Spatiotemporal dynamics of two generic predator–prey models (original) (raw)
Related papers
Analysis of two generic spatially extended predator-prey models
2006
We present the analysis of two reaction-diusion systems modeling predator-prey interactions with the Holling Type II functional response and logistic growth of the prey. Initially we undertake the local analysis of the systems, deriving conditions on the parameters that guarantee a stable limit cycle in the reaction kinetics, and construct arbitrary large invariant regions in the equal diusion coecien t case. We then provide an a priori estimate that leads to the global well-posedness of the classical (nonnegative) solutions, given any nonnegative L 1 - initial data. In order to verify the biological wave phenomena of solutions and the theoretical results, numerical experiments are undertaken in two space dimensions using a Galerkin nite element method with piecewise linear continuous basis functions.
Internal stabilizability for a reaction–diffusion problem modeling a predator–prey system
Nonlinear Analysis: Theory, Methods & Applications, 2005
In this work we consider a 2× 2 system of semilinear partial differential equations of parabolic-type describing interactions between a prey population and a predator population, featuring a Holling-type II functional response to predation. We address the question of stabilizing the predator population to zero, upon using a suitable internal control supported on a small subdomain of the whole spatial domain , and acting on predators. We give necessary and sufficient conditions for this stabilizability result to hold.
arXiv: Analysis of PDEs, 2017
We consider in this paper a microscopic model (that is, a system of three reaction-diffusion equations) incorporating the dynamics of handling and searching predators, and show that its solutions converge when a small parameter tends to 0 towards the solutions of a reaction-cross diffusion system of predator-prey type involving a Holling-type II or Beddington-DeAngelis functional response. We also provide a study of the Turing instability domain of the obtained equations and (in the case of the Beddington-DeAngelis functional response) compare it to the same instability domain when the cross diffusion is replaced by a standard diffusion.
Spatiotemporal dynamics of a diffusive predator-prey model with generalist predator
Discrete & Continuous Dynamical Systems - S, 2018
In this paper, we study the spatiotemporal dynamics of a diffusive predator-prey model with generalist predator subject to homogeneous Neumann boundary condition. Some basic dynamics including the dissipation, persistence and non-persistence(i.e., one species goes extinct), the local and global stability of non-negative constant steady states of the model are investigated. The conditions of Turing instability due to diffusion at positive constant steady states are presented. A critical value ρ of the ratio d 2 d 1 of diffusions of predator to prey is obtained, such that if d 2 d 1 > ρ, then along with other suitable conditions Turing bifurcation will emerge at a positive steady state, in particular so it is with the large diffusion rate of predator or the small diffusion rate of prey; while if d 2 d 1 < ρ, both the reaction-diffusion system and its corresponding ODE system are stable at the positive steady state. In addition, we provide some results on the existence and non-existence of positive non-constant steady states. These existence results indicate that the occurrence of Turing bifurcation, along with other suitable conditions, implies the existence of non-constant positive steady states bifurcating from the constant solution. At last, by numerical simulations, we demonstrate Turing pattern formation on the effect of the varied diffusive ratio d 2 d 1. As d 2 d 1 increases, Turing patterns change from spots pattern, stripes pattern into spots-stripes pattern. It indicates that the pattern formation of the model is rich and complex.
Numerische Mathematik, 2007
We study the numerical approximation of the solutions of a class of nonlinear reaction-diffusion systems modelling predator-prey interactions, where the local growth of prey is logistic and the predator displays the Holling type II functional response. The fully discrete scheme results from a finite element discretisation in space (with lumped mass) and a semi-implicit discretisation in time. We establish a priori estimates and error bounds for the semi discrete and fully discrete finite element approximations. Numerical results illustrating the theoretical results and spatiotemporal phenomena are presented in one and two space dimensions. The class of problems studied in this paper are real experimental systems where the parameters are associated with real kinetics, expressed in nondimensional form. The theoretical techniques were adapted from a previous study of an idealised reaction-diffusion system (Garvie and
2020
A thorough analysis is performed in a qualitative model inspired by three relevant complex dynamical ingredients: (a) a strong Allee effect; (b) ratio-dependent functional responses; and (c) transport attributes given by a diffusion process. As is well-known in the specialized literature, these aspects capture adverse survival conditions for the prey, predation search features and non-homogeneous spatial dynamical distribution of both populations. Our goal was to undertake a methodical investigation of traveling waves that may be found when the kinetic terms —based on qualitatively equivalent mathematical interactions of the predator and prey dynamics— are stated in their most simple, minimal form. We provide rigorous results coming from a standard local analysis, numerical bifurcation analysis, and relevant computations of invariant manifolds to exhibit homoclinic and heteroclinic connections and periodic orbits in the associated dynamical system in R4. In so doing, we present and ...
Diffusive instability in a prey-predator system with time-dependent diffusivity
International Journal of Mathematics and Mathematical Sciences, 2003
An ecological model for prey-predator planktonic species has been considered, in which the growth of prey has been assumed to follow a Holling type II function. The model consists of two reaction-diffusion equations and we extend it to time-varying diffusivity for plankton population. A comparative study of local stability in case of constant diffusivity and time varying diffusivity has been performed. It has been found that the system would be more stable with time varying diffusivity depending upon the values of system parameter.
Mathematical study of multispecies dynamics modeling predator–prey spatial interactions
Journal of Numerical Mathematics, 2017
In this work, we present analysis of a scaled time-dependent reaction–diffusion system modeling three competitive species dynamics that is of Lotka–Volterra type for coexistence, permanence and stability. The linear analysis is based on the application of qualitative theory of ordinary differential equations and dynamical systems. We consider two notable spatial discretization methods in conjunction with an adaptive time stepping method to verify the biological wave phenomena of the solutions and present the numerical results in one dimensional space. Adequate numerical resulting are provided in one and two dimensions to justify theoretical investigations. In addition, efficiency of the proposed numerical schemes are justified.
Spatiotemporal dynamics of reaction–diffusion models of interacting populations
Applied Mathematical Modelling, 2014
The present investigation deals with the necessary conditions for Turing instability with zero-flux boundary conditions that arise in a ratio-dependent predator-prey model involving the influence of logistic population growth in prey and intra-specific competition among predators described by a system of non-linear partial differential equations. The prime objective is to investigate the parametric space for which Turing spatial structure takes place and to perform extensive numerical simulation from both the mathematical and the biological points of view in order to examine the role of diffusion coefficients in Turing instability. Various spatiotemporal distributions of interacting species through Turing instability in two dimensional spatial domain are portrayed and analyzed at length in order to substantiate the applicability of the present model.