ◽  
Mark A. Lewis
Keyword(s):  
Reaction Diffusion ◽  
Direct Consequence ◽  
Reaction Diffusion Equations ◽  
Coupled Map Lattices ◽  
Ecological Data ◽  
Predator Prey ◽  
Modelling Framework ◽  
Frequent Absence ◽  
Spatiotemporal Oscillations ◽  
Parameter Values

A constant dilemma in theoretical ecology is knowing whether model predictions corrspond to real phenomena or whether they are artifacts of the modelling framework. The frequent absence of detailed ecological data against which models can be tested gives this issue particular importance. We address this question in the specific case of invasion in a predator–prey system with oscillatory population kinetics, in which both species exhibit local random movement. Given only these two basic qualitative features, we consider whether we can deduce any properties of the behaviour following invasion. To do this we study four different types of mathematical model, which have no formal relationship, but which all reflect our two qualitative ingredients. The models are: reaction–diffusion equations, coupled map lattices, deterministic cellular automata, and integrodifference equations. We present results of numerical simulations of the invasion of prey by predators for each model, and show that although there are certain differences, the main qualitative features of the behaviour behind invasion are the same for all the models. Specifically, there are either irregular spatiotemporal oscillations behind the invasion, or regular spatiotemporal oscillations with the form of a periodic travelling ‘wake’, depending on parameter values. The observation of this behaviour in all types of model strongly suggests that it is a direct consequence of our basic qualitative assumptions, and as such is an ecological reality which will always occur behind invasion in actual oscillatory predator–prey systems.

Pattern dynamics of a diffusive predator–prey model with delay effect

2017 ◽  
Vol 10 (04) ◽  
pp. 1750059 ◽  
Author(s):  
Guangping Hu ◽  
Xiaoling Li ◽  
Dongliang Li
Keyword(s):  
Time Delay ◽  
Dynamical Behavior ◽  
Reaction Diffusion ◽  
Wave Pattern ◽  
Turing Pattern ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Pattern Dynamics ◽  
Predator Prey Models ◽  
Parameter Values

We study the spatiotemporal dynamics in a diffusive predator–prey system with time delay. By investigating the dynamical behavior of the system in the presence of Turing–Hopf bifurcations, we present a classification of the pattern dynamics based on the dispersion relation for the two unstable modes. More specifically, we researched the existence of the Turing pattern when control parameters lie in the Turing space. Particularly, when parameter values are taken in Turing–Hopf domain, we numerically investigate the formation of all the possible patterns, including time-dependent wave pattern, persistent short-term competing dynamics and stationary Turing pattern. Furthermore, the effect of time delay on the formation of spatial pattern has also been analyzed from the aspects of theory and numerical simulation. We speculate that the interaction of spatial and temporal instabilities in the reaction–diffusion system might bring some insight to the finding of patterns in spatial predator–prey models.

Spatial Pattern of Ratio-Dependent Predator–Prey Model with Prey Harvesting and Cross-Diffusion

2019 ◽  
Vol 29 (03) ◽  
pp. 1950036 ◽  
Author(s):  
R. Sivasamy ◽  
M. Sivakumar ◽  
K. Balachandran ◽  
K. Sathiyanathan
Keyword(s):  
Temporal Dynamics ◽  
Intake Rate ◽  
Diffusion Term ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey Model ◽  
Nonlinear Harvesting ◽  
Ratio Dependent ◽  
The Cross

This study focuses on the spatial-temporal dynamics of predator–prey model with cross-diffusion where the intake rate of prey is per capita predator according to ratio-dependent functional response and the prey is harvested through nonlinear harvesting strategy. The permanence analysis and local stability analysis of the proposed model without cross-diffusion are analyzed. We derive the conditions for the appearance of diffusion-driven instability and global stability of the considered model. Also the parameter space for Turing region is specified by keeping the cross-diffusion coefficient as one of the crucial parameters. Numerical simulations are given to justify the proposed theoretical results and to show that the cross-diffusion term plays a significant role in the pattern formation.

scholarly journals Stabilization for a Periodic Predator-Prey System

2007 ◽  
Vol 2007 ◽  
pp. 1-17
Author(s):  
Sebastian Aniţa ◽  
Carmen Oana Tarniceriu
Keyword(s):  
Internal Control ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
System Modelling ◽  
Predator Prey ◽  
Periodic Environment ◽  
Prey System

A reaction-diffusion system modelling a predator-prey system in a periodic environment is considered. We are concerned in stabilization to zero of one of the components of the solution, via an internal control acting on a small subdomain, and in the preservation of the nonnegativity of both components.

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