Qualitative theory of the spread of a new gene into a resident population

This work addresses the problems that arise in the estimation of the risk of invader/unwanted GMO spread and the optimal release of desired genes into a population through the release of transgenic individuals. On the basis of a general model of the propagation of an advantageous allele through a population we analyze the thresholds and critical aggregations in gene frequency necessary for the spread of new gene carriers. It is shown that if the invader appears at one place in the ecosystem then it will not spread throughout the ecosystem unless it exceeds some critical threshold, where the critical threshold is defined in terms of both the amount and distribution of the invader. The value of the critical threshold will depend on the fitness of the invader relative to the fitness of the resident organisms in the ecosystem and the mechanism of its dispersion. It is also shown that typically an invader will not spread symmetrically, even if the environment is isotropic, but rather develops clusters that form filaments within the ecosystem. We also demonstrate that if the invader aggregation is sufficiently large then after an initial period the advance of the invader into the resident population takes the form of a traveling wave. The speed of this wave tends to a speed characteristic of the relative fitness and dispersive mechanisms of the invader.

[1]  R. Lande,et al.  Demographic stochasticity and Allee effect on a scale with isotropic noise , 1998 .

[2]  R. W. Hiorns,et al.  The mathematical theory of the dynamics of biological populations : based on a conference organised by the Institute of Mathematics and Its Applications in association with the Institute of Biology , 1975 .

[3]  Evolution under stabilizing selection through gene flow , 1998 .

[4]  Jonathan A. Sherratt,et al.  Algebraic decay and variable speeds in wavefront solutions of a scalar reaction-diffusion equation , 1996 .

[5]  Paul C. Fife,et al.  Mathematical Aspects of Reacting and Diffusing Systems , 1979 .

[6]  P Grewe,et al.  Engineered underdominance allows efficient and economical introgression of traits into pest populations. , 2001, Journal of theoretical biology.

[7]  Luigi Luca Cavalli-Sforza,et al.  The Neolithic Transition and the Genetics of Populations in Europe. , 2009 .

[8]  K. P. Hadeler,et al.  Travelling fronts in nonlinear diffusion equations , 1975 .

[9]  Brian Dennis,et al.  ALLEE EFFECTS: POPULATION GROWTH, CRITICAL DENSITY, AND THE CHANCE OF EXTINCTION , 1989 .

[10]  Viktor Mikhaĭlovich Glushkov,et al.  An Introduction to Cybernetics , 1957, The Mathematical Gazette.

[11]  F. C. Hoppensteadt.  Mathematical theories of populations : demographics, genetics and epidemics , 1975 .

[12]  M. Kot,et al.  Speeds of invasion in a model with strong or weak Allee effects. , 2001, Mathematical biosciences.

[13]  S. Petrovskii,et al.  Some exact solutions of a generalized Fisher equation related to the problem of biological invasion. , 2001, Mathematical biosciences.

[14]  W. Fagan,et al.  Invasion theory and biological control , 2002 .

[15]  N. Barton,et al.  The spread of an advantageous allele across a barrier: the effects of random drift and selection against heterozygotes. , 1997, Genetics.

[16]  Hugh P. Possingham,et al.  Optimal release strategies for biological control agents: an application of stochastic dynamic programming to population management , 2000 .

[17]  Alexei A. Sharov,et al.  MODEL OF SLOWING THE SPREAD OF GYPSY MOTH (LEPIDOPTERA: LYMANTRIIDAE) WITH A BARRIER ZONE , 1998 .

[18]  G. Fulford,et al.  Modelling the integration of a transgene by stocking. , 1999, Theoretical population biology.

[19]  S. Orszag,et al.  Nucleation and relaxation from meta-stability in spatial ecological models. , 1999, Journal of theoretical biology.

[20]  R. Fisher.  THE WAVE OF ADVANCE OF ADVANTAGEOUS GENES , 1937 .

[21]  Petra Schneider,et al.  The interaction between dispersal, the Allee effect and scramble competition affects population dynamics , 2002 .

[22]  J. C. Allen,et al.  Spatially explicit ecological models: a spatial convolution approach , 2001 .

[23]  Peter Kareiva,et al.  Can we use experiments and models in predicting the invasiveness of genetically engineered organisms , 1996 .

[24]  Stephens,et al.  Consequences of the Allee effect for behaviour, ecology and conservation. , 1999, Trends in ecology & evolution.

[25]  Periodic triggering of an inducible gene for control of a wild population. , 2000, Theoretical population biology.

[26]  Vicenç Méndez,et al.  Time-Delayed Theory of the Neolithic Transition in Europe , 1999 .

[27]  K. Aoki,et al.  Gene-culture waves of advance , 1987, Journal of mathematical biology.

[28]  Brian Dennis,et al.  Allee effects in stochastic populations , 2002 .

[29]  W. Landis,et al.  Dynamics of Horizontal Gene Transfer and the Ecological Risk Assessment of Genetically Engineered Organisms , 2000 .

[30]  W. C. Allee.  The social life of animals , 1938 .

[31]  T. Legovic.  A recent increase in jellyfish populations: A predator-prey model and its implications , 1987 .

[32]  K. Sznajd-Weron,et al.  Changes of a continuous character caused by gene flow. A Monte Carlo study , 1998 .

[33]  Response of predator-prey systems to nutrient enrichment and proportional harvesting , 1978 .

[34]  P. Kareiva,et al.  Allee Dynamics and the Spread of Invading Organisms , 1993 .

[35]  D. Aronson,et al.  Multidimensional nonlinear di u-sion arising in population genetics , 1978 .

[36]  Grenfell,et al.  Inverse density dependence and the Allee effect. , 1999, Trends in ecology & evolution.