• Title/Summary/Keyword: self and cross-diffusion

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INSTABILITY IN A PREDATOR-PREY MODEL WITH DIFFUSION

  • Aly, Shaban
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.13 no.1
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    • pp.21-29
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    • 2009
  • This paper treats the conditions for the existence and stability properties of stationary solutions of a predator-prey interaction with self and cross-diffusion. We show that at a certain critical value a diffusion driven instability occurs, i.e. the stationary solution stays stable with respect to the kinetic system (the system without diffusion) but becomes unstable with respect to the system with diffusion and that Turing instability takes place. We note that the cross-diffusion increase or decrease a Turing space (the space which the emergence of spatial patterns is holding) compared to the Turing space with self-diffusion, i.e. the cross-diffusion response is an important factor that should not be ignored when pattern emerges.

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TURING INSTABILITY IN A PREDATOR-PREY MODEL IN PATCHY SPACE WITH SELF AND CROSS DIFFUSION

  • Aly, Shaban
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.17 no.2
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    • pp.129-138
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    • 2013
  • A spatio-temporal models as systems of ODE which describe two-species Beddington - DeAngelis type predator-prey system living in a habitat of two identical patches linked by migration is investigated. It is assumed in the model that the per capita migration rate of each species is influenced not only by its own but also by the other one's density, i.e. there is cross diffusion present. We show that a standard (self-diffusion) system may be either stable or unstable, a cross-diffusion response can stabilize an unstable standard system and destabilize a stable standard system. For the diffusively stable model, numerical studies show that at a critical value of the bifurcation parameter the system undergoes a Turing bifurcation and the cross migration response is an important factor that should not be ignored when pattern emerges.

LONG-TIME PROPERTIES OF PREY-PREDATOR SYSTEM WITH CROSS-DIFFUSION

  • Shim Seong-A
    • Communications of the Korean Mathematical Society
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    • v.21 no.2
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    • pp.293-320
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    • 2006
  • Using calculus inequalities and embedding theorems in $R^1$, we establish $W^1_2$-estimates for the solutions of prey-predator population model with cross-diffusion and self-diffusion terms. Two cases are considered; (i) $d_1\;=\;d_2,\;{\alpha}_{12}\;=\;{\alpha}_{21}\;=\;0$, and (ii) $0\;<\;{\alpha}_{21}\;<\;8_{\alpha}_{11},\;0\;<\;{\alpha}_{12}\;<\;8_{\alpha}_{22}$. It is proved that solutions are bounded uniformly pointwise, and that the uniform bounds remain independent of the growth of the diffusion coefficient in the system. Also, convergence results are obtained when $t\;{\to}\;{\infty}$ via suitable Liapunov functionals.

GLOBAL EXISTENCE OF SOLUTIONS TO THE PREY-PREDATOR SYSTEM WITH A SINGLE CROSS-DIFFUSION

  • Shim, Seong-A
    • Bulletin of the Korean Mathematical Society
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    • v.43 no.2
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    • pp.443-459
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    • 2006
  • The prey-predator system with a single cross-diffusion pressure is known to possess a local solution with the maximal existence time $T\;{\leq}\;{\infty}$. By obtaining the bounds of $W\array_2^1$-norms of the local solution independent of T we establish the global existence of the solution. And the long-time behaviors of the global solution are analyzed when the diffusion rates $d_1\;and\;d_2$ are sufficiently large.

[ W12 ]-ESTIMATES ON THE PREY-PREDATOR SYSTEMS WITH CROSS-DIFFUSIONS AND FUNCTIONAL RESPONSES

  • Shim, Seong-A
    • Communications of the Korean Mathematical Society
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    • v.23 no.2
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    • pp.211-227
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    • 2008
  • As a mathematical model proposed to understand the behaviors of interacting species, cross-diffusion systems with functional responses of prey-predator type are considered. In order to obtain $W^{1_2}$-estimates of the solutions, we make use of several forms of calculus inequalities and embedding theorems. We consider the quasilinear parabolic systems with the cross-diffusion terms, and without the self-diffusion terms because of the simplicity of computations. As the main result we derive the uniform $W^{1_2}$-bound of the solutions and obtain the global existence in time.

UNIQUENESS OF POSITIVE STEADY STATES FOR WEAK COMPETITION MODELS WITH SELF-CROSS DIFFUSIONS

  • Ko, Won-Lyul;Ahn, In-Kyung
    • Bulletin of the Korean Mathematical Society
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    • v.41 no.2
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    • pp.371-385
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    • 2004
  • In this paper, we investigate the uniqueness of positive solutions to weak competition models with self-cross diffusion rates under homogeneous Dirichlet boundary conditions. The methods employed are upper-lower solution technique and the variational characterization of eigenvalues.

Tomographic Reconstruction of a Non-axisymmetric Diffusion Flame (자발광 확산 사각화염 내부 구조의 단층 진단)

  • Yang, In-Young;Ha, Kwang-Soon;Choi, Sang-Min
    • Journal of the Korean Society of Combustion
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    • v.4 no.1
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    • pp.105-115
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    • 1999
  • The structure of a non-axisymmetric propane diffusion flame was investigated. Tomographic reconstruction method to convert the line-integrated self-emission data of a fuel-rich diffusion flame with square cross-section was applied to get the spatially reconstructed emission data. Modified Shepp-Logan filter and concentric squares raster were chosen for reconstructing arbitrarily shaped object in this process. Spatially reconstructed emission data were then interpreted to several physical quantities, such as flame edge, FWHM, perimeter and 3-D flame temperature distribution. Necessary assumptions were discussed and the results were interpreted. In comparison with axisymmetric flame, flame edge was developed higher, and sooting region of upstream was broader than in this non-axisymmetric one. At some height, the flame was shrunk very rapidly and finally formed circular cross-section.

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Behavior of Non-buoyant Round Jet under Waves (파랑수역에서 비부력 원형 제트의 거동)

  • Ryu, Yong-Uk;Lee, Jong-In;Kim, Young-Taek
    • Journal of Korean Society of Coastal and Ocean Engineers
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    • v.19 no.6
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    • pp.596-605
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    • 2007
  • The behavior of a non-buoyant turbulent round jet discharging horizontally was investigated experimentally. The instantaneous velocity field of the jet was obtained using the particle image velocimetry (PIV) method and used to calculate the mean velocity field by phase-averaging. This study tested regular waves with a relatively small wave height for a wavy environmental flow. The centerline and cross-sectional velocity profiles were reported to demonstrate the effect of the waves on the jet diffusion in respect of wave height and wave phase. The wave phase effect was studied for three phases: zero-upcrossing point, zero-downcrossing point, trough. From the results, it is found that the centerline velocity decreases and width of the cross-sectional profile increases as the wave height increases. In addition, the self-similarity of the cross-sectional profile appears to break down although the width of each case along the axial distance does not vary significantly. The phase effect is found to be relatively small compared to the wave height effect.

MEMBRANES FOR GAS AND LIQUID SEPARATIONS

  • Golemme, Giovanni;Bove, Lucia;Clarizia, Gabriele;Muzzalupo, Rita;Ranieri, Giuseppe;Nam, Sang-Yong;Drioli, Enrico
    • Proceedings of the Membrane Society of Korea Conference
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    • 1999.07a
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    • pp.27-30
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    • 1999
  • Poly(phenylene oxide)s were used to prepare flat, integrally skinned self-supporthed asymmetric membranes by dry-wet phase separption. The intrinsic ideal gas selectivity of poly- (2,6-dimethy-1,4-phenylene oxide) (PMPO) was retained in the membranes, and improved by a coating with silicone rubber. Polymers of the same class were coated of UF supports with a silicon rubber gutter layer, yielding composite membranes with high flux but lower selectivity. The effect of th glutaraldehyde cross-linking of sodium alginate (SA) membranes on the mobility of water and ethanol has been studied with pfg nmr. Crosslinking reduces water self-diffusion, and does not seem to be stable on the timescale of weeks.

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Prediction of Creep Deformation and Short Time Rupture Life of AZ31 Magnesium Alloy below 0.5Tm (0.5Tm 이하에서의 AZ31 마그네슘합금의 크리이프 변형과 단시간 파단수명예측)

  • Kang, D.M.;An, J.O.;Jeon, S.H.;Koo, Y.;Sim, S.B.
    • Transactions of Materials Processing
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    • v.17 no.8
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    • pp.558-563
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    • 2008
  • The initial strain, the applied stress exponent, the activation energy, and rupture time in AZ31 magnesium alloy have been measured in order to predict the deformation mechanism and rupture life of creep over the temperature range of 423-443K. Creep tests were carried out under constant applied stress and temperature, and the lever type tester and automatic temperature controller was used for it, respectively. The experimental results showed that the applied stress exponent was about 9.74, and the activation energy for creep, 113.6KJ/mol was less than that of the self diffusion of Mg alloy including aluminum. From the results, the mechanism for creep deformation seems to be controlled by cross slip at the temperature range of 423-443K. Also the higher the applied stress and temperature, the higher the initial strain. And the rupture time for creep decreased as quadratic function with increasing the initial strain in double logarithmic axis.