On nonlinear dynamics of interacting populations: Coupled kink waves in a system of two populations

Vitanov, Nikolay K.; Jordanov, Ivan P.; Dimitrova, Zlatinka I.

doi:10.1016/j.cnsns.2008.07.015

Cited by 79 publications

(47 citation statements)

References 44 publications

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“…It was interestingly shown that the convergence time of random walks is related to the second largest eigenvalue of the transition matrix T. For a primitive stochastic matrix with (not necessarily real) eigenvalues 1 = λ 1 > |λ 2 | > |λ 3 | ≥ · · · ≥ |λ n |, it was shown that the number of steps k after which the total variation distance (π k ; π) between the visitation probabilities π k and the stationary distribution π of a random walk falls below ǫ is proportional to 1/ln(|λ 2 |). For a matrix T (2) capturing the statistics of two-paths in an empirical temporal network and a matrixT (2) representing the Markovian model derived from the symmetrized network, an analytical prediction for the change of convergence speed S * , due to non-Markovian properties can be derived as…”

Section: Slow-down or Speed-up Knowledge Diffusionmentioning

confidence: 99%

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Slow-down or speed-up of inter- and intra-cluster diffusion of controversial knowledge in stubborn communities based on a small world network

Ausloos

2015

Front. Phys.

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Citation:Ausloos M (2015) Slow-down or speed-up of inter-and intra-cluster diffusion of controversial knowledge in stubborn communities based on a small world network. Front. Phys. 3:43. doi: 10.3389/fphy.2015.00043 Slow-down or speed-up of inter-and intra-cluster diffusion of controversial knowledge in stubborn communities based on a small world network Diffusion of knowledge is expected to be huge when agents are open minded. The report concerns a more difficult diffusion case when communities are made of stubborn agents. Communities having markedly different opinions are for example the Neocreationist and Intelligent Design Proponents (IDP), on one hand, and the Darwinian Evolution Defenders (DED), on the other hand. The case of knowledge diffusion within such communities is studied here on a network based on an adjacency matrix built from time ordered selected quotations of agents, whence for inter-and intra-communities. The network is intrinsically directed and not necessarily reciprocal. Thus, the adjacency matrices have complex eigenvalues; the eigenvectors present complex components. A quantification of the slow-down or speed-up effects of information diffusion in such temporal networks, with non-Markovian contact sequences, can be made by comparing the real time dependent (directed) network to its counterpart, the time aggregated (undirected) network, -which has real eigenvalues. In order to do so, small world networks which both contain an odd number of nodes are studied and compared to similar networks with an even number of nodes. It is found that (i) the diffusion of knowledge is more difficult on the largest networks; (ii) the network size influences the slowing-down or speeding-up diffusion process. Interestingly, it is observed that (iii) the diffusion of knowledge is slower in IDP and faster in DED communities. It is suggested that the finding can be "rationalized," if some "scientific quality" and "publication habit" is attributed to the agents, as common sense would guess. This finding offers some opening discussion toward tying scientific knowledge to belief.

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Section: Slow-down or Speed-up Knowledge Diffusionmentioning

confidence: 99%

“…Thus, a diffusion slow-down exists if S * (T (2) ) ≥ 1. A diffusion speed-up exists if S * (T (2) ) ≤ 1.…”

Section: Slow-down or Speed-up Knowledge Diffusionmentioning

confidence: 99%

Slow-down or speed-up of inter- and intra-cluster diffusion of controversial knowledge in stubborn communities based on a small world network

Ausloos

2015

Front. Phys.

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“…It is demonstrated in [23] that the exact nonlinear kink and solitary wave solutions to a model system of partial differential equations for description of the spatio-temporal dynamics of interacting populations can be derived. Coupled kink waves in a system of two interacting populations in which the reproduction and intensity of interaction depend on their spatial density are constructed in [22]. Exact travelling-wave solutions to the reaction-diffusion and reaction-telegraph equations that are used in ecology and population dynamics are obtained in [20].…”

Section: Introductionmentioning

confidence: 99%

“…The insight into the structure of solitary solutions to (1) and the relationship between these solutions would be valuable for understanding nonlinear dynamical processes in biological systems coupled with multiplicative terms and extend the existing results on the dynamics of coupled population systems obtained by the modified simplest equation method [20,22,23,25]. This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

Direct and inverse relationships between Riccati systems coupled with multiplicative terms

Navickas

Vilkas

Telksnys

et al. 2016

Journal of Biological Dynamics

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An analytical and computational framework for the derivation of solitary solutions to biological systems describing the cooperation and competition of species and expressed by the system of Riccati equations coupled with multiplicative terms is presented in this paper. It is demonstrated that relationships between these solitary solutions can be either direct or inverse. Thus, an infinitesimal perturbation of one population would lead to an infinitesimal change in the other population -if only both solitary solutions are coupled with the direct relationship. But, in general, that is not true if solitary solutions are coupled with the inverse relationship -an infinitesimal perturbation of one population may result into a non-infinitesimal change in the other population. Necessary and sufficient conditions for the existence of solitary solutions are derived in the space of the system's parameters and initial conditions. ARTICLE HISTORY

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“…Because of all above, the nonlinear PDEs are widely applied in the theory of solitons [ 4,5 ], hydrodynamics and the theory of turbulence [6][7][8][9][10]; the theory of dynamical systems, chaos [11][12][13][14], etc. Sophisticated methods for obtaining exact solutions of nonlinear PDEs such as the inverse scattering transform or the method of Hirota [ 15 ] allow obtaining of soliton solutions of some equations.…”

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confidence: 99%

Discussion on Exp-function Method and Modified Method of Simplest Equation

Dimitrova¹

2013

Proceeding BAS

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We discuss the relation between the modified method of simplest equation and the exp-function method. First, on the basis of our experience from the application of the method of simplest equation we generalize the exp-function approach. Then we apply the method for obtaining exact solutions for members of a class of nonlinear PDEs which contains as particular cases several nonlinear PDEs that model the propagation of water waves.

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On nonlinear dynamics of interacting populations: Coupled kink waves in a system of two populations

Cited by 79 publications

References 44 publications

Slow-down or speed-up of inter- and intra-cluster diffusion of controversial knowledge in stubborn communities based on a small world network

Slow-down or speed-up of inter- and intra-cluster diffusion of controversial knowledge in stubborn communities based on a small world network

Direct and inverse relationships between Riccati systems coupled with multiplicative terms

Discussion on Exp-function Method and Modified Method of Simplest Equation

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