Exact nonclassical symmetry solutions of Lotka–Volterra-type population systems

Broadbridge, Philip; Cherniha, Roman; Goard, Joanna

doi:10.1017/s095679252200033x

Cited by 4 publications

(17 citation statements)

References 42 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Precise and rigorous definitions for these reductions can be found in [8,14,30]. A number of examples of non-classical reductions for diffusion-type systems are given in the book [31], and in the recent works [32][33][34][35][36]. We can refer to the results derived in this section as potential non-Lie operators for the systems…”

Section: Non-lie Operatorsmentioning

confidence: 99%

Non-Lie Reduction Operators and Potential Transformations for a Special System with Applications in Plasma Physics

Sophocleous

2024

Symmetry

View full text Add to dashboard Cite

Non-Lie reduction operators, also known as nonclassical symmetries, are derived for special systems that appear in Plasma Physics. These operators are used to construct similarity mappings, which reduce the systems under study into systems of ordinary differential equations. Furthermore, potential equivalence transformations are presented. Based on these results, a number of exact solutions are constructed.

show abstract

Section: Non-lie Operatorsmentioning

confidence: 99%

Non-Lie Reduction Operators and Potential Transformations for a Special System with Applications in Plasma Physics

Sophocleous

2024

Symmetry

View full text Add to dashboard Cite

show abstract

“…A reduction of the form u=eAtΦfalse(xfalse), where A is a M×M matrix and Φ is a M×1 vector, is compatible with the vector Helmholtz equation ∇2Φ+BΦ=0,where M×M matrix B is in the commutant of A (satisfying AB−BA=0), provided Rfalse(θfalse)=D−1Au+Bu.Explicit solutions have been constructed [60] for a two-component predator-prey system, with B=0 and with A the skew-symmetric generator of a rotation in the false(θ1,θ2false)-plane, expressed in terms of a Pauli spin matrix as A=aiσ2=a(…”

Section: Beyond Standard Diffusionmentioning

confidence: 99%

“…Explicit solutions have been constructed [60] for a two-component predator-prey system, with B = 0 and with A the skew-symmetric generator of a rotation in the (θ 1 , θ 2 )-plane, expressed in terms of a Pauli spin matrix as…”

Section: (C) Coupled Reaction-diffusion Systemsmentioning

confidence: 99%

Conditionally integrable PDEs, non-classical symmetries and applications

2023

View full text Add to dashboard Cite

Some multi-dimensional nonlinear partial differential equations reduce to a linear equation in fewer dimensions after imposing one additional constraint. A large class of useful conditionally integrable reaction–diffusion equations follows from a single non-classical symmetry reduction, yielding an infinite-dimensional but incomplete solution space. This solution device extends further to other nonlinear equations such as higher-order reaction–diffusion, fractional-order diffusion and diffusion–convection. Applications are shown for population dynamics with or without weak Allee effects, speed-limited hyperbolic diffusion, material phase field dynamics, soil–water–plant dynamics and calcium transport on the curved surface of an oocyte. Beyond non-classical symmetry analysis, examples of other conditionally integrable equations are given; nonlinear diffusion in 1 + 2 dimensions and the Madelung–Burgers quantum fluid in 1 + 3 dimensions.

show abstract

“…As is well known, the predators and prey distribute inhomogeneously in different locations; therefore, diffusion should be taken into account in ecological and biological models. Based on the fact that diffusion may destabilize the steady state and induce the occurrence of Turing instability, many scholars have investigated the diffusive predatorprey systems; see [13,[31][32][33][34], for example. Although the ratio-dependent predator-prey system has been extensively investigated, a study concerning the system incorporating the Allee effect, predator harvesting, and diffusion has not been seen yet.…”

Section: Introductionmentioning

confidence: 99%

Turing–Hopf Bifurcation Analysis in a Diffusive Ratio-Dependent Predator–Prey Model with Allee Effect and Predator Harvesting

Chen,

Xu,

Zhao

et al. 2023

Entropy

View full text Add to dashboard Cite

This paper investigates the complex dynamics of a ratio-dependent predator–prey model incorporating the Allee effect in prey and predator harvesting. To explore the joint effect of the harvesting effort and diffusion on the dynamics of the system, we perform the following analyses: (a) The stability of non-negative constant steady states; (b) The sufficient conditions for the occurrence of a Hopf bifurcation, Turing bifurcation, and Turing–Hopf bifurcation; (c) The derivation of the normal form near the Turing–Hopf singularity. Moreover, we provide numerical simulations to illustrate the theoretical results. The results demonstrate that the small change in harvesting effort and the ratio of the diffusion coefficients will destabilize the constant steady states and lead to the complex spatiotemporal behaviors, including homogeneous and inhomogeneous periodic solutions and nonconstant steady states. Moreover, the numerical simulations coincide with our theoretical results.

show abstract

Exact nonclassical symmetry solutions of Lotka–Volterra-type population systems

Cited by 4 publications

References 42 publications

Non-Lie Reduction Operators and Potential Transformations for a Special System with Applications in Plasma Physics

Non-Lie Reduction Operators and Potential Transformations for a Special System with Applications in Plasma Physics

Conditionally integrable PDEs, non-classical symmetries and applications

Turing–Hopf Bifurcation Analysis in a Diffusive Ratio-Dependent Predator–Prey Model with Allee Effect and Predator Harvesting

Contact Info

Product

Resources

About