Stabilization of a Predator-Prey System with Nonlocal Terms (original) (raw)

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.

A predator-prey reaction-diffusion system with nonlocal effects

Journal of Mathematical Biology, 1996

Weconsider apredator-prey systemin theform of a coupled system of reaction-diffusion equations with an integral term representing a weighted average of the values of the prey density function, both in past time and space. In a limiting case the system reduces to the Lotka Volterra diffusion system with logistic growth of the prey. We investigate the linear stability of the coexistence steady state and bifurcations occurring from it, and expressions for some of the bifurcating solutions are constructed. None of these bifurcations can occur in the degenerate case when the nonlocal term is in fact local.

Nonlocal generalized models of predator-prey systems

Discrete and Continuous Dynamical Systems - Series B, 2012

The method of generalized modeling has been applied successfully in many different contexts, particularly in ecology and systems biology. It can be used to analyze the stability and bifurcations of steady-state solutions. Although many dynamical systems in mathematical biology exhibit steady-state behaviour one also wants to understand nonlocal dynamics beyond equilibrium points. In this paper we analyze predatorprey dynamical systems and extend the method of generalized models to periodic solutions. First, we adapt the equilibrium generalized modeling approach and compute the unique Floquet multiplier of the periodic solution which depends upon so-called generalized elasticity and scale functions. We prove that these functions also have to satisfy a flow on parameter (or moduli) space. Then we use Fourier analysis to provide computable conditions for stability and the moduli space flow. The final stability analysis reduces to two discrete convolutions which can be interpreted to understand when the predator-prey system is stable and what factors enhance or prohibit stable oscillatory behaviour. Finally, we provide a sampling algorithm for parameter space based on nonlinear optimization and the Fast Fourier Transform which enables us to gain a statistical understanding of the stability properties of periodic predator-prey dynamics.

Zero-Stabilization for Some Diffusive Models with State Constraints

Mathematical Modelling of Natural Phenomena, 2014

We discuss the zero-controllability and the zero-stabilizability for the nonnegative solutions to some Fisher-like models with nonlocal terms describing the dynamics of biological populations with diffusion, logistic term and migration. A necessary and sufficient condition for the nonnegative zero-stabilizabiity for a linear integro-partial differential equation is obtained in terms of the sign of the principal eigenvalue to a certain non-selfadjoint operator. For a related semilinear problem a necessary condition and a sufficient condition for the local nonnegative zero-stabilizability are also derived in terms of the magnitude of the above mentioned principal eigenvalue. The rate of stabilization corresponding to a simple feedback stabilizing control is dictated by the principal eigenvalue. A large principal eigenvalue leads to a fast stabilization to zero. A necessary condition and a sufficient condition for the stabilization to zero of the predator population in a predator-prey system is also investigated. Finally, a method to approximate the above mentioned principal eigenvalues is indicated.

Stability of a one-predator two-prey system governed by nonautonomous differential equations

arXiv: Dynamical Systems, 2015

A non-periodic version of the one-predator two-prey system model presented in [L.T.H. Nguyen, Q.H. Ta, T.V. T\d{a}, Existence and stability of periodic solutions of a Lotka-Volterra system, SICE International Symposium on Control Systems, Tokyo, Japan, 712-4 (2015) 1-6] is considered. First, we prove existence of unique positive solutions to the model. Second, we show existence of an invariant set, which suggests the survival of all species in the system. On the other hand, we show that when the densities of two prey species are quite small, the predator falls into decay. Third, we explore global asymptotic stability of the system by using the Lyapunov function method. Finally, some numerical examples are given to illustrate our results.

Global well-posedness and stability analysis of prey-predator model with indirect prey-taxis

Journal of Differential Equations, 2019

This paper deals with a prey-predator model with indirect prey-taxis, which means chemical of prey causes the directional movement of the predator. We prove the global existence and uniform boundedness of solutions to the model for general functional responses in any spatial dimensions. Moreover, through linear stability analysis, it turns out that prey-taxis is an essential factor in generating pattern formations. This result differs in that the destabilizing effect of taxis does not occur in the direct prey-taxis case. In addition, we show the global stability of the semi-trivial steady state and coexistence steady state for some specific functional responses. We give numerical examples to support the analytic results.

A nonlocal swarm model for predators–prey interactions

Mathematical Models and Methods in Applied Sciences, 2016

We consider a two-species system of nonlocal interaction PDEs modeling the swarming dynamics of predators and prey, in which all agents interact through attractive/repulsive forces of gradient type. In order to model the predator–prey interaction, we prescribed proportional potentials (with opposite signs) for the cross-interaction part. The model has a particle-based discrete (ODE) version and a continuum PDE version. We investigate the structure of particle stationary solution and their stability in the ODE system in a systematic form, and then consider simple examples. We then prove that the stable particle steady states are locally stable for the fully nonlinear continuum model, provided a slight reinforcement of the particle condition is required. The latter result holds in one space dimension. We complement all the particle examples with simple numerical simulations, and we provide some two-dimensional examples to highlight the complexity in the large time behavior of the system.

Local and global stability analysis of a two prey one predator model with help

Communications in Nonlinear Science and Numerical Simulation, 2014

In this paper we propose and study a three dimensional continuous time dynamical system modelling a three team consists of two preys and one predator with the assumption that during predation the members of both teams of preys help each other and the rate of predation of both teams are different. In this work we establish the local asymptotic stability of various equilibrium points to understand the dynamics of the model system. Different conditions for the coexistence of equilibrium solutions are discussed. Persistence, permanence of the system and global stability of the positive interior equilibrium solution are discussed by constructing suitable Lyapunov functional. At the end, numerical simulations are performed to substantiate our analytical findings.

Stabilization for a Periodic Predator-Prey System

Abstract and Applied Analysis, 2007

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.