The Potential And Flux Landscape Theory Of Evolution

In this work, we established the potential and flux landscape theory for evolution.We found the conventional Wright’s fitness landscape and Fisher’s fundamentaltheorem of natural selection describe the equilibrium system. We found that thedetailed balance is break down and the system enters into non-equilibrium if there isbiotic interaction among individuals. The conventional evolutionary theory is basedon the equilibrium system, so that it cannot explain a lot ofphenomena caused bynon-equilibrium. The typical example of the non-equilibrium is Van Valen’s RedQueen hypothesis. This hypothesis indicates that the biotic interaction can driveendless evolutionary processes even if the physical environment is unchanged. AsVanValen himself remarked, this cannot be explained by traditionalevolutionary theory.The exploring of the non-equilibrium system is a great challenge in physics.The dynamical behavior of a system is inevitably affected by noise caused fromexternal or internal environment. So, the statistical description is necessary due to theindeterminacy. We can get Boltzmann potential and the corresponding flux ofevolutionary system, through obtaining the probability distribution in system’s statespace. The steady state probability flux breaks the detailed balance and led the systeminto non-equilibrium. We connect the equilibrium and non-equilibrium through thesteady state probability flux. Due to Boltzmann potential is non-Lyapunov function, itcannot be used to define the adaptive landscape for evolution. We found the adaptivelandscape is hidden behind theBoltzmann potential landscape, but the true topographyis distorted by the noise.We decrease infinitely the noise to obtain the true potential function withLyapunov function’s property from Boltzmann potential. In technical, we use WKBmethod to realize such treatment. At finite fluctuations, the probability distribution satisfies the Fokker-Planck equation. In the zero-fluctuation limit, we obtain thecorresponding Hamilton-Jacobi equation through WKB method. The solution of theHamilton-Jacobi equation has Lyapunov function’s property. We call such solutionintrinsic potential function. The intrinsic potential as a Lyapunov function can be usedto define a general adaptive landscape.In the zero-fluctuation limit, we also obtained the intrinsic flux velocity whichcorresponds to the steady state probability flux. We found the intrinsic flux velocity isdivergent free and perpendicular to the gradient of intrinsic potential. We take thegradient of intrinsic potential and intrinsic flux velocity as the bases to decompose thenatural selection. We measure the adaptive rate by the change rate of intrinsicpotential rather than the conventional mean fitness. In the framework of potential andflux landscape, we reinterpreted the fundamental theorem of natural selection whichcontains the additive geneticvariance and the two components of natural selection,intrinsic potential and flux. We found that the additive geneticvariance activate theadaptation while the intrinsic flux inhibits the adaptation. Through the newinterpretationof fundamental theorem of natural selection, we can obtain that there isadditional additive geneticvariance which is sustained by the flux even if the systemreaches its optima in which the gradient of intrinsic potential function is zero.We found the flux is caused by biotic interactions. The biotic interactions causethe additional additive genetic variance through the flux. Non-zero additive geneticvariance indicates the natural selection takes effects on the system. We obtain theresult as Van Valen’s Red Queen hypothesis predicted: the biotic interactions drive anendless evolutionary process, even if the external physical environment is unchanged.AsVan Valen himself remarked, this cannot be explained by traditionalevolutionarytheory, since the flux is the missing partof the traditional evolutionary theory. Thus,we offer a theoretical basis for explaining the Red Queen hypothesis.Based on the potential and flux landscape theory, we discuss the virulencemacroevolution induced by the extinction differential. We found there arevirulence-differential-dependent symbioses between the parent and daughter parasitespecies in their species longevity:’mutualism’ for small virulence-differential,and’parasitism’ for large virulence differential. The symbioses,whether’mutualism’ or’parasitism’, benefit species with higher virulence. Therefore,speciation towards lower virulence is an effective strategy for a parasite species toreduce its extinction risk, which is favored by species selection. Speciation towardslow virulence becomes the main trend of the long-term virulence evolution inducedby extinction differential.