Spatio-temporal pattern formation in coupled models of plankton dynamics and fish school motion

Zhang

et al. 2010

Ecological Modelling

In this paper, we investigate the emergence of a predator-prey system with Ivlevtype functional response and reaction-diffusion. We study how diffusion affects the stability of predator-prey coexistence equilibrium and derive the conditions for Hopf and Turing bifurcation in the spatial domain. Based on the bifurcation analysis, we give the spatial pattern formation, the evolution process of the system near the coexistence equilibrium point, via numerical simulation. We find that pure Hopf instability leads to the formation of spiral patterns and pure Turing instability destroys the spiral pattern and leads to the formation of chaotic spatial pattern. Furthermore, we perform three categories of initial perturbations which predators are introduced in a small domain to the coexistence equilibrium point to illustrate the emergence of spatiotemporal patterns, we also find that in the beginning of evolution of the spatial pattern, the special initial conditions have an effect on the formation of spatial patterns, though the effect is less and less with the more and more iterations. This indicates that for prey-dependent type predator-prey model, pattern formations do depend on the initial conditions, while for predator-dependent type they do not. Our results show that modeling by reaction-diffusion equations is an appropriate tool for investigating fundamental mechanisms of complex spatiotemporal dynamics.

Section: Discussionmentioning

confidence: 99%

Pattern formation of a predator–prey system with Ivlev-type functional response

Zhang

et al. 2010

Ecological Modelling

Journal of Biological Dynamics

“…Furthermore, the analysis in this paper can be extended to cover a class of problems that are sufficiently general to include various biological systems in addition to the predatorprey equations studied in this paper. For example, the setup covers reaction-diffusion equations modelling plankton-fish dynamics [19], biological pattern formation [23], competing species [5], and generic oscillatory phenomena [16]. With this simplified setup and a known Lyapunov-type condition, a single polynomial growth condition leads to global existence of the classical solutions.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Spatiotemporal dynamics of two generic predator–prey models

Garvie

Trenchea

2010

We present the analysis of two reaction-diffusion systems modelling predator-prey interactions, where the predator displays the Holling type II functional response, and in the absence of predators, the prey growth is logistic. The local analysis is based on the application of qualitative theory for ordinary differential equations and dynamical systems, while the global well-posedness depends on invariant sets and differential inequalities. The key result is an L(∞)-stability estimate, which depends on a polynomial growth condition for the kinetics. The existence of an a priori L(p)-estimate, uniform in time, for all p ≥ 1, implies L(∞)-uniform bounds, given any nonnegative L(∞)-initial data. The applicability of the L(∞)-estimate to general reaction-diffusion systems is discussed, and how the continuous results can be mimicked in the discrete case, leading to stability estimates for a Galerkin finite-element method with piecewise linear continuous basis functions. In order to verify the biological wave phenomena of solutions, numerical results are presented in two-space dimensions, which have interesting ecological implications as they demonstrate that solutions can be 'trapped' in an invariant region of phase space.

“…The necessary and sufficient condition for Turing instability, which leads to the formation of spatial patterns, has been derived [9,16,17,20] and very interesting patterns are also obtained from the numerical simulation results [9,21,22].…”

Section: Introductionmentioning

confidence: 86%

Pattern Formation in Tri-Trophic Ratio-Dependent Food Chain Model

Melese¹,

Gakkhar²

2011

In this paper, a spatial tri-trophic food chain model with ratio-dependent Michaelis-Menten type functional response under homogeneous Neumann boundary conditions is studied. Conditions for Hopf and Turing bifurcation are derived. Sufficient conditions for the emergence of spatial patterns are obtained. The results of numerical simulations reveal the formation of labyrinth patterns and the coexistence of spotted and stripelike patterns.