scholarly journals Global bounded solution of the higher-dimensional forager–exploiter model with/without growth sources

2020 ◽  
Vol 30 (07) ◽  
pp. 1297-1323 ◽  
Author(s):  
Jianping Wang ◽  
Mingxin Wang
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Classical Solution ◽  
Neumann Boundary Condition ◽  
Bounded Solution ◽  
Neumann Boundary ◽  
Homogeneous Neumann Boundary Condition ◽  
Higher Dimensional ◽  
Degradation Rates ◽  
Growth Restrictions

This paper concerns with the global existence and boundedness of classical solution of the higher-dimensional forager–exploiter model with homogeneous Neumann boundary condition and nonnegative initial data. For cases where there are no forager and exploiter growth sources, it will be shown that if either the initial data and the production rate of nutrient are small or the taxis effects are small, then the classical solution exists globally and is bounded. For the case that only the forager has growth source, when [Formula: see text], it will be shown that if the taxis effect of exploiter is small then the classical solution exists globally and is bounded. For the case that both the forager and exploiter have growth restrictions, when [Formula: see text], we find a condition for the logistic degradation rates that ensures the global existence and boundedness of the classical solution.

An n-dimensional chemotaxis system with signal-dependent motility and generalized logistic source: global existence and asymptotic stabilization

2020 ◽  
pp. 1-21 ◽  
Author(s):  
Wenbin Lv ◽  
Qingyuan Wang
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Boundary Conditions ◽  
Classical Solution ◽  
Neumann Boundary ◽  
Logistic Source ◽  
Global Classical Solution ◽  
Higher Dimensional ◽  
Image Position ◽  
Chemotaxis System

Abstract This paper deals with the global existence for a class of Keller–Segel model with signal-dependent motility and general logistic term under homogeneous Neumann boundary conditions in a higher-dimensional smoothly bounded domain, which can be written as $$\eqalign{& u_t = \Delta (\gamma (v)u) + \rho u-\mu u^l,\quad x\in \Omega ,\;t > 0, \cr & v_t = \Delta v-v + u,\quad x\in \Omega ,\;t > 0.} $$ It is shown that whenever ρ ∈ ℝ, μ > 0 and $$l > \max \left\{ {\displaystyle{{n + 2} \over 2},2} \right\},$$ then the considered system possesses a global classical solution for all sufficiently smooth initial data. Furthermore, the solution converges to the equilibrium $$\left( {{\left( {\displaystyle{{\rho _ + } \over \mu }} \right)}^{1/(l-1)},{\left( {\displaystyle{{\rho _ + } \over \mu }} \right)}^{1/(l-1)}} \right)$$ as t → ∞ under some extra hypotheses, where ρ+ = max{ρ, 0}.

scholarly journals Global Existence of a Chemotactic Movement with Singular Sensitivity by Two Stimuli

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Heping Ma
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Boundary Conditions ◽  
Classical Solution ◽  
Bounded Domain ◽  
Smooth Boundary ◽  
Neumann Boundary ◽  
Global Classical Solution ◽  
Singular Sensitivity ◽  
Chemotaxis System

In this study, we deal with the chemotaxis system with singular sensitivity by two stimuli under homogeneous Neumann boundary conditions in a bounded domain with smooth boundary. Under appropriate regularity assumptions on the initial data, we show that the system possesses global classical solution. Our results generalize and improve previously known ones.

scholarly journals Stability in a Simple Food Chain System with Michaelis-Menten Functional Response and Nonlocal Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-14
Author(s):  
Wenzhen Gan ◽  
Canrong Tian ◽  
Qunying Zhang ◽  
Zhigui Lin
Keyword(s):  
Steady State ◽  
Functional Response ◽  
Neumann Boundary Condition ◽  
Lyapunov Functional ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Food Ingestion ◽  
Nonlocal Delay ◽  
Homogeneous Neumann Boundary Condition

This paper is concerned with the asymptotical behavior of solutions to the reaction-diffusion system under homogeneous Neumann boundary condition. By taking food ingestion and species' moving into account, the model is further coupled with Michaelis-Menten type functional response and nonlocal delay. Sufficient conditions are derived for the global stability of the positive steady state and the semitrivial steady state of the proposed problem by using the Lyapunov functional. Our results show that intraspecific competition benefits the coexistence of prey and predator. Furthermore, the introduction of Michaelis-Menten type functional response positively affects the coexistence of prey and predator, and the nonlocal delay is harmless for stabilities of all nonnegative steady states of the system. Numerical simulations are carried out to illustrate the main results.

Positive steady states of the Holling–Tanner prey–predator model with diffusion

2005 ◽  
Vol 135 (1) ◽  
pp. 149-164 ◽  
Author(s):  
Rui Peng ◽  
Mingxin Wang
Keyword(s):  
Boundary Condition ◽  
Neumann Boundary Condition ◽  
Neumann Boundary ◽  
Steady States ◽  
Homogeneous Neumann Boundary Condition ◽  
Positive Steady States ◽  
Predator Model

This paper is concerned with the Holling–Tanner prey–predator model with diffusion subject to the homogeneous Neumann boundary condition. We obtain the existence and non-existence of positive non-constant steady states.

Asymptotics of solutions to a convection—diffusion equation on the half-line

2000 ◽  
Vol 130 (4) ◽  
pp. 837-853 ◽  
Author(s):  
G. Karch
Keyword(s):  
Boundary Condition ◽  
Asymptotic Expansion ◽  
Neumann Boundary Condition ◽  
Neumann Boundary ◽  
Half Line ◽  
Asymptotics Of Solutions ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Homogeneous Neumann Boundary Condition ◽  
Higher Order Terms

We study the behaviour, as t → ∞, of solutions to the convectiondiffusion equation on the half-line with the homogeneous Neumann boundary condition and with bounded initial data. The higher-order terms of the asymptotic expansion in Lp (R+) of solutions are derived.

On the set of eigenvalues of some PDEs with homogeneous Neumann boundary condition

2015 ◽  
Vol 116 ◽  
pp. 19-25 ◽  
Author(s):  
Maria Fărcăşeanu ◽  
Mihai Mihăilescu ◽  
Denisa Stancu-Dumitru
Keyword(s):  
Boundary Condition ◽  
Neumann Boundary Condition ◽  
Neumann Boundary ◽  
Homogeneous Neumann Boundary Condition

CONSTRAINED SCHRÖDINGER–POISSON SYSTEM WITH NON-CONSTANT INTERACTION

2013 ◽  
Vol 15 (01) ◽  
pp. 1250052 ◽  
Author(s):  
LORENZO PISANI ◽  
GAETANO SICILIANO
Keyword(s):  
Neumann Boundary Condition ◽  
Standing Waves ◽  
Neumann Boundary ◽  
The Other ◽  
Necessary Condition ◽  
Infinitely Many Solutions ◽  
Maxwell System ◽  
Poisson System ◽  
Homogeneous Neumann Boundary Condition ◽  
Variational Framework

In this paper we are dealing with a Schrödinger–Maxwell system in a bounded domain of R3; the unknowns are the charged standing waves ψ = e-iωtu(x) in equilibrium with a purely electrostatic potential ϕ. The system is not autonomous, in the sense that the coupling depends on a function q = q(x). The non-homogeneous Neumann boundary condition on ϕ prescribes the flux of the electric field 𝔉 and gives rise to a necessary condition. On the other hand we consider the usual normalizing condition in L2 for u. Under mild assumptions involving 𝔉 and the function q = q(x), we prove that this problem has a variational framework: its solutions can be characterized as constrained critical points. Then, by means of the Ljusternick–Schnirelmann theory, we get the existence of infinitely many solutions.

EFFECT OF CROSS DIFFUSION IN A COMPETITION MODEL WITH STAGE STRUCTURE

2012 ◽  
Vol 05 (06) ◽  
pp. 1250052 ◽  
Author(s):  
LINA ZHANG ◽  
SHENGMAO FU ◽  
PING HU
Keyword(s):  
Boundary Condition ◽  
Neumann Boundary Condition ◽  
Neumann Boundary ◽  
Stage Structure ◽  
Competition Model ◽  
Positive Equilibrium ◽  
Cross Diffusion ◽  
Homogeneous Neumann Boundary Condition

The purpose of this paper is to study the effect of cross diffusion in a competition model with stage structure, under homogeneous Neumann boundary condition. It will be shown that cross diffusion cannot only destabilize a uniform positive equilibrium, it can also help diffusion to induce instability of the uniform positive equilibrium. Moreover, stationary patterns can arise from the effect of cross diffusion.

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