On the Lotka–Volterra competition system with dynamical resources and density-dependent diffusion

Abstract

In this paper, we consider the following Lotka–Volterra competition system with dynamical resources and density-dependent diffusion

in a bounded smooth domain \(\Omega \subset {{\mathbb {R}}^2}\) with homogeneous Neumann boundary conditions, where the parameters \(\mu \), \(a_{i}\), \(b_{i}\), \( c_{i}\) (\(i=1,2\)) are positive constants, m(x) is the prey’s resource, and the dispersal rate function \(d_{i}(w)\) satisfies the the following hypothesis:

  • \(d_{i}(w)\in C^2([0,\infty ))\), \(d_{i}'(w)\le 0\) on \([0,\infty )\) and \(d(w)>0\).

When m(x) is constant, we show that the system (*) with has a unique global classical solution when the initial datum is in functional space \(W^{1,p}(\Omega )\) with \(p>2\). By constructing appropriate Lyapunov functionals and using LaSalle’s invariant principle, we further prove that the solution of (*) converges to the co-existence steady state exponentially or competitive exclusion steady state algebraically as time tends to infinity in different parameter regimes. Our results reveal that once the resource w has temporal dynamics, two competitors may coexist in the case of weak competition regardless of their dispersal rates and initial values no matter whether there is explicit dependence in dispersal or not. When the prey’s resource is spatially heterogeneous (i.e. m(x) is non-constant), we use numerical simulations to demonstrate that the striking phenomenon “slower diffuser always prevails” (cf. Dockery et al. in J Math Biol 37(1):61–83, 1998; Lou in J Differ Equ 223(2):400–426, 2006) fails to appear if the non-random dispersal strategy is employed by competing species (i.e. either \(d_1(w)\) or \(d_2(w)\) is non-constant) while it still holds true if both d(w) and \(d_2(w)\) are constant.

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Acknowledgements

The authors are grateful to referees for their valuable suggestions and comments which greatly improve the exposition of this paper. The research of Z.A. Wang was supported by the Hong Kong RGC GRF Grant No. 15303019 (Project ID P0030816) and an internal Grant No. UAH0 (Project ID P0031504) from the Hong Kong Polytechnic University. The research of J. Xu was supported by the Guangdong Basic and Applied Basic Research Foundation (No. 2020A151501140).

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Wang, ZA., Xu, J. On the Lotka–Volterra competition system with dynamical resources and density-dependent diffusion. J. Math. Biol. 82, 7 (2021). https://doi.org/10.1007/s00285-021-01562-w

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Keywords

  • Lotka–Volterra competition
  • Dynamical resources
  • Density-dependent diffusion
  • Homogeneous and heterogenous resource
  • Asymptotic dynamics

Mathematics Subject Classification

  • 35A01
  • 35B40
  • 35B44
  • 35K57
  • 35Q92
  • 92C17