Repulsive chemotaxis and predator evasion in predator–prey models with diffusion and prey-taxis (original) (raw)
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Predator–prey model with diffusion and indirect prey-taxis
Mathematical Models and Methods in Applied Sciences, 2016
We analyze predator–prey models in which the movement of predator searching for prey is the superposition of random dispersal and taxis directed toward the gradient of concentration of some chemical released by prey (e.g. pheromone), Model II, or released from damaged or injured prey due to predation (e.g. blood), Model I. The logistic O.D.E. describing the dynamics of prey population is coupled to a fully parabolic chemotaxis system describing the dispersion of chemoattractant and predator’s behavior. Global-in-time solutions are proved in any space dimension and stability of homogeneous steady states is shown by linearization for a range of parameters. For space dimension [Formula: see text] the basin of attraction of such a steady state is characterized by means of nonlinear analysis under some structural assumptions. In contrast to Model II, Model I possesses spatially inhomogeneous steady states at least in the case [Formula: see text].
Stability of a diffusive predator-prey model with Beddington-DeAngelis functional response
Our investigation deals with the stability and permanence of diffusive predator-prey model with modified Leslie-Gower and Holling-type II schemes and time-delay in two dimensions. Firstly, we prove that the solutions of this model are globally bounded and remain permanently in the positive quadrant. From this system, we obtain three trivial equilibrium points of which, under certain conditions, one is locally stable and the others unstable. We show that the unique point of positive internal equilibrium is locally stable. Then, by constructing an appropriate Lyapunov's functional, we establish the main result which is the global and asymptotic stability of this model. Finally, a numerical simulation is run to illustrate all these different theoretical results.
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.
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.
A reaction–diffusion system modeling predator–prey with prey-taxis
Nonlinear Analysis: Real World Applications, 2008
We are concerned with a system of nonlinear partial differential equations modeling the Lotka-Volterra interactions of predators and preys in the presence of prey-taxis and spatial diffusion. The spatial and temporal variations of the predator's velocity are determined by the prey gradient. We prove the existence of weak solutions by using Schauder fixed-point theorem and uniqueness via duality technique. The linearized stability around equilibrium is also studied. A finite volume scheme is build and numerical simulation show interesting phenomena of pattern formation.
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.
Chaos, Solitons & Fractals, 2015
In this paper, a diffusive predator-prey system, in which the prey species exhibits herd behavior and the predator species with quadratic mortality, has been studied. The stability of positive constant equilibrium, Hopf bifurcations, and diffusion-driven Turing instability are investigated under the Neumann boundary condition. The explicit condition for the occurrence of the diffusion-driven Turing instability is derived, which is determined by the relationship of the diffusion rates of two species. The formulas determining the direction and the stability of Hopf bifurcations depending on the parameters of the system are derived. Finally, numerical simulations are carried out to verify and extend the theoretical results and show the existence of spatially homogeneous periodic solutions and nonconstant steady states.
Predator–prey model with prey-taxis and diffusion
The objective of this paper is to investigate the effect of prey-taxis on predator-prey models with Paramecium aurelia as the prey and Didinium nasutum as its predator. The logistic Lotka-Volterra predator-prey models with prey-taxis are solved numerically with four different response functions, two initial conditions and one data set. Routh-Hurwitz's stability conditions are used to obtain the bifurcation values of the taxis coefficients for each model. We show that both response functions and initial conditions play important roles in the pattern formation. When the value of the taxis coefficient becomes considerably higher than the bifurcation value, chaotic dynamics develop. As diffusion in predator velocity is incorporated into the system, the system returns to a cyclic pattern.
Uniform boundedness for a predator-prey system with chemotaxis and dormancy of predators
Quarterly of Applied Mathematics
This paper deals with a nonlinear system of reaction-diffusion partial differential equations modelling the evolution of a prey-predator biological system with chemotaxis. The system is constituted by three coupled equations: a fully parabolic chemotaxis system describing the behavior of the active predators and preys and an ordinary equation, describing the dynamics of the dormant predators, coupled to it. Chemotaxis in this context affects the active predators so that they move towards the regions where the density of resting eggs (dormant predators) is higher. Under suitable assumptions on the initial data and the coefficients of the system, the global-in-time existence of classical solutions is proved in any space dimension. Besides, numerical simulations are performed to illustrate the behavior of the solutions of the system. The theoretical and numerical findings show that the model considered here can provide very interesting and complex dynamics.