scholarly journals Stability of synchronized steady state solution of diffusive Lotka–Volterra predator–prey model

2020 ◽  
Vol 105 ◽  
pp. 106331
Author(s):  
Yongyan Huang ◽  
Fuyi Li ◽  
Junping Shi
Keyword(s):  
Steady State ◽  
Steady State Solution ◽  
State Solution ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model

scholarly journals Spatiotemporal Complexity of a Leslie-Gower Predator-Prey Model with the Weak Allee Effect

2013 ◽  
Vol 2013 ◽  
pp. 1-16 ◽  
Author(s):  
Yongli Cai ◽  
Caidi Zhao ◽  
Weiming Wang
Keyword(s):  
Steady State ◽  
Allee Effect ◽  
Steady State Solution ◽  
Turing Instability ◽  
Dimensional System ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Nonexistence And Existence ◽  
The Impact

We investigate a diffusive Leslie-Gower predator-prey model with the additive Allee effect on prey subject to the zero-flux boundary conditions. Some results of solutions to this model and its corresponding steady-state problem are shown. More precisely, we give the stability of the positive constant steady-state solution, the refineda prioriestimates of positive solution, and the nonexistence and existence of the positive nonconstant solutions. We carry out the analytical study for two-dimensional system in detail and find out the certain conditions for Turing instability. Furthermore, we perform numerical simulations and show that the model exhibits a transition from stripe-spot mixtures growth to isolated spots and also to stripes. These results show that the impact of the Allee effect essentially increases the model spatiotemporal complexity.

Spatio-Temporal Patterns of Predator–Prey Model with Allee Effect and Constant Stocking Rate for Predator

2021 ◽  
Vol 31 (16) ◽  
Author(s):  
Xuan Tian ◽  
Shangjiang Guo
Keyword(s):  
Steady State ◽  
Allee Effect ◽  
Sufficient Conditions ◽  
Steady State Solution ◽  
Turing Pattern ◽  
Stocking Rate ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Spatio Temporal

A diffusive predator–prey model with Allee effect and constant stocking rate for predator is investigated and it is shown that Allee effect is the decisive factor driving the formation of Turing pattern. Furthermore, it is observed that Turing pattern appears only when the diffusion rate of the prey is faster than that of the predator, which is just opposite to the condition of Turing pattern in the classical predator–prey system. Some sufficient conditions are obtained to ensure the asymptotical stability of a spatially homogeneous steady-state solution. The existence and nonexistence of positive nonconstant steady-state solutions are investigated to understand the mechanisms of generating spatiotemporal patterns. Furthermore, Hopf and steady-state bifurcations are analyzed in detail by using Lyapunov–Schmidt reduction.

Spatial Patterns of a Predator–Prey Model with Beddington–DeAngelis Functional Response

2019 ◽  
Vol 29 (11) ◽  
pp. 1950145 ◽  
Author(s):  
Yu-Xia Wang ◽  
Wan-Tong Li
Keyword(s):  
Steady State ◽  
Functional Response ◽  
Spatial Patterns ◽  
Bifurcation Theory ◽  
Steady State Solution ◽  
State Solution ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Positive Steady State ◽  
Prey System

This paper is concerned with the spatial patterns of a predator–prey system with Beddington–DeAngelis functional response, in which the parameter [Formula: see text] measuring the mutual interference between predators can play an essential role. By using the bifurcation theory and implicit function theorem we first consider the positive steady state solution bifurcating from the semitrivial steady state solution set of the system and prove that the positive steady state solution is constant. Then we show that nonconstant positive steady state solution may bifurcate from the constant positive steady state solution when [Formula: see text] is neither small nor large. Finally, we show that spatially nonhomogeneous periodic orbits may also bifurcate from the constant positive steady state solution as [Formula: see text] is not large.

scholarly journals Spatially Nonhomogeneous Periodic Solutions in a Delayed Predator-Prey Model with Diffusion Effects

2012 ◽  
Vol 2012 ◽  
pp. 1-21
Author(s):  
Jia-Fang Zhang
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Positive Constant ◽  
Functional Differential Equations ◽  
Neumann Boundary ◽  
Steady State Solution ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Predator Prey Model

This paper is concerned with a delayed predator-prey diffusion model with Neumann boundary conditions. We study the asymptotic stability of the positive constant steady state and the conditions for the existence of Hopf bifurcation. In particular, we show that large diffusivity has no effect on the Hopf bifurcation, while small diffusivity can lead to the fact that spatially nonhomogeneous periodic solutions bifurcate from the positive constant steady-state solution when the system parameters are all spatially homogeneous. Meanwhile, we study the properties of the spatially nonhomogeneous periodic solutions applying normal form theory of partial functional differential equations (PFDEs).

The fundamentals of predator–prey interactions

2019 ◽  
pp. 81-95
Author(s):  
Gary G. Mittelbach ◽  
Brian J. McGill
Keyword(s):  
Steady State ◽  
Species Interactions ◽  
Predator Species ◽  
Trade Off ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Fundamental Building Block ◽  
Prey Model ◽  
Predator Prey Models ◽  
Consumer Resource

This chapter introduces the concept of the consumer-resource link, the idea that each species in a community consumes resources and is itself consumed by other species. The consumer–resource link is the fundamental building block from which more-complex food chains and food webs are constructed. The chapter continues by exploring what is arguably the simplest consumer–resource interaction—one predator species feeding on one species of prey. Important topics discussed in the context of predator–prey interactions are the predator’s functional response, the Lotka–Volterra predator–prey model, the Rosenzweig–MacArthur predator–prey model, and the suppression-stability trade-off. Isocline analysis is introduced as a method for visualizing the outcome of species interactions at steady-state or equilibrium. Herbivory and parasitism are briefly discussed within the context of general predator–prey models.

Bifurcations and Pattern Formation in a Predator–Prey Model

2018 ◽  
Vol 28 (11) ◽  
pp. 1850140 ◽  
Author(s):  
Yongli Cai ◽  
Zhanji Gui ◽  
Xuebing Zhang ◽  
Hongbo Shi ◽  
Weiming Wang
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Pattern Formation ◽  
Global Bifurcation ◽  
Periodic Pattern ◽  
Spatiotemporal Pattern ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Steady State Bifurcation

In this paper, we investigate the spatiotemporal dynamics of a Leslie–Gower predator–prey model incorporating a prey refuge subject to the Neumann boundary conditions. We mainly consider Hopf bifurcation and steady-state bifurcation which bifurcate from the constant positive steady-state of the model. In the case of Hopf bifurcation, by the center manifold theory and the normal form method, we establish the bifurcation direction and stability of bifurcating periodic solutions; in the case of steady-state bifurcation, by the local and global bifurcation theories, we prove the existence of the steady-state bifurcation, and find that there are two typical bifurcations, Turing bifurcation and Turing–Hopf bifurcation. Via numerical simulations, we find that the model exhibits not only stationary Turing pattern induced by diffusion which is dependent on space and independent of time, but also temporal periodic pattern induced by Hopf bifurcation which is dependent on time and independent of space, and spatiotemporal pattern induced by Turing–Hopf bifurcation which is dependent on both time and space. These results may enrich the pattern formation in the predator–prey model.

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