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.

The effects of obstacle size on periodic travelling waves in oscillatory reaction–diffusion equations

2007 ◽  
Vol 464 (2090) ◽  
pp. 365-390 ◽  
Author(s):  
Matthew J Smith ◽  
Jonathan A Sherratt ◽  
Nicola J Armstrong
Keyword(s):  
Large Scale ◽  
Order Approximation ◽  
Travelling Waves ◽  
Natural Populations ◽  
Reaction Diffusion ◽  
Field Studies ◽  
Reaction Diffusion Equations ◽  
Infinite Domain ◽  
Predator Prey ◽  
Predator Prey Model

Many natural populations undergo multi-year cycles, and field studies have shown that these can be organized into periodic travelling waves (PTWs). Mathematical studies have shown that large-scale landscape obstacles represent a natural mechanism for wave generation. Here, we investigate how the amplitude and wavelength of the selected waves depend on the obstacle size. We firstly consider a large circular obstacle in an infinite domain for a reaction–diffusion system of ‘ λ – ω ’ type. We use perturbation theory to derive a leading order approximation to the wave generated by the obstacle. This shows the dependence of the wave properties on both parameter values and obstacle size. We find that the limiting values of the amplitude and wavelength are approached algebraically with distance from the obstacle edge, rather than exponentially in the case of a flat boundary. We use our results to predict the properties of waves generated by a large circular obstacle for an oscillatory predator–prey system, via a reduction of the predator–prey model to normal form close to Hopf bifurcation. Our predictions compare well with numerical simulations. We also discuss the implications of these results for wave stability and briefly investigate the effects of obstacles with elliptical geometries.

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.

scholarly journals Instability Induced by Cross-Diffusion in a Predator-Prey Model with Sex Structure

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Shengmao Fu ◽  
Lina Zhang
Keyword(s):  
Reaction Diffusion ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Sex Structure ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Ode System ◽  
Prey Model ◽  
Weakly Coupled ◽  
Coupled Reaction

In this paper, we consider a cross-diffusion predator-prey model with sex structure. We prove that cross-diffusion can destabilize a uniform positive equilibrium which is stable for the ODE system and for the weakly coupled reaction-diffusion system. As a result, we find that stationary patterns arise solely from the effect of cross-diffusion.

Pattern Formation Controlled by External Forcing in a Spatial Harvesting Predator-Prey Model

2011 ◽  
pp. 214-221
Author(s):  
Feng Rao
Keyword(s):  
Reaction Diffusion ◽  
Euler Method ◽  
External Forcing ◽  
Periodic Forcing ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Flux Boundary Conditions ◽  
Predator Prey Models ◽  
The Stability

Predator–prey models in ecology serve a variety of purposes, which range from illustrating a scientific concept to representing a complex natural phenomenon. Due to the complexity and variability of the environment, the dynamic behavior obtained from existing predator–prey models often deviates from reality. Many factors remain to be considered, such as external forcing, harvesting and so on. In this chapter, we study a spatial version of the Ivlev-type predator-prey model that includes reaction-diffusion, external periodic forcing, and constant harvesting rate on prey. Using this model, we study how external periodic forcing affects the stability of predator-prey coexistence equilibrium. The results of spatial pattern analysis of the Ivlev-type predator-prey model with zero-flux boundary conditions, based on the Euler method and via numerical simulations in MATLAB, show that the model generates rich dynamics. Our results reveal that modeling by reaction-diffusion equations with external periodic forcing and nonzero constant prey harvesting could be used to make general predictions regarding predator-prey equilibrium,which may be used to guide management practice, and to provide a basis for the development of statistical tools and testable hypotheses.

A Semi-Analytical Method for Solving Problems on the Role of Prey Taxis in a Biological Control-Mathematical Model

2019 ◽  
Vol 10 (02) ◽  
pp. 1850009
Author(s):  
OPhir Nave ◽  
Yifat Baron ◽  
Manju Sharma
Keyword(s):  
Biological Control ◽  
Analytical Method ◽  
Reaction Diffusion ◽  
Prey Population ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Homotopy Analysis ◽  
The Stability ◽  
Pest Density

In this paper, we applied the well-known homotopy analysis methods (HAM), which is a semi-analytical method, perturbation method, to study a reaction–diffusion–advection model for the dynamics of populations under biological control. According to the predator–prey model, the advection expression represents the predator density movement in which the acceleration is proportional to the prey density gradient. The prey population reproduces logistically, and the interactions of prey population obey the Holling’s prey-dependent Type II functional response. The predation process splits into the following subdivided processes: random movement which is represented by diffusion, direct movement which is described by prey taxis, local prey interactions, and consumptions which are represented by the trophic function. In order to ensure a successful biological control, one should make the predator-pest population to stabilize at a very low level of pest density. One reason for this effect is the intermediate taxis activity. However, when the system loses stability, for example very intensive prey taxis destroys the stability, it leads to chaotic dynamics with pronounced outbreaks of pest density.

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