Stability of the positive Equilibrium Solution for a Class of Quasilinear Diffusion Equations

1977 ◽  
pp. 85-92
Author(s):  
Piero de Mottoni
Keyword(s):  
Equilibrium Solution ◽  
Diffusion Equations ◽  
Positive Equilibrium ◽  
Quasilinear Diffusion

scholarly journals The study of global stability of a diffusive Michaelis-Menten and tanner predator-prey model

Filomat ◽  
2019 ◽  
Vol 33 (17) ◽  
pp. 5651-5659
Author(s):  
Demou Luo
Keyword(s):  
General Type ◽  
Equilibrium Solution ◽  
Positive Equilibrium ◽  
Heterogeneous Environment ◽  
Smooth Bounded Domain ◽  
Flux Boundary Condition ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Random Diffusion

In this paper, we consider a parabolic predator-prey model of Michaelis-Menten and Tanner functional response with random diffusion: ut = d1?u + au-bu2- ?uv/?u + v', vt = d2?v + rv- ?v2/u with d1,d2,a,b,r,?,?,? > 0 under the no-flux boundary condition in a smooth bounded domain ? ? Rn (n = 1,2,3). By applying a new method, we establish much improved global asymptotic stability of the unique positive equilibrium solution than works in literature. We also show the result can be extended to more general type of systems with heterogeneous environment.

Exploring Complex Dynamics of Spatial Predator–Prey System: Role of Predator Interference and Additional Food

2020 ◽  
Vol 30 (07) ◽  
pp. 2050102
Author(s):  
Vandana Tiwari ◽  
Jai Prakash Tripathi ◽  
Debaldev Jana ◽  
Satish Kumar Tiwari ◽  
Ranjit Kumar Upadhyay
Keyword(s):  
Asymptotic Stability ◽  
Equilibrium Solution ◽  
Model System ◽  
Positive Equilibrium ◽  
Additional Food ◽  
Predator Prey ◽  
Local Asymptotic Stability ◽  
Diffusive Model ◽  
Predator Interference

In this paper, an attempt has been made to understand the role of predator’s interference and additional food on the dynamics of a diffusive population model. We have studied a predator–prey interaction system with mutually interfering predator by considering additional food and Crowley–Martin functional response (CMFR) for both the reaction–diffusion model and associated spatially homogeneous system. The local stability analysis ensures that as the quantity of alternative food decreases, predator-free equilibrium stabilizes. Moreover, we have also obtained a condition providing a threshold value of additional food for the global asymptotic stability of coexisting steady state. The nonspatial model system changes stability via transcritical bifurcation and switches its stability through Hopf-bifurcation with respect to certain ranges of parameter determining the quantity of additional food. Conditions obtained for local asymptotic stability of interior equilibrium solution of temporal system determines the local asymptotic stability of associated diffusive model. The global stability of positive equilibrium solution of diffusive model system has been established by constructing a suitable Lyapunov function and using Green’s first identity. Using Harnack inequality and maximum modulus principle, we have established the nonexistence of nonconstant positive equilibrium solution of the diffusive model system. A chain of patterns on increasing the strength of additional food as spots[Formula: see text][Formula: see text][Formula: see text]stripes[Formula: see text][Formula: see text][Formula: see text]spots has been obtained. Various kind of spatial-patterns have also been demonstrated via numerical simulations and the roles of predator interference and additional food are established.

scholarly journals Global stability of solutions in a reaction-diffusion system of predator-prey model

Filomat ◽  
2018 ◽  
Vol 32 (13) ◽  
pp. 4665-4672
Author(s):  
Demou Luo ◽  
Hailin Liu
Keyword(s):  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
Equilibrium Solution ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model

In this article, we investigate the global asymptotic stability of a reaction-diffusion system of predator-prey model. By applying the comparison principle and iteration method, we prove the global asymptotic stability of the unique positive equilibrium solution of (1.1).

scholarly journals Global Behavior of a New Rational Nonlinear Higher-Order Difference Equation

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-4 ◽  
Author(s):  
Wen-Xiu Ma
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Asymptotic Stability ◽  
Nonnegative Integer ◽  
Equilibrium Solution ◽  
Higher Order ◽  
Positive Equilibrium ◽  
Global Behavior ◽  
Order Difference ◽  
Order Difference Equation

Let k be a nonnegative integer and c a real number greater than or equal to 1. We present qualitative global behavior of solutions to a rational nonlinear higher-order difference equation zn+1=(czn+zn-k+c-1znzn-k)/(znzn-k+c),  n≥0, with positive initial values z-k,z-k+1,⋯,z0, and show the global asymptotic stability of its positive equilibrium solution.

scholarly journals Stability analysis of resource-consumer dynamic models

2006 ◽  
Vol 47 (3) ◽  
pp. 413-438
Author(s):  
V. Sree Hari Rao ◽  
P. Raja Sekhara Rao
Keyword(s):  
Equilibrium Solution ◽  
Growth Response ◽  
Dynamic Models ◽  
Sufficient Conditions ◽  
Distributed Delay ◽  
Positive Equilibrium ◽  
Discrete Delay ◽  
Material Recycling ◽  
Model Equations ◽  
Consumer Model

AbstractA nutrient-consumer model involving a distributed delay in material recycling and a discrete delay in growth response has been analysed. Various easily verifiable sets of sufficient conditions for global asymptotic stability of the positive equilibrium solution of the model equations have been obtained and the length of the delay in each case has been estimated.

scholarly journals Global Behavior of an Arbitrary-Order Nonlinear Difference Equation with a Nonnegative Function

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 825
Author(s):  
Wen-Xiu Ma
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Asymptotic Stability ◽  
Equilibrium Solution ◽  
Arbitrary Order ◽  
Nonnegative Function ◽  
Nonlinear Difference Equation ◽  
Positive Equilibrium ◽  
Global Behavior ◽  
Asymptotic Behaviors

Let k , l be two integers with k ≥ 0 and l ≥ 2 , c a real number greater than or equal to 1, and f a multivariable function satisfying f ( w 1 , w 2 , w 3 , ⋯ , w l ) ≥ 0 when w 1 , w 2 ≥ 0 . We consider an arbitrary order nonlinear difference equation with the indicated function f: z n + 1 = c ( z n + z n − k ) + ( c − 1 ) z n z n − k + c f ( z n , z n − k , w 3 , ⋯ , w l ) z n z n − k + f ( z n , z n − k , w 3 , ⋯ , w l ) + c , n ≥ 0 , where initial values z − k , z − k + 1 , ⋯ , z 0 are positive and w i , i ≥ 3 , are arbitrary functions of z j , n − k ≤ j ≤ n . We classify its solutions into three types with different asymptotic behaviors, and verify the global asymptotic stability of its positive equilibrium solution z ¯ = c .

Sign in / Sign up

Export Citation Format

Share Document