Zero-flux Boundary Conditions Research Articles

Boundedness of weak solutions to a 3D chemotaxis-Stokes system with slow [formula omitted]Laplacian diffusion and rotation

In this paper, we consider the following chemotaxis-Stokes system with general sensitivity and rotation nt+u⋅∇n=∇⋅(|∇n|p−2∇n)−∇⋅(nS(x,n,c)∇c),x∈Ω,t>0,ct+u⋅∇c=Δc−nc,x∈Ω,t>0,ut+∇P=Δu+n∇ϕ,x∈Ω,t>0,∇⋅u=0,x∈Ω,t>0in a smooth bounded domain Ω∈R3 with zero-flux boundary and no-slip boundary condition. We prove the boundedness of the weak solutions to the initial–boundary value problem of the 3D chemotaxis-Stokes system with p−Laplacian diffusion if 43p+α>72−63 and p>2, which improves the results of papers [Chen et al., Nonlinear Anal. Real World Appl., 76 (2024) 103996; Zhuang et al., Nonlinear Anal. Real World Appl., 56 (2020) 103163 and Tao et al., J. Differ. Equ., 268(11) (2020) 6879–6919] and extends the result of the paper [Jin, J. Differ. Equ. 287 (2021) 148–184] to the chemotaxis system with general sensitivity and rotation.

Modeling Study of Factors Determining Efficacy of Biological Control of Adventive Weeds

We model the spatiotemporal dynamics of a community consisting of competing weed and cultivated plant species and a population of specialized phytophagous insects used as the weed biocontrol agent. The model is formulated as a PDE system of taxis–diffusion–reaction type and computer-implemented for one-dimensional and two-dimensional cases of spatial habitat for the Neumann zero-flux boundary condition. In order to discretize the original continuous system, we applied the method of lines. The obtained system of ODEs is integrated using the Runge–Kutta method with a variable time step and control of the integration accuracy. The numerical simulations provide insights into the mechanism of formation of solitary population waves (SPWs) of the phytophage, revealing the factors that determine the efficacy of combined application of the phytophagous insect (classical biological method) and cultivated plant (phytocenotic method) to suppress weed foci. In particular, the presented results illustrate the stabilizing action of cultivated plants, which fix the SPW effect by occupying the free area behind the wave front so that the weed remains suppressed in the absence of a phytophage.

Lower-Dimensional Model of the Flow and Transport Processes in Thin Domains by Numerical Averaging Technique

In this work, we present a lower-dimensional model for flow and transport problems in thin domains with rough walls. The full-order model is given for a fully resolved geometry, wherein we consider Stokes flow and a time-dependent diffusion–convection equation with inlet and outlet boundary conditions and zero-flux boundary conditions for both the flow and transport problems on domain walls. Generally, discretizations of a full-order model by classical numerical schemes result in very large discrete problems, which are computationally expensive given that sufficiently fine grids are needed for the approximation. To construct a computationally efficient numerical method, we propose a model-order-reduction numerical technique to reduce the full-order model to a lower-dimensional model. The construction of the lower-dimensional model for the flow and the transport problem is based on the finite volume method and the concept of numerical averaging. Numerical results are presented for three test geometries with varying roughness of walls and thickness of the two-dimensional domain to show the accuracy and applicability of the proposed scheme. In our numerical simulations, we use solutions obtained from the finite element method on a fine grid that can resolve the complex geometry at the grid level as the reference solution to the problem.

Spatial pattern formation driven by the cross-diffusion in a predator–prey model with Holling type functional response

In this paper, we consider a cross-diffusion predator–prey system with Holling type functional response. We study the local stability, Turing instability, spatial pattern formation, Hopf and Turing–Hopf bifurcation of the equilibrium. Numerical simulation with zero-flux boundary conditions discloses that the system under consideration experiences the occurrence of cross-diffusion-driven instability. The dynamical system in Turing space emerges spots, stripe-spot mixtures and labyrinthine patterns, which reveals that the interaction of both self- and cross-diffusions play a significant role on the pattern formation of the present system in a way to enrich the pattern. We obtain the normal form of the Turing–Hopf bifurcation and observe that the system has stably spatially homogeneous periodic solutions, stable constant and nonconstant steady-state solutions, which indicates that the intrinsic growth rate coefficient and the environmental carrying capacity coefficient are two important factors for predator–prey system, and affect the stability of predator–prey system.

Global bounded solution of a 3D chemotaxis-Stokes system with nonlinear doubly degenerate diffusion

The paper considers the following chemotaxis-Stokes system with nonlinear doubly degenerate diffusion{nt+u⋅∇n=∇⋅(|∇nm|p−2∇nm)−χ∇⋅(n∇c),x∈Ω,t>0,ct+u⋅∇c=Δc−cn,x∈Ω,t>0,ut+∇P=Δu+n∇Φ,x∈Ω,t>0,∇⋅u=0,x∈Ω,t>0 in a bounded domain Ω⊂R3 with zero-flux boundary conditions and no-slip boundary condition. In this paper, we proved that global bounded weak solutions exist whenever m>1 and p≥2. It removes the restrict 8mp−8m+3p>15 and improves the result of paper Lin (2022) [15].

Entropy Formulation for Triply Nonlinear Degenerate Elliptic-Parabolic-Hyperbolic Equation with Zero-Flux Boundary Condition

In this note, we investigated existence and uniqueness of entropy solution for triply nonlinear degenerate parabolic problem with zero-flux boundary condition. Accordingly to the case of doubly nonlinear degenerate parabolic hyperbolic equation, we propose a generalization of entropy formulation and prove existence and uniqueness result without any structure condition.

Qualitative study of cross-diffusion and pattern formation in Leslie–Gower predator–prey model with fear and Allee effects

The present study is focused on interacting prey–predator reaction–diffusion model with modified Leslie–Gower type functional response incorporating Allee and fear effects. We introduce self as well as cross-diffusion in the model. The equilibrium points and their stability along with saddle–node and Hopf bifurcations around steady states are investigated for non-spatial system. The conditions for Turing instability and the critical line of Hopf and Turing bifurcation in a spatial domain with zero-flux boundary conditions are determined. The parametric space for different regions is depicted and numerical simulation in Turing space is carried out. It is observed that cross diffusion has significant role in forming patterns such as spots, stripes, mixture of spots and stripes. The issue of spatiotemporal pattern controllability is also examined. The availability of cross diffusion results a paradox to the prey density with increasing level of Allee.

Finite-time blow-up and boundedness in a 2D Keller–Segel system with rotation

This paper deals with the initial-boundary value problem for a Keller–Segel system with rotation with zero-flux boundary condition for u and zero-Neumann boundary condition for v, where Ω is a bounded domain in with smooth boundary , is a rotation matrix with . We show that:Let be a general smooth bounded domain.If , then there exists nonnegative initial data u 0 satisfying , such that the corresponding nonradial solution of system (*) blows up in finite time and the blow-up point lies in Ω.If and contains a line segment, then there exists nonnegative initial data u 0 satisfying , such that the corresponding nonradial solution of system (*) blows up in finite time and the blow-up point lies on the line segment of .Let be a disc in with radius R > 0 centered at origin. Although there is a rotation effect in system (*), solutions still preserve radial symmetry of initial data. If nonnegative radially symmetric initial data u 0 satisfies , then the corresponding radial solution of system (*) exists globally in time and is globally bounded.

Analytical modeling of temperature distribution in laser powder bed fusion with different scan strategies

This study proposes an analytical model to predict the transient temperature distribution and melt pool size in laser powder bed fusion (LPBF) process with different scan strategies. The scan strategies are divided into discontinuous scan path and continuous scan path. The moving Gaussian surface heat source solution and the instantaneous Gaussian surface heat source transient solution are applied for these two different scan strategies. The residual temperature linear superposition is adopted to consider the multi-track effect on the temperature distribution. The virtual negative power heat source method is utilized to consider the heat diffusion for the tracks that has been scanned of the build part. The image heat source method is used to consider the zero-flux boundary condition at the lateral surface of the build part. The proposed model is validated by the experimental data of the melt pool in single-track LPBF process. The analytical models of these two different scan strategies are verified by the comparison of the melt pool size. Based on the validated analytical model, it is extended for the multiple scan strategies combination processes. The temperature and melt pool predicted by the analytical model show good consistency with the FEM results. The analytical model method is much more efficient than FEM, which is about 120 times faster than FEM for the calculation of round-corner part. Due to the satisfied efficiency and accuracy in the prediction of melt pool size and the thermal history, the proposed analytical model has great potential application in optimization of the scan path planning and avoidance of the undesired residual temperature accumulation in the part.

Turing instability and Hopf bifurcation of a spatially discretized diffusive Brusselator model with zero-flux boundary conditions

Turing instability and Hopf bifurcation of a spatially discretized diffusive Brusselator model with zero-flux boundary conditions

Global classical solutions to a higher-dimensional doubly haptotactic cross-diffusion system modeling oncolytic virotherapy

This paper deals with the oncolytic virotherapy model{ut=DuΔu−ξu∇⋅(u∇v)+μuu(1−u)−ρuuz,x∈Ω,t>0,vt=−(αuu+αww)v+μvv(1−v),x∈Ω,t>0,wt=DwΔw−ξw∇⋅(w∇v)−δww+ρwuz,x∈Ω,t>0,zt=DzΔz−δzz−ρzuz+βw,x∈Ω,t>0, which was initially proposed by Alzahrani-Eftimie-Trucu (2019) [1] to model the process of oncolytic viral therapy, which is a therapeutic approach for cancer treatment. Here, Ω⊆RN(N≥1) is a bounded smooth domain with zero-flux boundary conditions, where Du,Dw,Dz,δw,δz,μu,ρu are positive parameters and ξu,αu,ξw,ρw,ρz,μv are nonnegative constants. The main results assert the global boundedness of solutions to an associated spatially N-dimensional initial-boundary value problems under suitably assumptions on the system parameters. We develop (refine) some integral estimate techniques and thereby improve previous results in [18,23,14]. To the best of our knowledge, there are the first results on boundedness of the system in higher dimension (N≥2).

Instability of all regular stationary solutions to reaction-diffusion-ODE systems

A general system of several ordinary differential equations coupled with a reaction-diffusion equation in a bounded domain with zero-flux boundary condition is studied in the context of pattern formation. These initial-boundary value problems may have regular (i.e. sufficiently smooth) stationary solutions. This class of close-to-equilibrium patterns includes stationary solutions that emerge due to the Turing instability of a spatially constant stationary solution. The main result of this work is instability of all regular patterns. It suggests that stable stationary solutions arising in models with non-diffusive components must be far-from-equilibrium exhibiting singularities. Such discontinuous stationary solutions have been considered in our parallel work (Cygan et al., 2021 [4]).

Boundary Conditions Cause Different Generic Bifurcation Structures in Turing Systems

Turing’s theory of morphogenesis is a generic mechanism to produce spatial patterning from near homogeneity. Although widely studied, we are still able to generate new results by returning to common dogmas. One such widely reported belief is that the Turing bifurcation occurs through a pitchfork bifurcation, which is true under zero-flux boundary conditions. However, under fixed boundary conditions, the Turing bifurcation becomes generically transcritical. We derive these algebraic results through weakly nonlinear analysis and apply them to the Schnakenberg kinetics. We observe that the combination of kinetics and boundary conditions produce their own uncommon boundary complexities that we explore numerically. Overall, this work demonstrates that it is not enough to only consider parameter perturbations in a sensitivity analysis of a specific application. Variations in boundary conditions should also be considered.

Predator-taxis creates spatial pattern of a predator-prey model

In this paper, we investigate the spatial pattern formation to a predator-prey model with the predator-taxis under the homogeneous zero-flux boundary conditions. Firstly, we employ the predator-taxis coefficient as the potential critical value of the Turing bifurcation to perform its role in forming the spatial pattern. One finds that there are multiple thresholds of the Turing bifurcation. Hereafter, we focus on the direction of the Turing bifurcation. To this end, the technique of the weakly nonlinear analysis is employed to deduce the amplitude equation near the threshold of the Turing bifurcation in one-dimensional space. It is found that the predator-taxis can create the subcritical or the supercritical Turing bifurcation. Finally, numerical experiments check the theoretical analysis well and obtain the spatial patterns with the different predator-taxis coefficients.

Large-data global existence in a higher-dimensional doubly degenerate nutrient system

This paper deals with a doubly degenerate nutrient taxis system given by(⋆){ut=∇⋅(uv∇u)−χ∇⋅(uαv∇v)+ℓuv,x∈Ω,t>0,vt=Δv−uv,x∈Ω,t>0, in a smoothly bounded convex domain Ω⊂Rn, n=2,3, with zero-flux boundary conditions, where α>0,χ>0 and ℓ≥0.It is known that when α=2, for which, in fact, the system was proposed by Leyva et al. (2013) to model the aggregation patterns of colonies of Bacillus subtilis identified on the surface of thin agar plates, a globally defined weak solution to (⋆) can be obtained, either in the case n=1, or for its corresponding spatially two-dimensional version, the latter case additionally requiring certain restrictions on the size of initial data. The present paper complements the results, asserting that under the assumptionα∈{(1,32)if n=2,and(76,139)if n=3, for all sufficiently smooth initial data, (⋆) possesses such a global weak solution (u,v).

Spatiotemporal dynamics of Leslie–Gower predator–prey model with Allee effect on both populations

In the present paper, we consider a reaction–diffusion system, based on the Leslie–Gower predator–prey model with Allee effect in both the predator and prey populations. The model is a generalization of the case where the individual population is subject to the Allee effect and it is flexible such that one can easily obtain the scenario with the Allee effect only on the prey population or, predator population. The non-spatial model was analyzed by using the positivity of solutions, uniformly boundedness and local stability of the interior equilibrium. The sufficient conditions for the instability with zero-flux boundary conditions are obtained for the spatial model. We have derived the analytical conditions for Hopf and Turing bifurcation on the spatial domain. We also observe that the system exhibits complex dynamics like Bogdanov–Takens (BT) bifurcation and generalized Hopf (GH) bifurcation. Our numerical simulations reveal that depending on the strength of Allee effects, the model dynamics exhibit stripes, spots, cold–hot spots–stripes coexistence and cold–hot spots patterns. We obtain very interesting characteristics of the reaction–diffusion model which reflects the fact that the system can develop patterns both inside and outside of the Turing parameter domain. Our study may enrich the field of the Allee effect and will help us to acquire a better understanding of the predator–prey interaction in a real environment.

Effect of a Membrane on Diffusion-Driven Turing Instability

Biological, physical, medical, and numerical applications involving membrane problems on different scales are numerous. We propose an extension of the standard Turing theory to the case of two domains separated by a permeable membrane. To this aim, we study a reaction–diffusion system with zero-flux boundary conditions on the external boundary and Kedem-Katchalsky membrane conditions on the inner membrane. We use the same approach as in the classical Turing analysis but applied to membrane operators. The introduction of a diagonalization theory for compact and self-adjoint membrane operators is needed. Here, Turing instability is proven with the addition of new constraints, due to the presence of membrane permeability coefficients. We perform an explicit one-dimensional analysis of the eigenvalue problem, combined with numerical simulations, to validate the theoretical results. Finally, we observe the formation of discontinuous patterns in a system which combines diffusion and dissipative membrane conditions, varying both diffusion and membrane permeability coefficients.

Dynamics of diffusive nutrient-microorganism model with spatially heterogeneous environment

A diffusive nutrient-microorganism model in a spatially heterogeneous environment with zero-flux boundary conditions is formulated. First, some fundamental properties of the related parabolic system are established. To be specific, some boundedness results are yielded by employing the comparison principle of parabolic equations, the theory of invariant regions and the method of Moser-Alikakos iteration. For the corresponding elliptic system, existence results of non-constant steady states are given by virtue of the Leray-Schauder degree theory. Profiles of the positive solutions are described when the diffusion rate of the nutrient is small or large, with the fixed diffusion rate of microorganisms. Different from the related previous models with spatial homogeneity, the obtained results suggest that such a diffusive nutrient-microorganism model in a spatially heterogeneous environment may be more realistic and perform more complex dynamical behaviors.

Nonconstant positive steady states and pattern formation of a diffusive epidemic model

It is our purpose in this paper to make a detailed description for the structure of the set of the nonconstant steady states for the two-dimensional epidemic S-I model with diffusion incorporating demographic and epidemiological processes with zero-flux boundary conditions. We first study the conditions of diffusion-driven instability occurrence, which induces spatial inhomogeneous patterns. The results will extend to the derivative of prey's functional response with prey is positive. Moreover, we establish the local and global structure of nonconstant positive steady state solutions. A priori estimates for steady state solutions will play a key role in the proof. Our results indicate that the diffusion has a great influence on the spread of the epidemic and extend well the finding of spatiotemporal dynamics in the epidemic model.

Cross-diffusion-driven instability in an interacting species model with prey refuge

The present study appertains to a reaction-diffusion system embracing two-dimensional continuous Beddington-DeAngelis predator-prey model incorporating intra-specific competition among predators and prey refuge in proportion to both the species as well. The existence of all conceivable ecologically significant equilibria is explored and consequently the diffusion-driven instability around the coexistence equilibrium position is reviewed. Numerical simulation with zero-flux boundary conditions discloses that the system under consideration experiences the occurrence of diffusion-driven instability. The dynamical system in Turing space emerges further to get influenced by prey refuge while it unveils diffusion controlled spatiotemporal pattern formation namely, growth of spots, stripe-spot mixtures, stripes, labyrinthine, stripe-hole mixtures and holes reproduction. The quantitative analysis reveals that the interaction of both self- and cross-diffusion plays a significant role on the pattern formation of the present system in a way to enrich the pattern at a greater height.