The notions of taxis and kinesis are introduced and used to describe two types of behaviour of an organism in non-uniform conditions: (i) Taxis means the guided movement to more favorable conditions; (ii) Kinesis is the non-directional change in space motion in response to the change of conditions. Migration and dispersal of animals has evolved under control of natural selection. In a simple formalisation, the strategy of dispersal should increase Darwinian fitness. We introduce new models of purposeful kinesis with diffusion coefficient dependent on fitness. The local and instant evaluation of Darwinian fitness is used, the reproduction coefficient. New models include one additional parameter, intensity of kinesis, and may be considered as the minimal models of purposeful kinesis. The properties of models are explored by a series of numerical experiments. It is demonstrated how kinesis could be beneficial for assimilation of patches of food or of periodic fluctuations. Kinesis based on local and instant estimations of fitness is not always beneficial: for species with the Allee effect it can delay invasion and spreading. It is proven that kinesis cannot modify stability of positive homogeneous steady states.

A new critical effect is predicted in population dispersal. It is based on the fact that a trade-off between the advantages of mobility and the cost of mobility breaks with a significant deterioration in living conditions. The recently developed model of purposeful kinesis (Gorban & Ç abukoǧlu, Ecological Complexity 33, 2018) is based on the "let well enough alone" idea: mobility decreases for high reproduction coefficient and, therefore, animals stay longer in good conditions and leave quicker bad conditions. Mobility has a cost, which should be measured in the changes of the reproduction coefficient. Introduction of the cost of mobility into the reproduction coefficient leads to an equation for mobility. It can be solved in a closed form using Lambert W-function. Surprisingly, the "let well enough alone" models with the simple linear cost of mobility have an intrinsic phase transition: when conditions worsen then the mobility increases up to some critical value of the reproduction coefficient. For worse conditions, there is no solution for mobility. We interpret this critical effect as the complete loss of mobility that is degeneration of diffusion. Qualitatively, this means that mobility increases with worsening of conditions up to some limit, and after that, mobility is nullified.

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