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We consider diffusion processes x_{t} on the unit interval.Doob-transformation techniques consist of a selection of x_{t}-paths procedure.The law of the transformed process is the one of a branching diffusion systemof particles, each diffusing like a new process tilde{x}_{t}, superposing anadditional drift to the one of x_{t}. Killing and/or branching oftilde{x}_{t}-particles occur at some space-dependent rate lambda. For thistransformed process, so in the class of branching diffusions, the questionarises as to whether the particle system is sub-critical, critical orsuper-critical. In the first two cases, extinction occurs with probability one.We apply this circle of ideas to diffusion processes arising in populationgenetics. In this setup, the process x_{t} is a Wright-Fisher (WF) diffusion,either neutral or with mutations. We study a particular Doob transform which isbased on the exponential function in the usual fitness parameter sigma. We havein mind that this is an alternative way to introduce selection or fitness inboth WF-like diffusions, leading to branching diffusion models ideas. For thisDoob-transform model of fitness, the usual selection drift sigma x(1-x) shouldbe superposed to the one of x_{t} to form tilde{x}_{t} which is the processthat can branch, binarily. In the first neutral case, there is a trade-offbetween branching events giving birth to new particles and absorption at theboundaries, killing the particles. Under our assumptions, the branchingdiffusion process gets eventually globally extinct in finite time withexponential tails. In the second case with mutations, there is a trade-offbetween killing events removing some particles from the system and reflectionat the boundaries where the particles survive. This branching diffusion processalso gets eventually globally extinct but in very long finite time withpower-law tails. Our approach relies on the spectral expansion of thetransition probability kernels of both x_{t} and tilde{x}_{t}.

A branching diffusion model of selection: from the neutral Wright-Fisher case to the one including mutations