• Title/Summary/Keyword: cross-diffusion

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POSITIVE COEXISTENCE FOR A SIMPLE FOOD CHAIN MODEL WITH RATIO-DEPENDENT FUNCTIONAL RESPONSE AND CROSS-DIFFUSION

  • Ko, Won-Lyul;Ahn, In-Kyung
    • Communications of the Korean Mathematical Society
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    • v.21 no.4
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    • pp.701-717
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    • 2006
  • The positive coexistence of a simple food chain model with ratio-dependent functional response and cross-diffusion is discussed. Especially, when a cross-diffusion is small enough, the existence of positive solutions of the system concerned can be expected. The extinction conditions for all three interacting species and for one or two of three species are studied. Moreover, when a cross-diffusion is sufficiently large, the extinction of prey species with cross-diffusion interaction to predator occurs. The method employed is the comparison argument for elliptic problem and fixed point theory in a positive cone on a Banach space.

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.

CONVERGENCE RESULTS FOR THE COOPERATIVE CROSS-DIFFUSION SYSTEM WITH WEAK COOPERATIONS

  • Shim, Seong-A
    • The Pure and Applied Mathematics
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    • v.24 no.4
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    • pp.201-209
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    • 2017
  • We prove convergence properties of the global solutions to the cooperative cross-diffusion system with the intra-specific cooperative pressures dominated by the inter-specific competition pressures and the inter-specific cooperative pressures dominated by intra-specific competition pressures. Under these conditions the $W^1_2-bound$ and the time global existence of the solution for the cooperative cross-diffusion system have been obtained in [10]. In the present paper the convergence of the global solution is established for the cooperative cross-diffusion system with large diffusion coefficients.

STATIONARY PATTERNS FOR A PREDATOR-PREY MODEL WITH HOLLING TYPE III RESPONSE FUNCTION AND CROSS-DIFFUSION

  • Liu, Jia;Lin, Zhigui
    • Bulletin of the Korean Mathematical Society
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    • v.47 no.2
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    • pp.251-261
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    • 2010
  • This paper deals with a predator-prey model with Holling type III response function and cross-diffusion subject to the homogeneous Neumann boundary condition. We first give a priori estimates (positive upper and lower bounds) of positive steady states. Then the non-existence and existence results of non-constant positive steady states are given as the cross-diffusion coefficient is varied, which means that stationary patterns arise from cross-diffusion.

PATTERN FORMATION FOR A RATIO-DEPENDENT PREDATOR-PREY MODEL WITH CROSS DIFFUSION

  • Sambath, M.;Balachandran, K.
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.16 no.4
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    • pp.249-256
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    • 2012
  • In this work, we analyze the spatial patterns of a predator-prey system with cross diffusion. First we get the critical lines of Hopf and Turing bifurcations in a spatial domain by using mathematical theory. More specifically, the exact Turing region is given in a two parameter space. Our results reveal that cross diffusion can induce stationary patterns which may be useful in understanding the dynamics of the real ecosystems better.

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.

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.

EXISTENCE OF GLOBAL SOLUTIONS FOR A PREY-PREDATOR MODEL WITH NON-MONOTONIC FUNCTIONAL RESPONSE AND CROSS-DIFFUSION

  • Xu, Shenghu
    • Journal of applied mathematics & informatics
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    • v.29 no.1_2
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    • pp.75-85
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    • 2011
  • In this paper, using the energy estimates and the bootstrap arguments, the global existence of classical solutions for a prey-predator model with non-monotonic functional response and cross-diffusion where the prey and predator both have linear density restriction is proved when the space dimension n < 10.

GLOBAL SOLUTIONS OF THE COOPERATIVE CROSS-DIFFUSION SYSTEMS

  • Shim, Seong-A
    • The Pure and Applied Mathematics
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    • v.22 no.1
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    • pp.75-90
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    • 2015
  • In this paper the existence of global solutions of the parabolic cross-diffusion systems with cooperative reactions is obtained under certain conditions. The uniform boundedness of $W_{1,2}$ norms of the local maximal solution is obtained by using interpolation inequalities and comparison results on differential inequalities.