On the dynamics of interacting populations in presence of state dependent fluctuations

We discuss several models of the dynamics of interacting populations. The models are constructed by nonlinear differential equations and have two sets of parameters: growth rates and coefficients of interaction between populations. We assume that the parameters depend on the densities of the populations. In addition the parameters can be influenced by different factors of the environment. This influence is modelled by noise terms in the equations for the growth rates and interaction coefficients. Thus the model differential equations become stochastic. In some particular cases these equations can be reduced to a Foker-Plancnk equation for the probability density function of the densities of the interacting populations.

[1]  R. Lande,et al.  Stochastic Population Dynamics in Ecology and Conservation , 2003 .

[2]  Nikolay K. Vitanov,et al.  Dynamical consequences of adaptation of the growth rates in a system of three competing populations , 2001 .

[3]  Holger Kantz,et al.  Modified method of simplest equation and its application to nonlinear PDEs , 2010, Appl. Math. Comput..

[4]  Nikolay K. Vitanov,et al.  Application of simplest equations of Bernoulli and Riccati kind for obtaining exact traveling-wave solutions for a class of PDEs with polynomial nonlinearity , 2010 .

[5]  Nikolay K. Vitanov,et al.  Application of the method of simplest equation for obtaining exact traveling-wave solutions for two classes of model PDEs from ecology and population dynamics , 2010 .

[6]  N. Vitanov,et al.  On Waves and Distributions in Population Dynamics , 2012 .

[7]  Nikolay K. Vitanov,et al.  Modified method of simplest equation: Powerful tool for obtaining exact and approximate traveling-wave solutions of nonlinear PDEs , 2011 .

[8]  H. Kantz,et al.  Markov chain model for turbulent wind speed data , 2004 .

[9]  Herbert W. Hethcote,et al.  The Mathematics of Infectious Diseases , 2000, SIAM Rev..

[10]  M. Cirone,et al.  Noise-induced effects in population dynamics , 2002 .

[11]  Marcel Ausloos,et al.  Verhulst-Lotka-Volterra (VLV) model of ideological struggles , 2010, ArXiv.

[12]  Nikolay K. Vitanov,et al.  Traveling waves and statistical distributions connected to systems of interacting populations , 2013, Comput. Math. Appl..

[13]  N. Goel,et al.  Stochastic models in biology , 1975 .

[14]  A. van Oudenaarden,et al.  Noise Propagation in Gene Networks , 2005, Science.

[15]  Nikolay K. Vitanov,et al.  On nonlinear population waves , 2009, Appl. Math. Comput..

[16]  J. Murray.  Lectures on nonlinear-differential-equation models in biology , 1977 .

[17]  M. Saunders,et al.  Plant-Provided Food for Carnivorous Insects: a Protective Mutualism and Its Applications , 2009 .

[18]  N. Vitanov,et al.  Adaptation and its impact on the dynamics of a system of three competing populations , 2001 .

[19]  N. Vitanov.  Upper bounds on the heat transport in a porous layer , 2000 .

[20]  R. May,et al.  Nonlinear Aspects of Competition Between Three Species , 1975 .

[21]  Nikolay K. Vitanov,et al.  On nonlinear dynamics of interacting populations: Coupled kink waves in a system of two populations , 2009 .

[22]  Vitanov.  Upper bounds on convective heat transport in a rotating fluid layer of infinite prandtl number: case of intermediate taylor numbers , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[23]  Nikolay K. Vitanov,et al.  Influence of adaptation on the nonlinear dynamics of a system of competing populations , 2000 .