Painlevé test, integrability, and exact solutions for density-dependent reaction–diffusion equations with polynomial reaction functions

Hearns, Jessica; Gorder, Robert A. Van; Choudhury, S. Roy

doi:10.1016/j.amc.2012.09.032

Cited by 6 publications

(2 citation statements)

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“…One method for analytically constructing a nonlinear wave solution to a reaction-diffusion equation is to exploit conditional integrability and construct a solution through dominant balance provided the equation passes the Painlevé test. This approach was used in the construction of an exact solution to Fisher's equation in (Ablowitz and Zeppetella, 1979), and has been applied to study a variety of other conditionally integrable reaction or reaction-diffusion equations; for more information, see (Goriely, 1996;Hearns et al, 2012;Ramani et al, 1989;Russo et al, 2013;Weiss et al, 1983;Weiss, 1983). To begin, consider the case of = 0, so that we have (13).…”

Section: Perturbation Approximation Of a Traveling Wave Solutionmentioning

confidence: 99%

A Non-local Cross-Diffusion Model of Population Dynamics II: Exact, Approximate, and Numerical Traveling Waves in Single- and Multi-species Populations

Krause

Gorder

2020

Bull Math Biol

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We study traveling waves in a non-local cross-diffusion type model, where organisms move along gradients in population densities. Such models are valuable for understanding waves of migration and invasion, and how directed motion can impact such scenarios. In this paper we demonstrate the emergence of traveling wave solutions, studying properties of both planar and radial wavefronts in one and two species variants of the model. We compute exact traveling wave solutions in the purely diffusive case, and then perturb these solutions to analytically capture the influence directed motion has on these exact solutions. Using linear stability analysis, we find that the minimum wavespeeds correspond to the purely diffusive case, but numerical simulations suggest that advection can in general increase or decrease the observed wavespeed substantially, which allows a single species to more rapidly move into unoccupied resource rich spatial regions, or modify the speed of an invasion for two populations. We also find interesting effects from the non-local interactions in the model, suggesting that single species invasions can be enhanced with stronger non-locality, but that invasion of a competitive species may be slowed due to this non-local effect. Finally we simulate pattern formation behind waves of invasion, showing that directed motion can have substantial impacts not only on wave speed but also on the existence and structure of emergent patterns, as predicted in the first part of our study (Taylor et al, 2020).

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Section: Perturbation Approximation Of a Traveling Wave Solutionmentioning

confidence: 99%

A Non-local Cross-Diffusion Model of Population Dynamics II: Exact, Approximate, and Numerical Traveling Waves in Single- and Multi-species Populations

Krause

Gorder

2020

Bull Math Biol

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“…Seeking exact solutions of these equations plays an important role in the study of these nonlinear physical phenomena. In the past several decades, many effective methods for obtaining exact solutions of these equations have been presented, such as the inverse scattering method (Ablowitz and Clarkson, 1991), Hirota bilinear method (Hirota, 1980), Backlund transformation (Hirota, 1980;Miura, 1978), Painleve expansion (Hearns et al, 2012;Kudryashov, 1988;Kudryashov, 1990;Kudryashov, 1991;Russo et al, 2012;Weiss et al, 1983), Sine-Cosine method (Wazwaz, 2004;, Jacobi elliptic function method (Lu, 2005;Liu et al, 2001), Tanh-function method (Fan, 2000;Yusufoglu and Bekir, 2008;Zayed and Abdel Rahman, 2010;Zhang and Xia, 2008), F-expansion method (Zhang and Xia, 2006), Exp-function method (Bekir, 2010; 2009; He and Wu, 2006), (G'/G)-expansion method (Bekir, 2008;Islan, 2010;Kudryashov, 2010;Kudryashov, 2009;Ma et al, 2011;Peng, 2008Peng, , 2009Reza and Rasoul, 2011;Wang et al, 2008;Zayed, 2009;Zayed and Al-Joudi, 2010;Zayed and Al-Joudi, 2009;Zhang et al, 2011), the modified (G'/G)-expansion method (Ma et al, 2011;Reza and Rasoul, 2011;Wang et al, 2008) and so on. …”

Section: Introductionmentioning

confidence: 99%

The modified (G/G)-expansion method for the (1+1) Hirota-Ramani and (2+1) breaking soliton equation

Zayed¹,

Arnous²

2013

Int. J. Phys. Sci.

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Painlevé test, integrability, and exact solutions for density-dependent reaction–diffusion equations with polynomial reaction functions

Cited by 6 publications

References 29 publications

A Non-local Cross-Diffusion Model of Population Dynamics II: Exact, Approximate, and Numerical Traveling Waves in Single- and Multi-species Populations

A Non-local Cross-Diffusion Model of Population Dynamics II: Exact, Approximate, and Numerical Traveling Waves in Single- and Multi-species Populations

The modified (G/G)-expansion method for the (1+1) Hirota-Ramani and (2+1) breaking soliton equation

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